A N ote To The Instructor .

The solutions here are somewhat brief, as they are designed for the instructor, not for the student.Check with the publishers before electronically posting any part of these solutions; website, ftp, orserver access must be restricted to your students.

I have been somewhat casual about subscripts whenever it is obvious that a problem is onedimensional, or that the choice of the coordinate system is irrelevant to the numerical solution.Although this does not change the validity of the answer, it wi11 sometimes obfuscate the approachif viewed by a novice.

There are some tmditional formula, such as

2- 2 2Vx -VOx + axx,

which are not used in the text. The worked solutions use only material from the text, so there maybe times when the solution here seems unnecessarily convoluted and drawn out. Yes, I know aneasier approach existed. But if it was not in the text, I did not use it here.

I also tried to avoid reinventing the wheel. There are some exercises and problems in the textwhich build upon previous exercises and problems. Instead of rederiving expressions, I simply referyou to the previous solution.

I adopt a different approach for rounding of significant figures than previous authors; in partic-ular , I usually round intermediate answers. As such, some of my answers will differ from those inthe back of the book.

Exercises and Problems which are enclosed in a box also appear in the Student's Solution Manualwith considerably more detail and, when appropriate, include discussion on any physical implicationsof the answer. These student solutions carefully discuss the steps required for solving problems, pointout the relevant equation numbers, or even specify where in the text additional information can befound. When two almost equivalent methods of solution exist, often both are presented. You areencouraged to refer students to the Student 's Solution Manual for these exercises and problems.However, the material from the Student's Solution Manual must not be copied.

Paul Stanley

Beloit College

stanley@clunet. edu

1

El-l (a) Megaphones; (b) Microphones; (c) Decacards (Deck of Cards); (d) Gigalows (Gigolos);(e) Terabulls (Terribles); (f) Decimates; (g) Centipedes; (h) Nanonanettes (?); (i) Picoboos (Peek-a-Boo); (j) Attoboys ('atta boy); (k) Two Hectowithits (To Heck With It); (1) Two Kilomockingbirds(To Kill A Mockingbird, or Tequila Mockingbird).

El-2 (a) $36,000152 week = $692¡week. (b) $10,000,0001(20 x 12 month) = $41, 700¡month. (c)30 x 109/8 = 3.75 x 109.

~ Multiply out the factors which make up a century.

365 days

1 year

24 hours

I day

60 minutes .

1 hour1 century = 100 years

This gives 5.256 x 107 minutes in a century, so a microcentury is 52.56 minutes.The percentage difference from Fermi's approximation is (2.56 min)/(50 min) x 100% or 5.12%.

El-4 (3000 mi)/(3 hr) = 1000 mi/timezone-hour.is approximately 24 x 1000 mi = 24,000 miles.

There are 24 time-zones, so the circumference

El-5 Actual number oí seconds in a year is

24 hr

1 day

60 minlhr

608'

1 min

= 3.1558 x 107 s.{365.25 days)

The percentage error of the approximation is then

(a) 10-8 seconds per shake means 108 shakes per second. There areEl-6

365 days

1 year

24 hr

1 day

60 min

I hr

60 s

1 min= 3.1536 x 107 s/year.

This means there are more shakes in a second.(b) Humans have existed for a fraction of

106 years/101O years = 10-4.

That fraction of a day is

60 minlhr

60 s

1 min10-4 (24 hr) = 8.64 s.

~ We'll 8Ssume, for convenience only, that the runner with the longer time ran exactly onemile. Let the speed of the runner with the shorter time be given by Vl , and call the distance actuallyran by this runner d1. Then Vl = dl/tl. Similarly, V2 = d2/t2 for the other runner, and d2 = 1 mile.

We want to know when Vl > V2. Substitute our expressions for speed, and get dl/tl > d2/t2.Rearrange, and d1/d2 > tl/t2 or d1/d2 > 0.99937. Then d1 > 0.99937 mile x (5280 feet/l mile) ord1 > 5276.7 feet is the condition that the first runner W8S indeed f8Ster. The first track can be nomore than 3.3 feet too short to guarantee that the first runner W8S f8Ster .

2

~ The volume of Antarctica is approximated by the area of the base time the height; thearea of the base is the area of a semicircle. Then

'1 2-7rr2

V=Ah= h.

The volume is

v -

El-16 The volume is (77x104m2)(26m) = 2.00x107m3. This is equivalent to

(2.00 x 107 m3)(10-3 km/m)3 = 0.02 km3

El-17 (a) C = 27rr = 27r(6.37 x 103 km) = 4.00 x 104 km. (b) A = 47rr2 = 47r(6.37 x 103 km)2 =

5.10 x 108 km. (c) V = ~7r(6.37 x 103 km)3 = 1.08 x 1012 km3.

El-18 The conversions: squirrel, 19 km/hr(1000 m/km)/(3600 s/hr) = 5.3m/s;rabbit, 30 knots(1.688ft/s/knot)(0.3048 m/ft)= 15m/s;snail, 0.030 mi/hr(1609 m/mi)/(3600 s/hr) = O.013m/s;spider, 1.8 ft/s(0.3048 m/ft) = 0.55m/s;cheetah, 1.9 km/min(1000 m/km)/(60 s/min) = 32m/s;human, 1000 cm/s/(100 cm/m) = 10m/s;fox, 1100 m/min/(60 s/min) =18m/s;lion, 1900 km/day(1000 m/km)/(86,400 s/day} = 22m/s.The order is snail, spider, squirrel, human, rabbit, fox, lion, cheetah.

~ One light-year is the distance traveled by light in one year, or (3 x 108 mis) x (1 year).Then

1609 m

1 mi100 year

1 century ,

El-20 Start with the British units inverted,

1.639 x 10-2 L

in3

~ )(gal= 7.84 x 10-2 L/km.gal

30.0 mimi

1.609 km

El-21 (b) A light-year is

3600 s'

1 hr

24 hr

1 day,

(365 days) = 9.46 x 1012 km.(3.00 x 105 km/s)

A parsec is

360°

27r rad= 3.09 x 1013 km.

1.50 x 108 km

0° 0' 1"

(a) (1.5 x 108 km)/(3.09 x 1013 km/pc = 4.85 x 10-6 pc. (1.5 x 108 km)/(9.46 x 1012 km/ly) =1.59 x 10-5 ly.

4

El-22 First find the "logarithmic average" by

log dav

~

12 (log(2 x 1026) + log(l x 10-15)) ,

121og (2 x 1026 x 1 x 10-15) ,

~ log2 x 1011 = log ( /2XWU) .

Salve, and dav = 450 km.

El-24 (a) (2 x 1.0 + 16)u(1:661 x 10-27kg) = 3.0 x 10-26kg.(b) (1.4 x 1021kg)/(3.0 X 10-26kg) = 4.7 x 1046 molecules.

El-25

It is cheaper to buy coffee in New York (at least according to the physics textbook, that is.)

El-26 The room volume is (21 x 13 x 12)ft3(0.3048 m/ft)3 = 92.8m3. The mass contained in the

room is(92.8m3)(1.2Lkg/m3) = 112 kg.

~ One mole ofsugar cubes would have avolume ofNA x 1.0 cm3, where NA is the Avogadroconstant. Since the volume of a cube is equal to the length cubed, V = l3, then l = ~ cm= 8.4 x 107 cm.

El-28 The number of 8econd8 in a week i8 60 x 60 x 24 x 7 = 6.05 x 105, The "weight" 1088 per

8econd i8 then(0.23kg)/(6.05 x 105 s) = 3.80 x 10-1 mg/s.

~ The definition of the meter WBS wavelengths per meter; the question BSks for meters perwavelength, so we want to take the reciprocal. The definition is accurate to 9 figures, so the reciprocalshould be written BS 1/1,650,763.73 = 6.05780211 x 10-7 m = 605.780211 nm.

El-30 (a) 37.76 + 0.132 = 37.89. (b) 16.264- 16.26325 = 0.001,

~ The easiest appraach is ta first salve Darcy's Law far K, and then substitute the knawnSI units far the ather quantities. Then

(m3) (~)~ has units of (m2) (m) (S)AHt

K=

which can be simplified to m/s.

fi

El-32 The Planck length, lp, is found from

[lp]

L

=lci][Gj][hk],

(LT-l)i(L3T-2M-l)j(ML2T-1)k,

Li+3j+2kT-i-2j-kM-j+k.

=

Equate powers on each sirle,

L: 1 = i + 3j + 2k,

T: O = -i -2j -k,

M: O = -j + k.

Then j = k"and i = -3k, and 1 = 2k; so k = 1/2, j = 1/2, and i = -3/2. Then

[lp}= [C-3/2][G1/2][h1/2],

{3.00 x 108 m/8)-3/2{6.67 x 10-11 m3/82 .kg)1/2{6.63 x 10-34 kg .m2/8)1/2,

4.05 x 10-35 m.

El-33 The Planck mass, mp, is found fron

[mp]

M

= [Ci][Gj][hk],

(LT-l)i(L3T-2M-l)j(ML2T-l)k,

Li+3j+2kT-i-2j-kM-j+k.

=

Equate powers on each side,

L: O

T: O

M: I

i + 3j + 2k,

-i -2j -k,

-j+k.

Then k = j + 1, and i = -3j -1, and O = -1 + 2kj so k = 1/2, and j = -1/2, and i = 1/2. Then

[mp] = [C1/2][G-1/2][h1/2],

= {3.00 x 108 m/s)1/2{6.67 X 10-11 m3 /S2. kg)-1/2{6.63 X 10-34 kg .m2/s)1/2,

= 5.46 x 10-8 kg.

~ There are 24 x 60 = 1440 traditional minutes in a day. The conversion plan is then fairly

straightforward" 1440trad. min

822.8 dec. min ( = 1184.8 trad. min., lUUU dec. min )

This is traditional minutes since midnight, the time in traditional hours can be found by dividingby 60 min/hr , the integer part of the quotient is the hours, while the remainder is the minutes. Sothe time is 19 hours, 45 minutes, which would be 7:45 pm.

Pl-2 (a) By similar triangles, the ratio ofthe distances is the same as the ratio ofthe diameters-390:1.

(b) Volume is proportionalto the radius (diameter) cubed, or 3903 = 5.93 x 107.(c) 0.52°(27r/360°) = 9.1 x 10-3 rad. The diameter is then (9.1 x 10-3 rad)(3.82 x 105 km) =

3500 km.

6

-

-

Pl-3 (a) The circumference of the Earth is approximately 40,000 km; 0.5 seconds of an arc is0.5/(60 x 60 x 360) = 3.9 x 10-7 of a circumference, so the north-south error is ::1:(3.9 x 10-7)(4 x107 m) = ::1:15.6 m. This is a range of 31 m.

(b) The east-west range is smaller, because the distance measured along a latitude is smaller thanthe circumference by a factor of the cosine of the latitude. Then the range is 31 cos 43.6° = 22 m.

(c) The tanker is in Lake Ontario, s(jme 20 km off the coast of Hamlin?

Pl-4 Your position is determined by the time it takes for your longitude to rotate "underneath"the sun (in fact, that's the way longitude was measured originally as in 5 hours west ofthe Azores...)the rate the sun sweep over at equator is 25,000 miles/86,400 s = 0.29 miles/second. The correctionfactor because of latitude is the cosine of the latitude, so the sun sweeps overhead near England atapproximately 0.19 mi/s. Consequently a 30 mile accuracy requires an error in time of no more than(30 mi)/(0.19 lI!i/s) = 158 seconds.

Trip takes about 6 m~nths, so clock accuracy needs to be within (158s)/(180 day) = 1.2 sec-

onds/day.(b) Same, except 0.5 miles accuracy requires 2.6 s accuracy, so clock needs to be within 0.007

s/day!

Pl-5 Let B be breaths/minute while sleeping. Each breath takes in (1.43 g/L)(0.3 L) = 0.429 gjand lets out (1.96 g/L)(0.3 L) = 0.288 g. The net loss is 0.141 g. Multiply by the number ofbreaths,(8 hr)(60 min./hr)B(0.141 g) = B(67.68 g). I'll take a short nap, and count my breaths, then finishthe problem.

I'm back now, and I foundmy breaths to be 8/minute. So I lose 541 g/night, or about 1 pound.

~ Let the radius of the grain be given by r 9 .Then the surface area of the grain is Ag = 47rr~ ,and the volume is given by Vg = (4/3)7rr:.

If N grains of sand have a total surface area equal to that of a cube 1 m on a edge, thenN Ag = 6 m2. The total volume Vt of this number of grains of sand is NVg. Eliminate N from these

two expressions and get

~Vg = ~.3

Yt = NVg =Ag

Then Yt = (2 m2)(50 X 10-6 m) = 1 x 10-4 m3.

The mass oí a volume Yt is given by

= 0.26 kg.

Pl-8 For a cylinder V = 7rr2h, and A = 27rr2 + 27rrh.

changes in r, soWe want to minimize A with respect to

dA

dr1; ( 27rr2 + 27rr~ )=

v47rr -22.

r

Set this equal to zero; then V = 2?rr3. Notice that h = 2r in this expression.

7

Pl-9 (a) The volume per particle is

{9.27 x 10-26kg)/{7870kg/m3) = 1.178 x 10-28m3.

The radius of the corresponding sphere is

r = 3/3(1.178 x lO-28m3)V. 47r .= 1.41 x 10-1Om.

Double this, and the spacing is 282 pm.(b) The volume per particle is

{3.82 x lO-26kg)/{lOl3kg/m3) = 3.77 x lO-29m3.

The r~ius of the corresponding sphere is

r = 3/3(3.77 x 10-29m3)V. 411" ,

Double this, and the spacing is 416 pm.

= 2.08 x 10-1om.

PI-I0 (a) The area of the plate is (8.43 cm)(5.12 cm) = 43.2 cm2. (b) (3.14)(3.7 cm)2 = 43 cm2.

8

~ Add the vectors as is shown in Fig. 2-4. If a has length a = 4 m and b has length b = 3 m

then the sum is given by s. The cosine law can be used to find the magnitude s of s,

82 = a2 + b2 -2abcos(},

where () is the angle between sides a and b in the figure.(a) (7 m)2 = (4 m)2 + (3 m)2 -2(4 m)(3 m) cos(), so cos() = -1.0, and () = 180°. This means

that a and b are pointing in the same direction.(b) (1 m)2 = (4 m)2 + (3 m)2 -2(4 m)(3 m)cos(), so cos() = 1.0, and () = 0°. This means that

a and b are pointing in the opposite direction.(c) (5 m)2 = (4 m)2 + (3 m)2 -2(4 m)(3 m) cos(), so cos(J = 0, and () = 90°. This means that a

and b are pointing at right angles to each other .

( & ) Consider the figures below.

= 2.1) km south. A bird would By{b) Net displacement is 2.4 km west, {5.2 -3.

.¡2.42 + 2.12 km = 3.2 km.

Consider the figure below.

E2-4 (a) The components are (7.34) cos(252°) = -2.27i and (7.34) sin(252°)= -6.98j.(b) The magnitude is V( -25)2 + (43)2 = 50; the direction is (} = tan-l(43/- 25) = 120°. We

did need to choose the correct quadrant.

~ The components are given by the trigonometry relations

o =Hsin{} = {3.42 km) sin 35.0° = 1.96 km

A = H cos (} = (3.42 km) cos 35.0° = 2.80 km.

9

The stated angle is measured from the east-west axis, counter clockwise from east. So O is measuredagainst the north-south axis, with north being positive; A is measured against east-west with east

being positive.Since her individual steps are displacement vectors which are only north-south or east-west, she

must eventually take enough north-south steps to equal 1.96 km, and enough east-west steps toequal 2.80 km. Any individual step can only be along one or the other direction, so the minimumtotal will be 4.76 km.

Let rr = 124¡ km and ri = (72.6¡ + 31.4]) km. Then the ship needs to travelE2-6

~r = rf -ri = (51.4i + 31.4]) km.

Ship needs to travel v51.42 + 31.42 km = 60.2 km in a direction (} = tan-l(31.4/51.4) = 31.4° west

of north.

~ (a) In unit vector notation we need only add the components; a+b = (5¡+3j)+( -3¡+2j) =

(5- 3)¡ + (3 + 2)j = 2¡ + 5j.

(b) If we define c = a + b and write the magnitude of c as c, then c = ~ = V22 + 52 =

5.39. The direction is given by tan() = Cy/cx which gives an angle of 68.2°, measured counterclock-

wise from the positive x-axis.

...A A A A A AE2-8 (a) a + b = (4- l)i + ( -3 + l)j + (1 + 4)k = 3i -2j + 5k.

...A A A A A A(b) a- b = (4- -1)i + ( -3- l)j + (1- 4)k = 5i -4j -3k.

A A A A A A(c) Rearrange, and c= b -a, or b -a= (-1-4)i+ (1- -3)j + (4 -1)k = -5i+4j +3k.

E2-9 (a) The magnitude of a is V4.02 + (-3.0)2 = 5.0j the direction is 9 = tan-l(-3.0/4.0) =

323°.(b) The magnitude of b is V6.02 + 8.03 = 10.0j the direction is 9 = tan-l(6.0/8.0) = 36.9°.

-A A -(c) The resultant vector is a + b = (4.0 + 6.0)i + ( -3.0 + 8.0)j. The magnitude of a + b is

V(10.0)2 + (5.0)2 = 11.2j the direction is 9 = tan-l(5.0/10.0) = 26.6°.-A A -

(d) The resultant vector is a- b = (4.0- 6.0)i + (-3.0- 8.0)j. The magnitude of a- b is

V( -2.0)2 + ( -11.0)2 = 11.2j the direction is 9 = tan-l( -11.0/ -2.0) = 260°.-A A -

(e) The resultant vector is b- a = (6.0- 4.0)i + (8.0- -3.0)j. The magnitude of b -a isV(2-:0)2 + (11.0)2 = 11.2j the direction is 9 = tan-l(11.0/2.0) = 79.7°.

E2-10 (a) Find components of a; ax = (12.7) cos(28.2°) = 11.2, ay = (12.7) sin(28.2°) =

Find components of b; bx = (12.7) cos(133°) = -8.66, by = (12.7) sin(133°) = 9.29. Then

...A A A Ar = a + b = {11.2 -8.66)i + {6.00 + 9.29)j = 2.54i + 15.29j.

(b) The magnitude oí r is v2.542 + 15.292 = 15.5.(c) The angle is (} = tan-l(15.29/2.54) = 80.6°.

E2-11 Consider the figure below.

10

E2-12 Consider the figure below.

~ Our axes will be chosen so that i points toward 3 O'clock and j points toward 12 O'clock.

(a)The two relevant positions are ri = (11.3 cm)i and rf = (11.3 cm)j. Then

~i! = i!f -i!i

= {11.3 cm)j -{11.3 cm)i.

{b)The two relevant positions are now ri = {11.3 cm)] and rr = { -11.3 cm)]. Then

j;}.r = rf -ri

= {11.3 cm)j -{ -11.3 cm)j

-{22.6 cm)j.

{c) , ,The two relevant positions are now rj = (-11.3 cm)j and rf = {-11.3 cm)j. Then

~i = if -ii

= {-11.3 cm)] -{-11.3 cm)]

= {0 cm)].

(a) The components of rl areE2-14

Tlx = (4.13 m) cos(225°) = -2.92 m

andrly = (4.13 m) sin(225°) = -2.92 m.

11

The components oí r2 arerlx = {5.26 m) cos{OO) = 5.26 m

and

rly = (5.26m)sin(OO) = Om.

The components of r3 arerlx = (5.94m)cos(64.00) = 2.60m

and

rly = (5.94 m) sin(64.00) = 5.34 m.

(b) The resulting displacement is

[ ( -2.92 + 5.26 + 2.60)i + ( -2.92 + O + 5.34)1] m = ( 4.94i + 2.421) m.

( c ) The magnitude of the re8ulting di8placement i8 v 4.942 + 2.422 m = 5.5 m. The direction ofthe re8ulting di8placement i8 (J = tan-l(2.42/4.94) = 26.1°. (d) To bring the particle back to the8tarting point we need only rever8e the an8Wer to (c); the magnitude will be the 8ame, but the anglewill be 206°.

E2-15 The components oí the initial position are

rlx = (12,000 ft) cos(40°) = 9200 ft

rly = (12,000 ft) sin(40°) = 7700 ft.

The co~ponents of the final position are

r2x = (25,8000 ft) cos(163°) = -24,700 ft

r2y = (25,800 ft) sin(163°) = 7540 ft.

The displacement is

r = r2 -r1 = [ ( -24,700 -9, 200)i + (7,540 -9,200)]) ] = ( -33900i -1660]) ft.

E2-16 (a) The displacement vector is r = (410i -820j) mi, where positive x is ea.st and positivey is north. The magnitude ofthe displacement is V(410)2 + (-820)2 mi = 920 mi. The direction is(} = tan-l(-820/410) = 300°.

(b) The average velocity is the displacement divided by the total time, 2.25 hours. Then

"av = (1801- 360j) mi/hr.

(c) The average speed is total distance over total time, or (410+820)/(2.25) mi/hr = 550 mi/hr.

12

~ (a) Evaluate r when t = 2 s.

r - [(2 m/s3)t~ -(5 m/s)t]i+ [(6 m) -(7 m/s4)t4]]

[(2 m/s3)(2 8)3- (5 m/s)(2 s»)i+ 1(6 m)- (7 m/s4)(2 8)4]]

[(16 m) -(10 m)]i + {(6 m) -(112 m)]]

[(6 m)]i + [-(106 m)]].

(b) Evaluate:

drv- dt [(2 m/s3)3t2 -(5 m/s)]i +

[(6 m/s3)t2 -(5 m/s)]i + [-

[-(7 m/s4)4t3]j

-(28 m/s4)t3Jj.

Into this last expression we now evaluate v(t = 2 s) and get

[(6 m/s3)(2 sf -(5 m/s)]i + [-(28 m/s4)(2 s)3]j

[(24 m/s) -(5 m/s)]i + [-(224 m/s)]j

[(19 m/s)]i + [-(224 m/s)ti,

v =

for the velocity v when t = 2 s.

(c) Evaluate

dv---a- dt [(6 m/s3)2t]i + l-(28 m/s4)3t2]j

[(12 m/s3)t]i +{-(84 m/s4)t2]j,

~

Into this last expression we now evaluate a( t = 2 s) and get

[(12 mis3)(2 s)]i + [-(84 mis4)(2 2)2]j2 A 2 A

[(24 mis )]i + [-(336 mis )]j.

a =

E2-18 (a) Let ui point north, j point e8St, and k point up. The displacement is (8.7i + 9.7j +2.9k) km. The average velocity is found by dividing each term by 3.4 hr; then

Vav = (2.6i + 2.9j + 0.85) km/hr .

The magnitude of the average velocity is v2.62 + 2.92 + 0.852 km/hr = 4.0 km/hr.(b) The horizontal velocity has a magnitude of v2.62 + 2.92 km/hr = 3.9 km/hr. The angle with

the horizontal is given by (} = tan-l(0.85/3.9) = 13°.

E2-19 (a) The derivative of the velocity is

a = [(6,Om/s2) -(8.Om/s3)t]i

so the acceleration at t = 3 s is a = ( -18.0 m/s2)i. (b) The acceleration is zero when (6.0 m/s2) -(8.0m/s3)t = 0, or t = 0.75s. (c) The velocity is never zero; there is no way to "cancel" out they component. (d) The speed equals 10m/s when 10 = ~, or Vx = :!::6.0m/s. This happenswhen (6.0m/s2) -(8.0m/s3)t = :!::6.0m/s, or when t = Os.

13

E2-20 If v is constant then so is V2 = V~ + V~. Take the derivative;

d d2vxdtvx + 2v1ldtv1l = 2(vxax + v1la1l).

But if the value is constant the derivative is zero.

~ Let the actual flight time, as measured by the passengers, be T. There is some timedifference between the two cities, call it ~T = Namulevu time -Los Angeles time. The ~T will bepositive ií Namulevu is east oí Los Angeles. The time in Los Angeles can then be íound from the

time in Namulevu by subtracting ~T.The actual time oí flight from Los Angeles to Namulev\¡ is then the difference between when the

plane lands (LA times) and when the plane takes off (LA time) :

T = (18:50 -~T) -(12:50)

= 6:00-~T,

where we have written times in 24 hour format to avoid the AM/PM issue. The return flight time

can be found from

T = (18:50) -(1:50- 6.T)

= 17:00 + 6.T,

where we have again changed to LA time for the purpose of the calculation.(b) Now we just need to solve the two equations and two unknowns.

17:00 + 6.T = 6:00 -6.T

26.T = 6:00 -17:00

6.T = -5:30.

Since this is a negative number, Namulevu is located west of Los Angeles.(a) T = 6:00- 6.T = 11: 30, or eleven and a half hours.(c) The distance traveled by the plane is given by d = vt = (520 mi/hr)(11.5 hr) = 5980 mi.

We'll draw a circle around Los Angeles with a radius of 5980 mi, and then we look for where itintersects with longitudes that would belong to a time zone 6.T away from Los Angeles. Since theEarth rotates once every 24 hours and there are 360 longitude degrees, then each hour correspondsto 15 longitude degrees, and then Namulevu must be located approximately 15° x 5.5 = 83° west ofLos Angeles, or at about longitude 160 east. The location on the globe is then latitude 5°, in the

vicinity of Vanuatu.When this exercise was originally typeset the times for the outbound and the inbound fiights were

inadvertently switched. I suppose that we could blame this on the airlines; nonetheless, when the answerswere prepared for the back of the book the reversed numbers put Namulevu east of Los Angeles. That wouldput it in either the North Atlantic or Brazil.

E2-22 There is a three hour time zone difference. So the flight is seven hours long, but it takes3 hr 51 min for the sun to travel same distance. Look forwhen the sunset distance has caught up

with plane:

= dplane'= Vplanet,= t/7:00,

dsunset

Vsunset(t- 1:35)

(t- 1:35)/3:51

so t = 3:31 into flight.

14

E2-23 The distance is

d = vt = (112 km/hi)(1 8)/(3600 8/hr) = 31 m.

E2-24 The time taken for the hall to reach the plate is

~ Speed is distance traveled divided by time taken; this is equivalent to the inverse of theslope of the line in Fig. 2-32. The line appears to pass through the origin and through the point(1600 km, 80 x 106 y), so the speed is v = 1600 km/80 x 106 y= 2 x 10-5 km/y. Converting,

1000 m

1km

100 cm

1mv = 2 x lO-5km/y = 2 cm/y

E2-26 {a) For Maurice Greene vav = {100m)/{9.81 m) = 10.2m/.s. For Khalid Khannouchi,

{26.219 mi)vav = {2.0950 hr)1609 m1 mi

lhr )3600"8 = 5.594 m/s.

(b) Ir Maurice Greene ran tbe maratbon witb an average speed equal to bis average sprint speedtben it would take bim

(26.219 mi) 1609 mI mi

lhr

36008t= = 1.149 hr,

or 1 hour, 9 minutes.

E2-27 The time saved is the difference,

{700 km) {700 km)A.t = {88.5 km/hr) -{104.6 km/hr) = 1.22 hr,

which is about 1 hour 13 minutes.

E2-28 The ground elevation will increase by 35 m in a horizontal distance of

x = {35.0m)/tan{4.3°) = 465 m.

The plane will cover that distance in

(0.465 km)

(1300 km/hr)t= ~ ) = 1.38.

lhr

~ Let Vl = 40 km/hr be the speed up the hill, tl be the time taken, and d1 be the distancetraveled in that time. We similarly define V2 = 60 km/hr for the down hill trip, as well as t2 and d2.Note that d2 = d1.

15

Vl = d1/tl and V2 = d2/t2. vav = d/t, where d total distance and t is the total time. The totaldistance is d1 + d2 = 2d1. The total time t is just the sum of tl and t2, so

vav = d

t

2dl

tl + t2

2dl

dl/Vl + d2/V2

2l/v1'+ 1/v2 ,

Take the reciprocal of both sides to get a simpler looking expression

2 1 1-= -+ -,vav VI V2

Then the average speed is 48 km/hr .

E2-30 (a) Average speed is total distance divided by total time. Then

(b) Same approach, but different information given, so

E2-31 The distance traveled is the total area under the curve. The "curve" has four regions: (I)a triangle from O to 2 s; (II) a rectangle from 2 to 10 s; (III) a trapezoid from 10 to 12 s; and (IV)a rectangle from 12 to 16 s.

The area underneath the curve is the sum of the areas of the four regions.

E2-32 The acceleration is the slope of a velocity-time curve,

(8m/s) -(4m/s)= -2 m/s2,a=

(IOs) -(12s)

~ The initial velocity is Vi = (18 m/s)i, the final velocity is Vf = ( -30 m/s)i. The averageacceleration is then

6.v .( -30 m!s)i -(18 m!s)i--

Vf -Vi

~t

Rav-At

-20.0 m/s2)i.

=2.4 s

which gives aav =

16

E2-34 Consider the figure below.

E2-35 (a) Up to A Vx > O and is constant. From A to B Vx is decreasing, but still positive. FromB to C Vx = O. From C to D Vx < 0, but Ivxl is decreasing.

(b) No. Constant acceleration would appear as (part of) a parabola; but it would be challengingto distinguish between a parabola and an almost parabola.

E2-36 (a) Up to A Vx > O and is decreasing. From A to B Vx = O. From B to C Vx > 0 and isincreasing. From C to D Vx > 0 and is constant.

(b) No. Constant acceleration would appear as (part of) a parabola; but it would be challengingto distinguish between a parabola and an almost parabola.

~ Consider the figure below.

E2-38 Consider the figure below.

17

The acceleration

is a constant

2 cm/s/s during

the entire time

interval.

E2-39 (a) A must have units of m/s2, B must have units of m/s3.(b) The maximum positive x position occurs when v"' = O, so

dx

dt= 2At -3Bt2v:& =

implies Vx = O when either t = O or t = 2A/3B = 2(3.0m/s2)/3(1.0m/s3) = 2.0s.( c ) Particle starts from rest, then travels in positive direction until t = 2 s, a distance of

x = (3.0m/s2)(2.0S)2 -(1.0m/s3)(2.0s)3 = 4.0m.

Then the particle moves back to a final position of

x = (3.0 m/s2)(4.0S)2 -(1.0m/s3)(4.0s)3 = -16.0m.

The total path followed was 4.0 m + 4.0 m + 16.0 m = 24.0 m.(d) The displacement is -16.0m as was found in part (c).(e) The velocity is Vx = (6.0m/s2)t -(3.0m/s3)t2. When t = 0, Vx = O.Om/s. When t = 1.0s,

Vx = 3.0m/s. When t = 2.0s, Vx = O.Om/s. When t = 3.0s, Vx = -9.0m/s. When t = 4.0s,Vx = -24.0m/s.

(f) The acceleration is the time derivative of the velocity,

dvx

dt= (6.0m/s2) -(6.0m/s3)t.ax =

When t = Os, ax = 6.Om/s2. When t = 1.Os, ax = O.Om/s2. When t = 2.Os, ax = -6.Om/s2.When t = 3.Os, ax = -12.Om/s2. When t = 4.Os, ax = -18.Om/s2.

(g) The distance traveled was found in part (a) to be -2Om The average speed during the timeinterval is then Vx,av = (-2Om)/(2.Os) = -lOm/s.

E2-40 VOz = O, Vz = 360 km/hr = 100 m/s. Assuming constant acceleration the average velocitywill be

Vx,av = i(lOOm/s + O) = 50m/s.

The time to travel the distance oí the runway at this average velocity is

t = (1800m)/(50m/8) = 368.

The acceleration isax = 2x/t2 = 2(1800m)/(36.0s)2 = 2.78m/s2.

18

~ (a) Apply Eq. 2-26,

Vz = VOz + azt,

{3.0 x 107 mis) = {0) + {9.8 mis2)t,

3.1 x 106 s = t.

(b) Apply Eq. 2-28 using an initial position of Xo = O,

1 2X = Xo + VOx + 2axt ,

x = (0) + (0) + ~(9.8 m/s~)(3.1 x 106 S)2,

X = 4.7 x 1013 m.

E2-42 VOx = o and Vx = 27.8m/s. Then

t = (vx -vox)/a = ((27.8m/s) -(O)) /(50m/s2) = 0~56s.

I want that car .

E2-43 The muon will travel for t seconds before it comes to a rest, where t is given by

t = (vx -vox)/a = ((O) -(5.20 x 106m/s)) /(-1.30 x 1014m/s2) = 4 x 10-8s.

The distance traveled will be

1 1x = 2axt2 + vOxt = 2( -1.30 x 1014m/s2)(4 X 10-8s)2 + (5.20 x 106m/s)(4 x 10-8s) = 0.104m.

E2-44 The average velocity of the electron was

1V:1:,av = 2(1.5 x 105 m/s + 5.8 x 106 m/s) = 3.0 x 106 m/s.

The time to travel the distance of the runway at this average velocity is

t = (0.012m)/(3.0 x 106m/s) = 4.0 x 10-9s.

The acceleration is

ax = (v x -vOx)/t = ((5.8 x 106 m/s) -(1.5 x 105 m/s))/(4.0 x 10-9 s) = 1.4 x 1015m/s2,

~ It will be easier to solve the problem ir we change the units ror the initial velocity,

1000 m

km

and then applying Eq. 2-26,

v x :

(O) :

1 2

-202 m s :

The problem asks íor this in terms oí g, so

= VOx + axt,

= (283 mis) + ax(1.4 s),

= ax.

9

9:8-;;;¡;;--202 m/s2 = 21g.

19

~«34m) -(16 m(s)(4.0 s)) = -3.75m/s2.

E2-50 (a) The deceleration is found from

2 2ax = "i2(x -vot) = ---

(b) The impact speed is

v x = VOx + axt = (16m/s) + (-3.75m/s2)(4.0s) = 1.0m/s.

E2-51 Assuming the drops fall from rest, the time to fall is

- {if; 2y - J ~S2(-1700m) -19t ---s.

ay (-9.8 m/s2)

The velocity of the falling drops would be

Vy = ayt = (-9.8m/s2)(19s) = 190m/s,

or about 2/3 the speed of sound.

E2-52 Solve the problem out of order .(b) The time to fall is

-fiY- 12(-120m)-t- V ~ -V (-9.8m/s2) -4.9s.

(a) The speed at which the elevator hits the ground is

Vy = ayt = (-9.8m/s2)(4.9s) = 48m/s.

( d) The time to fall half-way is

- {jj.2y - J ~s2(-60m) - 3 5t s.ay ( -9.8 m/s2)

(C) The speed at the half-way point is

Vy = ayt = (-9.8m/s2)(3.5s) = 34m/s.

~ The initial velocity of the "dropped" wrench would be zero. I choose vertical to be alongthe y axis with up as positive, which is the convention of Eq. 2-29 and Eq. 2-30. It turns out thatit is much easier to solve part (b) before solving part (a).

(b) We solve Eq. 2-29 for the time of the fall.

Vy = VOy -gt,

( -24.0 m/s) = (0) -(9.8 m/s2)t,

2.45 s = t.

(a) Now we can use Eq. 2-30 to find the height from which the wrench fell.

1 2y = Yo + VOyt --gt ,

21 2 2

(0) = Yo + (0)(2.45 s) --(9.8 m/s )(2.45 s) ,2

0 = Yo -29.4 m

We have set y = 0 to correspond to the final position ofthe wrench: on the ground. This results inan initial position of Yo = 29.4 m; it is positive because the wrench was dropped from a point above

where it landed.

21

E2-54 (a) It is easier to solve the problem from the point of view of an object which falls from thehighest point. The time to fall from the highest point is

- fj;2y - V~~2(-53.7m) - 331t s.ay ( -9.81 m/s2)

The speed at which the object hits the ground is

Vy = ayt = ( -9.81 m/s2)(3.31 s) = -32.5 m/s.

But the motion is symmetric, so the object must have been launched up with a velocity of Vy =32.5 m/s.

(b) Double the previous answer; the time of flight is 6.62 s.

E2-55 (a) The time to fall the first 50 meters is

E2-56 The rock returns to the ground with an equal, but opposite, velocity. The acceleration isthen

ay = (( -14.6m/s) -(14.6 m/s))/(7.72s) = 3.78m/s2,

That would put them on Mercury.

~ (a) Solve Eq. 2-30 for the initial velocity. Let the distances be measured from the groundso that Yo = 0.

1 2y = Yo + VOyt -2gt ,

(36.8 m) == (O) + voy(2.25 s) -~(9.8 mis2)(2.25 8)2,

36.8 m = voy(2.25 s) -24.8 m,

27.4 mis = VOy.

(b) Salve Eq. 2-29 far the velacity, using the result fram part (a).

Vy = VOy -gt,

Vy = (27.4 m/s) -(9.8 m/s2)(2.25 s),

Vy = 5,4 m/s.

(c) We need to solve Eq. 2-30 to find the height to which the ball rises, but we don't know howlong it takes to get there. So we first solve Eq. 2-29, because we do know the velocity at the highestpoint (Vy = 0).

VOy -gt,

(27.4 mis) -(9.8 mis2)t,f

v y =

(O) =

2.8 s =

22

And then we find the height to which the object rises,

1 2y = Yo + vOyt -2gt ,

Iy = {0) + {27.4 m/s){2.8 s) -2{9.8 m/s2){2.8 8)2,

y = 38.3m.

This is the height as measured from the ground; so the hall rises 38.3- 36.8 = 1.5 m ahove the pointspecified in the prohlem.

The hall hits the ground with a velocity of

Vy = ayt = (-9.8m/s2)(0.67s) = -6.6m/s.

The ball then bounces up to a height of 1.9 m. It is easier to solve the falling part of the motion,and then apply symmetry. The time is would take to fall is

/2Y I, 2(-1.9m) = 0.628.= v ( -9.8 m/s2)

t"=v~

The hall hits the ground with a velocity oí

V1l = a1lt = (-9.8m/s2)(0.62s) = -6.1m/s.

But we are interested in when the ball moves up, so v1l = 6.1 m/s.

The"acceleration while in contact with the ground is

ay = ((6.1 m/s) -( -6.6m/s))/(O.O96s) = 130m/s2.

E2-59 The position as a function of time for the first object is

1 2yl = -2gt ,

The position as a function of time for the second object is

y2 = -.!g(t -1 s)22

The difference,~y= Y2 -yl = frac12g ((2s)t -1) ,

is the set equal to 10 m, so t = 1.52 s.

E2-60 Answer part (b) first.(b) Use the quadratic equation to solve

1(-81.3m) = 2(-9.81m/s2)t2 + (12.4m/s)t

for time. Get t = -3.0 s and t = 5.53 s. Keep the positive answer .

(a) Now find final velocity from

Vy = (-9.8m/s2)(5.53s) + (12.4m/s) = -41.8m/s.

23

E2-58 The time it takes for the hall to fall 2.2 m is

~ The total time the pot is visible is 0.54 s; the pot is visible for 0.27 s on the way down.We'll define the initial position as the highest point and make our measurements from there. ThenYo = 0 and VOy = 0. Define tl to be the time at which the falling pot passes the top of the windowYl, then t2 = tl + 0.27 s is the time the pot passes the bottom of the window y2 = yl -1.1 m. Wehave two equations we can write, both based on Eq. 2-30,

Yl = I 2Yo + VOytl -29tl'

I 2(O) + (O)tl -29tl'Yl =

and

y2 = 1 2Yo + VOyt2 -2gt2'

1{0) + {0)t2 -29{tl + 0.278)2,yl -1.1 m =

Isolate Yl in this last equation and then set the two expressions equal to each other so that we cansolve for tl,

1.) 2

1.1 m- 29(tl + 0.27 s ,=

1 m- ~g(t~ + [0.54 8]tl + 0.07382),

1--

g t22 1

1--

g t22 1

=

~g([0.54 8]tl + 0.073 82).o 1.1 m=

This last line can be directly solved to yield tl = 0.28 s as the time when the falling pot passes thetop ofthe window. Use this value in the first equation above and we can find Yl = -~(9.8 m/s2)(0.28S)2 = -0.38 m. The negative sign is because the top of the window is beneath the highest point, sothe pot must have risen to 0.38 m above the top of the window.

P2-1 (a) The net shift is v(22m)2 + (17m)2) = 28m.(b) The vertical displacement is (17m) sin(52°) = 13m.

The vertical displacement is 2r =P2-2 Wheel "rolls" through halí oí a turn, or 1rr = 1.41 m.

0.90 m. The net displacement is

The angle is(} = tan-l(O.90m)/(1.41m) = 33°.

~ We align the coordinate system so that the origin corresponds to the starting position ofthe fl.y and that all positions inside the room are given by positive coordinates.

(a) The displacement vector can just be written,

~r = {10 ft)i + (12 ft)j + (14 ft)k.

(b) The magnitude of the displacement vector is 16.r¡ = y102 + 122 + 142 ft= 21 ft.

'>A

( c ) The straight line distance between two points is the shortest possible distance, so the lengthof the path taken by the fiy must be greater than or equal to 21 ft.

(d) If the fiy walks it will need to cross two faces. The shortest path will be the diagonal acrossthese two faces. If the lengths of sides of the room are ll, l2, and l3, then the diagonallength acrosstwo faces will be given by

v (ll + l2)2 + l~,

where we want to choose the li from the set of 10 ft, 12 ft, and 14 ft that will minimize the length.The minimum distance is when II = 10 ft, l2 = 12 ft, and l3 = 14. Then the minimal distance thefly would walk is 26 ft.

P2-4 Choose vector a to lie on the x axis. Then a = ai and b = bxi + byj where bx = b cos (} andby = bsin(}. The sum then has components

T", = a + bcos{} and Ty = bsin{}.

Then

2 2(a+ bcosO) + (bsinO) ,

a2+ 2abcosO + b2.

r2 =

P2-5 (a) Average speed is total distance divided by total time. Then

(b) Average speed is total distance divided by total time. Then

vav = (d/2) + {d/2) .(d/2)/(35.0 mi/hr) + (d/2)/(55.0 mi/hr) = 42.8 ml/hr.

P2-6 (a) We'll do jU8t one together. How about t = 2.08?

x = (3.0 m/s)(2.0 s) + -4.0m/s2){2.0S)2 + {1.Om/s3){2.0s)3 = -2.0m.

and

25

~ (a) Assume the bird has no size, the trains have some separation, and the bird is justleaving one of the trains. The bird will be able to fiy from one train to the other be/ore the twotrains collide, regardless of how close together the trains are. After doing so, the bird is now on theother train, the trains are still separated, so once again the bird can fiy between the trains beforethey collide. This process can be repeated every time the bird touches one of the trains, so the birdwill make an infinite number of trips between the trains.

(b) The trains collide in the middle; therefore the trains collide after (51 km)/(34 km/hr) = 1.5hr. The bird was fiying with constant speed this entire time, so the distance fiown by the bird is (58km/hr)(1.5 hr) = 87 km.

P2-8 (a) Start with a perfect square:

(v¡ -V2)2

V~ + V~

(V~ + V~)t¡t2

d~ + d~ + (v~ + V~)t¡t2

(V~t¡ + V~t2)(t¡ + t2)

v~t¡ + V~t2

>

o,

2vlv2,

2V¡V2t¡t2,

d~ + d~ + 2V¡V2t¡t2,

(d¡ + d2)2,

d¡ + d2

>>

>

>

d1 + d2

V1 d1 + v2d2

d1 + d2

f;:+t;'v¡t¡ + V2t2

t¡ + t2

Actually, it only works ir d¡ + d2 > O!(b) Ir v¡ = V2.

P2-9 (a) The average velocity during the time interval is vav = 6.x/6.t, or

vav = (A+B(3s)3) =(A+B(2s)3) = (1.50 cm/s3)(19s3)/(ls) = 28.5 cm/s.

(38) -(28)

(b) v = dx/dt = 3Bt2 = 3(1.50 cm/s3)(2s)2 = 18 cm/s.(c) v = dx/dt = 3Bt2 = 3(1.50 cm/s3)(3s)2 = 40.5 cm/s.(d) v = dx/dt = 3Bt2 = 3(1.50 cm/s3)(2.5s)2 = 28.1 cm/s.

(e) The midway position is (Xf + xi)/2, or

Xmid = A + B[(3s)3 + (2s)3)]/2 = A + (17.5s3)B.

This occurs when t = ~. The instantaneous velocity at this point is

v = dx/dt = 3Bt2 = 3(1.50 cm/s3)(~)2 = 30.3 cm/s.

P2-10 Consider the figure below.

x x

""'--Po;; 1 t t

(a) (b) (c) (d)

26

~ (a) The average velocity is displacement divided by change in time,

(2.0 m/s3)(2.0 8)3- (2.0 m/s3){1.0 8)3'Vav = =

(2.0 s) -(1.0 s)

14.0 m

1.0 s

= 14.0 m/s.

The average acceleration is the change in velocity. So we need an expression for the velocity, whichis the time derivative of the position,

18.0 m/s1.0 s

= 18.0 m/s2.

(b) The instantaneous velocities can be íound directly from v = (6.0 m/s2)t2, so v(2.0 s) = 24.0m/s and v(1.0 s) = 6.0 m/s. We can get an expression íor the instantaneous acceleration by takingthe time derivative oí the velocity

Then the instantaneous accelerations are a(2.0 s) = 24.0 m/s2 and a(1.0 s) = 12.0 m/s2( C ) Since the motion is monotonic we expect the average quantities to be somewhere between

the instantaneous values at the endpoints of the time interval. Indeed, that is the case.

x(m)

o 2 4 6 8 10

t(s)

o 2 4 6 8 10

t(s)

o 2 4

t(s)

6 8 10

P2-13 Start with Vf = Vi + at, but Vf = O, so Vi = -at, then

1 1 1x = -at2 + Vit = -at2 -at2 = --at2

2 2 2

so t = p¡x¡a = J=2(19.2 ft)/(-32 ft/S2) = 1.10s. Then Vi = -(-32 ft/s2)(1.10s) = 35.2 ft/s.Converting,

35.2 ft/s(1/5280 mi/ft)(3600 s/h) = 24 mi/h.

27

From this we find average acceleration

P2-14 (b) The average speed during while traveling the 160 m is

vav = (33.0m/s + 54.0m/s)/2 = 43.5m/s.

The time to travel the 160 m is t(a) The acceleration is

(160 m)/(43.5m/8) =;= 3.688.

2(160m) 2(33.0 m/8 }

(3.688}2 -(3.688}= 5.69m/s2,

(c) The time required to get up to a speed of 33 mis is

t = v/a = (33.0m/8)/(5.69m/82) = 5.808.

( d ) The distance moved from start is

P2-15 (a) The distance traveled during the reaction time happens at constant speed; treac = d/v =(15m)/(20m/s) = 0.75s.

(b) The braking distance is proportional to the speed squared (look at the numbers!) and inthis case is dbrake = v2/(20m/s2). Then dbrake = (25m/s)2/(20m/s2) = 31.25m. The reaction timedistance is dreac = (25m/s)(0.75s) = 18.75m. The stopping distance is then 50 m.

P2-16 (a) For the car Xc = act2/2. For the truck Xt = Vtt. Set both Xi to the same value, andthen substitute the time from the truck expression:

act2/2 = ac(x/vt)2/2,x

or

x = 2Vt2/ac =2(9.5m/s)2/(2.2m/s) = 82m.

(b) The speed oí the car will be given by Vc = act, or

Vc = act = acx/vt = (2.2 m/s)(82 m)/(9.5m/s) = 19m/s.

~ The runner covered a distance d1 in a time interval tl during the acceleration phase anda distance d2 in a time interval t2 during the constant speed phase. Since the runner started fromrest we know that the constant speed is given by v = atl, where a is the runner's acceleration.The distance covered during the acceleration phase is given by

1 2dl = 2atl.

The distance covered during the constant speed phase can also be found from

d2 = Vt2 = atlt2.

We want to use these two expressions, along with d¡ + d2 = 100 m and t2 = (12.2 s) -t¡, to get

100 m1 2= d¡ + d2 = 2at¡ + at¡(12.2 S -t¡),

1 2 ( )= -2at¡+ a 12.2 s t¡,

= -(1.40 m/s2)t~ + (34.2 m/s)t¡.

28

This 185t expression is quadratic in tl, and is solved to give tl = 3.40 s or tl = 21.0 s. Since the raceonly 185ted 12.2 s we can ignore the second answer .

(b) The distance traveled during the acceleration ph85e is then

1d¡ = 2at~ = {1.40 m/s2){3.40s)2 = 16.2 m.

P2-18 (a) The ball will return to the ground with the same speed it was launched. Then the totaltime of flight is given by

t = {Vf -Vi)/g = ( -25m/s -25 m/s)/(9.8m/s2) = 5.1 s.

(b) For small quantities we can think in terms of derivatives, so

8t = (8vr -8vj)/g,

orT=2t:/g.

P2-19 Use y = -gt2/2, but only keep the absolute value. Then Y50 = (9.8m/s2)(0.05s)2/21.2 cm; YI00 = (9.8m/s2)(0.10s)2/2 = 4.9 cm; Y150 = (9.8m/s2)(0.15s)2/2 = 11 cm; Y200(9.8m/s2)(0.20s)2/2 = 20 cm; Y250 = (9.8m/s2)(0.25s)2/2 = 31 cm.

P2-20 The truck wi}} move 12 m in (12 m)/(55 km/h) = 0.785 s. The app}e wi}} fa}} y = -gt2 /2 =-(9.81m/s2)(0.785s)2/2 = -3.02 m.

~ The rocket travels a distance d1 = !at~ = !(20 m/s2)(60 S)2 = 36,000 m during the accel-eration phase; the rocket velocity at the end of the acceleration phase is v = at = (20 m/s2)(60 s) =1200 m/s. The second half of the trajectory can be found from Eqs. 2-29 and 2-30, with Yo = 36,000m and VOy = 1200 m/s.

(a) The highest point of the trajectory occurs when Vy = 0, so

v y

(O)

122 s

= VOy -gt,

= {1200 mis) -{9.8 mis2)t,

This time is used to find the height to which the rocket rises,

1 2y = Yo + VOyt -2gt ,

1(36000 m) + (1200 m/8)(1228) -2(9.8 m/82)(122 8)2 = 110000 m.

(b) The easiest way to find the total time of flight is to solve Eq. 2-30 for the time when therocket has returned to the ground. Then

1 2Yo + VOyt -2gt ,y =

(O) -

This quadratic expression has two solutions for t; one is negative so we don 't need to worry aboutit, the other is t = 270 s. This is the free-fall part of the problem, to find the total time we need toadd on the 60 seconds of accelerated motion. The total time is then 330 seconds.

')(\

P2-22 (a) The time required for the player to "fall" from the highest point a distance ofy = 15 cmis V2Y79j the total time spent in the top 15 cm is twice this, or 2V2Y79 = 2V2(0.15 m)/(9.81 m/s) =0.350 s.

(b The time required for the player to "fall" from the highest point a distance of 76 cm is2(0.76m)/(9.81m/s) = 0.394s, the time required for the player to fall from the highest point a

distance of (76 -15 = 61) cm is V2(0.61m)/g = 0.353s. The time required to fall the bottom 15cm is the difference, or 0.041 s. The time spent in the bottom 15 cm is twice this, or 0.081 s.

P2-23 (a) The average speed between A and B is vav = (v + v/2)/2 = 3v/4. We can also writevav = (3.0m)/~t = 3v/4. Finally, v/2 = v -g~t. Rearranging, v/2 = g~t. Combining all of the

above,v - (~ ) or V2 = {8.0 m)9.

2-9 3v

Then v = v(8.0m)(9.8:iñ:/s2) = 8.85m/s.(b) The time to the highest point above B is v /2 = gt, so the distance above B is

2

y=-~t2+~t=-~(~). +~(~)=~.

Then y = (8.85 m/s)2/(8(9.8m/s2)) = 1.00m.

P2-24 (a) The time in free fall is t = F'iiiTii= V-2(-145m)/(9.81m/s2)= 5.44s.(b) The speed at the bottom is v = -gt = -(9.81m/s2)(5.44s) = -53.4m/s.(c) The time for deceleration is given by v = -25gt, or t = -( -53.4m/s)/(25 x 9.81 /S2) =

0.218 s. The distance through which deceleration occurred is

25gy = -t2 + vt = (123m/s2)(0.218s)2 + (-53.4m/s)(0.218s) = -5.80m.2

~ Find the time she fell from Eq. 2-30,

(O ft) = (144 ft) + (o)t -~(32 ft/S2)t2,

which is a simple quadratic with solutions t = :1:3.0 s. Only the positive solution is of interest. Usethis time in Eq. 2-29 to find her speed when she hit the ventilator box,

Vy = (0) -(32 ft/s2)(3.0 s) = -96 ft/s.

This becomes the initial velocity for the deceleration motion, so her average speed during decelerationis given by Eq. 2-27,

1 1Vav,y = 2 (Vy + VOy) = 2 ((O) + ( -96 ft/s)) = -48 ft/s.

This average speed, used with the distance of 18 in (1.5 ft), can be used to find the time of deceleration

Vav,y = ~y/ ~t,

and putting numbers into the expression gives ~t = 0.031 s. We actually used ~y = -1.5 ft, wherethe negative sign indicated that she was still moving downward. Finally, we use this in Eq. 2-26 tofind the acceleration,

(O) = ( -96 ft/s) + a(0.031 s),

which gives a = +3100 ft/S2. In terms oí g this is a = 97g, which can be íound by multiplyingthrough by 1 = g/(32 ft/S2).

30

P2-26 Let the speed of the disk when it comes into contact with the ground be Vlj then theaverage speed during the deceleration process is vl/2j so the time taken for the deceleration processis tl = 2d/Vl, where d = -2mm. Butd is also give by d = at~/2 + Vltl, so

1009

2

d=2d

) 2

~ +VI

or v~ = -200gd. The negative signs are necessary!.The disk was dropped from a height h = -y and it first came into contact with the ground when

it had a speed of Vl. Then the average speed is vl/2, and we can repeat most of the above (excepta = -g instead of 100g), and then the time to fall is t2 = 2y/Vl,

or v~ = -2gy. The negative signs are necessary!.Equating, y = 100d = 100(-2mm) = -0.2m, so h = O.2m. Note that although 100g's sounds

like plenty, you still shouldn't be dropping your hard disk drive!

P2-27 Measure from the feet! Jim is 2.8 cm tall in the photo, so 1 cm on the photo is 60.7cm in real-life. Then Jim has fallen a distance Yl = -3.04m while Clare has fallen a distanceY2 = -5.77 m. Clare jumped first, and the time she has been falling is t2j Jim jumped seconds, thetime he has been falling is tl = t2 -6.t. Then Y2 = -gt~/2 and Yl = -gt~/2, or t2 = piii;Tii =

..¡-2(-5.77m)/(9.81m/s2) = 1.08s and tl = pi;j;Jij = ..¡-2(3.04m)/(9.81m/s2) = 0.79s. So

Jim waited 0.29 s.

P2-28 (a) Assuming she starts from rest and has a speed of Vl when she opens her chute, thenher average speed while falling freely is vl/2, and the time taken to fall yl = -52.0m is tl = 2yl Vl.Her speed Vl is given by Vl = -gtl, or v~ = -2gyl. Then Vl = --2(9.81 m/s2)( -52.0m) =-31.9m/s. We must use the negative answer, because she fall down! The time in the air is thentl = ,pii¡J¡j = V-2(52.0m)/(9.81 m/s2) = 3.26s.

Her final speed is V2 = -2.90m/s, so the time for the decelera:tion is t2 = (V2 -vl)/a, wherea = 2.10m/s2. Then t2 = (-2.90m/s --31.9m/s)/(2.10m/s2) = 13.8s.

Finally, the total time of fiight is t = tl + t2 = 3.26 s + 13.8 s = 17.1 s.(b) The distance fallen during the deceleration phase is

9 2y2 = --t2 + Vlt2 = -.2

The total distance fallen is y = Yl + y2 = -52.0 m -240 m = -292 m. It is negative because she was

falling down.

~(13.8s)2 + (-31.9 mjs)(13.8s) = -240m.2

P2-29 Let the speed of the bearing be Vl at the top of the windows and V2 at the bottom.These speeds are related by V2 = Vl -gt12, where t12 = 0.125 s is the time between when thebearing is at the top of the window and at the bottom of the window. The average speed isvav = (Vl + v2)/2 = Vl -gt12/2. The distance traveled in the time t12 is y12 = -1.20m, so

2Vltl2 -gtl2/2,

y12 = Vavt12

and expression that can be solved for Vl to yield

31

Now that we know Vl we can find the height ofthe building above the top ofthe window. The timethe object has fallen to get to the top ofthe window is tl = -Vl/g = -(-8.99m/s)/(9.81m/s2) =0.916 m.

The total time for falling is then (0.916s) + (0.125s) + (1.0s) ;= 2.04s. Note that we rememberedto divide the last time by two! The total distance from the top of the building to the bottom is then

y = -gt2/2 = -(9.81m/s2)(2.04s)2/2 = 20.4 m.

P2-30 Each hall falls from a highest point through a distance of 2.0m in

t = V-2(2.0m)/(9.8m/82) = 0.6398.

The time each ball spends in the air is twice this, or 1.28 s. The frequency of tosses per ball is thereciprocal, f = l/T = 0.781 s-l. There are five ball, so we multiply this by 5, but there are twohands, so we then divide that by 2. The tosses per hand per second then requires a factor 5/2, andthe tosses per hand per minute is 60 times this, or 117.

~ Assume each hand can toss n objects per second. Let 7" be the amount of time that anyone object is in the air. Then 2n7" is the number of objects that are in the air at any time, wherethe "2" comes from the fact that (most?) jugglers have two hands. We'll estimate n, but 7" can befound from Eq. 2-30 for an object which falls a distance h from rest:

o = h + (O)t -~gt2,

solving, t = .JihTg. But T is twice this, because the object had to go up before it coulcl come clown.So the number of objects that can be jugglecl is

4 n .j2h[¡j

We estímate n = 2 tosses/second. So the maxímum number of objects one could juggle to a heíghth would be

3.6yh/meters.

P2-32 ( a) We need to look up the height of the leaning tower to solve this! If the height is h = 56 m,then the time taken to fall a distance h = -Yl is tl = ,piij;yij = V-2(-56m)/(9.81m/s2) = 3.4s.The second object, however, has only fallen a a time t2 = tl -~t = 3.3 s, so the distance the secondobject falls is y2 = -gt~/2 = -(9.81m/s2)(3.3s)2/2 = 53.4. The difference is yl -y2 = 2.9m.

(b) If the vertical separation is ~y = 0.01 m, then we can approach this problem in terms ofdifferentials,

t5y = at t5t,

so t5t = (0.01 m)/[((9.81 m/s2)(3.4s)] = 3 x 10-4 s.

P2-33 Use symmetry, and focus on the path from the highest point downward. Then ~tu = 2tu ,where tu is the time from the highest point to the upper level. A similar expression exists forthe lower level, but replace U with L. The distance from the highest point to the upper level isYu = -gtt /2 = -g(~tu /2)2/2. The distance from the highest point to the lower level is YL =-gtl/2 = -g(~tL/2)2/2. Now H = Yu -YL = -g~tt/8 --g~tl/8, which can be written as

8H9 = ~ti -~tt .

32

E3-1 The Earth orbits the sun with a speed of 29.8 km/s. The distance to Pluto is 5900xl06 km.The time it would take the Earth to reach the orbit of Pluto is

t = (5900xlO6 km)/(29.8 km/s) = 2.0xlO8s,

or 6.3 years!

E3-2 (a) a = F/m = (3.8N)/(5.5kg) = O.69m/82,(b) t = Vf/a = (5.2 m/8)/(0.69m/82) =7.58.(c) x = at2/2 = (0.69m/82)(7.58)2/2 = 20m.

~ Assuming constant acceleration we can find the average speed during the interval from Eq.2-27

1 1 ( 6 ) 6vav,x = 2 (vx +vOx) = 2 (5.8x 10 m/s) + (0) = 2.9x 10 m/s.

From this we can find the time spent accelerating from Eq. 2-22, since ~x = vav,x~t. Putting inthe numbers ~t = 5.17 x 10-9s. The acceleration is then

-.~vx (5.8x 106 m/s) -(0) - 11 1015 / 2ax- -.X ms.

~

--

~t -{5.l7xlO-9s)

The net force on the electron is from Eq. 3-5,

LFx = max = (9.llxlO-31kg)(l.lxlO15m/s2) = l.OxlO-15N.

E3-4 The average speed while decelerating is vav = 0.7 x 107 m/s. The time of deceleration ist = x/vav = (1.0x10-14m)/(0.7x107m/s) = 1.4x10-21S. The deceleration is a = ~v/t = (-1.4x107m/s)/(1.4x10-21 S) = -1.0x1028m/s2. The force is F = ma = (1.67x10-27kg)(1.0x1028m/s2) =

17N.

~ The net force on the sled is 92 N-90 N= 2 N; subtract because the forces are in oppositedirections. Then

-¿Fx -(2N)ax m I = 8.0x lO-2m/s2,

~

E3-6 53 km/hr is 14.7m/s. The average speed while decelerating is v..v = 7.4m/s. The timeof deceleration is t = x/v..v = (0.65m)/(7.4m/s) = 8.8x 10-2s. The deceleration is a = f).v/t =(-14.7m/s)/(8.8x10-2s) = -17x102m/s2. The force is F = ma = (39kg)(1.7x102m/s2) = 6600N.

E3-7 Vertical acceleration is a = F/m = (4.5 x 10-15N)/(9.11 x 10-31kg) = 4.9 x 1015m/s2. Theelectron moves horizontally 33 mm in a time t = x/vx = (0.033m)/(1.2x107m/s) = 2.8x10-9s.The vertical distance deflected is y = at2/2 = (4.9 x 1015m/s2)(2.8 X 10-9 s)2 /2 = 1.9 x 10-2m.

E3-8 (a) a = F/m = (29N)/(930kg) = 3.1x10-2m/s2.(b) x = at2/2 = (3.1 x 10-2 m/s2)(86400s)2 /2 = 1.2x 108 m.(c) v = at = (3.1 x 10-2 m/s2)(86400s) = 2700m/s.

33

~ Write the expression for the motion of the first object as ¿::; Fx = mlalx and that of thesecond object as ¿ Fx = m2a2x. In both cases there is onlyone force, F, on the object, so ¿Fx = F.We will solve these for the mass as ml = F / al and m2 = F / a2 .Since al > a2 we can conclude thatm2 > ml

(a) The acceleration of and object with mass m2 -ml under the influence of a single force ofmagnitude F would be

F F 1a==-m2 -ml F/a2 -F/al 1/(3.30m/s2) -1/(12.0m/s2) ,

which has a numerical value of a = 4.55 m/s2.(b) Similarly, the acceleration of an object of mass m2 +ml under the influence of a force of

magnitude F would be

1 1a---1/a2 + r/al -1/(3.30m/s2) + 1/(12.0m/s2) ,

which is the same as part (a) except for the sign change. Then a = 2.59 m/s2.

E3-10

Then

(a) The required acceleration is a = v/t O.lc/t. The required force is F = ma = O.lmc/t.

F = O.l(l200xlO3kg)(3.00xlO8m/s)/(2.59xlO5S) = l.4xlO8N,

andF = 0.1(1200x103kg)(3.00x108m/s)/(5.18x106s) = ~.9x106N,

(b) The distance traveled during the acceleration phase is Xl = at~/2, the time required totravel the remaining distance is t2 = X2/V where X2 = d- Xl. d is 5 light-months, or d = (3.00 x108m/s)(1.30x107s) = 3.90x1015m. Then

2d-at~2v

2d-vti.

2v

d-Xl

Vt = t¡ + t2 = t¡ + =t¡ + =tl +

If t¡ is 3 days, then

2(3.90xlO15m) -(3.00xlO7m/s)(2.59xlO5s)t = (2.59xlO5s) + = 1.30x108s = 4.12 yr,2(3.00 x 107m/s)

if t¡ is 2 months, then

2(3.90 x 1015m) -(3.00 x 107 m/8)(5.18 x 1068)t = {5.18 x 1068) + = 1.33x108s = 4.20 yr,

2(3.00 x lO7ffi7S)

~ (a) The net force on the second block is given by

LFx = m2a2x = (3.8kg)(2.6m/s2) = 9.9N.

There is only one (relevant) force on the block, the force of block 1 on block 2.(b) There is only one (relevant) force on block 1, the force of block 2 on block 1. By Newton's

third law this force has a magnitude of 9.9, N. Then Newton's second law gives EFx = -9.9N= mlalx = (4.6 kg)alx. So alx = -2.2 m/s2 at the instant that a2x = 2.6 m/s2.

E3-12 (a) W = (5.00 lb)(4.448N/lb) = 22.2N; m = W/g = (22.2N)/(9.81m/s2) = 2.26kg.(b) W = (240 lb)(4.448N/lb) = 1070N; m = W/g = (1070N)/(9.81m/s2) = 109kg.(c) W = (3600 lb)(4.448N/lb) = 16000N; m = W/g = (16000N)/(9.81m/s2) = 1630kg.

34

E3-13 (a) W = (1420.00 lb) (4.448 N/lb) = 6320Nj m = W/g = (6320N)/(9.81 m/s2) = 644kg.(b) m = 412kg; W = mg = (412kg)(9.81m/s2) = 4040 N.

E3-14 (a) W = mg = (75.0kg)(9.81m/s2) = 736N.(b) W = mg = (75.0kg)(3.72m/s2) = 279N.(c) W = mg = (75.0kg)(Om/s2) = ON.

(d) The mass is 75.0 kg at alllocations.

~ If 9 = 9.81 m/s2, then m = W/g = (26.0 N)/(9.81 m/s2) = 2.65 kg.(a) Apply W = mg again, but now 9 = 4.60 m/s2, so at this point W = (2.65 kg)(4.60 m/s2) =

12.2 N.(b) If there is no gravitational force, there is no weight, because 9 = 0. There is still mass,

however, and that mass is still 2.65 kg.

E3-16 Upward force balances weight, so F = W = mg = (l2000kg)(9.8lm/s2) = l.2x lO5N.

E3-17 Mass is m = W/g; net force is F= ma, or F = Wa/g. Then

(3900 lb)(13 ft/s2)/(32 ft/s2) = 1600 lb.F

(523 kg)(247 m/s2) =E3-18 a = ~v/~t = (450 m/s)/(l.82s) = 247m/s2. Net force is F = ma

l.29x lO5N.

~ ¿ Fx = 2(1.4 x 105 N) = max. Then m 1.22 x 105 kg and

w = mg = (l.22x lO5kg)(9.8l m!s2) = l.20x lO6N.

E3-20 Do part (b) first; there must be a 10 lb force to support the mass. Now do part (a), butcover up the left hand side of both pictures. If you can 't telJ which picture is which, then they mustboth be 10 lb!

E3-21 (b) Average speed during deceleration is 40 km/h, or 11 m/s. The time taken to stop thecar is then t = X/vav = (61 m)/(11 m/s) = 5.6s.

(a) The deceleration is a = Av/~t = (22m/s)/(5.6s) = 3.9m/s2. The braking force is F =ma = Wa/g = (13,000N)(3.9m/s2)/(9.81m/s2) = 5200N.

(d) The deceleration is same; the time to stop the car is then ~t = ~v/a = (11 m/s)/(3.9m/s2) =2.8s.

(c) The distance traveled during stopping is x= vavt = (5.6m/s)(2.8s) = 16m.

E3-22 Assume acceleration of free fall is 9.81 m/s2 at the altitude of the meteor. The net force isFnet = ma = (0.25kg)(9.2m/s2) = 2.30N. The weight is W = mg = (0.25kg)(9.81m/s2) = 2.45N.The retarding force is Fnet -W = (2.3N) -(2.45N) = -0.15N.

~ (a) Fincl the time cluring the "jump clown" phase from Eq. 2-30.

1(O m) = (0.48 m) + (o)t -2(9.8 m/s2)t2,

which is a simple quadratic with solutions t = :!:0.31 s. Use this time in Eq. 2-29 to find his speedwhen he hit ground,

v y = (O) -(9.8 m/s2)(0.31 s) = -3.1 m/s.

35

This becomes the initial velocity for the deceleration motion, so his average speed during decelerationis given by Eq. 2-27,

1 1Vav,y = 2 (v y + VOy) = 2 ((O) + ( -3.1 mis)) = -1.6 mis.

Vav,y = 6.y/6.t,

and putting numbers into the expression gives 6.t = 0.014 s. Finally, we use this in Eq. 2-26 to findthe acceleration,

(O) = ( -3.1 mis) + a(0.014 s),

E3-24 The average speed of the salmon while decelerating is 4.6 ft/s. The time required to stopthe salmon is then t = X/Vav = {0.38 ft)/{4.6 ft/s) = 8.3 x 10-2s. The deceleration of the salmonis a = Av/At = {9.2 ft/s)/{8.2-2s) = 110 ft/s2. The force on the salmon is then F = Wa/g ={19 lb){110 ft/s2)/{32 ft/s2) = 65 lb.

~ From appendix G we find 11b = 4.448 N; so the weight is (100 lb)(4.448 N/11b) = 445N; similarly the cord will break if it pulls upward on the object with a force greater than 387 N.The mass of the object is m = W/g = (445 N)/(9.8 m/s2) = 45 kg.

There are two vertical forces on the 45 kg object, an upward force from the cord Foc (which has amaximum value of387 N) and a downward force from gravity FoG. Then EFy = Foc-FoG = (387N) -(445 N) = -58 N. Since the net force is negative, the object must be accelerating downwardaccording to

E3-26 (a) Constant speed means no acceleration, hence no net forcej this means the weight isbalanced by the force from the scale, so the scale reads 65 N .

(b) Net force on mass is Fnet = ma = Wa/g = (65N)(-2.4m/s2)/(9.81m/s2) = -16N.. Sincethe weight is 65 N, the scale must be exerting a force of ( -16 N) -I -65 N) = 49 N .

E3-27 The magnitude of the net force is W- R = (1600kg)(9.81 m/s2) -(3700N) = 12000N.The acceleration is then a = F/m = (12000N)/(1600kg) = 7.5m/s2. The time to fall is

t = V~ = V2(-72m)/(-7.5m/s2) = 4.4s.

The final speed is v = at = (-7.5m/s2)(4.4s) = 33m/s. Get better brakes, eh?

E3-28 The average speed during the acceleration is 140 ft/s. The time for the plane to travel 300ft is

t = X/Vav = (300 ft)/(140 ft/8)= 2.148.

The acceleration is then

a = ~v/~t = (280 ft/s)/(2.14s) = 130 ft/S2.

The net force on the plane is F = ma = Wa/g = (52000 Ib)(130 ft/s2)/(32 ft/s2) = 2.1x105Ib.The force exerted by the catapult is then 2.1 x 105lb -2.4 x 104 lb = 1.86 x 105 lb.

36

This average speed, used with the distance of -2.2 cm (-0.022 m), can be used to find the time ofdeceleration

which gives a = 220 m/s2.

(b) The average net force on the man is

E3-29 (a) The acceleration of a hovering rocket is O, so the net force is zero; hence the thrust mustequal the weight. Then T = W = mg = (51000kg)(9.81m/s2) = 5.0x105N.

(b) If the rocket accelerates upward then the net force is F = ma = (51000kg)(18m/s2) =9.2x 105 N. Now Fnet = T- W, so T = 9.2x 105 N + 5.0x 105N = 1.42x 106N.

E3-30 (a) Net force on parachute + person system is Fnet = ma = (77kg+5.2kg)(-2.5s2) =-210N. The weight ofthe system is W = mg = (77kg+5.2kg)(9.81s2) = 810N. IfP is the upwardforce of the air on the system (parachute) then P = F net + W = ( -210 N) + (810 N) = 600 N .

(b) The net force on the parachute is Fnet = ma = (5.2kg)(-2.5s2) = -13N. The weight oftheparachute is W = mg = (5.2 kg)(9.81 m/s2) = 51 N. If D is the downward force of the person on theparachute then D = -Fnet -W + P = -(-13N) -(51N) + 600N = 560N.

~ (a) The total mass of the helicopter+car system is 19,500 kg; and the only other forceacting on the system is the force of gravity, which is

w = mg = (l9,500kg)(9.8m/s2) = l.9lxlOsN.

The force of gravity is directed down, so the net force on the system is ¿ Fy = FBA -(1.91 x 105N). The net force can also be found from Newton's second law: ¿ Fy = may = (19,500 kg)(1.4m/s2) = 2.7x104 N. Equate the two expressions for the net force, FBA -(1.91x105 N) = 2.7x104N, and solve; FBA = 2.2x105 N.

(b) Repeat the above steps except: (1) the system will consist only ofthe car, and (2) the upwardforce on the car comes from the supporting cable only F cc .The weight of the car is W = mg = ( 4500kg)(9.8 m/s2) = 4.4 x 104 N. The net force is ¿ Fy = Fcc -( 4.4 x 104 N), it can also be written as¿Fy = may = (4500 kg)(1.4 m/s2) = 6300 N. Equating, Fcc = 50,000 N.

P3-1 (a) The acceleration is a = F/m = (2.7xlO-5N)/(280kg) = 9.64xlO-8m/s2. The displace-ment (from the original trajectory) is

y = at2/2 = (9.64x lO-8m/s2)(2.4s)2 /2 = 2.8x lO-7m.

(b) The acceleration is a = F/m = (2.7xlO-5N)/(2.lkg) = l.3xlO-5m/s2. The displacement

(from the original trajectory) is

y = at2/2 = (l.3x lO-5m/s2)(2.4s)2 /2 = 3. 7x lO-5m.

P3-2 (a) The acceleration ofthe sled is a = F/m = (5.2N)/(8.4kg) = 0.62m/s2.(b) The acceleration ofthe girl is a = F/m = (5.2N)/(40kg) =0.13m/s2.(c) The distance traveled by girl is Xl = alt2/2; the distance traveled by the sled is X2 = a2t2/2.

The two meet when Xl + X2 = 15 m. This happens when ( al + a2)t2 = 30 m. They then meet whent = ..¡(30m)/(0.13m/s2 + 0.62m/s2) = 6.3s. The girl moves Xl = (0.13m/s2)(6.3s)2/2 = 2.6m.

~ (a) Start with block one. It starts from rest, accelerating through a distance of 16 m in atime of 4.2 s. Applying Eq. 2-28,

1 2Xo + vOxt + 2axt ,x =

(O) + (0)(4.2 s) + ~ax(4.2 8)2,-16 m =

find the acceleration to be ax = -1.8 m/s2,

37

Now for the second block. The acceleration of the second block is identical to the first for muchthe same reason that all objects fall with approximately the same acceleration.

(b) The initial and final velocities are related by a sign, then Vx = -VOx and Eq. 2-26 becomes

= (-1.8 m/s2)(4.2 s).

Vx

-VOx

-2vox

which gives an initial velocity of Vo", = 3.8 m/s.(c) Half of the time is spent coming down from the highest point, so the time to "fall" is 2.1 s.

The distance traveled is found from Eq. 2-28,

1x = (O) + (0)(2.1s) + 2(-1.8m/s2)(2.1sf = -4.0 m.

P3-4 (a) The weight ofthe engine is W = mg = (1400kg)(9.81m/s2) = 1.37x104N. Ifeach boltsupports 1/3 of this, then the force on a bolt is 4600 N .

(b) If engine accelerates up at 2.60m/s2, then net force on the engine is

z 3F net = ma = (1400kg)(2.60m/s ) = 3.64x 10 N.

The upward force from the bolts must then be

B = Fnet + W = (3.64x103N) + (1.37x104N) = 1.73x104N.

The force per bolt is one third this, or 5800 N .

P3-5 (a) If craft descends with constant speed then net force is zero, so thrust balances weight.The weight is then 3260 N .

(b) If the thrust is 2200 N the net force is 2200N -3260N = -1060N. The mass is thenm = F/a = (-1060N)/(-0.390m/s2) = 2720kg.

(c) The acceleration due to gravity is 9 = W/m = (3260N)/(2720kg) = 1.20m/s2.

P3-6 The weight is origina1ly Mg. The net force is origina1ly -Ma. The upward force is thenorigina1ly B = Mg -Ma. The goal is for a net force of (M -m)a and a weight (M- m)g. Then

(M-m)a = B- (M-m)g = Mg-Ma- Mg+mg =mg -Ma,

or m = 2Ma/(a + g).

~ ( a) Consider all three carts as one system. Then

¿ Fx = mtotalax,

6.5N = (3.1kg+2.4kg+1.2kg)ax,

0.97 m/s2 = ax.

(b) Now choose your system so that it only contains the third car. Then

¿Fx = F23 = m3ax = (1.2kg)(0.97m/s2).

The unknown can be solved to give F23 = 1.2 N directed to the right.(c) Consider a system involving the second and thirdcarts. Then LFx = F12, so Newton's law

applied to the system gives

F12 = (m2 +m3)ax = (2.4kg+ 1.2 kg)(0.97m/s2) =3.5N.

.qR

=' VOx + axt,

= VOx + axt,

P3-8 (a) F =ma = (45.2kg + 22.8kg + 34.3kg)(1.32m/s2) = 135N.(b) Consider only m3. Then F = ma = (34.3kg)(1.32m/s2) = 45.3N.(c) Consider m2 and m3. Then F = ma = (22.8kg + 34.3kg)(1.32m/s2) = 75.4N.

P3-9 (c) The net force on each link is the same, Fnet = ma = (0.100kg)(2.50m/s2) = 0.250N.(a) The weight ofeach link is W = mg = (0.100kg)(9.81m/s2) = 0.981N. On each link (except

the top or bottom link) there is a weight, an upward force from the link above, and a downward forcefromthe linkbelow. Then Fnet = U -D-W. Then U = Fnet+W+D = (0.250N)+(0.981N)+D =1.231 N + D. For the bottom link D = o. For the bottom link, U = 1.23N. For the link above,U = 1.23N+ 1.23N = 2.46N. For the link above, U = 1.23N+2.46N= 3.69N. For the link above,U = 1.23N + 3.69N = 4.92N.

(b) For the top link, the upward force is U = 1.23N + 4.92N = 6.15N.

P3-10 (a) The acceleration ofthe two blocks is a = F/(ml +m2) The net force on block 2 is from

the force of contact, and is

p = m2a = Fm2/(ml +m2) = (3.2N)(1.2kg)/(2.3kg+ 1.2kg) = 1.lN.

(b) The acceleration of the two blocks is a =F/(ml +'m2) The net force on block I is from theforce of contact, and is

p = m la= Fm~/{m;+ m2) = {3.2N)(2.3kg)/{2.3kg +1.2kg) =2.1N.

Not a third law pair, eh?

~ (a) Treat the system as including both the block and the rope, so that the mass ofthe system is M + m. There is one (relevant) force which acts on the system, so LFx = P. ThenNewton's second law would be written as P = (M +m)ax. Solve this for ax and get ax = P/(M +m).

(b) Now consider only the block. The horizontal force doesn 't act on the block; instead, there isthe force of the rope on the block. We'll assume that force has a magnitude R, and this iS: the only(relevant) force on the block, so E Fx = R for the net fo:rce on the block.. In this case Newton'ssecond law would be written R = Max. Yes, ax is the same in p~ (a) and (b); the acceleration ofthe block is the same as the acceleration oÍ the block + rope. Substituting in the results from part(a) we find

MpR=M+m .

39

E4-1 (a) The time to pass between the plates is t = x/vx = (2.3 cm)/(9.6x108 cm/s) = 2.4x10-9s.(b) The vertical displacement of the beam is then y = ayt2/2 == (9.4 x 1016 cm/s2)(2.4 X

10-9s)2/2 = 0.27 cm.(c) The velocity components are Vx = 9.6 x 108 cm/s and Vy = ayt = (9.4 x 1016 cm/s2)(2.4 X

10-9s) = 2.3x108cm/s.

E4-2 a = ~v! ~t = -(6.30i -8.42j)(m!s)!(3 s) = ( -2.10i + 2.81j)(m!s2).

~ (a) The velocity is given by

dr

dt

v

~ (Al) + ~ (Bfj) + ~ (ctk) ,

(O) + 2Btj + ck.

=

=

(b) The acceleration is given by

dv~ ( 2Btj) + ~ ( ck ) ,

(O) + 2Bj + (0).

-dt

--a -

(c) Nothing exciting happens in the x direction, so we will focus on the yz plane. The trajectoryin this plane is a parabola.

E4-4 (a) Maximum x is when Vz = O. Since Vz = azt + vz,o, Vz = 0 when t = -vz,o/az =-(3.6m/s)/( -1.2m/s2) = 3.0s. :

(b) Since Vz = 0 we have IvJ = Iv!ll. But v!I = a!lt + v!I,O = -(1.4m/s)(3.0s) + (Q) = -4.2m/s.

Then Ivl = 4.2m/s.

(c) r = at2/2 + vot, so

r= [-(0.6m/s2)i -(0.7m/s2)j](3.0S)2 + [(3.6m/s)i](3.0s) = (5.4m)i -(6.3m)j.

E4-5 F = Fl + F2 = (3.7N)j + (4.3N)i. Then a = F/'m = (0.71 m/s2)j + (0.83m/s2)i.

E4-6

or

{a) The acceleration is a = F /m = {2.2 m/s2)j. The yelocity after 15 seconds is v = at + Yo,

v = [(2.2 m/s2)j](15s) + [(42m/s)i] = (42m/s)i + (33m/s)j.

(b) r = 8t2/2 + vot, so

r = [(1.1 m/82)]](15 8)2 + [(42 m/8)i](158) = (630m)i + (250 m)].

~ The block h8B a weight W = mg = (5.1 kg)(9.8 m/s2) = 50 N.(a) Initially P = 12 N, so Py = (12N) sin(25°) = 5.1 N and Px = (12N) cos(25°) = 11 N. Since

the upward component is less than the weight, the block doesn't leave the floor, and a normal forcewill be present which will make E Fy = 0. There is on1y one contribution to the horizontal force,so E Fx = Px. Newton's second law then gives ax = Px/m = (11 N)/(5.1 kg) = 2.2 m/s2.

(b) As P is incre8Bed, so is P y; eventually P y will be large enough to overcome the weight of theblock. This happens just after Py = W = 50 N, which occurs when P = Py/sin(} = 120 N.

(c) Repeat part (a), except now P = 120 N. Then Px = 110 N, and the acceleration ofthe blockis ax = Px/m = 22 m/s2.

40

E4-8 (a) The block has weight W = mg = (96.0kg)(9.81m/s2) = 942N. Px = (450N)cos(38°) =355N; Py = (450N)sin(38°) = 277N. Since Py < W the crate stays on the fioor and there is anormal force N = W -Py. The net force in the x direction is Fx = Px- (125N) = 230N. Theacceleration is ax = Fx/m = (230N)/(96.0kg) = 2.40m/s2.

(b) The block has mass m = W/g = (96.0N)/(9.81m/s2) = 9.79kg. Px = (450N)cos(38°) =355N; Py = (450N) sin(38°) = 277N. Since Py > W the crate lifts off of the fioor! The netforce in the x direction is Fx = Px- (125N) = 230N. The x acceleration is ax = Fx/m =(230N)/(9.79kg) = 23.5m/s2. The net force in the y direction is Fy = Py -W = 181N. The yacceleration is ay = Fy/m = (181N)/(9.79kg) = 18.5m/s2. Wow.

E4-9 Let y be perpendicular and x be parallel to the incline. Then p = 4600 N ;

Px = (4600 N) cos(27°) = 4100N;

Py = (4600 N) sin(27°) = 2090 N.

The weight ofthe car is W = mg = (1200kg)(9.81m/s2) = 11800Nj

Wx = (11800N) sin(18°) = 3650 N;

Wy = (11800N) cos(18°) = 11200N.

Since Wy > Py the car stays on the incline. The net force in the x direction is Fx = Px- Wx = 450N.The acceleration in the x direction is ax = Fx/m = (450N)/(1200kg) = O.375m/s2. The distancetraveled in 7.5s is x = axt2/2 = (0.375m/s2)(7.5s)2/2 = 10.5m.

E4-10 Constant speed means zero acceleration, so net force is zero. Let y be perpendicular and xbe parallel to the incline. The weight is W = mg = (110kg)(9.81 m/s2) = 1080N; Wx = W sin(34°);Wy = Wcos(34°). The push F has components Fx = Fcos(34°) and Fy = -Fsin(34°). The

y components will balance after a normal force is introduced; the x components will balance ifFx = Wx, or F = Wtan(34°) = (1080N)tan(34°) = 730N.

~ If the x axi8 i8 parallel to the river and the y axi8 i8 perpendicular , then a = 0.121 mi 82.

The net force on the barge i8

LF = ma = (9500kg)(0.12im/s2) = 110oiN.

The force exerted on the barge by the horse has components in both the x and y direction. Ifp = 7900 N is the magnitude of the pull and (} = 18° is the direction, then p = p cos (}i + p sin (}j =(7500i + 2400j) N.

Let the force exerted on the barge by the water be F: = Fw,xi + Fw,yj. Then E Fx = (7500N) + Fw,x and E Fy = (2400 N) + Fw,y. But we already found E F, so

Fx = 1100 N = 7500 N + Fw,x,

Fy = 0 = 2400 N + Fw,yo

Solving, Fw,x = -6400 N and Fw,y = -2400 N. The magnitude is found by Fw = VF;',x + F;',y

6800 N .

=

41

E4-12 (a) Let y be perpendicular and x be parallel to the direction of motion of the plane.Then Wx = mgsine; Wy = mgcose; m = W/g. The plane is accelerating in the x direction, soax = 2.62m/s2; the net force is in the x direction, where Fx = max. But Fx = T- Wx, so

ax 4T"= Fx + Wx = W + Wsin{} = (7.93 x 10 N)

9

(b) There is no motion in the y direction, so

L = Wy = (7.93xlO4N)cos(27°) = 7.07xlO4N.

~ (a) The hall rolled off horizontally so VOy = O. Then

y =1 2

VOyt- 2gt ,

(O)t- ~(32.2 ft/S2)t2,( -4.23 ft)

~

which can be solved to yield t = 0.514 s.(b) The initial velocity in the x direction can be found from x = vOxt; rearranging, VOx = x/t =

(5.11 ft)/(0.514 s) = 9.94 ft/s. Since there is no y component to the velocity, then the initial speedis Vo = 9.94 ft/s.

E4-14 The electron travels for a time t = x/vx. The electron "falls" vertically through a distancey = -gt2/2 in that time. Then

2

)2(9.81 m/s2) (1.0 m)

(3.0x107m/s)

9Y=-2

x

v"'= -5.5 x 10-15 m.=-

2

E4-15 (a) The dart "falls" vertically through a distance y = -gt2/2 = -(9.81 m/s2)(0.19s)2 /2 =-0.18m.

(b) The dart travels horizontally x = v:z:t = (10m/s)(0.19s) = 1.9m.

E4-16 The initial velocity components are

Vx,Q = (15m/s) cos(200) = 14m/s

andVy,Q = -(15 m/s) sin(200) = -5.1 m/s.

(a) The horizontal displacement is x = vxt = {14m/s)(2.3s) = 32m.(b) The vertical displacement is

y = -gt2/2 + Vy,ot = -(9.81 m/s2)(2.3s)2 /2 +( -5.1 m/s)(2.3s) = -38m.

~ Find the time in terms of the the initial y component of the velocity:

v y = VOy -gt,

(O) = VOy -gt,

t = VOy/g.

42

Use this time to find the highest point:

2

= (vosin<t>o)2/2g.Finally, we know the initial y component oí the velocity írom Eq. 2-6, SO Yrnax

E4-18 The horizpntal displacement is x = v:¡;t. The vertical displacement is y = -gt2/2. Combin-ing, y = -g(x/v:¡;)2/2. The edge of the nth step is located at y = -nw, x = nw, where w = 2/3 ft.

If Iyl > nw when x = nw then the ball hasn't hit the step. Solving,

g(nw/vx)2/2

gnw/v;

< nw,

2 ,

2v;/(gw) = 2(5.0 ft/s)2/[(32 ft/s2)(2/3 ft)] = 2.34.

Then the balllands on the third step.

E4-19 (a) Start from the observation point 9.1 m above the ground. The ball will reach thehighest point when Vy = O, this will happen at a time t after the observation such that t = vy,olg =(6.1 m/s)/(9.81 m/s2) = 0.62 s. The vertical displacement (from the ground) will be

y= -gt2/2+Vy,ot+yo = -(9.81m/s2)(0.62s)2/2+ (6.1 m/s)(0.62 s) + (9.1 m) = llm.

(b) The time for the ball to return to the ground from the highest point is t = ,j'iii;;:;Tii =

V2(11 m)/(9.81 m7s2) = 1.5 s. The total time of flight is twice this, or 3.0 s. The horizontal distancetraveled is x = vxt = (7.6m/s)(3.0s) = 23m.

( c ) The velocity of the ball just prior to hitting the ground is

= at + yo = ( -9.81 m/s2)j(1.5s) + (7.6m/s)i = 7.6 m/si -15m/suj.

The magnitude is V7.62 + 152(m/s) = 17m/s. The direction is

(} = arctan(-15/7.6} = -63°.

E4-20 Focus on the time it takes the ball to get to the plate, assuming it traveled in a straightline. The ball has a "horizontal" velocity of 135 ft/s. Then t = x/vx = (60.5 ft)/(135 ft/s) = 0.448s.The ball will "fall" a vertical distance of y = -gt2/2 = -(32 ft/s2)(0.448 s)2 /2 = -3.2 ft. That's inthe strike zone.

E4-21 Since R cx: 1/9 one can write R2/R1 = 91/92, or

1- !!:!.92

6.R = R2 -Rl = Rl ) = (8.09 m)

4~

ort2 = [(13 ft) tan(55°) -(3 ft)][2/(32m/82)] = 0.97382,

or t = 0.9868. Then v = (13 ft)/(co8(55°)(0.9868)) = 23ft/8.

E4-23 Vx = x/t = (150 ft)/(4.50s) = 33.3 ft/s. The time to the highest point is half the hangtime, or 2.25s. The vertical speed when the ball hits the ground is Vy = -gt = -(32 ft/s2)(2.25s) =72.0 ft/s. Then the initial vertical velocity is 72.0 ft/s. The magnitude of the initial velocity isv722 + 332(ft/s) = 79 ft/s. The direction is

e = arctan(72/33) = 65°.

E4-24 (a) The magnitude of the initial velocity of the projectile is v = 264 ft/s. The projectilewas in the air for a time t where

x x ~t = ;;: = -;;c-¡;s O. ,,~ ~ , ";,!C,, tf,c;

(b) The height of the plane was -y where

-y = gt2/2 -Vy,ot = (32 ft/s)(9.8 S)2 /2- (264 ft/s) sin( -27°)(9.8s) = 2700 ft.

-(2300ft)

-(264 ft/s) cos( -27°)9.88

~ Define the point the ballleaves the racquet as r = o.(a) The initial conditions are given as VOx = 23.6 m/s and VOy = 0. The time it takes for the ball

to reach the horizontallocation of the net is found from

x

(12 m)

0.51 s

= vOxt,

= {23.6 m/s)t,

= t,

Find how far the ball has moved horizontally in this time:

1 2 1 2 2y = VOyt -2gt = (0)(0.51 s) -2(9.8 m/s )(0.51 s) = -1.3 m.

Did the ball clear the net? The ball started 2.37 m above the ground and "fell" through a distanceof 1.3 m by the time it arrived at the net. So it is still1.1 m above the ground and 0.2 m above thenet.

(b) The initial conditions are now given by VOx = (23.6 m/s)(cos[-5.0°]) = 23.5 m/s and VOy =(23.6 m/s)(sin[-5.0°]) = -2.1 m/s. Now find the time to reach the net just as done in part (a):

t = x/vox = (12.0 m)/(23.5 m/s) = 0.51 s.

Find the vertical position of the ball when it arrives at the net:

1 1 2 ,y = VOyt --gt2 = ( -2.1 m/s)(0.51 s) --(9.8 m/s )(0.51 s)2 = -2.3 m.

2 2Did the ball clear the net? Not this time; it started 2.37 m above the ground and then passed thenet 2.3 m lower, or only 0.07 m above the ground.

44

E4-26 The initial speed of the ball is given by v = J9R = J (32 ft/s2)(350 ft) = 106 ft/s. The

time of flight from the batter to the wall is

x/vx = (320 ft)/[(106 ft/8) co8(45°)] = 4.38.

The height of the ball at that time is given by y = Yo + VOyt -~gt2, or

y = (4 ft) + (106 ft/s)sin(45°)(4.3s) -(16 ft/s2)(4.3s)2 = 31 ft,

clearing the fence by 7 feet.

E4-27 The balllands x = (358 ft) + (39 ft) cos(28°) = 392 ft from the initial position. The balllands y = (39 ft) sin(28°) -(4.60 ft) = 14 ft above the initial position. The horizontal equationis (392 ft) = vcos(}tj the vertical equation is (14 ft) = -(g/2)t2 + vsin(}t. Rearrange the verticalequation and then divide by the horizontal equation to get

,---"-; , = tan(},14 ft + (g/2)t2

ort2 = [(392 ft) tan(52°) -(14 ft)][2/(32m/82)] = 30.582,

or t = 5.528. Then v = (392 ft)/(co8(52°)(5.528)) = 115 ft/8.

E4-28 Since ball is traveling at 45° when it returns to the same level from which it was thrownfor maximum range, then we can assume it actually traveled ~ 1.6 m. farther than it would havehad it been launched from the ground level. This won 't make a big difference, but that means thatR = 60.0 m -1.6 m = 58.4 m. If Vo is initial speed of ball thrown directly up, the ball rises to thehighest point in a time t = vo/g, and that point is Ymax = gt2/2 = v5/(2g) above the launch point.But v5 = gR, so Ymax = R/2 = (58.4m)/2 = 29.2m. To this we add the 1.60m point of release toget 30.8 m.

~ The net force on the pebble is zero, so E Fy = o. There are only two forces on thepebble, the force of gravity W and the force of the water on the pebble Fpw. These point inopposite directions, so 0 = Fpw -W. But W = mg = (0.150 kg)(9.81 m/s2) = 1.47 N. SinceFpw = W in this problem, the force of the water on the pebble must also be 1.47 N.

E4-30 Terminal speed is when drag force equal weight, or mg = bvT2, Then VT = Jiñii7b.

~ Eq. 4-22 is

?~~~ Vy(t) = VT ( 1- e-bt/m) ,

where we have used Eq. 4-24 to substitute for the terminal speed. We want to solve this equationfor time when Vy(t) = vT/2, so

~VT = VT (1- e-bt/m) ,

~ = (1- e-bt/m),

e -bt/m -.!-2

bt/m = -ln(1/2)

t=~ln2

45

E4-32 The terminal speed is 7m/s for a raindrop with r = 0.15 cm. The mass of this drop ism = 41Tpr3/3, so

47r(l.O x lO-3kg/cm3)(O.l5 cm)3(9.Sl m/s2)mgVT

b= = 2.0 x 10-5kg/s.= -:- ..3(7m/s)

E4-33 (a) The speed of the train is v = 9.58m/s. The drag force on one car is f = 243(9.58) N =2330N. The total drag force is 23(2330N) = 5.36x104N. The net force exerted on the cars (treatedas a single entity) is F = ma = 23(48.6x103kg)(0.182m/s2) = 2.03x105N. The pull ofthe locomotiveis then p = 2.03 x 105N + 5.36 x 104N = 2.57 x 105N .

(b) If the locomotive is pulling the cars at constant speed up an incline then it must exert aforce on the cars equal to the sum of the drag force and the parallel component of the weight. Thedrag force is the same in each case, so the parallel component of the weight is WII = W sin (} =2.03x105N = ma, where a is the acceleration from part (a). Then

(} = arcsin(a/g) = arcsin[(0.182m/s2)/(9.81m/s2)] = 1.06°,

E4-34 (a) a = v2fr = (2.18 x 106mfs)2f(5.29 x 10-11m) = 8.98 x 1022mfs2.(b) F = ma = (9.11 x 10-31kg)(8.98 x 1022mfs2) = 8.18x 10-8N, toward the center.

~ (a) v = ~ = v (5.2 m)(6.8)(9.8 m/s2) = 19 m/s.

(b) Use the fact that one revolution corresponds to a length of 27rr:

19~ ( 1 rev) ( ~ ) = 35~s 27!"(5.2 m) 1 min min .

E4-36 (a) v = 27rr/T = 27r(15m)/(12s) = 7.85m/s. Then a = v2fr = (7.85m/s)2/(15m) =4.11 m/s2, directed toward center, which is down.

(b) Same arithmetic as in (a); direction is still toward center, which is now up.(c) The magnitude of the net force in both (a) and (b) is F = ma = (75kg)(4.11 m/s2) = 310N.

The weight of the person is the same in both parts: W = mg = (75kg)(9.81m/s2) = 740N. Atthe top the net force is down, the weight is down, so the Ferris wheel is pushing up with a force ofp = W -F = (740N) -(310N) = 430N. At the bottom the net force is up, the weight is down, sothe Ferris wheel is pushing up with a force of p = W + F = (740N) + (310N) = 1050N.

E4-37 (a) v = 21rr/T = 21r(20xlO3m)/(l.Os) = l.26xlO5m/s.(b) a = v2/r = (l.26xlO5m/s)2/(20xlO3m) = 7.9xlO5m/s2,

E4-38 (a) v = 27rr/T = 27r(6.37x 106m)/(86400s) = 463m/s. a = v2/r = (463m/s)2/(6.37x106m) = 0.034 m/s2.

(b) The net force on the object is F = ma = (25.0kg)(0.034m/s2) = 0.85N. There are two forceson the object: a force up from the scale (S), and the weight down, W = mg = (25.0 kg)(9.80 m/s2) =245N. Then S = F + W = 245N + 0.85N = 246N.

~ Let Ax = 15 m be the length; tw = 90 s, the time to walk the stalled Escalatorj t8 = 60s, the time to ride the moving Escalatorj and tm, the time to walk up the moving Escalator .

The walking speed of the person relative to a fixed Escalator is Vwe = Ax/tw; the speed of theEscalator relative to the ground is Veg = Ax/t8j and the speed of the walking person relative to the

46

ground on a moving Escalator is Vwg = 6.x/tmo But these three speeds are related by Vwg

Combine all the above:

= Vwe+Veg.

Vwe + Veg,

¡lx ¡lx

Vwg~x

tm-+-,=

tstw 1

1 1-+-tw ts

-tm

Putting in the numbers, tm = 36 s.

E4-40 Let Vw be the walking speed, Vs be the sidewalk speed, and Vm = Vw + Vs be Mary's speedwhile walking on the moving sidewalk. All three cover the same distance x, so Vi = x/ti, where i isone of w, s, or m. Put this into the Mary equation, and

l/tm = l/tw + l/ts = 1/(1508) + 1/(708) = 1/488.

E4-41 If it takes longer to fly westward then the speed of the plane (relative to the ground)westward must be less than the speed of the plane eastward. We conclude that the jet-stream mustbe blowing east. The speed of the plane relative to the ground is Ve = Vp + Vj when going east andVw = Vp -Vj when going west. In either case the distance is the same, so x = Viti, where i is e orw. Since tw -te is given, we can write

x x 2v.tw-te=-=x J .

vp-'/Jj vp+Vj Vp2-Vj2

Solve the quadratic if you want, but since Vj « Vp we can neglect it in the denominator and

Vj = vp2(0.83 h)/(2x) = (600 mi/h)2(0.417 h)/(2700 mi) = 56 mi/hr.

E4-42 The horizontal component of the rain drop's velocity is 28m/s. Since v"' = v sin(}, v =(28 m/s)/ sin(64°) = 31 m/s.

~ (a) The position of the bolt relative to the elevator is Ybe, the position of the bolt relativeto the shaft is yb8, and the position of the elevator relative to the shaft is Ye8. Zero all three positionsat t = Oj at this time VO,b8 = VO,e8 = 8.0 ft/s.

The three equations describing the positions are

ybs

1 2= VO,bst- 2gt ,

1 2= VO,est + 2at ,

Yes

y be + res = rbs,

where a = 4.0 m/s2 is the upward acceleration of the elevator. Rearrange the last equation andsolve for Ybe; get ybe = -1 (g + a )t2 , where advantage was taken of the fact that the initial velocitiesare the same.

Thent = v -2ybe/(g + a) = V -2( -9.0 ft)/(32 ft/S2 + 4 ft/S2) = 0.71 s

(b) Use the expression for yb8 to find how the bolt moved relative to the shaft:

1 2 1 2 2ybs = VO,bst -2gt = (8.0 ft)(0.71 s) -2(32 ft/s )(0.71 s) = -2.4 ft.

A7

E4-44 The speed of the plane relative to the ground is Vpg = (810 km)/(1.9 h) = 426 km/h. Thevelocity components ofthe plane relative to the air are VN = (480 km/h)cos(21°) = 448 km/h andVE = (480 km/h)sin(21°) = 172 km/h. The wind must be blowing with a component of 172 km/hto the west and a component of 448 -426 = 22 km/h to the south.

E4-45 (a) Let i point east andj point north. The velocityofthe torpedo is v = (50 km/h)isin(}+, ,

(50 km/h)jcos(}. The initial coordinates of the battleship are then ro = (4.0 km)isin(20°) +(4.0 km)jcos(20°) = (1.37 km)i+ (3.76 km)j. The final position ofthe battleship is r= (1.37 km+24 km/ht)i + (3.76 km)j, where t is the time of impact. The final position of the torpedo is thesame, so

[(50 km/h)isine + (50 km/h)]cosejt = (1.37 km + 24 km/ht)i + (3.76 km)],

or

[(50 km/h) sinO]t -24 km/ht = 1.37 km

and

[(50 km/h)cos{}]t = 3.76 km.

Dividing the top equation by the bottom and rearranging,

50sin{} -24 = 18.2cos{}.

Use any trick you want to solve this. I used Maple and found {} = 46.8°.(b) The time to impact is then t = 3.76 km/[(50 km/h) cos(46.8O)1= 0.110 h, or 6.6 minutes.

~ Let rA be the position of particle of particle A, and rB be the position of particle B. Theequations for the motion of the two particles are then

rA = rO,A + vt,

= dj + vti;

...1-2rB = 2at ,

= ~a(sinei + cosej)t2.

A collision will occur if there is atime when rA = rB. Then

dj + vti = ~a(sinei + cosej)t2,

but this is really two equations: d = !at2cose and vt = !at2sine.Solve the second one for t and get t = 2v / ( a sin e) .Substitute that into the first equation, and

then rearrange,

d =

'2

d = cos (} ,

sin2 (} =

1 -COS2 (} =

2v2

ado = COS2 () + cosO -

48

This last expression is quadratic in cosO. It simplifies the solution if we define b = 2v/(ad) = 2(3.0m/s)2/([0.4 m/s2][30 m]) = 1.5, then

Then cos () = 0.5 and () = 60° .

P4-2 (a) The acceleration of the ball is a = (1.20 m/s2)i -(9.81 m/s2)j. Since a is constant thetrajectory is given by r = at2/2, since Vo = 0 and we choose ro = 0. This is a straight line trajectory,with a direction given by a. Then

(J = arctan(9.81/1.20) = 83.0°,

and R = (39.0m)/tan(83.00) = 4.79m. It will be useful to find H = (39.0m)/sin(83.00) = 39.3m.(b) The magnitude ofthe acceleration ofthe ball is a = v9.812 + 1.202(m/s2) = 9.88m/s2. The

time for the ball to travel down the hypotenuse of the figure is then t = ~-39.3m)/(9.88m/s2) =

2.82s.( c ) The magnitude of the speed of the ball at the bottom will then be

v = at = (9.88 m/s2)(2.82s) = 27.9m/s.

P4-3 (a) The rocket thrust is T = (61.2 kN) cos(58.00)i + (61.2 kN) sin(58.00)i = 32.4 kNi +~ 51.9 kNj. The net force on the rocket is the F = T + W, or

F = 32.4 k Ni + 51.9 kNj- (3030 kg)(9.81 m/s2)j = 32.4 kNi + 22.2 kNj.

The acceleration ( until rocket cut-off) is this net force divided by the mass, or

a = 10.7m/s2¡ + 7.33m/s2j.

The position at rocket cut-off is given by

= at2/2 = (10.7m/s2i + 7.33m/s2j)(48.0s)2 /2,

= 1.23 x 104mi + 8.44 x 103mj.r

The altitude at rocket cut-off is then 8.44 km.(b) The velocity at rocket cut-off is

v = at = (10.7m/s2i + 7.33m/s2j)(48.0 s) = 514 m/si + 352m/sj,

this becomes the initial velocity for the "free fal1" part of the journey. The rocket will hit the groundafter t seconds, where t is the solution to

o = -(9.81 m/s2)t2 /2 + (352 m/s)t + 8.44 x 103 m.

The solution is t = 90.7s. The rocket lands a horizontal distance of x = vxt = (514m/s)(90.7s) =

4.66xl04 m beyond the rocket cut-off; the total horizontal distance covered by the rocket is 46.6 km+12.3 km = 58.9 km.

49

P4-4 (a) The horizontal speed of the hall is Vx = 135 ft/s. It takes

x/vx = {30.0 ft)/(135 ft/8) = 0.2228

to travel the 30 feet horizontally, whether the first 30 feet, the last 30 feet, or 30 feet somewhere inthe middle.

(b) The ball "falls" y = -gt2/2 = -(32 ft/s2)(0.222s)2/2 = -0.789 ft while traveling the first30 feet.

(c) The ball "falls" a total of y = -gt2/2 = -(32 ft/s2)(0;444s)2/2 = -3.15 ft while travelingthe first 60 feet, so during the last 30 feet it must have fallen (-3.15 ft) -(-0.789 ft) = -2.36 ft.

( d) The distance fallen because of acceleration is not linear in time; the distance moved horizon-tally is linear in time.

P4-5 (a) The initial velocity of the hall has components

v:¡;,o =(25.3m/s)cos(42.00) 18.8mjs

andVy,O = (25.3m/s)sin(42.0~) = 16.9m/s.

The ball is in the air for t = x/vx = (21.8m)/(18.8m/s) = 1.16s before it hits the wall.(b) y = -gt2/2 + Vy,ot = -(4.91 m/s2)(1.16s)2 + (16.9m/s)(1.16s) = 13.0m.(c) Vx = vx,o = 18.8m/s. Vy = -gt + Vy,O = -(9.81 m/s2)(1.16s) + (16.9m/s) = 5.52m/s.

(d) Since Vy > O the ball is still heading up.

P4-6 (a) Theinitial vertical velocity is Vy,a = Va sin<t>a. The time to the highest point is t = Vy,a/g.The highest point is H = gt2/2. Combining,

H = y( Vo sin </>o/ y )2/2 = V~ sin2 4>O/ (2y ) .

The range is R = (v~/g)sin2<t>o = 2(v~/g)sin<t>ocos<t>o. Since tan(} = H/(R/2), we have

2H

R

tan(} =

2. 2,¡, !-Vo Slll '!'0 g 1-2(v5!g) sin </>o COS</>o = 2 tan </>o.

(b) (} = arctan(O.5tan45°) = 26.6°,

~ The components of the initial velocity are given by VOx = Vo cos(} = 56 ft/s and VOyvosin(} = 106 ft/s where we used Vo =120 ft/s and (} = 62°.

(a) To find h we need only find out the vertical position of the stone when t = 5.5 s.

(106 ft/8)(5.5 8)- ~(32 ft/81{5.5 8)2 = 99 ft.

(b) Look at this as a vector problem:

v

~

Vo + at,

( voxi + voyj) -gjt,

voxi + (VOy -gt)j,

(56 ft/s)i+ ( (106 ft/s -(32 ft/s2)(5.5 s) ) j,

(56 ft/s)i+ (-70.0 ft/s)j.

50

The magnitude of this vector gives the speed when t = 5.5 s; v = V562 + ( -70)2 ft/s= 90 ft/s.(c) Highest point occurs when Vy = 0. Solving Eq. 4-9(b) for time; Vy = 0 = VOy -gt =

(106 ft/s) -(32 ft/s2)t; t = 3.31 s. Use this time in Eq. 4-10(b),

1 1y = VOyt -29t2 = (106 ft/s)(3.31 s) -2(32 ft/s2)(3.31 8)2 = 176 ft.

P4-8 (a) Since R = (v5/g)sin 2</>0, it is suflicient to prove that sin2</>o is the same for both

sin 2(45° + a) and sin 2(45° -a).

sin 2(45° :1:: a) = sin(90° :1:: 2a) = cos(:1::2a) = cos(2a).

Since the :f: dropped out, the two quaJ1tities are equal.(b) <1>0 = (1/2) arcsin(Rg/v5) = (1/2) arcsin«20.0m)(9.81 m/s2)/(30.0m/s)2) = 6.3°. The other

choice is 90° -6.3° = 83.7° .

~ To score the ball must pass the horizontal distance of 50 m with an altitude of no less than3.44 m. The initial velocity components are VOx = Vo cos () and VOy = Vo sin () where Vo = 25 m/s,and () is the unknown.

The time to the goal post is t = x/vOx = x/(vo cos()).The vertical motion is given by

21 ( x ) 1 ( x )VOy t -2 gt2 = ( 'IlO sin (}) ;;;;-oos-9 -2 g ;;;;-oos-9y

2~ (tan2 (j + I) ,y = x tan(} -2vo

which can be combined with numbers and constraints to give

(9.8 m/s2)(50 m)2 2(3.44 m) 5 (50 m) tan(} -2(25 m/s)2 (tan (} + 1) ,

3.44 5 50 tan (} -20 ( tan2 (} + 1) ,

0 5 -20tan2 (} + 50tan(} -23

Solve, and tan (} = 1.25:!: 0.65, so the allowed kicking angles are between (} = 31° and (} = 62°,

P4-10 (a) The height of the projectile at the highest point is H = Lsin(j. The amount of timebefore the projectile hits the ground is t = ,j'iiiTii = v2Lsin(j/g. The horizontal distance covered

by the projectile in this time is x = vxt = vv2Lsin(j/g. The horizontal distance to the projectilewhen it is at the highest point is x' = Lcos(j. The projectile lands at

D = x -x' = vV2Lsin(j/g -Lcos(j.

(b) The projectile will pass overhead ir D > O.

51

P4-11 V2 =V; +V;. For a projectile Vx is constant, so we need only evaluate ~(V;)/dt2. The firstderivative is 2vy dvy / dt = -2vyg. The derivative of this ( the second derivative) is -2g dvy / dt = 2g2.

P4-12 1f1 is a maximum when r2 is a maximum. r2 = X2 + y2, or

2 2 2 2r = (vx,ot) + ( -gt /2 + Vy,ot) ,

= (votcOS<t>O)2+ (votsip<t>o -gt2/2)2 ,

= v5t2 -vogt3 sin <t>o + g2t4 /4.

We want to look for the condition which will allow dr2 / dt to vanish. Since

dr2 / dt = 2v~t -3vOgt2 sin <t>O + g2t3

we can focus on the quadratic discriminant, b2 -4ac, which is

9 22 .2,¡, 8 22voY Slll 'l'a -voY,

a quantity which will only be greater than zero ir 9 sin2 <t>o > 8. The critica! angle is then

<l>c = arcsin( J879) = 70.5°,

f-p;¡:i3l There is a downward force on the balloon of 10.8 kN from gravity and an upward forceof 10.3 kN from the buoyant force of the air. The resultant of these two forces is 500 N down, butsince the balloon is descending at constant speed so the net force onthe balloon must be zero. Thisis possible because there is a drag force on the balloon of D = bv2, this force is directed upward.The magnitude must be 500 N, so the constant b is

(500 N)

(1.88m/s)2b= = 141 kg/m.

If the crew drops 26.5 kg of ballast they are "lightening" the balloon by

{26.5kg){9.81 m/s2) = 260 N.

This reduced the weight, but not the buoyant force, so the drag force at constant speed will now be500N- 260N = 240N.

The new constant downward speed will be

v = .JD7b= v(240N)X141kg7m) = 1.30m/s.

P4-14 The constant b isb = (500N)/(1.88m/s) = 266N .s/m.

The drag force after "lightening" the load will still be 240 N. The new downward speed will be

v= Dlb = (240 N)/(266N .s/m) = O.902m/s.

P4-15 (a) Initially vo= O, so D = 0, the only force is the weight, so a =-g.(b) After some time the acceleration is zero, then W = D, or bvT2 = mg, or VT = v'ñi97b.(c) When v = vT/2 the drag force is D = bvT2/4 = mg/4, so the net force is F = D- W =

-3mg/4. The acceleration is the a = -3g/4.

52

P4-16 (a) The net force on the barge is F = -D = -bv, this results in a differential equationmdv/dt = -bv, which can be written as

dv/v -(b/m)dt,

-(b/m) J dt,

=

JdV/Vln(vf/vJ = -bt/m.

=

Then t = (m/b) ln(vi/vf).(b) t= I(970kg)/(68N .8/m)] ln(32/8.3) = 198.

~ (a) The acceleration is the time derivative of the velocity,

a = ~ = i (~ (1- e-bt/m )) = ~~e-bt/my dt dt b b m ,

which can be simplified as ay = ge-bt/m. For large t this expression approaches O; for small t the

exponent can be expanded to give

bt'

1--

m

= g -VTt,ay ~g

P4-18 (a) We have v y = vT(l -e-bt/m) from Eq. 4-22; this can be substituted into the last line

oí the solution íor P4-17 to give

Y95=VT(t-;).

We can also rearrange Eq. 4-22 to get t = -(m/b) ln(l- Vy/VT), SO

y95 = VT2/y

But Vy/VT ~ 0.95, so

VT2/g (In 20 -19/20)y95 = VT2 / 9 ( -ln(0.05) -0.95)

(b) y95 = (42m/s)2/(9.81m/s2)(2.05) = 370 m.

P4-19 (a) Convert units first. v = 86.1 m/s, a = 0.05(9.81 m/s2) = 0.491 m/s2. The minimumradius is r = v2/a = (86.1m/s)/(0.491m/s2) = 15 km.

(b) v = .¡ar = ..¡(0.491 m/s2)(940m) = 21.5m/s. That's 77 km/hr.

53

P4-20 (a) The position is given by r= Rsin(¡)ti+R(l-cos(¡)t)j, where (¡) = 27r/(20s) = 0.314s-1and R = 3.0m. When t = 5.0s r = (3.0m)i + (3.0m)jj when t = 7.5s r = (2.1m)i + (5.1m)j;when t = 10 s r = (6.0 m)j. These vectors have magnitude 4.3 m, 5.5 m and 6.0 m, respectively. The

vectors have direction 45°, 68° and 90° respectively.(b) ~r = ( -3.0 m)i + (3.0 m)j, which has magnitude 4.3 m and direction 135° .(c) vav = ~r/~t = (4.3m)/(5.0s) = 0.86m/s. The direction is the same as ~r.

(d) The velocity is given by v = &cOS(¡)ti+&sin(¡)tj. At t = 5.0s v = (0.94m/s)j; at t = lOsv = (-0.94m/s)i.

(e) The acceleration is given by a = -&2 sin(¡)ti + &2 COS(¡)tj. At t = 5.0s a =( -0.30m/s2)ijat t = lOs a = ( -0.30m/s2)j.

~ Start fram where the stane lands; in arder ta get the~e the stane fell thraugh a verticaldistance af 1.9 m while maving 11 m harizantally. Then

1 F2Yy = -2gt2 which can be written as t = V yO

Putting in the numbers, t = 0.62 s is the time of flight from the moment the string breaks. Fromthis time find the horizontal velocity,

P4-22 (a) The path traced out by her feet ha.s circumference Cl = 27rrcos50°, where r is theradius of the earth; the path traced out by her head ha.s circumference C2 = 27r( r + h) cos 50°, whereh is her height. The difference is Ac = 27rh cos 50° = 27r(1.6 m) cos 50° = 6.46 m.

(b) a = v2/r = (27rr /T)2 /r = 47r2r /T2. Then Aa = 47r2 Ar /T2. Note that Ar = h cos (J! Then

Aa = 47r2(1.6 m) cos 50° /(86400 S)2 = 5.44 x 10-9m/s2.

, ,/

~ (a) A cycloid looks something like this:

~~~)

(b) The position of the particle is given by

r = (RsinúJt + úJRt)i + (RcosúJt + R)j.

The maximum value of y occurs whenever coswt = 1. The minimum value of y occurs whenevercoswt = -1. At either of those times sinwt = O.

The velocity is the derivative of the displacement vector ,

v = (&coswt+wR)i+ I -&sinúJt)j.

When y is a maximum the velocity simplifies to

v = (2f.¡)R)i + (O)].

54

When y is a minimum the velocity simplifies to

v = (0)1 + (O)].

The acceleration is the derivative of the velocity vector ,

a = ( -&;2 sin(¡)t)i + ( -&;2 COS(¡)t)j.

When y is a maximum the acceleration simplifies to

a = (0)1 + { -&2)1.

When y is a minimum the acceleration simplifies to

a = (0)1 + (Rr.JJ2)j,

(b)'The'apparent speed is V(15.3)2 + (7.8)2(m/s) = 17.2m/s.

P4-25 (a) The decimal angles are 89.994250° and 89.994278°. The earth moves in the orbitaround the sun with a speed of v = 2.98x 104m/s (Appendix C). The speed of light is then betweenc = (2.98 x 104m/s) tan(89.994250°) = 2.97 x 108m/s and c = (2.98 x 104m/s) tan(89.994278°) =2.98 x 108m/s. This method is highly sensitive to rounding. Calculating the orbital speed from theradius and period of the Earth 's orbit willlikely result in different answers!

P4-26 (a) Total distance is 21, so to = 21/v.(b) Assume wind blows east. Time to travel out is tl = l/(v + u), time to travel back is

t2 = l/(v -u). Total time is sum, or

If wind blows west the times reverse, but the result is otherwise the same.( c ) Assume wind blows i1orth. The airplane will still have a speed of v relative to the wind, but

it will need to fiy with a heading away from east. The speed of the plane relative to the ground willbe Vv2 -u2. This will be the speed even when it fiies west, so

-to

-~.

2[t N = .¡;;.-=-t;'

(d) If u > v the wind sweeps the plane along in one general direction only; it can never By back.Sort of like a black hole event horizon.

~ The velocity of the police car with respect to the ground is v pg = -76km/h¡. The velocityof the motorist with respect the ground is Vmg = -62 km/hj.The velocity of the motorist with respect to the police car is given by solving

---Vmg = Vmp + Vpg,

so Vmp = 76km/hi -62 km/hj. This velocity has magnitude

Vmp = v (76km/h)2 + ( -62 km/h)2 = 98 km/h.

55

P4-24 (a) The speed of the car is Vc = 15,3m/s. The snow appears to fall with an angle (}arctan(15,3/7.8) = 63°,

The direction isf) = arctan( -62 km/h)/(76km/h) = -39°,

but that is relative to i. We want to know the direction relative to the line oí sight. The line oí sightis

a = arctan(57m)/(41m) = -54°

relative to i, so the answer must be 15°.

P4-28 (a) The velocity of the plane with respect to the air is vpa; the velocity of the air withrespect to the ground is Vag, the velocity of the plane with respect to the ground is Vpg. ThenVpg = vpa + Vag. This can be represented by a triangle; since the sides are given we can find theangle between Vag and Vpg (points north) using the cosine law

(} = arccos (\135)2 -(135)2 -(70)2

-2(135)(70)

(b) The direction of v pa can also be found using the cosine law,

(70)2 -(135)2 -(135)2

-2(135)(135)(J = arccos 30°.

56

~ There are three forces which act on the charged sphere-- an electric force, FE, the force ofgravity, W, and the tension in the string, T. All arranged as shown in the figure on the right below.

(a) Write the vectors so that they geometrically show that the sum is zero, as in the figureon the left below. Now W = mg = (2.8 x 10-4 kg)(9.8 m/s2) = 2.7 x 10-3 N. The magnitudeof the electric force can be found from the tangent relationship, so FE = W tan (} = (2.7 x 10-3N) tan(33°) = 1.8 x 10-3 N.

\

w

(a) (b)

(b) The tension can be found from the cosine relation, so

T = W/cos(} = (2;7 x lO-3N)/cos(33°) =;: 3.2 x lO-3N.

E5-2 (a) The net force on the elevator is F = ma = Wa/g = (6200 Ib)(3.8 ft/s2)/(32 ft/S2) =740 lb. Positive means up. There are two force on the elevator: a weight W down and a tensionfrom the cable T up. Then F = T- W or T = F + W = (740 lb) + (6200 lb) = 6940 lb.

(b) If the elevator acceleration is down then F = -740 lb; consequently T = F + W = ( -740 lb) +(6200 lb) = 5460 lb.

E5-3 (a) The tension Tis up, the weight W is down, and the net force F is in the direction of theacceleration (up). Then F =T -W. But F = ma and W = mg, so

m = T/(a + y) = (89N)/[(2.4m/s2) + (9.8m/s2)] = 7.3kg.

(b) T = 89N. The direction of velocity is unimportant. In both (a) and (b) the acceleration is

up.

E5-4 The average speed of the elevator during deceleration is vav = 6.0m/s. The time to stop theelevator is then t = (42.0m)/(6.0m/s) = 7.0s. The deceleration is then a = (12.0m/s)/(7.0s) =1.7 m/s2. Since the elevator is moving downward but slowing down, then the acceleration is up,which will be positive.

The net force on the elevator is F = ma; this is equal to the tension T minus the weight W.Then

T = F + W = ma + mg = (l600kg)[(l.7m/s2) + (9.8m/s2)] = l.8x lO4N.

~ (a) The magnitude o~,the man's acceleration is given by

m2 -mlI

m2+ml' "a=

and is directed down. The time which elapses while he falls is found by solving y = VOyt + ~ayt2,or, with numbers, (-12 m) = (o)t + ~( -0.2g)t2 which has the Bolutions t = :1::3.5 s. The velocitywith which he hits the ground is then v = VOy + ayt = (O) + ( -0.2g)(3.5 s) = -6.9 m/s.

57

(b) Reducing the speed can be accomplished by reducing the acceleration. We can 't change Eq.5-4 without also changing one of the assumptions that went into it. Since the man is hoping toreduce the speed with which he hits the ground, it makes sense that he might want to climb up the

rope.

E5-6 (a) Although it might be the monkey which does the work, the upward force to lift him stillcomes from the tension in the rope. The minimum tension to lift the log is T = Wl = mlg. The netforce on the monkey is T- W m = mma. The acceleration of the monkey is then

a = (m¡ -mm)g/mm = [(15kg) -(11 kg)](9.8m/s2)/(11 kg) = 3.6m/s2.

(b) Atwood 's machine!

a = (mi -mm)g/(ml + mm) = [(15 kg) -(11 kg)](9.8 m/s2)/[(15 kg) + (11 kg)] = 1.5 m/s.

(c) Atwood's machine!

T = 2mlmm9/(ml + mm) = 2(15kg)(11kg)(9.8m/s2)/[(15kg) + (llkg)] = 120N.

E5- 7 The weight of each car has two compOnent8: a component parallel to the cable8 WII = W 8in (}and a component normal to the cable8 W J... The normal component i8 "balanced" by the 8upportingcable. The parallel component act8 with the pull cable.

In order to accelerate a car up the incline there mU8t be a net force up with magnitude F = ma.Then F = T above -Tbelow -WII' or

~T = ma + mgsin(J = (2800 kg)[(0.81 m/s2) + (9.8m/s2) sin(35°)}= 1.8x 104N.

E5-8 The tension in the cable is T, the weight ofthe man + platform system is W = mg, and thenet force on the man + platform system is F = ma = Wa/g = T- W. Then

Wa/g + W = W(a/g + 1) = (180 lb + 43 lb)[(1.2 ft/s2)/(32 ft/S2) + 1] = 2311b.T

~ See Sample Problem 5-8. We need only apply the (unlabeled!) equation

JLs = tan (}

to find the egg angle. In this case (} = tan-l(O.O4) = 2.3°,

E5-10 (a) The maximum force offriction is F = JJ.sN. Ifthe rear wheels support halfofthe weightof the automobile then N = W/2 = mg/2. The maximum acceleration is then

a = F/m = JJ.sN/m = JJ.s9/2.

2 2{b) a T {0.56){9.8mls )12 = 2.7mls .

E5-11 The maximum force of friction is F = J1,sN. Since there is no motion in the y direction themagnitude of the normal force must equal the weight, N = W = mg. The maximum acceleration isthen

a = F/m = JLsN/m = JLsg = (0.95)(9.8m/s2) = 9.3m/s2.

E5-12 There is no motion in the vertical direction, so N = W = mg. Then JLk = FIN =(470N)I[(9.8mls2)(79kg)] = 0.61.

58

~ A 75 kg mass has a weight of W = (75kg)(9.8m/s2) = 735 N, so the force of friction oneach end of the bar must be 368 N. Then

E5-14 (a) There is no motion in the vertical direction, so N = W = mg.To get the box moving you must overcome static friction and push with a force of p ~ JLsN =

(0.41)(240N) = 98N.(b) To keep the box moving at constant speed you must push with a force equal to the kinetic

friction, p = JLkN = (0.32)(240N) = 77N.(c) If you push with a force of 98 N on a box that experiences a (kinetic) friction of 77 N, then

the net force on the box is 21 N. The box will accelerate at

a = F/m = Fg/W = (21 N)(9.8m/s2)/(240N) = O.86m/s2.

E5-15 (a) The maximum braking force is F = J1sN. There is nomotion in the vertical direction,so N = W = mg. Then F = J1smg = (0.62)(1500kg)(9.8m/s2) = 9100N.

(b) Although we still use F = J1sN, N # W on an incline! The weight has two components;one which is parallel to the surface and the other which is perpendicular .Since there is no motionperpendicular to the surface we must have N = W.l = W cos () .Then

F= JLsmg cosO = (0.62)(1500 kg)(9.8 m/s2) cos(8;6°) = 9000 N .

E5-16 JLs = tan (} iB the condition for an object to Bit without Blipping on an incline.(} = arctan(O.55} = 29°. The angle Bhould bereduced by 13°.

Then

~ (a) The force of static friction is less than .UsN, where N is the normal force. Since thecrat,eisn't moving up or down, ¿Fy =O = N- W. So in this case N = W = mg = (136 kg)(9.81m/s2) = 1330 N. The force of static friction is less than or equal to (0.37)(1330N) = 492Nj movingthe crate will require a force greater than or equal to 492 N .

(b) The second worker could lift upward with a force L, reducing the normal force, and hencereducing the force of friction. If the first worker can move the block with a 412 N force, then412 ?; .UsN. Solving for N, the normal force needs to be less than 1110 N. The crate doesn't moveoff the table, so then N + L = W, or L = W -N = (1330 N) -(1110 N) = 220 N.

(c) Or the second worker can help by adding a push sothat the total force of both workers isequal to 492 N. If the first worker pushes with a force of 412 N, the second would need to push witha force of 80 N.

E5-18 The coefficient of static friction is J1s = tan(28.0°) = 0.532. The acceleration is a =2(2.53m)/(3.92s)2 = .329m/s2. We will need to insert a negative sign since this is downward.

The weight has two components: a component parallel to the plane, WII = mg sin f}j and acomponent perpendicular to the plane, W-L = mg cos f} .There is no motion perpendicular to theplane, so N = W -L. The kinetic friction is then f = JLkN = JLkmgcosf}. The net force parallel tothe plane is F = ma = f -WII = JLkmgcosf} -mgsinf}. Solving this for JLk,

Jl,k- (a + y sin (})/(y COS (}),

[( -0.329 m/s2) + (9.81 m/s2) sin(28.0°)]/[((9.81 m/s2) cos(28.0°)} = 0.494.

59

E5-19 The acceleration is a = -2d/t2, where d = 203m is the distance down the slope and t is

the time to make the run.The weight has two components: a component parallel to the incline, WII = mgsin(}; and a

component perpendicular to the incline, W.l = mg cos (} .There is no motion perpendicular tot heplane, so N = W.l. The kinetic friction is then f = J.tkN = J.tkmgcos(}. The net force parallel tothe plane is F = ma = f -WII = J.tkmgcos(} -mgsin(}. Solving this for J.tk,

J.tk= (a + 9 sin 9)/(g cos 9),

(gsin9 -2d/t2)/(gcos9).

If t = 61 s, then

JLk =

if t = 42 s, then

JLk =

E5-20 (a) If the block slides down with constant velocity then a = O and J.tk = tanO. Not onlythat, but the force of kinetic friction must be equal to the parallel component of the weight, f = WII.If the block is projected up the ramp then the net force is now 2WII = 2mgsinO. The decelerationis a = 2gsinO; the block will travel a time t = vo/a before stopping, and travel a distance

d = -at2/2 + vot = -a(vo/a)2/2 + vo(vo/a) = v5/(2a) = v5/(4gsinO)

before stopping.(b) SÍnce Jl,k < Jl,s, the incline is not steep enough to get the block moving again once it stops.

~ Let al be acceleration down frictionless incline of length l, and tl the time taken. The a2is acceleration down "rough" incline, and t2 = 2tl is the time taken. Then

1 2 1 ( )2l = 2altl and l = 2a2 2tl .

Equate and find al/a2 = 4.There are two force which act on the ice when it sits on the frictionless incline. The normal force

acts perpendicular to the surface, so it doesn 't contribute any components parallel to the surface.The force of gravity has a component parallel to the surface, given by

WII = mgsin O,

and a component perpendicular to the surface given by

W.1. = mgcos(}.

The acceleration clown the frictionless ramp is then

al = ~ = gsinO.m

When friction is present the force of kinetic friction is fk = JLkN j since the ice doesn 't moveperpendicular to the surface we also have N = W.l ; and finally the acceleration down the ramp is

Wll-fka2 = = g(sin9 -JLcos9).m

60

Previously we found the ratio of al/a2, so we now have

sin (}

sin 33°

= 4sin8 -4JLcOS 8,

= 4 sin 33° -4JL cos 33° ,

= 0.49.11

E5-22 (a) The static friction between A and the table must be equal to the weight of block B tokeep A from sliding. This means mBg = JLs(mA + mc)g, or mc = mB/JLs -mA = (2.6kg)/(0.18)-(4.4kg) = 10kg.

(b) There is no up/down motion for block A, so f = JLkN = JLkmAg. The net force on the systemcontaining blocks A and B is F = W B -f = mBg -JLkmAg; the acceleration of this system is then

a = gmB -JLkmA ={~.m/s2)(2.6kg) -(0.1~)(4.4kg) = 2.7m/s2.mA + mB (2.6kg) +(4.4kg)

~ There are four forces on the block- the force of gravity, W = mg; the normal force,N; the horizontal push, P, and the force of friction, f. Choose the coordinate system so thatcomponents are either parallel (x-axis) to the plane or perpendicular (y-axis) to it. (J = 39°. Referto the figure below.

?1

The magnitudes of the x components of the forces are W x = W sin () , p x = p cos () and f; themagnitudes of the y components of the forces are W y = W cos () , p y = p sin () .

(a) We consider the first the case of the block moving up the ramp; then f is directed down.Newton 's second law for each set of components then reads as

¿Fx

¿Fy

~

-

Px -f -Wx = Pcos() -f -Wsin() = max,

N- Py -Wy = N- Psin() -Wcos() = may

Then the secand equatian is easy ta salve far N

N = Psin(} + Wcos(} = (46 N) sin(39°) + (4.8 kg)(9.8 m/s2) cos(39°) = 66 N.

The force offriction is found from f = JLkN = (0.33)(66 N) = 22 N. This is directed down the inclinewhile the block is moving up. We can now find the acceleration in the x direction.

max = Pcos(}-f-Wsin(},

= (46N)cos(39°) -(22N) -(4.8kg)(9.8m/s2)sin(39°) = -16N.

So the block is slowing down, with an acceleration of magnitude 3.3 m/s2.(b) The block has an initial speed ofvox = 4,3 m/s; it will rise until it stops; so we can use

Vy = O = VOy+ayt to find the time to the highest point. Then t = (Vy -voy)/ay = -( -4.3 m/s)/(3.3m/s2 = 1.3 s. Now that we know the time we can use the other kinematic relation to find thedistance

y = VOyt + 4ayt2 ~ (4.3 m/s)(1.3 s) + ~(-3.3 m/s2)(¡.3 8)2 = 2.8 m

61

( c ) When the block gets to the top it might slide back down. But in order to do so the frictionalforce, which is now directed up the ramp, must be sufficiently small so that f + p x .$: W x. Solvingfor f we find f .$: Wx -Px or, using our numbers from above, f.$: -6 N. Is this possible? No, sothe block will not slide back down the ramp, even if the ramp were frictionless, while the horizontalforce is applied.

E5-24 (a) The horizontal force needs to overcome the maximum static friction, so P ~ .U8N ~.U8mg ~ (0.52)(12 kg)(9.8 m/s2) = 61 N .

(b) If the force acts upward from the horizontal then there are two components: a horizontalcomponent Px = Pcos(J and a vertical component Py = Psin(J. The normal force is now given byW = Py + N; consequently the maximum force of static friction is now .U8N = .U8(mg -Psin(J).The block will move only if Pcos(J ~ .U8(mg -Psin(J), or

(c) If the force acts downward from the horizontal then e = -62°, so

E5-25 (a) If the tension acts upward from the horizontal then there are two components: a hori-zontal component T x = T cos () and a vertical component T y = T sin () .The normal force is now givenby W = Ty + N; consequently the maximum force of static friction is now JLsN = JLs(W -Tsin()).

The crate will move only if T cos () ?; JLs (W -T sin ()) , or

(b) Once the crate starts to move then the net force on the crate is F = Tx -f. The acceleration

is then

a = .JL¡TcosfJ -Jlk(W -Tsin fJ)],W

(3( 2 ft/S2) ){(70 Ib)cos(17°) -(0.35)¡(150 lb) -(70 Ib)sin(17°)]},

150 lb

4.6 ft/S2,

E5-26 If the tension acts upward from the horizontal then there are two components: a horizontalcomponent T x = T cos (} and a vertical component T y = T sin (} .The normal force is now given byW = Ty +N; consequently the maximum force ofstatic friction is now J.tsN = J.ts(W -Tsin(}). The

crate will move only if T cos (} ?: J.ts(W -T sin (}), or

w :::; Tcos(}/JLs + Tsin(}.

We want the maximum, so we find dW/d(},

dW/dO = -(T/JLs) sinO + TcosO,

which equals zero when IJ.s = tan(J. For this problem (J = arctan(O.35) = 19°, so

w ~ (1220N)cos(19°)/(0.35) + (1220N)sin(19°) = 3690 N.

62

E5-27 The three force on the know above A must add to zero, Construct a vector diagram:TA + TB + Td = O, where Td refers to the diagonal rope. TA and TB must be related by TA =

TB tan(}, where (} = 41°,

T

There is no up/down motion of block B, so N = WB and f = JJ.sWB. Since block B is at restf = TB. Since block A is at rest W A = TA. Then

WA = WB(JLstan()) = (712N)(0.25)tan(41°) = 155N.

E5-28 (a) Block 2 doesn't move up/down, so N = W2 = m2g and the force of friction on block 2is f = J1,km2g. Block 1 is on a frictionless incline; only the component of the weight parallel to thesurface contributes to the motjon, and WII = mlg sin (} .There are two relevant forces on the twomass system. The effective net force is the of magnitude WII -f, so the acceleration is

= 1.25 m/s2.a=gm¡ sin () -¡¡'km2

The blocks accelerate clown the ramp.(b) The net force on block 2 is F = m2a = T -f. The tension in the cable is then

T = m2a + /lkm29 = {2.30 kg)[{1.25 m/s2) + {0.47){9.81 m/s2)] = 13.5 N.

~ This problem is similar to Sample Problem 5- 7, except now there is friction which can acton block E. The relevant equations are now for block E

-mBgcoslJ = O

andT- mBgsin(} ::I: f = mBa,

where the sign in front of f depends on the direction in which block B is moving. If the block ismoving up the ramp then friction is directed down the ramp, and we would use the negative sign.If the block is moving down the ramp then friction will be directed up the ramp, and then we willuse the positive sign. Finally, if the block is stationary then friction we be in such a direction as tomake a = O.

For block A the relevant equation is

mAg -T =,: mAa.

Combine the first two equations with f = JJ.N to get

T- mEgsin (} ::!:: J1,mEgcOS(} = mEa.

63

Take some care when interpreting friction for the static case, since the static value of .u yields themaximum possible static friction force, which is not necessarily the actual static frictional force.

Combine this last equation with the block A equation,

mAg -mAa -mEgsin(J :!:: ¡¡,mEgcos(J = mEa,

mB sin() = (13.2kg) -(42.6 kg)(0.669) = 15.3 kgmA

where the negative sign indicates that block B would slide downhill if there were no friction.H the blocks are originally at rest we need to consider static friction, so the last term can be as

large asf1,mB COS(} = (.56)(42.6kg)(0.743) = 17.7kg.

Since this quantity is larger than the first static friction would be large enough to stop the blocksfrom accelerating if they are at rest.

(b) If block B is moving up the ramp we use the negative sign, and the acceleration is

~ ~4.08m/s2.

where the negative sign means down the ramp. The block, originally moving up the ramp, willslow down and stop. Once it stops the static friction takes over and the results of part (a) becomerelevant.

( c) If block B is moving down the ramp we use the positive sign, and the acceleration is

=-1.30m/s2.

where the negative sign means clown the ramp. This means that if the block is moving clown theramp it will continue to move clown the ramp, faster ancl faster .

E5-30 The weight can be resolved into a component parallel to the incline, WII = Wsin(} and acomponent that is perpendicular , W.L = W cos (} .There are two normal forces on the crate, one fromeach side of the trough. By symmetry we expect them to have equal magnitudes; since they bothact per:pendicular to their respective surfaces we expect them to be perpendicular to each other .They must add to equal the perpendicular component of the weight. Since they are at right anglesand equal in magnitude, this yields N2 + N2 = W.L 2, or N= W .L/V2.

Each surface contributes a frictional force f = J1,kN = J1,k W .L/ V2; the total frictional force isthen twice this, or V2J1,k W.L .The net force on the crate is F = W sin (} -V2J1,k W cos (} down theramp. The acceleration is then

a = g(sinO -V2'Uk cosO).

64

E5-31 The normal force between the top slab and the bottom slab is N = W t = mtg. The forceof friction between the top and the bottom slab is f ~ .uN = .umtg. We don't yet know if the slabsslip relative to each other, so we don 't yet know what kind of friction to consider .

The acceleration of the top slab is

at = (110N)/(9.7kg) -J1,(9.8m/s2) = 11.3m/s2 -J1,(9.8m/s2).

The acceleration of the bottom slab is

ab = JL(9.8m/s2)(9.7,kg)/(42kg) = JL(2.3m/s2).

Can these two be equal? Only if f1, ?; 0.93. Since the static coefficient is less than this, the blockslide. Then at = 7.6m/s2 and ab = 0.S7m/s2.

E5-32 (a) Convert the speed to ft/s: v = 88 ft/s. The acceleration is

a = v2jr = (88 ftjs)2j(25 ft) = 310 ftjs2.

(b) a = 310 ft/S29/(32 ft/S2) = 9.79.

~ (a) The force required to keep the car in the turn is F = mv2 Ir = Wv2 I(rg), or

F = (10700N)(13.4m/s)2/[(61.0m)(9.81m/s2)] = 3210N.

(b) The coefficient of friction required is 1Ls =F/W = (3210N)/(10700N) = 0.300.

E5-34 (a) The proper banking angle is given by

V2 (16.7m/s)2 = 11°.= arctan (150m)(9.8m/s2)(} = arctan "Rg

(b) If the road is not banked then the force required to keep the car in the turn is F = mv2/r =Wv2/(Rg) and the required coefficient of friction is

V2

Rg

JLs = F/W =

E5-35 (a) This conical pendulum makes an angle (J = arcsin(0.25/1.4) = 10° with the vertical.The pebble has a speed of

v = ~ = v(O.25m)(9.8m/s2)tan(10o) = O.66m/s.

(b) a = v2/r = (0.66m/s)2/(0.25m) = 1.7m/s2.(c} T = mg/cosO = (0.053 kg)(9.8m/s2)/cos(100) = O.53N.

E5-36 Ignoring air friction (there must be a forward component to the friction!), we have a nor-mal force upward which is equal to the weight: N = mg = (85kg)(9.8m/s2) = 833N. Thereis a sideways component to the friction which is equal tot eh centripetal force, F = mv2/r =(85kg)(8. 7m/s)2 /(25m) = 257N. The magnitudeof the net force of the road on the person is

F = V(833N)2 + (257N)2 = 870N,

and the direction is (J = arctan(257/833) = 17° off of vertical.

65

E5-37 (a) The speed is v = 27rrf = 27r(5.3x10-11m)(6.6x1015/s) = 2.2x106m/s.(b) The acceleration is a = v2/r = (2.2 x 106m/s)2 /(5.3x 10-11m) = 9.1 x 1022 m/s2.(c) The net force is F = ma = (9.1 x 10-31 kg)(9.1 x 1022 m/s2) = 8.3x 10-8N.

E5-38 The bask~t has speed v = 27rr /t. The basket experiences a frictional force F = mv2/r =m(27rr/t)2/r = 47r2mr/t2. The coefficient of static friction is Jl,s = F/N = F/W. Combining,

47r2r

gt2

47r2(4.6m)JLs = .

~ There are two forces on the hanging cylinder: the force of the cord pulling up T and theforce of gravity W = Mg. The cylinder is at rest, so these two forces must balance, or T = W.There are three forces on the disk, but only the force of the cord on the disk T is relevant here, sincethere is no friction or vertical motion.

The disk undergoes circular motion, so T = mv2 /r .We want to solve this for v and then expressthe answer in terms of m, M, r, and G.

v = I!J.= {!1!.

E5-40 (a) The frictional force stopping the caris then a = JLsg. If the car is moving at v = 13.is half this, or vav = 6.7 m/s. The time requiredThe deceleration is a = (13.3m/s)/(3.1s) = 4.3(4.3m/s2)/(9.8m/s2) = 0.44.

(b) The acceleration is the same as in part ( a) .

~ There are three forces to consider: the normal force of the road on the car N; the force ofgravity on the car W; and the frictional force on the car f. The acceleration of the car in circularmotion is toward the center of the circle; this means the net force on the car is horizontal, towardthe center. We will arrange our coordinate system so that r is horizontal and z is vertical. Then thecomponents ofthe normal force are Nr = Nsin(} and Nz = Ncos(}; the components ofthe frictionalforce are fr = f cos (} and f z = f sin(}.

The direction of the friction depends on the speed of the car .The figure below shows the twoforce diagrams.

1:1 ¡1

y 7

w

The turn is designed for 95 km/hr , at this speed a car should require no friction to stay on theroad. Using Eq. 5-17 we find that the banking angle is given by

tan(}b = ~ = (26 m/s)2 = 0.33,rg (210 m)(9.8 m/s2)

66

is F = JLsN = JLsmg. The deceleration of the car

3 m/s then the average speed while deceleratingto stop is t = x/Vav = (21m)/(6.7m/s) = 3.1s.m/s2. The coefficient of friction is JLs = a/g =

, so r = v2/a = (13.3m/s)2/(4.3m/s2) = 41m.

for a bank angle of ()b = 18° .(a) On the rainy day traffic is moving at 14 m/s. This is slower than the rated speed, so any

frictional force must be directed up the incline. Newton 's second law is then

mv2¿Fr Nr -fr = Nsin() -fcos() =

¿Fz

~

rN z + f z -w = N cos () + f sin () -mg = O.=

We can substitute f = J1,8N to find the minimum value of J1,8 which will keep the cars from slipping.There will then be two equations and two unknowns, J1,8 and N. Solving for N ,

2N(sin()-J1,8COS()) = mv and N(COS()+J1,8Sin()) =mg.

r

Combining,

Rearrange,gr sin (J -V2 cos (JJ1,8 = gr COS (J + V2 sin (J .

We know all the numbers. Put them in and we'll find J1,8 = 0.22(b) Now the frictional force will point the other way, so Newton's second law is now

mv2LFr = Nr+fr=Nsin(J+fcos(J=

LFz =

The bottom equation can be rearranged to show that

mg

r

Nz -fz -w = Ncos{} -fsin{} -mg = o.

N= ,cos o -J.l;s sin o

This can be combined with the top equation to give

sinO + J.l;s cosO n~-,

/'V2

',,¡,g .cos f} -JL8 S1ll f} r

We can solve this final expression for v using all our previous numbers and get v = 35 m/s. That's

about 130 km/hr .

-=

E5-42 (a) The net force on the person at the top of the Ferris wheel is mv2/r = W -N t, pointingdown. The net force on the bottom is still mv2 / r , but this quantity now equals N b -W and is pointup. Then Nb = 2W -Nt = 2(150 lb) -(125 lb) = 175 lb.

(b) Doubling the speed would quadruple the net force, so the new scale reading at the top wouldbe (150 lb) -4[(150 lb) -(125 lb)] = 50 lb. Wee!

E5-43 The net force on the object when it is not sliding is F = mv2/r; the speed of the object isv = 27rrf (f is rotational frequency here), so F = 47r2mrf2. The coefficient of friction is then atleast Jls = F /W = 47r2r f2 / g. If the object stays put when the table rotates at 331 rev /min then

JLs ?: 47r2(0.13 m) (33.3/60 18)2/(9.8 m/s2) = 0.16.

If the object slips when the table rotates at 45.0 rev /min then

JLs ::; 47r2(0.13m)(45.0/60/s)2/(9.8m/s2) = 0.30.

67

E5-44

E5-45 (a) Assume that frigate bird fiies as we1l as a pilot. Then this is a banked highway problem.The speed of the bird is given by V2 = gr tan (J , Butthere is also vt = 27rr, so 27rV2 = gvt tan (J , or

E5-46 (a) The radius ofthe turn is r = v(33m)2 -(18m)2 = 28m. The speed ofthe plane is v =27rrf = 27r(28m)(4.4/60/s) = 13m/s. The acceleration is a = v2/r = (13m/s)2/(28m) = 6.0m/s2.

(b) The tension has two components: Tx = Tcos(} and Ty = Tsin(}. In this case (} =arcsin(18/33) = 33°. All of the centripetal force is provided for by Tx, so

T= (0.75kg)(6.0m/s2)/cos(33°) = 5.4N.

( c ) The lift is balanced by the weight and T y. The lift is then

Ty+W= (5.4N)sin(33°)+(O.75kg)(9.8m/s2) = ION.

E5-47 (IJ.) The acceleration is a = v2/r = 47r2r/t2 = 47r2(6.37x 106m)/(8.64x 104s)2 = 3.37x10-2m/s2. The centripetal force on the standard kilogram is F = ma = (1.00kg)(3.37x10-2m/s2) =0.0337 N .

(b) The tension in the balance would be T = W -F =(9.80N) -(0.0337N) = 9.77N.

E5-48 (a) v = 4(0.179m/s4)(7.18s)3 -2(2.08 m/s2)(7.18s) = 235m/s.(b) a = 12(0.179m/s4)(7.18s)2 -2(2.08m/s2) = 107m/s2.(c) F = ma = (2.17kg)(107m/s2) = 232N.

~ The force only has an x component, so we can use Eq. 5-19 to find the velocity.

Integrating,

v x = Vo + ao1 'I

t -271"t2, )

Now put this expression into Eq. 5-20 to find the position a.s a functio~ of time

t t .x = Xo + 1 Vx dt = 11

Integrating,T2

ao-.3

/~T2 -,2

Now We can put t = T into the expression for v.

x = 'IJoT + do (

~"' =,va + aa ('1;'- ~T2) = va + aaT/2.1

68

P5-1 (a) There are two forces which accelerate block I: the tension, T, and the parallel componentofthe weight, W",l = mlgsin()lo Assuming the block accelerates to the right,

m la = mlgsin 01 -T.

There are two forces which accelerate block 2: the tension, T, and the parallel component of theweight, WII,2 = m2g sin (}2. Assuming the block 1 accelerates to the right, block 2 must also accelerate

to the right, andm2a = T- m2gsin(}2.

Add these two equations,(m¡ + m2)a = m¡gsin(J¡ -m2gsin (J2,

and then rearrange:m19 sin 01 -m29 sin 02

a=m¡ + m2

Or, take the two net force equations, divide each side by the mass, and set them equal to each other:

9sin {}1 -T/m1 = T/m2 -gsin{}2°

Rearrange,

= 9sin 01 + gsinO2,T1

m¡

1 ,

m2)and then rearrange again:

mlm29 ( . (} .(} )SIn 1 +SIn 2.

ml + m2

(b) The negative sign we get in the answer means that block 1 accelerates up the ramp.

T=

P5-2 (a) Since the pulley is massless, F = 2T. The largest value of T that will allow block 2 toremain on the fioorisT ~ W2 = m2g. So F ~ 2(1.9kg)(9.8m/s2) = 37N.

(b) T = F/2 = (110N)/2 = 55N.(c) The net force on block 1 is T- W1 = (55N) -(1.2kg)(9.8m/s2) = 43N. This will result in

an acceleration of a = (43N)/(1.2kg) = 36m/s2.

~ As the string is pulled the two m8SSeS will move together so that the configuration willlooklike the figure below. The point where the force is applied to the string is m8Ss1ess, so ¿ F = O atthat point. We can take advantage of this fact and the figure below to find the tension in the cords,F/2 = Tcos(}. The factor of 1/2 occurs because only 1/2 ofF is contained in the right triangle thath8S T 8S the hypotenuse. From the figure we can find the x component of the force on one m8Ss tobe Tx = Tsin(}. Combining,

Tx = ~ sin{} F2 cose = 2 tan{}.

69

(c) No acceleration happens when m2 = (3.70kg)sin(28°)/sin(42°) = 2.60kg. If m2 is moremassive than this ml will accelerate up the plane; if m2 is less massive than this ml will acceleratedown the plane.

But the tangent is equal to

Opposite

Adjacent

tan (} =x-

-VL2 -X2And now we have the answer in the book.

!

What happens when x = L ? Well, ax is infinite according to this expression. Since that couldonly happen if the tension in the string were infinite, then there must be some other physics thatwe had previously ignored.

P5-4 (a) The force of static friction can be as large as f ~ JLsN = (0.60)(12 lb) = 7.2 lb. That ismore than enough to hold the block up.

(b) The force of static friction is actually only large enough to hold up the block: f = 5.0 lb.The magnitude of the force of the wall on the block is then Fbw = V(5.0)2 + (12.0)2lb = 13 lb.

P5-5 (a) The weight has two components: normal to the incline, W.L = mg cos (} and parallel to theincline, WII = mg sin (} .There is no motion perpendicular to the incline, so N = W.L = mg cos (} .Theforce of friction on the block is then f = JLN = JLmg cos (} , where we use whichever JL is appropriate.The net force on the block is F -f -WII = F :!: JLmg cos (} -mg sin (} .

To hold the block in place we use JLs and friction will point up the ramp so the :!: is +, and

F = (7.96 kg)(9.81 mjs2)[sin(22.0°) -(0.25) cos(22.0°)] = 11.2N.

(b) To find the minimum force to begin sliding the block up the ramp we still have static friction,but now friction points down, so

F = (7.96kg)(9.81m/s2)[sin(22.00) + (0.25)cos(22.00)] =47.4N.

( c ) To keep the block sliding up at constant speed we have kinetic friction, so

F = (7.96 kg}(9.81 m/s2}[sin(22.0°} + (0.15} cos(22.0°}] = 40.1 N.

P5-6 The sand will slide if the cone makes an angle greater than (} where .u8 = tan (}.tan(} = h/R or h = Rtan(}. The volume ofthe cone is then

But

Ah/3 = 7rR2h/3 = 7rR3tan(}/3 = 7rJ1,sR3/3.

~ There are four forces on the broom: the force of gravity W = mg; the normal force of thefloor N; the force offriction f; and the applied force from the person p (the book calls it F). Then

LFx

LFy

= P", -f = PsinO -f = ma""

N- Py -w = N-PcosO-mg = may = O=

Salve the secand equatian far N,N = PcosO + mg.

70

(a) If the mop slides at constant speed f = JLkN. Then

Psin(j -f = Psin(j -J.l;k (PCOS(j + mg) = O.

We can solve this for P (which was called F in the book);

P= J.l;mgsin (} -/Lk COS (} .

This is the force required to push the broom at constant speed.(b) Note that p becomes negative (or infinite) ifsin(} .$: /LkCOS(}. This occurs when tan(}c .$: /Lk.

If this happens the mop stops moving, to get it started again you must overcome the static friction,but this is impossible if tan (}o .$: /Ls

P5-8 (a) The condition to slide is J.l;s ~ tanO. In this case, (0.63) > tan(24°) = 0.445.(b) The normal force on the slab is N = W l. = mg cos O. There are three forces parallel to the

surface: friction, f = J.l;sN = J.l;smgcosOj the parallel component of the weight, WII = mgsinO, and

the force F. The block will slide if these don 't balance, or

F > J.l;smgcosO -mgsinO = (1.8x 107kg)(9.8m/s2)[(0.63) cos(24°) -sin(24°)] = 3.0x 107N.

~ To hold up the smaller block the frictional force between the larger block and smaller blockmust be as large as the weight of the smaller block. This can be written as f = mg. The normalforce of the larger block on the smaller block is N, and the frictional force is given by f ~ JLsN. Sothe smaller block won't fall if mg ~ JLsN.

There is only one horizontal force on the large block, which is the normal force of the small blockon the large block. Newton 's third law says this force has a magnitude N, so the acceleration of thelarge block is N = M a.

There is only one horizontal force on the two block system, the force F. So the acceleration ofthis system is given by F = (M + m)a. The two accelerations are equal, otherwise the blocks won'tstick together. Equating, then, gives N/M = F/(M + m).

We can combine this last expression with mg ~ JLsN and get

Mmg ~ JLsF ~

or

g(M +m)m

J1sM

F?:

P5-10 The normal force on the ith block is Ni = migcos(); the force of friction on the ith blockis then fi = J1,imigcos(). The parallel component of the weight on the ith block is WII,i = migsin().

(a) The netiorce on the system is

F = }:= mig(sin 0- J.Li cos O).i

Then

(9.81a ={1.65 kg) + {3.22 kg)

3.46m/s2,

(b) The net force on the lower mass is m2a = WII,2 -!2- T, so the tension is

T = (9.81 m/s2)(3.22kg)(sin29.5° -O.127cos29.5°) -(3.22 kg)(3.46m/s2) = 0.922 N.

71

( c ) The acceleration will stay the same, since the system is still the same. Reversing the orderof the masses can only result in a reversing of the tension: it is still 0.992 N, but is now negative,meaning compression.

rL\e

F'",

~

~ The rope wraps around the dowel and there is a contribution to the frictional force ~ffrom each small segment of the rope where it touches the dowel. There is also a normal force ~N ateach point where the contact occurs. We can find ~N much the same way that we solve the circularmotion problem.

In the figure on the left below we see that we can form a triangle with long side T and short side~N. In the figure on the right below we see a triangle with long side r and short side r~(}. Thesetriangles are similar, so r~(}/r = ~N/T.

<J-~~ N

íi::~ / r\

Ir

,1

J \\""' """ ," ,

~

Now ~f = JL~N and T(()) + ~f ~ T(() + ~()). Combining, and taking the limit as ~() -O,

dT=df

J ~~

JLT

= Id(}

integrating both sides of this expression,

J ~~

Ji,T

1

-lnT¡T2Ji, Tl

T2

= 7r,

In this case Tl is the weight and T2 is the downward force.

P5-12 Answer this out of order!(b) The maximum static friction between the blocks is 12.0 N; the maximum acceleration of the

top block is then a = F/m = (12.0N)/(4.40kg) = 2.73m/s2.(a) The net force on a system of two blocks that will accelerate them at 2.73m/s2 is F =

(4.40kg + 5.50kg)(2.73m/s2) = 27.0N.(c) The coefficient of friction is JLs = F/N = F/mg = (12.0N)/[(4.40kg)(9.81m/s2)] = 0.278.

P5-13 The speed is v = 23.6m/s.(a) The average speed while stopping is half the initial speed, or vav = 11.8 m/s. The time to stop

is t = (62m)/(11.8m/s) = 5.25s. The rate of deceleration is a = (23.6m/s)/(5.25s) = 4.50m/s2.The stopping force is F = ma; this is related to the frictional force by F = JLsmg, so JLs = a/ 9 =(4.50m/s2)/(9.81 m/s2) = 0.46.

72

(b) Thrning,a = v2/r = (23.6 m/s)2/(62 m) = 8.98 m/s2,

Then Jl,s = a/g = (8.98m/s2)/(9.81 m/s2) = 0.92.

P5-14 (a) The net force on car as it travels at the top of a circular hill is F net = mv2/r = W -N;in this case we are told N = W/2, so Fnet = W/2 = (16000N)/2 = 8000N. When the car travelsthrough the bottom valley the net force at the bottom is F net = mv2/r = N -W. Since themagnitude of v, r, and hence F net is the same in both cases,

N = W/2 + W = 3W/2 = 3(16000N)/2 = 24000N

at the bottom of the valley.(b) You leave the hill when N = O, or

v = .¡r:g ~ J(250m)(9.81 m/s2) = 50m/s.

(c) At this speed Fnet = W, so at the bottom of the valley N= 2W = 32000 N.

P5-15 (a) v = 27rr/t = 27r(0.052m)(3/3.3s) = 0.30m/s.(b) a = v2/r = (0.30m/s)2/(0.052m) = 1.7m/s2, toward center.(c) F = ma = (0.0017kg)(1.7m/s2) = 2.9xI0-3N.

(d) If the coin can be as far awayas r before slipping, then

J1,s = F:/mg = (27rr/t)2/(rg) = 47r2r/(t2g) = 47r2(0.12m)/[(3/3.3s)2(9.8m/s2)] = 0.59.

P5-16 (a) Whether you as8ume constant 8peed or con8tant energy, the ten8ion in the 8tring mu8tbe the greate8t at the bottom of the circle, 80 that'8 where the 8tring will break.

(b) The net force on the 8tone at the bottom i8 T- W = mv2/r. Then

v = Vrg[T/W -iJ= J (2.9 ft)(32 ft~s2)[(9.2 lb)/(0.82 lb) -1];;:= 31 ft/s.

~ (a) In order to keep the ball moving in a circle there must be a net centripetal force Fcdirected horizontally toward the rod. There are only three forces which act on the ball: the force ofgravity, W = mg = (1.34kg)(9.81 m/s2) = 13.1 N; the tension in the top string T1 = 35;ON, and thetension in the bottom string, T2.

The components of the force from the tension In the top string are

T1,x = (35.0N) cos30° = 30.3N and T1,1/ = (35.0N) sin30° = 17.5N.

The vertical components do balance, so

T!,y + T2,y = W,

or T2,11~ (13.1N) -(17.5N) = -4;4N. From this we can find the tensionin the bottom string,

T2 = T2,y/sin(-300) = 8.8N.

(b) The net force on the object wil1 be the sum of the two horizontal components,

Fc = (30.3N) +(8.8N)cos30° = 37.9N.

(c) The speed will be found from

~ = ,¡F;;:¡m ,

v(37.9m)(1.70m) sin 60° /(1.34kg) = 6.45m/s.

73

P5-18 The net force on the cube is F = mv2 /r .The speed is 27rr(U. (Note that we are using (U ina non-standard way!) Then F = 47r2mr(U2. There are three forces to consider: the normal force of

the funnel on the cube N; the force of gravity on the cube W; and the frictional force on the cubef. The acceleration of the cube in circular motion is toward the center of the circle; this means thenet force on the cube is horizontal, toward the center. We will arrange our coordinate system sothat r is horizontal and z is vertical. Then the components of the normal force are N r = N sin (}and N z = N cos (}; the components of the frictional force are f r = f cos (} and f z = f sin (} .

The direction of the friction depends on the speed of the cube; it will point up if (U is small anddown if (U is large.

(a) If(U is small, Newton's second law is

LFr

LFz

Nr -Ir = N sin {} -Icos{} = 411"2mr(.;2,=

Nz + fz -w = NcosB +fsinB -mg = o.=

We can substitute f = JLsN. Solving for N

N ( cos o + 11.8 sin O) = mg.

Combining,sin () -JLs cos ()

47r2r(¿)2 = 9:cos (} + ¡¡;:s¡;;-¡¡ .

Rearrange,

1 !gsin(}-JLscos(}(.;= 2;V~ cos(}+JLssin(}'

This is the minimum value.(b) Now the frictional force will point the other way, so Newton's second law is now

LFr

LFz

Nr + Ir = N sin {} + Icos{} = 47r2mr(¡)2,=

Nz -fz -w = Ncos() -fsin() -mg = o.-

We've swapped + and -signs, so

(U=.!.- 1:?: sin (} + /1.8 COS (}

27r V ~ cos (} -JLs sin (}

is the maximum value.

P5-19 (a) The radial distance from the axis of rotation at a latitude L is RcosL. The speedin the circle is then v = 27rRcosL/T. The net force on a hanging object is F = mv2/(RcosL) =47r2mRcosL/T2. This net force is notdirected toward the center ofthe earth, but is instead directedtoward the axis of rotation. It makes an angle L with the Earth 's vertical. The tension in the cablemust then have two components: one which is vertical ( compared to the Earth) and the other whichis horizontal. If the cable makes an angle f) with the vertical, then 111 = Tsinf) and T.L = Tcosf).Then TII = FII and W -T.L = F.L. Written with a little more detail,

Tsin{} = 47r2mRcosL sinL/T2 ~ T{},

andTcos(} = 47r2mRcOS2 L/T2 + mg ~ T.

74

But 47r2 R COS2 L/T2 « y, so it can be ignored in the last equation compared to y, and T ~ my.Then from the first equation,

9 = 211"2 Rsin 2L/(gT2).

(b) This is a maximum when sin2L is a maximum, which happens when L = 45°. Then

(} = 27r2(6.37x 106m)/[(9.8m/s2)(86400S)2] = 1.7x 10-3 rad.

(c) The deflection at both the equator and the poles would be zero.

P5-20 a = (Fo/m)e-t/T. Then v = J~ adt = (FoT/m)e-t/T, and x = J~ vdt = (FoT2/m)e-t/T.(a) When t = T v = (FoT/m)e-l = O.368(FoT/m).(b) When t = T x = (FoT2/m)e-l = O.368(FoT2/m).

75

E6-1 (a) VI = (m2/mI)V2 = (2650kg/816kg)(16.0 km/h) = 52.0 km/h.(b) VI = (m2/mI)v2 = (9080 kg/816kg)(16.0 km/h) = 178 km/h.

E6-2 Pi = (2000kg)(40 km/h)j = 8.00x 104kg .km/bj. Pf = (2000kg)(50 km/h)i = 1.00x 105kg .km/hi. ~P = Pf -Pi = 1.00 x 105kg .km/hi -8.00 x 104kg .km/bj. ~P = V(~Px)2 + (~py)2 =1.28 x 105kg .km/h. The direction is 38.7° south of east.

~ The figure below shows the initial and final momentum vectors arranged to geometricallyshow Pf -Pi = ~p. We can use the cosine law to find the length of ~p.

""' -pi

~ , ,y -- ---

e

pf

~p

//

The angle a = 42° + 42°, Pi = m v = (4.88kg)(31.4m/s) = 153kg.m/s. Then the magnitude oí~p is

6.p = (153kg.m/s)2 + (153kg.m/s)2 -2(153kg.m/s)2 cos(84°) = 205kg.m/s,

directed up from the plate. By symmetry it must be perpendicular .

E6-4 The change in momentum is ~p = -mv = -(2300kg)(15m/s) = -3.5xl04kg .m/s. Theaverage force is F = ~p/ ~t = ( -3.5 x 104kg .m/s)/(0.54s) = -6.5 x 104N.

E6-5 (a) The change in momentum i8 ~p = (-mv) -mvj the average force i8 F = ~p/~t =

-2mv/~t.(b) F = -2(0.14 kg)(7.8m/8)/(3.9x 10-38) = 560N.

E6-6 (a) J = ~p = (O.O46kg)(52.2m/s) -O = 2.4N .s.

(b) The impulse imparted to the club is opposite that imparted to the ball.(c) F = ~p/ ~t = (2.4N .s)/(1.20x 10-3s) = 2000N.

~ Choose the coordinate system so that the ball is only moving along the x axis, with awayfrom the batter as positive. Then Pfx = mvfx = (0.150kg)(61.5m/s) = 9.23 kg.m/s and Pix =mvix = (0.150 kg)( -41.6 m/s) = -6.24 kg.m/s. The impu1se is given by Jx = Pfx -Pix = 15.47kg.m/s. We can find the average force by application of Eq. 6-7:

Jx

~tF, av,:!: =

76

E6-8 The magnitude of the impulse is J = Ft5t = ( -984 N)(O.O270s) = -26.6N .s. Then Pf =

Pi + ~P, so

The hall moves hackward!

E6-9 The change in momentum ofthe ball is ~p = (mv) -( -mv) = 2mv = 2(O.O58kg)(32m/s) =

3.7 kg .m/ s. The impulse is the area under a force -time graph; for the trapezoid in the figure thisarea is J = Fmax(2ms + 6ms)/2 = (4ms)Fmax. Then Fmax = (3.7kg .m/s)/(4ms) = 930N.

E6-10 The final speed of each object is given by Vi = J/mi, where i refers to which object (asopposed to "initial"). The object are going in different directions, so the relative speed will be thesum. Then

Vrel = V¡ + V2 = (300 N. s)[1/(1200 kg) + 1/(1800 kg)] O.42m/s.

~ Use Simpson 's rule. Then the area is given by

1-h(/o +4!1 + 2!2 +4!3 + ...+4!13+!14) ,31-(0.2ms) (200 + 4.800 + 2 .1200...N)3

Jx :r.

which gives Jx = 4.28 kg.m/s.Since the impulse is the change in momentum, and the ball started from rest, Pfx = Jx+Pix = 4.28

kg.m/s. The final velocity is then found from Vx = px/m = 8.6 m/s.

E6-12 ( a) The average speed during the the time the hand is in contact with the board is half of theinitial speed, or vav = 4.8m/s. The time of contact is then t = y/vav = (O.O28m)/(4.8m/s) = 5.8ms.

(b) The impulse given to the board is the same as the magnitude in the change in momentum ofthe hand, or J = (0.54kg)(9.5m/s) = 5.1 N. s. Then Fav = (5.1 N. s)/(5.8ms) = 880N.

E6-13 Ap = J = FAt = (3000N)(65.0s) = l.95x lO5N .s. The direction of the thrust relative tothe velocity doesn 't matter in this exercise.

E6-14 (a) p = m v = (2.14x 10-3kg)(483m/s) = 1.03 kg .m/s.(b) The impulse imparted to the wall in one second is ten times the above momentum, or

J = 10.3kg .m/s. The average force is then F av = (10.3kg .m/s)/(1.0s) = 10.3N.(c) The average force for each individual particle is F av = (1.03kg. m/s)/(1.25x10-3 s) = 830N.

~ A transverse direction means at right angles, so the thrusters have imparted a momentumsufficient to direct the spacecraft 100+3400 = 3500 km to the side of the original path. The spacecraftis half-way through the six-month journey, so it has three months to move the 3500 km to the side.This corresponds to a transverse speed of v = (3500x103m)/(90x86400s) = 0.45m/s. The requiredtime for the rocket to fire is ~t = (5400kg)(0.45m/s)/(1200N) = 2.0s.

E6-16 Total initial momentum is zero, so

~~m m

.!!!:!!iL~

mmg

fJs = - lIs = -,Vm=-

77

~ Conservation of momentum:

Pf,m + Pf,c

mmVf,m + mcVf,c

= Pi,m +pi,c,

= mmVi,m + mcVi,c,

~~!,m -mmVf,mVf,c -Vi,c

~

mc

{75.2 kg){~.33 mis) -{75.2 kg){O)~vc =

(38.6 kg)

4.54m/s.

The answer is positive; the cart speed increases.

E6-18 Conservation of momentum:

Pf,m +Pf,c = Pi,m +Pi,c,

m m (Vf,c -Vrel) + mcVf,c = (m m + mc)Vi,C,

(m m + mc)Vf,c -1nmVrel = (m m + mc)Vi,c,

Avc = mmVrel/(mm + mc),

= WVrel/(W + W).

E6-19 Conservation of momentum. Let m refer to motor and c refer to command module:

Pf,m + Pf,c

m m (Vf,c -Vrel) + mcVf,c

(m m + mc)Vf,c -mmVrel

= Pi,m +Pi,c,

= (m m + mc)Vi,c,

= (mm+mc)Vi,c,

= .mmVrel + (mm+ mc)Vi,CVf,c

(m m + mc)

4mc(125 km/h}+ (4mc + mc)(3860 km/h)= 3960 km/h.

(4mc + mc)

E6-20 Conservation of momentum. The block on the left is 1, the other is 2.

mlVl f + m2V2 f, ,

Vl,f

= m¡V¡,i + m2V2,i,m2= V¡ i + -(V2 i -V2 f),, m¡ , ,

= (5.5m/s) + ~[(2.5m/S) -(4.9m/s)],

1.9m/s,

E6-21 Conservation oí momentum. The block on the left is I, the other is 2.

mi V1,f + m2V2,f

Vl,f

= mlVl,i + m2V2.i,m2= Vl i + - (V2 i -V2 f ) ,, m ..

1

(2.4 kg)= (5.5m/s) + -¡i:¡j"kg)[(-2.5m/s) -(4.9m/s)],

-5.6 m/s.

78

E6-22 Assume a completely inelastic collision. Can the Earth I and the meteorite 2. Then

That 's 2 mm/y!

~ Canservatian af mamentum is used ta salve the prablem:

Pf = Pi,

Pf,bl + Pf,bu = Pi,bl + Pi,bu,

mblVf,bl + mbuVf,bu = mblvi,bl + mbuVi,bu.

(715 g)Vf,bl + (5.18 g)(428 mis) = (715 g)(O) + (5.18 g)(672 mis),

which has solution Vf bl = 1.77m/s.,

E6-24 The y component oí the initial momentum is zero; thereíore the magnitudes oí the y com-ponents oí the two particles must be equal after the collision. Then

mQvQ sin()Q= movo sin (}o,

movo sin (}oVa = mQvQ sin(}Q ,

.(16 u)(1.20x105m/s)~~ = 4.15 x 105m/s.=

E6-25 The total momentum is

¡3 = (2.0 kg)[(15m/s)i + (30m/s)jj + (3.0kg)[( -10m/s)i + (5m/s)jj,= 75kg. m/sj.

The final velocity of B is

VBf1 ( -- )p-mAVAf ,=

mB

E6-26 Assume electron travels in +x direction while neutrino travels in +y direction. Conservation

of momentum requires that

p = -{1.2 x 10-22kg .mis)i -{6.4 x 10-23kg .mis)j

be the momentum oí the nucleus after the decay. This has a magnitude oí p = 1.4 x 10-22kg .mis

and be directed 152° from the electron.

79

E6-27 What we know:

Pl,i

P2,i

P2,f

= (1.50x 105kg)(6.20m/s)i = 9.30x 105kg .m/si,

= (2. 78x 105kg)(4.30m/s)j = 1.20x 106kg .m/sj,

= (2.78 x 105kg)(5.10 m/s)[sin(18°)i + cos(18°)j],

= 4.38 x 105kg .m/si + 1.35 x 106kg .m/sj.

Conservation of momentum then requires

Pl,f = (9.30x105kg.m/si)-(4.38x105kg.m/si)

+(1.20 x 106kg .m/sj) -(1.35 x 106kg .m/sj),

= 4.92 x 105kg .m/s i -1.50 x 105kg .m/s i.

This corresponds to a velocity of

Vl,f = 3.28m/si -1.00m/sj,

which has a magnitude oí 3.43 mis directed 17° to the right.

E6-28 Vf = -2.1 m/s.

Making these~ We want ta salve Eq. 6-24 far m2 given that Vl,f = O and Vl,i = -V2,i.substitutians

so m2 = 100 g.

E6-30 (a) Rearrange Eq. 6-27:

E6-31 Rearrange Eq. 6-27:

Vli '- Vlfm2 = ml , = (2.0

Vli + VlfI~Vli + Vli/4 = 1.2kg.

E6-32 I'll multiply all momentum equations by g, then I can use weight directly without convertingto mass.

(a) Vf = [(31.8 T)(5.20 ft/s) + (24.2 T)(2.90 ft/s)]/(31.8 T + 24.2 T) = 4.21 ft/s.(b) Evaluate:

80

~ Let the initial momentum of the first object be Pl,i = mVl,i, that of the second object beP2,i = mV2,i, and that of the combined final object be Pf = 2mVf. Then

-+ -+ -+Pl,i + P2,¡;;: Pf,

implies that we can find a triangle with sides of length Pl,i, P2,i, and Pf. These lengths are

Pl,i

P2,i

Pf

= mvj,

= mvj,

2mvf = 2mvj/2 = mvj,

so this is an equilateral triangle. This meansthe angle between the initial velocities is 120°.

E6-34 We need to change to the center of mass system. Since both particles have the samemass, the conservation of momentum problem is effectively the same as a ( vector ) conservation ofvelocity problem. Since one of the particles is originally at rest, the center of mass moves withspeed Vcm = Vli/2. In the figure below the center of mass velocities are primed; the transformationvelocity is Vt.

v t

Note that since Vt = v' li = v' 2i = v' lf = v' 2f the entire problem can be inscribed in a rhombus.The diagonals of the rhombus are the directions of Vlf and V2f j note that the diagonals of a rhombusare necessarily at right angles!

( a) The target proton moves off at 90° to the direction the incident proton moves after thecollision, or 26° away from the incident pr9tons original direction.

(b) The y components ofthe final momenta must be equal, so v2fsin(26°) = Vlf8in(64°), or V2f =Vlf tan(64°). The x components must add to the original momentum, so (514m/s) = V2f cos(26°) +

Vlfcos(64°), or(514m/s)/{tan(64°) cos(26°) + cos(64°)} = 225m/s,

Vlf

andV2f = (225m/s)tan(64°) = 461m/s.

E6-35 v cm = {(3.16 kg)(15.6m/s)+(2.84 kg)(-12.2m/s)}/{(3.16kg)+(2.84 kg)} = 2.44m/s, pos-

itive means to the left.

81

~ The force is the change in momentum over change in time; the momentum is the mass timevelocity, so

dp mdv mF= -= -=dv- =2u.u

At At At ,

since J.L is the mass per unit time.

P6-2 (a) The initial momentum is Pi = (1420kg)(5.28m/s)j = 7500kg .m/sj. After making theright hand turn the final momentum is Pf = 7500 kg .m/s ¡. The impulse is j = 7500 kg .m/s ¡ -

7500kg .m/sj, which has magnitude J = 10600kg .m/s.

(b) During the collision the impulse is j = 0-7500kg.m/s¡. The magnitude is J = 7500kg.m/s.(c) The average force is F = J/t = (10600kg .m/s)/(4.60s) = 2300N.(d) The average force is F = J/t = (7500kg .m/s)/(0.350s) = 21400N.

P6-3 (a) Only the component ofthe momentum which is perpendicular to the wall changes. Then

.1 = ~p = -2(0.325 kg)(6.22 mis) sin(33°)j = -2.20 kg .misj.

A(b) F = -J/t = -( -2.20kg .m/sj)/(O.O104s) = 212N.

P6-4 The change in momentum ofone bullet is ~p = 2mv = 2(O.OO30kg)(500m/s) = 3.0kg.m/s.The average force is the total impulse in one minute divided by one minute, or

F av = lOO(3.0kg .m/s)/(60s) = 5.0N.

P6-5 (a) The volume of a hailstone is V = 47rr3/3 = 47r(0.5 cm)3/3 = 0.524 cm3. The mass of ahailstone is m = pV = (9.2 x 10-4kg/cm3)(0.524 cm3) = 4.8x 10-4kg.

(b) The change in momentum of one hai1stone when it hits the ground is

~p = (4.8 x 10-4kg)(25 m/s) = 1.2 x 10-2kg .m/s.

The hailstones fall at 25 m/s, which means that in one second the hailstones in a column 25 m highhit the ground. Over an area of 10mx 20m then there would be (25m)(10m)(20m) = 500m3 worthof hailstones, or 6.00 x 105 hailstones per second striking the surface. Then

Fav = 6.00xlO5(l.2xlO-2kg .m/s)/(ls) = 7200 N.

P6-6 Assume the links are not connected once the top link is released. Consider the link thatstarts h above the table; it falls a distance h in a time t = J2h79 and hits the table with a speedv = gt = ..¡2Tí¡j. When the link hits the table h of the chain is already on the table, and L- h is yetto come. The linear mass density of the chain is M / L, so when this link strikes the table the mass ishitting the table at a rate dm/dt = (M/L)v = (M/L)..¡2Tí¡j. The average force required to stop thefalling link is then vdm/dt = (M/L)2hg = 2(M/L)hg. But the weight ofthe chain that is alreadyon the table is (M/L)hg, so the net force on the table is the sum ofthese two terms, or F = 3W.

~ The weight of the marbles in the box after a time t is mgRt because Rt is the number ofmarbles in the box.

The marbles fa11 a distance h from rest; the time required to fal1 this distance is t = .j2ií7Y,the speed of the marbles when they strike the box is v = gt = .¡2g7i". The momentum each marble

imparts on the box is then m.¡2g7i". If the marbles strike at a rate R then the force required to stopthem is Rm.¡2g7i".

82

The reading on the scale is then

w = mR( J29h + gt).

This will give a numerical result of

(4.60x 10-3kg)(115 S-I ) ( .¡2(9.81 m/s2)(9.62 m) + (9.81 m/s2)(6.50s) ) = 41.0N.

P6-8 (a) v = (108kg)(9.74m/s)/(108kg+ 1930kg) = 0.516m/s.(b) Label the person as object 1 and the car as object 2. Then mlVl+m2V2 = (108kg)(9.74m/s)

and Vl = V2 + 0.520 m/s. Combining,

V2 = [1050 kg .m/s -(0.520 m/s)(108 kg)1/(108 kg + 1930 kg) = 0.488 m/s.

P6-9 (a) It takes a time tl = J2h79 to fall h = 6.5 ft. An object will be moving at a speedVl = gtl = V2fig after falling this distance. If there is an inelastic collision with the pile then thetwo will move together with a speed of V2 = MV1/(M + m) after the collision.

Ifthe pile then stops within d = 1.5 inches, then the time ofstopping is given by t2 = d/(V2/2) =

2d/V2.For inelastic collisions this corresponds to an average force of

F -..= .(M + m)v2 -(M + m)v~ -M2v~ (gM)2 ~.=-t2 2d 2(M+m)d g(M+m) do

Note that we multiply through by 9 to get weights. The numerical result is F av = 130 t.(b) For an elastic collision V2 = 2Mvl/(M + m); the time of stopping is still expressed by

t2 = 2d/V2, but we now know F av instead of d. Then

F -mv2 -mv~ 4Mmv~ -2(gM)(gm) ~av- g(M+m) d'=-==-

(M+m)d2dt2

or

2(gM)(gm)~d= ,g(M+m) Fav

which has a numerical result of d = 0.51 inches.But wait! The weight, which just had an elastic collision, "bounced" off of the pile, and then hit

it again. This drives the pile deeper into the earth. The weight hits the pile a second time with aspeed of V3 = (M -m)/(M +m)vl; the pile will (in this second elastic collision) then have a speed ofV4 = 2M(M + m)v3 = [(M -m)/(M + m)]v2. In other words, we have an infinite series of distancestraveled by the pile, and if a = [(M -m)/(M + m)] = 0.71, the depth driven by the pile is

df = d(l + a2 + a4 + a6d

) ,= l-a2

or d = 1.03.

P6-10

O, or

The catjumps offofsled A; conservationofmomentum requires that MVA,l +m(VA,l +Vc) =

VA,l = -mvc/(m + M) = -(3.63kg)(3.05m/s)/(22.7kg + 3.63kg) = -0.420m/s.

The cat lands on sled B; conservation of momentum requires VB,l = m(VA,l + vc)/(m + M). The

cat jumps off of sled B; conservation of momentum is now

MVB,2 + m(VB,2 -Vc) = m(VA.l + Vc),

83

or

VB,2 = 2mvc/(m + M) = (3.63kg)[(-0.420m/s) + 2(3.05m/s)]/(22.7kg + 3.63kg) = O.783m/s.

The cat then lands on cart A; conservation of momentum requires that (M + m)vA,2 = -MVB,2, or

VA,2 = -(22.7kg)(0.783m/s)/(22.7kg+3.63kg) = -0.675m/s.

~ We align the coordinate system so that west is +x and south is +y. The each carcontributes the following to the initial momentum

A: (2720 lb/g)(38.5 mi/h)i = 1.05x 105 lb. mi/h/gi,

B : (3640 lb/g)(58.0 mi/h)j = 2.11 x 105 lb. mi/h/gj.

These become the components of the final momentum. The direction is then

2.11 x 105 lb. mi/h/g o(} = arctan 1.05 x 105 lb. mi/h/ 9 = 63.5 ,

south of west. The magnitude is the square root of the sum of the squares,

2.36 x 105 lb. mi/h/ g,

and we divide this by the mass (6360 lb/g) to get the final speed after the collision: 37.1 mi/h.

P6-12 (a) Hall A must carry offamomentumofp = mBvi-mBv/2j, whichwould be in adirectionf) = arctan(-0.5/1) = 27° from the original direction of E, or 117° from the final direction.

(b) No.

P6-13 (a) We assume all balls have a mass m. The collision imparts a "sideways" momentum tothe cue ball ofm(3.50m/s) sin(65°) = m(3.17m/s). The other ball must have an equal, but opposite"sideways" momentum, so -m(3.17m/s) = m(6.75m/s)sin(}, or (} = -28.0°.

(b) The final "forward" momentum is

m(3.50m/s) cos(65°) + m(6.75m/s) cos( -28°) = m(7.44m/s),

so the initial speed of the cue ball would have been 7.44m/s.

P6-14 Assuming M » m, Eq. 6-25 becomes

V2f = 2Vli -Vli = 2(13 km/s) -(-12 km/s) = 38 km/s.

~ (a) We get

2(220 g)V2,f = (220 g) + (46.0 g) (45.0m/s) = 74.4m/s.

(b) Doubling the m88S of the clubhead we get

2L440 g) I .-~ r- .--'~ 1-U2,f = (440 g) + {46.0 g) t45.0mjs) = dl.5m¡s.

{c) Tripling the mass of the clubhead we get

2{660 g)V2,f = (66üg) + (46.0 g) t45.0m¡s) = 84.1m/s.

84

Although the heavier club helps some, the maximum speed to get out of the ball will be less thantwice the speed of the club.

The balls are a, b, and c from left to right.P6-16 There will always be at least two collisiollS.After the first collision between a and b one has

Vb,l = va and Va,l = O.

After the first collision between b and c one has

Vc,l = 2mvo/(m+ M) and Vb,2 = (m- M)vo/(m + M).

(a) Ifm ;:::: M then ball b continue to move to the right (or stops) and there are no more collisions.(b) If m < M then ball b bounces back and strikes ball a which was at rest. Then

Va,2 = (m -M)vo/(m + M) and Vb,3 = O.

P6-17 All three balls are identical in mass and radii? Then balls 2 and 3 will move off at 30° tothe initial direction of the first ball. By symmetry we expect balls 2 and 3 to have the same speed.

The problem now is to define an elastic three body collision. It is no longer the case that theballs bounce off with the same speed in the center of mass. One can 't even treat the problem as twoseparate collisions, one right after the other. No amount of momentum conservation laws will helpsolve the problem; we need some additional physics, but at this point in the text we don 't have it.

P6-18 The original speed is Vo in the lab frame. Let a be the angle of deflection in the cm ffameand v~ be the initial velocity in the cm frame. Then the velocity after the collision in the cm frame

A A A Ais v~ cos a i + v~ sin aj and the velocity in the lab frame is ( v~ cos a + v )i + v~ sin aj, where v is thespeed of the cm frame. The deflection angle in the lab frame is

() = arctan[(v~ sina)/(v~ cosa + v)],

but v = mlVO/(ml + m2) andv~ = Vo -v so v~ = m2VO/(ml + m2) and

(}= arctan[(m2 sina)/(m2 cosa + ml)].

(c) (} is a maximum when (cosa + ml/m2)/sina is a minimum, which happens when cosa =-ml/m2 if ml S m2. Then [(m2 sina)/(m2 cosa + ml)] can have any value between -00 and 00,so (} can be between O and 7r .

(a) Ifml > m2 then (cosa+ml/m2)/sina is a minimum when cosa = -m2/ml, then

[(m2 sina)/(m2 cosa + ml)] = m2/Vm~ -m~.

If tan(} = m2/ vm~ -m~ then m¡ is like a hypotenuse and m2 the opposite side. Then

(b) We need to change to the center of m8SS system. Since both particles have the same m8Ss,the conservation of momentum problem is effectively the same 8S a (vector) conservation of velocityproblem. Since one of the particles is originally at rest, the center of m8SS moves with speedVcm = Vli/2. In the figure below the center of m8Ss velocities are primed; the transformationvelocity is Vt .

85

Vt

Note that since Vt = v' li = v' 2i = v' lf = v' 2f the entire problem can be inscribed in a rhombus.The diagonals of the rhombus are the directions of Vlf and V2fj note that the diagonals of a rhombusare necessarily at right angles!

~ ( a) The speed of the bullet after leaving the first block but before entering the second canbe determined by momentum conservation.

(l.78kg)(l.48m/s)+(3.54x lO-3kg)(l.48m/s) = (l.78kg)(0)+(3.54x lO-3kg)Vi,bu,

which has solution Vi,bl = 746m/s.(b) We do the same steps again, except applied to the first block,

Pf =

(l.22kg)(0.63mis)+ (3.54 x lO-3kg)(746 mis)

==

(l.22kg)(0)+(3.54x lO-3kg)Vi,bu,=

which has solution Vi,bl = 963 m/s.

P6-20 The acceleration of the block down the ramp is al = gsin(22°). The ramp has a length ofd = h/sin(22°), so it takes a time tl = J2d7a;- = J2h79/sin(22°) to reach the bottom. The speedwhen it reaches the bottom is Vl = altl = V2g7i:. Notice that it is independent of the angle!

The collision is inelastic, so the two stick together and move with an initial speed of V2 =mlVl/(ml + m2). They slide a distance x before stopping; the average speed while decelerating isvav = v2/2, so the stopping time is t2= 2X/V2 and the deceleration is a2 = V2/t2 = v~/(2x). Iftheretarding force is f = (ml + m2)a2, then f = JLk(ml + m2)g. Glue it aIl together and

86

Pf = Pi,

Pf,bl + Pf,bu = Pi,bl + Pi,bu,

mblVf,bl + mbuVf,bu = mblVi,bl + mbuVi.bu,

Pf,bl + Pf,bu

mblVf,bl + mbuVf,bu

Pi,

Pi,bl +Pi,bu,

mblVi,bl + mbuVi,bu,

P6-21 (a) For an object with initial speed v and deceleration -a which travels a distance x beforestopping, the time t to stop is t = v/a, the average speed while stopping is v/2, and d = at2/2.Combining, v = V2ax. The deceleration in this case is given by a = Jl,kg.

Then just after the collision

VA = V2(0.130)(9.81m/s2)(8.20m)=4.57m/s,

whileVE = V2(0.130)(9.81 m/s2)(6.10m) = 3.94m/s,

(b) Va = [(1100 kg)(4.57m/s) + (1400kg)(3.94m/s)]/(1400kg) = 7.53m/s.

87

E7-1 Xcm = {7.36x lO22kg){3.82 x lO8m)/{7.36x lO22kg + 5.98x lO34kg) = 4.64x lO6m. This is lessthan the radius of the Earth.

E7-2 Ir the particles are l apart then

Xl = mll(ml + m2)

is the distance from particle 1 to the center oí mass and

X2 = m2l(ml + m2)

is the distance from particle 2 to the center of mass. Divide the top equation by the bottom and

Xl/X2 = ml/m2.

~ The center of mass velocity is given by Eq. 7-1,

mlVl +m2V2"cm = ,

mi + m2

.(2210 kg)(105 km/h) + (2080 kg)(43.5 km/h)= 75.2 km/h.

(2210 kg) + (2080 kg)

E7-4 They will meet at the center of mass, so

Xcm = (65kg)(9.7m)/(65kg+42kg) = 5.9m.

E7-5 (a) No external forces, center of mass doesn't move.(b) The collide at the center of mass,

Xcm = (4.29 kg)(1.64m)/(4.29kg + 1.43kg) = 1.23m.

E7-6 The range of the center of ma.ss is

R = v~sin2(}/g = (466m/s)2sin(2 x 57.4°)/(9.8lm/s2) = 2.0lxlO4m.

Half lands directly underneath the highest point, or 1.00 x 104 m. The other piece must land at x,such that

2.01 x 104 m = (1.00 x 104 m + x)/2;

then x = 3.02 x 104 m.

~ The center of mass of the boat + dog doesn 't move because there are no external forces onthe system. Define the coordinate system so that distances are measured from the shore, so towardthe shore is in the negative x direction. The change in position of the center of mass is given by

~Xcm = ' = O,

-1-

md~Xd + mb~Xb

Both ~Xd and ~Xb are measured with respect to the shore; we are given ~Xdb = -8.50 ft, thedisplacement of the dog with respect to the boat. But

~Xd = ~Xdb + ~Xb.

88

Since we want to find out about the dog, we'll substitute for the boat's displacementt

O = ~~~b (~Xd -~Xdb)md + mb

Rearrange and solve for ~Xd. Use W = m y and multiply the top and bottom of the expression byy. Then

6.90 ft.

E1-8 Richardhag too much time on his hands.The center of mags of the system is Xcm away from the center of the boat. Switching seats is

effectively the same thing ag rotating the canoe through 180°, so the center of mags of the systemhag moved through a distance of 2xcm = 0.412 m. Then Xcm = 0.206 m. Then

Xcm = (Ml- ml)/(M + m + mc) = 0.206m,

where 1 = 1.47m, M = 78.4kg, mc = 31.6kg, and m is Judy's mags. Rearrange,

Ml- (M + mc)Xcm (78.4kg)(1.47m) ~ (78.4kg+ 31.6kg) (0.206m )m = .-= 55.2 kg.

~

+Xcm (1.47m) + (0.206 m)

E7-9 It takes the man t = (18.2m)/(2.08m/s) = 8.75s to walk to the front of the boat. Duringthis time the center ofmass ofthe system has moved forward x = (4.16m/s)(8.75s) = 36.4m. Butin walking forward to the front of the boat the man shifted the center of mass by a distance of(84.4kg)(18.2m)/(84.4kg + 425kg) = 3.02m, so the boat only traveled 36.4m -3.02m = 33.4m.

E7-10 Do each coordinate separately.

(3kg)(O) + (8kg)(lm) + (4kg)(2m)07mXcm =

(3kg) + (8kg) + (4kg)

and

E7-12 The velocity components of the center of mass at t = 1.42s are Vcm,x = 7.3m/s andVcm,y = (10.0m/s) -(9.81m/s)(1.42s) = -3.93m/s. Then the velocity components ofthe "other"piece are

V2,x = [(9.6kg)(7.3m/s) -(6.5kg)(11.4m:/s)]/(3.1 kg) = -1.30m/s.

andV2,y = [(9.6kg)(-3.9m/s) -(6.5kg)(-4.6m/s)]/(3.1kg) = -2.4m/s.

89

~ The center of mass should lie on the perpendicular bisector of the rod of mass 3M. Wecan view the system as having two parts: the heavy rod of mass 3M and the two light rods each ofmass M. The two light rods have a center of mass at the center of the square.

Both of these center of masses are located along the vertical line of symmetry for the object.The center of mass of the heavy bar is at Yh,cm = O, while the combined center of mass of the twolight bars is at Yl,cm = L/2, where down is positive. The center of mass of the system is then at

~

5=L/5.y cm = ':' .r .~ .r -

2Myl,cm + 3Myh,cm

E7-14 The two slabs have the same volume and have mass mi = PiV. The center oí mass islocated at

- {5 5 ) {7.85 g/cm3) -{2.70 g/cm3) - 2 68-.cm 3 3 -.cm{7.85 g/cm ) + {2.70 g/cm )

from the boundary inside the iron; it is centered in the y and z directions.

E7-15 Treat the íour sides oí the box as one thing oí mass 4m with a mass located l/2 above thebase. Then the center oí mass is

Zcm = ([/2) (4m)/(4m + m) = 2[/5 = 2(0.4m)/5 = O.16m,

Xcm = y cm = O.2m.

E7-16 One piece moves off with momentum m(31.4m/s)i, another moves off with momentum2m(31.4m/s)j. The third piece must then have momentum -m(31.4m/s)i -2m(31.4m/s)j andvelocity -(1/3)(31.4m/s)i -2/3(31.4m/s)j = -10.5m/si -20.9m/sj. The magnitude of V3 is23.4m/s and direction 63.3° away from the lighter piece.

E7-17 It will take an impulse of (84.7kg)(3.87mis) = 328kg.mis to stop the animal. This wouldcome from firing n bullets where n = (328 kg .mis)i[(O.O126 kg)(975 mis) = 27.

E7-18 Conservation of momentum for firing one cannon ball of mags m with muzzle speed Vcforward out of a cannon on a trolley of original total mags M moving forward with original speedVo is

Mvo = (M -m)Vl + m(vc + Vl) = MVl + mvc,

where Vl is the speed of the trolley after the cannonball is fired. Then to stop the trolley we requiren cannonballs be fired so that

n = (Mvo)/(mvc) = [(3500 kg)(45m/s)]/[(65kg)(625m/s)] = 3.88,

so n = 4.

~ Label the velocities of the various containers as Vk where k is an integer between one andtwelve. The mass of each container is m. The subscript "g" refers to the goo; the subscript k refersto the kth container .

The total momentum before the collision is given by

...~ ...p = L.., mVk,

k

+ mgVg,i = 12mvcont"cm + mgVg,i'

90

We are told, however, that the initial velocity of the center of mass of the containers is at rest, sothe initial momentum simplifies to p = mgVg,i, and has a magnitude of 4000 kg.m/s.

(a) Thenp (4000kg.m/s) = 3.2m/s.

12m+mg 12(100.0kg) + (50kg)(b) It doesn 't matter if the cord breaks, we'll get the same answer for the motion of the center

of mass.

v~~ = =

E7-20 (a) F = (3270 m/s)(480 kg/s) = l.57x lO6N.(b) m = 2.55x lO5kg -(480 kg/s)(250s) = l.35x lO5kg.

(C) Eq. 7-32:

Vf = ( -3270 m/s) ln(1.35 x 105kg/2.55 x 105kg) = 2080 m/s.

E7-22 Eq. 7-32 rearranged:

Mf~e e-l~tJ/tJrell :;::¡ e-(22.6m/s)/(1230m/s) = 0.982.

Mi

The fraction of the initial mass which is discarded is 0.0182.

E7-23 The loaded rocket has a weight of (1.11xl05kg)(9.81 m/s2) = 1.09xl06N; the thrust must beat least this large to get the rocket offthe ground. Then v 2:: (1.09xl06N)/(820kg/s) = 1.33xl03m/8is the minimum exhaust speed.

E7-24 The acceleration down the incline is (9.8 m/s2) sin(26°) = 4.3 m/s2. It will take t =v2(93m)/(4.3m/s2) = 6.6s. The sand doesn't affect the problem, so long as it only "leaks" out.

~ We'll use Eq. 7-4 to solve this problem, but since we are given weights instead of masswe'll multiply the top and bottom by 9 like we did in Exercise 7-7. Then

-mlVl + m2V2 9 W1Vl + W2V2Vcm =- =9 Wl + W2m¡ + m2

Now for the numbers

P7-1 (a) The balloon moves clown so that the center of mass is stationary;

o = MVb +mvm = MVb +m(v + Vb),

or Vb = -mv/(m + M).(b) When the man stops so does the balloon.

91

P7-2 (a) The center oí ma.ss is midway between them.(b) Mea.sure from the heavier ma.ss.

Xcm = (0.0560 m)(0.816kg)/(1.700kg) = 0.0269 m,

which is 1.12 m m closer to the heavier mass than in part (a).( c ) Think Atwood 's machine. The acceleration of the two masses is

a = 2Am9/(ml + m2) = 2(0.034kg)9/(1.700kg) = 0.04009,

the heavier going clown while the lighter moves up. The acceleration of the center of mass is

acm = (am¡ -am2)/(m¡ + m2) = (O.O400g)2(O.O34kg)g/(1.700kg)0.001609.

~ This is a glorified Atwood's machine problem. The total mass on the right side is the massper unit length times the length, mr = >.x; similarly the mass on the left is given by ml = >.(L -x) .Then

m2 -m¡ ).x- ).(L- x) 2x -Lg = ,-

, .,- , -a= g- g

P7-4 (a) Assume the car is massless. Then moving the cannonballs is moving the center of mass,unless the cannonballs don 't move but instead the car does. How far? L.

(b) Once the cannonballs stop moving so does the rail car .

~ By symmetry, the center of mass of the empty storage tank should be in the very center ,along the axis at a height Yt,cm = H /2. We can pretend that the entire mass of the tank, mt = M ,is located at this point.

The center of mass of the gasoline is also, by symmetry, located along the axis at half the height ofthe gasoline, Yg,cm = x /2. The mass, if the tank were filled to a height H, is m; assuming a uniformdensity for the gasoline, the mass present when the level of gas reaches a height x is mg = mx / H .

(a) The center of mass of the entire system is at the center of the cylinder when the tank is fulland when the tank is empty. When the tank is half full the center of mass is below the center. So asthe tank changes from full to empty the center of mass drops, reaches some lowest point, and thenrises back to the center of the tank.

(b) The center of mass of the entire system is found from

mx2+MH2

2mx+2MH.

Take the derivative:m (mx2 + 2xMH -MH2)

~ =dx (mx+MH)2

Set this equal to zero to find the minimum; this means we want the numerator to vanish, or mx2 +2xMH- MH2 = O. Then

-M+vM2+mM~ H.x= Ccm

92

m2 + ml AX -¡- A~1.1 -X) 1.1

which salves the prablem. The acceleratian is in the rlirectian ar the sirle ar length X ir X > L/2.

The center oí masswill be located along symmetry axis. Call this the x axis. Then

Xcm = :b J xdm,

4R

3;-'

P1-1 (a) The components of the shell velocity with respect to the cannon are

v~ = (556m/s)cos(39.00) = 432m/s and v~ = (556m/s)sin(39.00) = 350m/s.

The vertical component with respect to the ground is the same, Vy = v~ ' but the horizontal compo-nent is found from conservation of momentum:

M(vx -v~) + m(vx) = O,

so Vx = (1400kg)(432m/s)(70.0kg+ 1400kg) = 411m/s. The resulting speed is v = 540m/s.

(b) The direction is (} = arctan(350/411) = 40.4°.

P7-8 v = (2870 kg)(252 m/s)/(2870kg + 917kg) = 191 m/s.

P7-9 It takes (1.5m/s)(20kg) = 30N to accelerate the luggage to the speed of the belt. Thepeople when taking the luggage off will ( on average) a1so need to exert a 30 N force to remove it;this force (because of friction) will be exerted on the belt. So the belt requires 60 N of additionalforce.

P7-10 (a) The thrust must be at least equal to the weight, so

dmldt = (5860kg)(9.81m/s2)/(1170m/s) = 49.1kg/s.

(b) The net force on the rocket will need to be F = (5860kg)(18.3m/s2) = 107000N. Add thisto the weight to find the thrust, so

dmldt = [107000N+ (5860kg)(9.81m/s2)]/(1170m/s) = 141kg/s

~ Consider Eq. 7-31. We want the barges to continue at constant speed, so the left handside of that equation vanishes. Then

93

We are told that the frictional force is independent ofthe weight, since the speed doesn't change thefrictional force should be constant and equal in magnitude to the force exerted by the engine beforethe shoveling happens. Then ¿ F ext is equal to the additional force required from the engines. We'llcall it P.

The relative speed of the coal to the faster moving cart has magnitude: 21.2- 9.65 = 11.6km/h= 3.22 m/s. The mass flux is 15.4 kg/s, so p = (3.22 m/s)(15.4kg/s) = 49.6 N. The fastermoving cart will need to increase the engine force by 49.6 N. The slower cart won't need to doanything, because the coal left the slower barge with a relative speed of zero according to ourapproximation.

P7-12 (a) Nothing is ejected from the string, so Vrel = O. Then Eq. 7-31 reduces to mdv/dt = F ext.(b) Since F ext is from the weight of the hanging string, and the fraction that is hanging is y / L ,

F ext = mgy/ L. The equation of motion is then ~y/dt2 = gy/ L.(c) Take first derivative:

~- Yodt -2( J97L) ( e.¡g¡¡;t -e-ViiTit) ,

and then second derivative,

d2y = ~ ( V97i)2 (eat +e-at ) .dt2 2

Substitute into equation of motion. It works! Note that when t = O we have y = Yo.

94

~ An n-dimensional object can be oriented by stating the position of n different carefullychosen points Pi inside the body. Since each point has n coordinates, one might think there are n2coordinates required to completely specify the position of the body. But if the body is rigid thenthe distances between the points are fixed. There is a distance dij for every pair of points Pi andPj. For each distance dij we need one fewer coordinate to specify the position of the body. Thereare n(n -1)/2 ways to connect n objects in pairs, so n2 -n(n -1)/2 = n(n + 1)/2 is the numberof coordinates required.

E8-2 (1 rev/min)(27r rad/rev)/(60 s/min) = 0.105 rad/s.

E8-3 (a) úJ = a + 3bt2 -4ct3.

(b) a = 6bt -12t2.

E8-4 (a) The radius is r = (2.3 x 1041y)(3.0 x 108m/s) = 6.9 x 1012m .y/s. The time to make onerevolution is t = (27r6.9 x 1012m .y/s)/(250 x 103m/s) = 1. 7x 108y.

(b) The Sun has made 4.5 x 109y /1.7 x 108y = 26 revolutions.

~ (a) Integrate.

(Alz = "'o + lt ( 4at3 -3bt2) dt = (Alo + at4 -bt3

(b) Integrate, again.

[t [t I IA(} = lo (LIzdt = lo ((LIO + at4 -bt3) dt = (LIot + sat5 -¡bt4

E8-6 (a) (1 rev/min)(27r rad/rev)/(60 s/min) = 0.105 rad/s.(b) (1 rev/h)(27r rad/rev)/(3600 s/h) = 1.75xI0-3 rad/s.(c) (1/12 rev/h)(27r rad/rev)/(3600 s/h) = 1.45x 10-3 rad/s.

E8-7 85 mi/h = 125 ft/8. The hall take8 t = (60 ft)/(125 ft/8) = 0.488 to reach the plate. Itmake8 (30 rev/8)(0.488) = 14 revolution8 in that time.

E8-8 It take8 t = V2(10iñ)/(9.81m/82) = 1.438 to falll0 m.then iJ) = (2.5)(27r rad)/(1.438) = 11 rad/8.

The average angular velocity is

~ (a) Since there are eight spokes, this means the wheel can make no more than 1/8 of arevolution while the arrow traverses the plane of the wheel. The wheel rotates at 2.5 rev/s; itmakes one revolution every 1/2.5 = 0.4 s; so the arrow must pass through the wheel in less than0.4/8 = 0.05 s.

The arrow is 0.24 m long, and it must move at least one arrow length in 0.05 s. The correspondingminimum speed is (0.24 m)/(0.05 s) = 4.8 m/s.

(b) It does not matter where you aim, because the wheel is rigid. It is the angle thiough whichthe spokes have turned, not the distance, which matters here.

95

E8-10 We look for the times when the Sun, the Earth, and the other planet are collinear in somespecified order .

Since the outer planets revolve around the Sun more slowly than Earth, after one year the Earthhas returned to the original position, but the outer planet has completed less than one revolution.The Earth will then "catch up" with the outer planet before the planet has completed a revolution.If (}E is the angle through which Earth moved and (}p is the angle through which the planet moved,then (}E = (}p + 211' , since the Earth completed one more revolution than the planet.

If IJJp is the angular velocity of the planet, then the angle through which it moves during thetime Ts (the time for the planet to line up with the Earth). Then

(}E = (}p + 27r ,

c.JETs = c.JpTs + 27r ,

c.JE = c.Jp+27r/Ts

The angular velocity of a planet is w = 27r /T , where T is the period of revolution. Substituting thisinto the last equation above yields

l/TE = l/Tp + l/Ts.

~ We look for the times when the Sun, the Earth, and the other planet are collinear in somespecified order .

Since the inner planets revolve around the Sun more quickly than Earth, after one year theEarthhas returned to the original position, but the inner planet has completed more than one revolution.The inner planet must then have "caught-up" with the Earth before the Earth has completed arevolution. If (}E is the angle through which Earth moved and (}p is the angle through which theplanet moved, then (}p = (}E + 27r , since the inner planet completed one more revolution than theEarth.

96

If IA)p is the angular velocity of the planet, then the angle through which it moves during thetime Ts (the time for the planet to line up with the Earth). Then

(}p = (}E+27r,

IA)pTs = IA)ETs + 27r ,

IA)p = IA)E + 27r /Ts

The angular velocity of a planet is lA) = 27r /T , where T is the period of revolution. Substituting thisinto the last equation above yields

l/Tp = l/TE + l/Ts.

E8-12 (a) a = (-78 rev/min)/(0.533 min) = -150 rev/min2.(b) Average angular speecl while slowing clown is 39 rev/min, so (39 rev/min)(0.533 min) =

21 rev.

E8-13 (a) a = (2880 rev/min -1170 rev/min)/(0.210 min) = 8140 rev/min2.(b) Average angular speed while accelerating is 2030 rev/min, so (2030 rev/min)(0.210 min) =

425 rev.

E8-14 Find area under curye.

~(5 min+2.5 min)(3000 rey/min) = l.l3xlO4 rey.

~ (a) iJ.JOz = 25.2 rad/s; iJ.Jz = O; t 19.7 s; and Qz and <P are unknown. From Eq. 8-6,

97

(.,)z = (.,)Oz + azt,

(O) = (25.2 rad/s) + az(19. 7 s),

az = -1.28 rad/s2

(b) We use Eq. 8- 7 to find the angle through which the wheel rotates.

(c) <P = 248 rad~ = 39.5 rey.21r rau

E8-16 (a) a = (225 rev/min -315 rev/min)/(1.00 min) = -90.0 rev/min2

(b) t= (0- 225 rev/min)/( -90.0 rev/min2) = 2.50 min.(c) (-90.0 rev/min2)(2.50min)2/2 + (225 rev/min)(2.50 min) = 281 rev.

E8-17 (a) The average angular speed was (90 rev)/(15s) = 6.0 rev/s. The angular speed at thebeginning of the interval was then 2(6.0 rev/s) -(10 rev/s) = 2.0 rev/s.

(b) The angular acceleration was (10 rev/s-2.0 rev/s)/(15s) = 0.533 rev/s2. The time requiredto get the wheel to 2.0 rev/s was t = (2.0 rev/s)/(0.533 rev/s2) = 3.8s.

E8-18 (a) The wheel will rotate through an angle <I> where

t/> = (563 cm)/(8.14 cm/2) = 138 rad.

~ (a) We are given I/> = 42.3 rev= 266 rad, (AlOz = 1.44 rad/s, and (Alz = O. Assuming a

uniform deceleration, the average angular velocity during the interval is

1lA)av ,z = 2 ( lA)oz + IA)z) = 0.72 rad/ s.

Then the time taken for deceleration is given by <t> = lA)av ,zt, so t = 369 s.(b) The angular acceleration can be found from Eq. 8-6,

IAJz

(O)

Ctz

= (AJOz + azt,

= (1.44 rad/s) + az(369 s),

= -3.9 x 10-3 rad/s2.

(c) We'll solve Eq. 8-7 for t,

1 2<1> = <1>0 + IJJozt + ?Qzt ,

1{0) + {1.44 rad/s)t + 2{ -3.9 x 10-3 rad/s2)t2,

-133+ {1.44s-1)t -{-1.95 x 10-3s-2)f.

(133 rad) ;,;,

o =

Solving this quadratic expression yields two answers: t = 108 s and = 630 s.

E8-20 The angular acceleration is a = (4.96 rad/s)/(2.33s) = 2.13 rad/s2. The angle throughwhich the wheel turned while accelerating is <t> = (2.13 rad/s2)(23.0s)2/2 = 563 rad. The angular

speed at this time is (¿) = (2.13 rad/s2)(23.0s) = 49.0 rad/s. The wheel spins through an additionalangle of (49.0 rad/s)(46s -23s) = 1130 rad, for a total angle of 1690 rad.

E8-21 /JJ = (14.6 m/s)/(110m) = 0.133 rad/s.

E8-22 The linear acceleration is (25m/s -12m/s)/(6.2s) =2.1 m/s2, The angular acceleration isa = (2.1 m/s2)/(0.75m/2) = 5.6 rad/s.

98

~ (a) The angular speed is given by VT = i.l)r. So i.I) = vT/r = (28,700 km/hr)/(3220km) = 8.91 rad/hr. That's the same thing as 2.48x10-3 rad/s.

(b) aR = i.l)2r = (8.91 rad/h)2(3220 km) = 256000 km/h2, or

aR = 256000 km/h2(1/3600 h/s)2(1000 m/km) = 19.8m/s2.

( c) If the speed is constant then the tangential acceleration is zero, regardless of the shape of the

trajectory!

E8-24 The bar needs to make

1.50 cm)(12.0 turns/cm) = 18 turns.

Thi8 will happen i8 (18 rev)/(237 rev/min) = 4.568.

E8-25 (a) The angular speed is úJ = (21r rad)/(86400s) = 7.27x10-5 rad/s.(b) The distance from the polar axis is r = (6.37x106m) cos(40°) = 4.88x106m. The linear speed

is then v = (7.27x 10-5 rad/s)(4.88x 106m) = 355 m/s.(c) The angular speed is the same as part (a). The distance from the polar axis is r = (6.37x

106m) cos(O°) = 6.37x106m. The linear speed is then v = (7.27x10-5 rad/s)(6.37x106m) = 463m/s.

E8-26 (a) aT = (14.2 rad/s2)(O.O283m) = O.402m/s2.(b) F\111 speed is iA) = 289 rad/s. aR = (289 rad/s)2(O.O283m) = 2360m/s2.

(c) It takest = (289 rad/8)/(14.2 rad/82) = 20.48

to get up to full 8peed. Then x = (0.402m/82)(20.48)2/2 = 83.6m i8 the di8tance through which a

point on the rim mOVe8.

~ (a) The pilot sees the propeller rotate, no more. So the tip ofthe propeller is moving witha tangential velocity of VT = wr = (2000 rev/min)(27r rad/rev)(1.5 m) = 18900 m/min. This is thesame thing as 315 m/s.

(b) The observer on the ground sees this tangential motion and sees the forward motion ofthe plane. These two velocity components are perpendicular , so the magnitude of the sum is.¡(315m/s)2 + (133m/s)2 = 342m/s.

E8-28 aT = aR when ra = r(¿)2 = r(at)2, or t = Jl/(0.236 rad/82) = 2.068.

E8-29 (a) aR = rúJ2 = ra:2t2.(b) arr = ra:.(c) Since aR = aTtan(57.00), t = ..¡tan(57.00)/a:. Then

0.77 rad = 44.1°.

E8-30 (a) The tangential speed of the edge of the wheel relative axle is v = 27m/s. úJ =(27m/s)/(0.38m) = 71 rad/s.

(b) The average angular speed while slowing is 71 rad/s/2, the time required to stop is thent = (30 x 27r rad)/(71 rad/s/2) = 5.3s. The angular acceleration is then a = (-71 rad/s)/(5.3s) =-13 rad/s.

(c) The car moves forward (27m/s/2)(5.3s) = 72m.

99

Vt/rs,E8-31 Ves, the speed would be wrong. The angular velocity ofthe small wheel would be 1.1) :but the reported velocity would be v = l.l)rl = Vtrl/rs. This would be in error by a fraction

~v

Vt

(72 cm)

~-1 = 0.16.=

E8-32 (a) Square both equations and then add them:

X2 + y2 = (RcoS('¡)t)2 + (Rsin(,¡)t)2 = R2,

which is the equation for a circle of radius R.(b) Vx = -RJoIJsin(¿)t = -(¿)y; Vy = RJoIJcOS(¿)t = (¿)x. Square and add, v = (¿)R. The direction is

tangent to the circle.(b) ax = -RJoIJ2cOS(¿)t = -(¿)2X; ay = -RJoIJ2sin(¿)t = -(¿)2y. Square and add, a = (¿)2R. The

direction is toward the center .

~ (a) The object is "slowing down", so a. = ( -2.66 rad/s2)k. We know the directionbecause it is rotating about the z axis and we are given the direction of ¡;j. Then from Eq. 8-19,

-A A A A AV = ¡;j X R = (14.3 rad/s)k x [(1.83 m)j + (1.26 m)k]. But only the cross term k x j survives, sov = (-26.2m/s)i.

(b) We find the acceleration from Eq. 8-21,

R a = Q x +r..J x v,

= ( -2.66 rad/s2)k x [(1.83 m)j + (1.26 m)k] + (14.3 rad/s)k x ( -26.2 m/s)i,

= ( 4.87 m/s2)i + ( -375 m/s2)j.

E8-34 (a) F = -2miiJ x v = -2Tn¡,}vcos8, where 8 is the latitude. Then

F = 2(12kg)(27r rad/86400s)(35m/s) cos(45°) = 0.043 N,

and is directed west.(b) Reversing the velocity will reverse the direction, so east.(c) No. The Coriolis force pushes it to the west on the way up and gives it a westerly velocity;

on the way down the Coriolis force slows down the westerly motion, but does not push it back east.The object lands to the west of the starting point.

P8-1 (a) (¡.J = (4.0 rad/s) -(6.0 rad/s2)t+ (3.0 rad/s)t2. Then (¡.J(2.0s) = 4.0 rad/s and (¡.J(4.0s) =

28.0 rad/s.(b) a&v = (28.0 rad/s -4.0 rad/s)/(4.0s -2.0s) = 12 rad/s2.(C) a = -(6.0 rad/s2) + (6.0 rad/s)t. Then a(2.0s) = 6.0 rad/s2 and a(4.0s) = 18.0 rad/s2.

P8-2 If the wheel really does moye counterclockwise at 4.0 rey /min, then it turns through

(4.0 rev/min)/[(60 s/min)(24 frames/s)] = 2.78x10-3 rev/frame.

This means that a spoke has moved 2.78 x 10-3 rev. There are 16 spokes each located 1/16 of arevolution around the wheel. If instead of moving counterclockwise the wheel was instead movingclockwise so that a different spoke had moved 1/16 rev -2.78 x 10-3 rev = 0.0597 rev, then thesame effect would be present. The wheel then would be turning clockwise with a speed of YJ =(0.0597 rev)(60 s/min)(24 frames/s) = 86 rev/min.

100

P8-3 (a) In the diagram below the Earth is shown at two locations a day apart. The Earth rotatesclockwise in this figure.

Note that the Earth rotates through 27r rad in order to be correctly oriented for a completesidereal day, but because the Earth has moved in the orbit it needs to go farther through an angle(} in order to complete a solar day. By the time the Earth has gone all of the way around the sunthe total angle (} will be 27r rad, which means that there was one more sidereal day than solar day.

(b) There are (365.25 d)(24.000 h/d) = 8. 7660x 103 hours in a year with 265.25 solar days. Butthere are 366.25 sidereal days, so each one has a length of 8. 7660x 103/366.25 = 23.934 hours, or 23hours and 56 minutes and 4 seconds.

P8-4 (a) The period is time per complete rotation, so (,¡J = 21r/T.(b) a = ~(,¡J/~t, so

a =

~

= -2.30xlO-9rad/s2.(3.16x 1078) (0.0338)2

(c) t = (21r/0.033s)/(2.30x10-9rad/s2) = 8.28x101os, or 2600 years.(d) 21r/To = 21r/T -at, or

= 0.024 s.

~ The final angular velocity during the acceleration pha8e is (.cJz = azt = (3.0 rad/s)(4.0s) = 12.0 rad/s. Since both the acceleration and deceleration pha8es are uniform with endpoints(.cJz = 0, the average angular velocity for both pha8es is the same, and given by half of the maximum:(.cJav,z = 6.0 rad/s.

lO1

The angle through which the wheel turns is then

<I> = ú)av,zt = (6.0 rad/s)(4.1 s) = 24.6 rad.

The time is the total for both phases.(a) The first student sees the wheel rotate through the smallest angle less than one revolution;

this student would have no idea that the disk had rotated more than once. Since the disk movedthrough 3.92 revolutions, the first student will either assume the disk moved forward through 0.92revolutions or backward through 0.08 revolutions.

(b) According to whom? We've already answered from the perspective of the second student.

P8-6 !.A) = (0.652 rad/s2)t and a = (0.652 rad/s2).(a) !.A) = (0.652 rad/s2)(5.60s) = 3.65 rad/s(b) VT = !.A)r = (3.65rad/s)(10.4m) = 38m/s.(c) aT = ar = (0.652 rad/s2)(10.4m) = 6.78m/s2.(d) aR = !.A)2r = (3.65 rad/s)2(10.4m) = 139m/s2.

P8-7 {a) (¿) = {27r rad)/{3.16x 107 s) = 1.99x10-7 rad/s.{b) VT = (¿)R = {1.99x 10-7 rad/s){1.50 x 1011m) = 2.99x 104m/s.{c) aR = (¿)2R = {1.99x10-7 rad/s)2{1.50x 1011m) = 5.94x10-3m/s2.

P8-8 (a) a = ( -156 rev/min)/(2.2 x 60 min) = -1.18 rev/min2.(b) The average angular speed while slowing down is 78 rev /min, so the wheel turns through

(78 rev/min)(2.2 x 60 min) = 10300 revolutions.(c) aT = (27r rad/rev)(-1.18 rev/min2)(0.524m) = -3.89 m/min2. That's the same as -1.08x

10-3m/s2.(d) aR = (27r rad/rev)(72.5 rev/min)2(0.524m) = 1.73 x 104 m/min2. That's the same as

4.81m/s2. This is so much larger than the aT term that the magnitude of the total linear ac-celeration is simply 4.81m/s2.

~ (a) There are 500 teeth (and 500 spaces between these teeth); so disk rotates 27r/500 radbetween the outgoing light pulse and the incoming light pulse. The light traveled 1000 m, so theelapsed time is t = (1000 m)/(3 x 108 m/s) = 3.33 x 10-6s.

Then the angular speed of the disk is (¡)z = 4>/t = 1.26 x 10-2 rad)/(3.33 x 10-6s) = 3800 rad/s.(b) The linear speed of a point on the edge of the would be

VT = (¿)R = (3800 rad/s)(0.05 m) = 190m/s.

P8-10 The linear acceleration of the belt is a = aAr A. The angular acceleration of C i8 ac =a/rc = aA(rA/rc). The time required for C to get up to speed is

t = (27r rad/rev)(100 rev/min)(1/60 min/8) = 16.4s.

(1;60 rad/s2)(10.0/25.0)

P8-11 {a) The final angular speed is LLIo= {130 cm/s)/{5.80 cm) = 22.4 rad/s.{b) The recording area is 7I"{Ro2 -Ri2), the recorded track has a length 1 and width w, so

1 = 71"[{5.80 cm)2 -{2.50 cm2)J = 5 38 x 105{1.60 x 10-4 cm) .cm.

(c) Playing time is t = (5.38x 105 cm)/(130 cm/s) = 41408, or 69 minutes.

102

P8-12 The angular position is given by 4> = arctan(vt/b). The derivative (Maple!) is

vb(¿}--b2 + v2t2 ,

and is directed up. Take the derivative again,

2bv3ta= (b2 + v2t2)2 ,

but is clirectecl clown.

~ (a) Let the rocket sled move along the line x = b. The observer is at the origin andsees the rocket move with a constant angular speed, so the angle made with the x axis increasesaccording to (} = ú.lt. The observer, rocket, and starting point form a right triangle; the position yof the rocket is the opposite side of this triangle, so

tan(} = y/b implies y = b/tanr..Jt.

We want to take the derivative oí this with respect to time and get

v(t) = i.J)b/cos2(i.J)t).

(b) The speed becomes infinite (which is clearly unphysical) when t = 7r/2(AJo

103

{a) First, F = {5.0 N)i.

f' [yFz -zF1I]i + [zF:I; -xFz]j + [xF1I -yF:1;]k,

[y(O) -(0)(0)]i + [(O)F:I; -x(o)Jj + [x(o) -yF:1;]k,

[-yF:1;]k = -(3.0 m)(5.0 N)k = -(15.0 N. m)k.

(b) Now F = (5.0 N)]. Ignoring all zero terms,

f = [xFy]k = (2.0 m)(5.0 N)k = (10 N. m)k.

(c) Finally, F = ( -5.0 N)i.

T= [-yFx]k = -{3.0 m){-5.0 N)k = {15.0 N. m)k.

E9-2 (a) Everything is in the plane of the page, so the net torque will either be directed normalto the page. Let out be positive, then the net torque is T = rlF1 sin (}1 -r2F2 sin (}2 .

(b) T = (1.30m)(4.20N) sin(75.00) -(2.15 m)(4.90N) sin(58.00) = -3.66N .m.

E9-4 Everything is in the plane of the page, so the net torque will either be directed normal tothe page. Let out be positive, then the net torque is

T = (8.0m)(10N)sin(45°) -(4.0m)(16N)sin(900) + (3.0m)(19N)sin(200) = 12N .m.

~ Since r and s lie in the xy plane, then t = r x s must be perpendicular to that plane, andcan only point along the z axis.

The angle between rand sis 320°- 85° = 235°. So Itl = r8lsin(JI = (4.5)(7.3)lsin(235°)1 = 27.Now for the direction of t. The smaller rotation to bring r into s is through a counterclockwise

rotation; the right hand rule would then show that the cross product points along the positive zdirection.

E9-6 a = {3.20)[cos{63.00)j + sin{63.00)k] and b = {1.40)[cos{48.00)i + sin{48.00)k]. Then

axb (3.20) cos(63.0°)(1.40) sin(48.0°)¡

+(3.20) sin(63.0°)(1.40) cos( 48.0°)]

-(3.20) cos(63.0°)(1.40)cos(48.0°)k

1.51¡ + 2.67] -1.36k.

=

E9-7 b x a has magnitude absin<1> and points in the negative z direction. It is then perpendicularto a, so c has magnitude a2bsin<1>. The direction of c is perpendicular to a but lies in the planecontaining vectors a and b. Then it makes an angle 7r /2 -<I> with b.

(a) In unit vector notation,

c [( -3)( -3) -( -2)(1)]¡ + [(1)(4) -(2)( -3)]] + [(2)( -2) -( -3)(4)]k,

11¡ + 10] + 8k.

(b) Evaluate arcsin[li x bl/(ab)], finding magnitudes with the Pythagoras relationship:

<I> = arcsin (16.8)/[(3.74)(5.39)] = 56°.

104

~ This exercise is a three dimensional generalization of Ex. 9-1, except nothing is zero.

f = [yF% -zFy]¡ + [zFx -xF%]j + [xFy -yFx]k,

= [( -2.0 m)(4.3 N) -(1.6 m)( -2.4 N)]¡ + [(1.6 m)(3.5 N) -(1.5 m)(4.3 N)]j

+[(1.5 m)( -2.4 N) -( -2.0 m)(3.5 N)]k,

= [-4.8N.m]¡+[-0.85N.m]j+[3.4N.m]k.

-A A A A AE9-10 (a) F = (2.6 N)i, then f = (0.85 m)(2.6 N)j -( -0.36 m)(2.6 N)k = 2.2 N. mj + 0.94 N. mk.

-A A A A A{b) F = (-2.6N)k, then f = (-0.36m)(-2.6N)i -(0.54m)(-2.6N)j = 0.93N .mi + 1.4N .mj.

E9-11 (a) The rotational inertia about an axis through the origin is

I=mr2 = (0.025kg)(0.74m)2 = 1.4x10-2kg.m2.

(b) a = (0.74 m)(22 N) sin(50°)/(1.4 x 10-2kg .m2) = 890 rad/s.

E9-12 {a) lo = {0.052kg){0.27m)2+{0.035kg){0.45m)2+{0.024kg){0.65m)2 = 2.1x10-2kg.m2.

{b) The center of mass is located at

-{0.052kg){0.27m) + {0.035kg){0.45m) + {0.024kg){0.65m)Xcm -{0.052 kg) + {0.035 kg) + {0.024 kg) = 0.41 m.

Applying the parallel axis theorem yields lcm = 2.1x10-2kg.m2 -{0.11 kg){0.41 m)2 = 2.5x10-3kg.

m2.

I = Lmnr~ = (0.075 kg)(0.42m)2 + (O.O30kg)(0.65m)2 = O.O26kg.m2.

(b) No change.

E9-14 f = [(0.42m)(2.5N) -(0.65m)(3.6N)]k = -1.29N .mk. Using the result from E9-13,

a. = ( -1.29N .m k)/(0.026kg.m2) = 50 rad/s2k. That's clockwise if viewed from above.

E9-15 (a) F = mi.lJ2r = (110kg)(33.5 rad/s)2(3.90m) = 4.81x105N.(b) The angular acceleration is a = (33.5 rad/s)/(6.70s) = 5.00 rad/s2. The rotational inertia

about the axis of rotation is I = (110 kg)(7 .80 m)2 /3 = 2.23 x 103kg .m2. T = I a = (2.23 x 103kg .

m2)(5.00 rad/s2) = 1.12x 104N .m.

E9-16

~ The diagonal distance from the axis through the center of mass and the axis through theedge is h = v(a/2)2 + (b/2)2, so

I = Icm + Mh2 = bM (a2 + b2) + M ((a/2)2 + (b/2)2) = (b + ¡) M (a2 + b2) .

Simplifying, I = ~M (a2 + b2).

105

~ (a) Rotational inertia is additive so long as we consider the inertia about the same a;xis.

We can use Eq. 9-10:

We can add the inertia.s for the three rods together ,

E9-18 I = Icm + Mh2 = (0.56 kg)(1.0m)2/12 + (0.56)(0.30m)2 = 9.7x 10-2kg .m2.

~ For particle one /1 = mr2 = mL2; for particle two 12 = mr2 = m(2L)2 = 4mL2. Therotational inertia of the rod is Irod = l(2M)(2L)2 = ~ML2. Add the three inertias:

( 8 ,

= 5m+ 3M~

E9-20 la) I = M R2 /2 = M(R/ J2)2.(b) Let I be the rotational inertia. Assuming that k is the radius of a hoop with an equivalent

rotational inertia, then I = M k2 , or k = JI7M.

E9-21 Note the mistakes in the equation in the part (b) of the exercise text.(a) mn = M/N.(b) Each piece has a thickness t = L / N, the distance from the end to the nth piece is Xn =

(n- 1/2)t = (n- 1/2)L/N. The axis of rotation is the center, so the distance from the center isrn = Xn -L/2 = nL/N -(1 + 1/2N)L.

(c) The rotational inertia is

T

N

L mnr~,n=l

ML2 N

N3.L(n-1/2-N)2,n=l

ML2 N

N3 .L (n2 -(2N + l)n + (N + 1/2)2) ,

n=l ,

ML2. ( .N(N + 1)(2N +i} ...' j

N3

ML2,"

-~2N+ 1)~ + (N+ 1/2)2N6

2N3

2

2N3,~ -::r - + N 3n

~ N3 I

ML2/3.

E9-22 F = (46 N)(2.6 cm)/(13 cm) = 9.2N.

E9-23 Tower topples when center of gravity is no longer above base. Assuming center of gravityis located at the very center of the tower, then when the tower leans 7.0 m then the tower falls. Thisis 2.5 m farther than the present.

(b) f) = arcsin(7.0m/55m) = 7.3°.

E9-24 If the torque from the force is sufficient to lift edge the cube then the cube will tip. Thenet torque about the edge which stays in contact with the ground will be T = Fd -mgd/2 if F issufficiently large. Then F ;::: mg /2 is the minimum force which will cause the cube to tip.

The minimum force to get the cube to slide is F ;::: JLsmg = (0.46)mg. The cube will slide first.

106

~ The ladder slips if the force of static friction required to keep the ladder up exceeds J.t8N .Equations 9-31 give us the normal force in terms of the masses of the ladder and the firefighter,N = (m + M)g, and is independent of the location of the firefighter on the ladder. Also from Eq.9-31 is the relationship between the force from the wall and the force of friction; the condition at

which slipping occurs is Fw ?; J.t8(m + M)g.Now go straight to Eq. 9-32. The a/2 in the second term is the location of the firefighter, who in

the example was halfway between the base of the ladder and the top of the ladder. In the exercisewe don't know where the firefighter is, so we'll replace a/2 with x. Then

mga-Fwh+Mgx+-=O 3

Substitute in the condition of Fw when slipping justis an expression for rotational equilibrium.starts, and we get

mga

3

=o.-(JLs(m + M)g) h + Mgx +

Salve this far x,

I(9.3 m) -0:- ,-- .\ = 6.6 m(45 kg)(7.6 m)

This is the horizontal distance; the fraction of the total length along the ladder is then given byx/a = (6.6 m)/(7.6 m) = 0.87. The firefighter can climb (0.87)(12m) = 10.4m up the ladder.

E9-26 (a) The net torque about the rear axle is (1360kg)(9.8m/s2)(3.05m-l.78m)-Ff(3.05m) =O, which hassolution Ff = 5.55 x 103N. Each of the front tires support half of this, or 2.77x 103N.

(b) The net torque about the front axle is (1360kg)(9.8m/s2)(1.78m) -Ff(3.05m) = 0, whichhas solution F f = 7.78 x 103N. Each of the front tires support half of this, or 3.89 x 103N .

E9-27 The net torque on the bridge about the end closest to the person is

(160 lb)L/4 + (600 lb)L/2 -FfL = 0,

which has a solution for the supporting force on the far end of F f = 340 lb.The net force on the bridge is (160 lb)L/4+ (600 lb)L/2 -(340 lb) -F c = 0, so the force on the

close end of the bridge is F c = 420 lb.

E9-28 The net torque on the board about the left end is

Fr(1.55m) -(142N)(2.24m) -(582 N)(4.48m) = O,

which has a solution for the supporting force for the right pedestal of F r = 1890 N. The force on theboard from the pedestal is up, so the force on the pedestal from the board is down (compression).

The net force on the board is Fl+(1890N) -(142N) -(582 N) = 0, so the force from the pedestalon the left is Fl = -1170N. The negative sign means up, so the pedestal is under tension.

~ We can assume that both the force F and the force of gravity W act on the center of thewheel. Then the wheel will just start to lift when

w x r + F x r = o,

orWsin() = F cos(),

107

where (} is the angle between the vertical (pointing down) and the line between the center of thewheel and the point of contact with the step. The use of the sine on the left is a straightforwardapplication of Eq. 9-2. Why the cosine on the right? Because

sin(90° -e) = cose.

Then F = W tan f} .We can express the angle f} in terms of trig functions, h, and r. r cos f} is the

vertical distance from the center of the wheel to the top of the step, or r -h. Then

Finally by combining the above we get

=w~r-h

E9-30 (a) Assume that each of the two support points for the square sign experience the sametension, equal to half of the weight of the sign. The net torque on the rod about an axis throughthe hinge is

(52.3kg/2)(9.81m/s2)(0.95m) + (52.3kg/2)(9.81m/s2)(2.88m) -(2.88m)TsinO = O,

where T is the tension in the cable and f} is the angle between the cable and the rod. The angle canbe found from f} = arctan(4.12m/2.88m) = 55.00, so T = 416N.

(b) There are two components to the tension, one which is vertical, ( 416 N) sin(55.0°) = 341 N ,and another which is horizontal, (416N) cos(55.0°) = 239N. The horizontal force exerted by the wallmust then be 239N. The net vertical force on the rod is F+ (341N) -(52.3kg/2)(9.81m/s2) = 0,which has solution F = 172 N as the vertical upward force of the wall on the rod.

E9-31 (a) The net torque on the rod about an axis through the hinge is

T = W(L/2) cos(54.0°) -TLsin(153.0°) = O.

or T = (52.7 lb/2)(sin 54.0° / sin 153.0°) = 47.0 lb.

(b) The vertical upward force of the wire on the rod is Ty = Tcos(27.0°). The vertical upwardforce of the wall on the rod is Py = W -Tcos(27.0°), where W is the weight of the rod. Then

Py = (52.71b) -(47.0 lb)cos(27.0°) = 10.81b

The horizontal force from the wall is balanced by the horizontal force from the wire.(47.0 lb) sin(27.0°) = 21.3 lb.

Then Px =

E9-32 Ií the ladder is not slipping then the torque about an axis through the point oí contact withthe ground is

r = (W L/2) cos() -Nh/sin() = O,

where N is the normal force ofthe edge on the ladder. Then N = WLcos() sin()/(2h).N has two componentsj one which is vertically up, N y = N cos () , and another which is horizontal,

N x = N sin (). The horizontal force must be offset by the static friction.

The normal force on the ladder from the ground is given by

Ng = W -NcosB = W[l- LCOS2 B sinB/(2h)].

108

The force of static friction can be as large as f = .Us N g, so

L cos () sin2 ()WLcos(j sin2(j/(2h)JLs = W[l -L COS2 (j sin (j /(2h)] =

Put in the numbers and (j = 68.0°. Then .Us = 0.407.

'2h -Lcos2 (} sin(}"

E9-33 Let out be positive. The net torque about the axis is then

T = (0.118m)(5.88N) -(0.118m)(4.13m) -(O.O493m)(2.12N) = O.102N .m.

The rotational inertia of the disk is I = (1.92 kg)(0.118 m)2/2 = 1.34 x 10-2kg .m2. Then a =

(0.102N .m)/(1.34x 10-2kg .m2) = 7.61 rad/s2.

E9-34 (a) I = Tia = (960 N .m)I(6.23 radls2) = 154kg .m2.(b) m = (312)Ilr2 = (1.5)(154kg .m2)1(1.88m)2 = 65.4kg.

~ (a) The angular ~celeration is

~(.,; 6.20 rad/s = 28.2 rad/s2-a = "Xt" -0.22 s

{b) From Eq. 9-11, 'T = la = {12.0kg. m2){28.2 rad/s2) = 338N.m.

E9-36 The angular acceleration is a = 2(7r/2 rad)/(30s)2 = 3.5xlO-3 rad/s2. The required force

is thenF = 7"/r = Ia/r = (8.7xlO4kg .m2)(3.5xlO-3 rad/s2)/(2.4m) = l27N.

Don't let the door slam...

E9-37 The torque is T = r F , the angular acceleration is a = T / I = r F / I. The angular velocity is

rAt2

21

rBt3

3I'1.1.1 = lt a dt = +

so when t = 3.60 s,

ú.i = (9.88x10-2m)(0.496Nfs)(3.60s)2 + (9.88x10-2m)(0.305Nfs2)(3.60s)3 = 690 radfs.2(1.14 x 10-3kg .m2) 3(1.14 x 10-3kg .m2)

E9-38 (a) a = 2()lt2.(b) a = aR = 2()Rlt2.(c) T1 and T2 are not equal. Instead, (Tl -T2)R = la. For the hanging block Mg -T1 = Ma.

ThenTI = Mg -2MR(}/t2,

andT2 = Mg -2MR(}/t2 -2(I/R)(}/t2.

109

~ Apply a kinematic equation from chapter 2 to find the acceleration:

1 2y = VOyt + 2ayt ,

= ~ = 2(0.165 m) = 0 0586 / 2ay t2 (5.11 S)2 .m S

Closely follow the approach in Sample Problem 9-10. For the heavier block, ml = 0.512 kg, and

Newton 's second law givesmlg -Tl = mlay,

where ay is positive and down. For the lighter block, m2 = 0.463 kg, and Newton's second law gives

T2 -m2g = m2ay,

where ay is positive and up. We do know that Tl > T2; the net force on the pulley creates a torquewhich results in the pulley rotating toward the heavier mass. That net force is Tl -T2; so therotational form of Newton's second law gives

(TI -T2) R = Iaz = IaT/R,

JL-lay

where R = 0.049 m is the radius of the pulley and aT is the tangential acceleration. But this

acceleration is equal to ay, because everything- both blocks and the pulley- are moving together.We then have three equations and three unknowns. We'll add the first two together ,

mlg -Tl + T2 -m2g = mla1l + m2ay,

T1 -T2 = (g- ay)ml -(g + ay)m2,

and then combine this with the third equation by substituting for T1 -T2,

(g -ay)ml -(g + ay)m2 = Iay/R2,

.!. + ; m2] R2 = I.ay.

ml-

(0.512 kg) - 7.23 kg,

Now for the numbers:

( {9.81 m/s2){0.0586 m/s2) -1 =

= O.O174kg.m2,

E9-40 The wheel turns with an initial angular speed of ú)o = 88.0 rad/s. The average speed whiledecelerating is Ú)av = ú)o/2. The wheel stops turning in a time t = <t>/Ú)av = 2<t>/ú)o. The decelerationis then a = -ú)o/t = -ú)~/(2<t».

The rotational inertia is I = M R2 /2, so the torque required to stop the disk is T = I a =-M R2ú)~ / ( 4<t> ) .The force of friction on the disk is f = JLN, so T = Rf. Then

JL = ~ = .(1.40kg)(~(88.0 rad/s)2 = 0.272.

4N<t> 4(130N)(17.6 rad)

E9-41 (a) The automobile h88 an initial speed of Vo = 21.8 m/s. The angular speed is then(¡)o = (21.8m/s)/(0.385m) = 56.6 rad/s.

(b) The average speed while decelerating is (¡)av = (¡)o/2. ,

t = 4>/(¡)av = 24>/(¡)o. The deceleration is then

a = -(¡)0/t = -(¡)~/(24» = -(56.6 rad/s)2/[2(180 rad)] = -8.90 rad/s.

(c) The automobile traveled x = <Pr = (180 rad)(0.385m) = 69.3m.

The wheel stops turning in a time

110

(a) The angular acceleration is derived in Sample Problem 9-13,E9-42

The initial rotational speed was Wo = (1.30m/s)/(3.2x10-3m) = 406 rad/s. Then

ú) = ú)o + at = (406 rad/s) + (39.1 rad/s2)(0.945s) = 443 rad/s,

which is the same as 70.5 rev/s.

~ ( a) Assuming a perfect hinge at B, the only two vertical forces on the tire will be thenormal force from the belt and the force of gravity. Then N = W = mg, or N = (15.0 kg)(9.8m/s2) = 147 N.

While the tire skids we have kinetic friction, so f = Jl.kN = (0.600)(147 N) = 88.2 N. The forceof gravity and and the pull from the holding rod AB both act at the axis of rotation, so can 'tcontribute to the net torque. The normal force acts at a point which is parallel to the displacementfrom the axis of rotation, so it doesn 't contribute to the torque either (because the cross productwould vanish); so the only contribution to the torque is from the frictional force.

The frictional force is perpendicular to the radial vector, so the magnitude of the torque is justT = rf = (0.300 m)(88.2 N) = 26.5 N.m. This means the angular acceleration will be a = T/I =(26.5 N.m)/(0.750 kg.m2) =35.3 rad/s2.

When ",-,R = VT = 12.0 m/s the tire is no longer slipping. We solve for "'-' and get "'-' = 40 rad/s.Now we solve "'-' = "'-'o + at for the time. The wheel started from rest, so t = 1.13 s.(b) The length of the skid is x = vt = (12.0 m/s)(1.13 s) = 13.6 m long.

~ The problem of sliding down the ramp hBB been solved (see Sample Problem 5-8); thecritical angle (}8 is given by tan(}8 = JL8.

The problem of tipping is actually not that much harder: an object tips when the center ofgravity is no longer over the bBBe. The important angle for tipping is shown in the figure below; wecan find that by trigonometry to be

otan(}t = A =

so (Jt = 34°,

111

(a) If J1s = 0.60 then (}s = 31° and the crate slides.(b) If J1s = 0.70 then (}s = 35° and the crate tips before sliding; it tips at 34°.

P9-2 (a) The total force up on the chain needs to be equal to the total force downj the forcedown is W. Assuming the tension at the end points is T then T sin () is the upward component, soT = W/(2sin()).

(b) There is a horizontal component to the tension T cos () at the wallj this must be the tensionat the horizontal point at the bottom of the cable. Then Tbottom = W/(2tan()).

P9-3 (a) The rope exerts a force on the sphere which has horizontal Tsin(} and vertical Tcos(}components, where (} = arctan(r/L). The weight ofthe sphere is balanced by the upward force fromthe rope, so T cos (} = W. But cos (} = L / .¡;;.-:¡:7;'l, so T = W Vl + r2 / L2 .

(b) The wall pushes outward against the sphere equal to the inward push on the sphere from therope, or p = Tsin(} = Wtan(} = Wr/L.

P9-4 'll:eat the problem as having two forces: the man at one end lifting with force F = W /3

and the two men acting together a distance x away from the first man and lifting with a force2F = 2W /3. Then the torque about an axis through the end of the beam where the first man islifting is T = 2xW/3 -WL/2, where L is the length ofthe beam. This expression equal zero whenx = 3L/4.

~ (a) We can solve this problem with Eq. 9-32 after a few modifications. We'll assume thecenter of mass of the ladder is at the center, then the third term of Eq. 9-32 is mga/2. The cleanerdidn't climb half-way, he climbed 3.10/5.12 = 60.5% of the way, 80 the second term of Eq. 9-32becomes Mga(0.605). h, L, and a are related by L2 = a2+h2, so h = V(5.12 m)2 -(2.45 m)2 = 4.5m. Then, putting the correction into Eq. 9-32,

Fw =

(b) The vertical component of the force of the ground on the ground is the sum of the weight ofthe window cleaner and the weight of the ladder, or 833 N .

112

.1 l 2(4:5"tñ) (74.6 kg)(9.81 m/s )(2.45 m)(0.605) ,

+ (10.3 kg)(9.81 m/s2)(2.45 m)/2] ,

269 N

The horizontal component is equal in magnitude to the force of the ladder on the window. Thenthe net force of the ground on the ladder has magnitude

V(269N)2 + (833N)2 = 875N

and direction= arctan(833j269) = 72° above the horizontal.I}

P9-6 (a) There are no upward forces other than the normal force on the bottom ball, so the forceexerted on the bottom ball by the container is 2W.

( c ) The bottom ball must exert a force on the top ball which has a vertical component equal to theweight of the top ball. Then W = N sin (J or the force of contact between the balls is N = W / sin (J .

(b) The force of contact between the balls has a horizontal component p = N cos (J = W / tan (J ,this must also be the force of the walls on the balls.

--P9-7 (a) There are three forces on the ball: weight W, the normal force from the lower plane N1,

and the normal force from the upper plane Ñ 2. The force from the lower plane has componentsN1,x = -Nl sin(}l and Nl,y = Nl COS(}l. The force from the upper plane has components N2,x =N2 sin(}2 and N2,y = -N2 COS(}2. Then N1 sin(}l = N2 sin(}2 and N1 COS(}l = W + N2 COS(}2.

Solving for N2 by dividing one expression by the other,

cos (}1 W Cos (}2-=.+-sin (}1

-,sin (}2N2 Sin(}2

or

N2 =

sin (}1 sin (}2sin 02

W sin 01sin(02 -01) .

Then solve for Nl,w sin (}2

Sin((}2 -(}1) ,NI =

(b) Friction changes everything.

P9-8 (a) The net torque about a line through A is

r = w x .:.- T L sin () = O,

so T = Wx/(Lsin()).(b) The horizontal force on the pin is equal in magnitude to the horizontal component of the

tension: Tcos() = Wx/(Ltan()). The vertical component balances the weight: W- Wx/L.(c) x = (520N)(2.75m)sin(32.00)/(315N) = 2.41m.

P9-9 (a) As long as the center of gravity of an object (even if combined) is above the base, thenthe object will not tip.

Stack the bricks from the top down. The center of gravity of the top brick is L /2 from the edgeof the top brick. This top brick can be offset nor more than L /2 from the one beneath. The centerof gravity of the top two bricks is located at

Xcm = [(L/2) + (L)]/2 = 3L/4.

113

These top two bricks can be offset no more than L /4 from the brick beneath. The center of gravityof the top three bricks is located at

Xcm = [(L/2) + 2(L)]/3 = 5L/6.

These top three bricks can be offset no more than LI6 from the brick beneath. The total offset isthen LI2 + LI4 + LI6 = llL112.

(b) Actually, we never need to know the location of the center of gravity; we now realize thateach brick is located with an offset LI(2n) with the brick beneath, where n is the number of thebrick counting from the top. The series is then of the form

(L/2)[(l/l) + (1/2) + (1/3) + (1/4) +

a series (harmonic, for those of you who care) which does not converge.( c ) The center of gravity would be half way between the ends of the two ex:treme bricks. This

would be at N L/n; the pile will topple when this value exceeds L, or when N = n.

P9-10 (a) For a planar object which lies in the x -y plane, I", = J x2dm and Iy = J y2dm. ThenI", + ly = J(X2 + y2)dm = J r2 dm. But this is the rotational inertia about the z axis, sent r is the

distance from the z axis.(b) Since the rotational inertia about one diameter (I",) should be the same as the rotational

inertia about anyother (ly) then I", = ly and I", =Iz/2 = MR2/4.

~ Problem 9-10 says that Ix + Iy = Iz for any thin, flat object which lies only in thex -y plane.lt doesn 't matter in which direction the x and y axes are chosen, so long as they areperpendicular .We can then orient our square as in either of the pictures below:

y

--{>

x

By symmetry Ix = 11I in either picture. Consequently, Ix = 11I = Iz/2 for either picture. It isthe same square, so Iz is the same for both picture. Then Ix is also the same for both orientations.

P9-12 Let Mo be the mass of the plate before the holes are cut out. Then M1 = Mo(al L)2 is themass of the part cut out of each hole and M = Mo -9M1 is the mass of the plate. The rotationalinertia ( about an axis perpendicular to the plane through the center of the square) for the largeuncut square is MoL2/6 and for each smaller cut out is M1a2/6.

From the large uncut square's inertia we need to remove M1a2/6 for the center cut-out, M1a2/6+M1(LI3)2 for each of the four edge cut-outs, and M1a2/6 + M1(.J2LI3)2 for each of the comersections.

114

Then

MoL2

6

MoL2

6

-Mla2

6

-3~

2L2

2M1L2

,9

-9 -4 MlL2

-g-4

-4~ .1 .

~ (a) From Eq. 9-15, I = J r2dm about some axis of rotation when r is measured from thataxis. If we consider the x axis as our axis of rotation, then r = ~ , since the distance to thex axis depends only on the y and z coordinates. We have similar equations for the y and z axes, so

J (y2 +"Z2) dm,Ix -

I" =

lz

~

These three equations can be added together to give

J ( 2 2 2 )I x + I y + lz = 2 x + y + z dm,

so if we now define r to be measured from the origin (which is not the definition used above), thenwe end up with the answer in the text.

(b) The right hand side of the equation is integrated over the entire body, regardless of howthe axes are defined. So the integral should be the same, no matter how the coordinate system isrotated.

P9-14 (a) Since the shell is spherically symmetric I", = Iy = Iz, so I", = (2/3) J r2dm =(2R2/3) J dm = 2M R2 /3.

(b) Since the solid ball is spherically symmetric I", = Iy = Iz, so

P9-15 (a) A simple ratio will suffice:

dm

27rr dr

M 2Mr-R2 .dr.7rR2 or dm ==

(b) dI = r2dm = (2Mr3/R2)dr.

(c) I = JOR(2Mr3/R2)dr = MR2/2.

P9-16 (a) Another simple ratio will suffice:

dm

~

M{4/3)7rR3 or dm =

~

(b) dI = r2dm/2 = [3M(R2 -Z2)2/8R3]dz.

115

( c ) There are a few steps to do here:

I =

~ The rotational acceleration will be given by az = ¿r/I.The torque about the pivot comes frorn the force of gravity on each block. This forces will both

originally be at right angles to the pivot arm, so the net torque will be ¿ r = mgL2 -mgLl , whereclockwise as seen on the page is positive.

The rotational inertia about the pivot is given by I = ¿mnr~ = m(L~ + L~). So we can nowfind the rotational acceleration,

a = ~ = mgL2 -mgL1 -L2 -L1I m(L~ + L~) -g~ = 8.66 rad/s2.

The linear acceleration is the tangential acceleration, aT = aR. For the left block, aT = 1.73 m/s2;for the right block aT = 6.93 m/s2.

P9-18 (a) The force of friction on the hub is JLkMg. The torque is T = JLkMga. The angularacceleration has magnitude a = T/I = JLkga/k2. The time it takes to stop the wheel will bet = ú)o/a = ú)ok2/(JLkga).

(b) The average rotational speed while slowing is ú)o/2. The angle through which it turns whileslowing is ú)ot/2 radians, or ú)ot/(47r) = ú)~k2/(47rJLkga)

P9-19 (a) Consider a diíf.erential ring of the disk. The torque on the ring because of friction is

The net torque is thenJ rR 2JLkM gr2 2

T = dT = JO -¡¡;¡---dr = 3JLkMgR.

(b) The rotational acceleration has magnitude a = T / I = ~ JLkg / R. Then it will take a time

3&0

4JLkgt = ú)o/a =

to stop.

P9-20 We need only show that the two objects have the same acceleration.Consider first the hoop. There is a force WII = W sin (} = mg sin (} pulling it down the ramp

and a frictional force f pulling it up the ramp. The frictional force is just large enough to causea torque that will allow the hoop to roll without slipping. This means a = aR; consequently,fR = al = aI/R. In this case I = mR2.

The acceleration down the plane is

ma = mgsin(} -f = mgsin(} -maI/R2 = mgsin(} -ma.

116

{R 3M(R2 -Z2)2l-R 8R3 dz,

3M {R4R3 lo (R4 -2R2Z2 + z4)dz,

Then a = 9 sin (} /2. The mass and radius are irrelevant!For a block sliding with friction there are also two forces: WII = W sin (} = mg sin (} and f =

.Ukmg cos (}. Then the acceleration down the plane will be given by

a = gsin{} -Jl,kgCOS{},

which will be equal to that or the hoop ir

1= 2 tan(}.

~ This problem is equivalent to Sample Problem 9-11, except that we have a sphere insteadof a cylinder. We'll have the same two equations for Newton 's second law,

MgsinO- f = Macm and N -MgcosO = O.

Newton's second law for rotation willlook like

-fR = Icma.

The conditions for accelerating without slipping are acm = aR, rearrange the rotational equation to

getf = -~ = -Icm(-acm)

.-R2 ,R

and then.Icm(acm)Mgsm() -R2 = Macm,

and solve for acm. For fun, let's write the rotational inertia as 1 = {3M R2, where {3 = 2/5 for the

sphere. Then, upon some mild rearranging, we get

sin f)acm = gU"!i

For the sphere, acm = 5/7gsin().(a) If acm = 0.133g, then sin() = 7/5(0.133) = 0.186, and () = 10.7°.(b) A frictionless block has no rotational propertiesj in this case {3 = 0! Then acm = 9 sin () =

0.186g.

P9-22 (a) There are three forces on the cylinder: gravity W and the tension from each cable T.The downward acceleration of the cylinder is then given by ma = W -2T .

The ropes unwind according to Q = a/ R, but Q = T / I and I = mR2/2. Then

a = TR/I = (2TR)R/(mR2/2) = 4T/m.

Combining the above, 4T = W -2T, or T = W/6.(b) a = 4(mg/6)/m = 2g/3.

~ The force of friction required to keep the cylinder rolling is given by

f = ~Mgsin(};

the normal force is giveIi to be N = M 9 cos (}; so the coefficient of static friction is given by

f 1JLB ~ N = 3 tan(}.

117

P9-24 a = F / M, since F is the net force on the disk. The torque about the center of mass is F R,so the disk has an angular acceleration of

FR FR

MW72

2F

MR'

a= = =

~9-25 This problem is equivalent to Sample Problem 9-11, except that we have an unknown rollingobject. We'll have the same two equations for Newton 's second law,

Mgsin{} -f = Macm and N -Mgcos{} = O.

Newton's second law for rotation willlook like

-IR = Icma.

The conditions for accelerating without slipping are acm = aR, rearrange the rotational equation toget

f = -~ = -Icm(-acm)R R2 ,

and then

~Mgsinf} -R2

and solve for acm. Write the rotational inertia as I = ,8MR2, where ,8 = 2/5 for a sphere, fJ =!= 1/2for a cylinder, and ,8 = 1 for a hoop. Then, upon some mild rearranging, we get

= Macm

sin f}acm = g"f+ft

Note that a is largest when ,8 is smallest; consequently the cylinder wins. Neither M nor R enteredinto the final equation.

118

ElO-l l = rp = mvr = {13.7x 10-3kg){380 m/s){0.12m) = 0.62 kg .m2/s.

ElO-2 (a) r= mrx V, or

-A A A1 = m(yvz -zvy)i + m(zvx -x'!!z)j + m(xvy -yvx)k.

(b) Ir v and r exist only in the xy plane then z = Vz = O, so only the u k term survives.

dr

dt

Now the cross product of a vector with itself is zero, so the first term vanishes. But in the exercisewe are t-.old the particle has constant velocity, so a = O, and consequently the second term vanishes.Hence, 1 is constant for a single particle if v is constant.

EI0-4 (a) L = ¿ li; li = rimipi. Putting the numbers in for each planet and then summing (Iwon't bore you withthe arithmetic details) yields L = 3.15 x 1043kg .m2/s.

(b) Jupiter has l = 1.94 x 1043kg .m2/s, which is 61.6% of the total.

E 10-5 I = mvr = m(27rr/T)r = 27r(84.3kg)(6.37xl06m)2 /(86400s) = 2.49x lOllkg .m2/s.

ElO-6 (a) Substitute and expand:

L = L(rcm + ~) x (miVcm + p~),

~ ( , "" - + "" -' )LI mircm X Vcm + rcm X Pi +miri X Vcm ri X Pi ,

Mrcm x vcm + rcm x (Lp~) + (Lmi~) x Vcm + L~ x p~.

(b) But L: p~ = O and L: mi~ = O, because these two quantities are in the center of momentumand center of mass. Then

ElO-7 (a) Substitute and expand:

-I d~ dri drcm --Pi = mi-¡¡¡ = mi-¡¡¡ -mi-¡¡¡- = Pi -miVcmo

(b) Substitute and expand:

di:'

dt

The first term vanished because v~ is parallel to p~ .

119

di-. -.dv= m- xv+mrx -dt dt ,

= m v x v+mix a.

( c ) Substitute and expand:

di' d{Pi -miV cm)

dL' = Lr: x Fi,dt

--where F i is the net force on the ith particle. The force F i may include both internal and externalcomponents. If there is an internal component, say between the ith and jth particles, then thetorques from these two third law components will cancel out. Consequently,

di'

dt'\:"""'-- --= L-ITi = Texto

ElO-8 (a) Integrate.

f f' dt = f *dt = f dL = ~L.

(b) If I is fixed, ~L = I~(Li. Not only that,

fTdt = fFrdt = rfFdt = rFav~t,

where we use the definition of average that depends on time.

~ (a) T~t = ~r. The disk starts from rest, so ~r = r -lo = I.

ourselves with the magnitudes, soWe need only concern

l = Al = TAt = {15.8 N.m){O.O33s) = O.521kg.m2/s.

(b) (¿) = 1/1 = (0.521 kg.m2/s)/(1.22x 10-3kg.m2) = 427 rad/s.

EI0-I0 (a) Let Vo be the initial speed; the average speed while slowing to a stop is vo/2; the timerequired to stop is t = 2x/vo; the acceleration is a = -vo/t = -v3/(2x). Then

a = -(43.3m/s)2/[2(225m/s)] = -4.17m/s2.

(b) a = a/r = (-4.17m/s2)/(0.247m) = -16.9 rad/s2.

(c) T = la = (0.155kg .m2)(-16.9 rad/s2) = -2.62N .m.

EI0-ll Let ri = Z+ ~. From the figure, Pl = -P2 and r'¡ = -~. Then

L = r1 + t = r1 X Pl + r2 X P2,(-- ) - = rl -r2 X Pl,

( "" "" ) -= rl -r2 X Pl,

2 "" --rl X Pl.

Since r'¡ and Pl both lie in the xy plane then L must be along the z axis.

120

""""""' ,dt = L...riX- dt

Lr: x (mi~ -mi8cm),

= L~ x miai + (Lmi~) x 8cm)

The second term vanishes because of the definition of the center of mass. Then

ElO-l2 Expand:

L = ¿:: ri = ¿:: ri X Pi,

'""' '""' ... ( )L miri X Vi = L miri X W X ri

¿::mi[(ri .ri)¡;J -(ri .¡;J)i!)i],'""' 2 2A A A

L mi [ri ¡;J -(Zi W )k -(ZiXiW)i -(ZiYiW )j],

but if the body is symmetric about the z axis then the last two terms vanish, leaving

~ An impulse of 12.8 N.s will change the linear momentum by 12.8 N.s; the stick startsfrom rest, so the final momentum must be 12.8 N.s. Since p = mv, we then can find v = p/m = (12.8N.s)/(4.42 kg) = 2.90 m/s.

Impulse is a vector, given by f Fdt. We can take the cross product of the impulse with thedisplacement vector r (measured from the axis of rotation to the point where the force is applied)and get

r x f Fdt ~ f r x Fdt,

The two sides of the above expression are only equal if r has a constant magnitude and direction. Thiswon 't be true, but if the force is of sufficiently short duration then it hopefully won 't change much.The right hand side is an integral over a torque, and will equal the change in angular momentum ofthe stick.

The exercise states that the force is perpendicular to the stick, then Irx FI = rF, and the "torqueimpulse" is then (0.464 m)(12.8 N.s) = 5.94 kg.m/s. This "torque impulse" is equal to the changein the angular momentum, but the stick started from rest, so the final angular momentum of thestick is 5.94 kg.m/s.

But how fast is it rotating? We can use Fig. 9-15 to find the rotational inertia about the centerofthe stick: I = ~ML2 = ~(4.42 kg)(1.23 m)2 = 0.557 kg.m2. The angular velocity ofthe stick isw = l/I = (5.94 kg.m/s)/(0.557 kg.m2) = 10.7 rad/s.

EI0-14 The point of rotation is the point of contact with the plane; the torque about that pointis T = rmgsin(). The angular momentum is I(J.J, so T = la. In this case I = mr2/2 + mr2, thesecond term from the parallel axis theorem. Then

a = ra = rT/I = mr2gsin(}/(3mr2/2) = ~gsin(}.3

ElO-l5 From Exercise 8 we can immediately write

Il(U)l -U)o)/rl = 12(U)2 -O)/r2,

but we also have rl(¿)l = -r2(¿)2. Then

rlr2I11.1}o1.1} --2 .2- r~I2 -r2Il

121

EI0-16 (a) ~1..)/1..) = (l/Tl -1/T2)/(1/Tl) = -(T2 -T1)/T2 = -~T/T, which in this case is-(6.0x 10-3s)(8.64x 104s) = -6.9X 10-8.

(b) Assuming conservation of angular momentum, ~I/ I = -~1..) /I..). Then the fractional changewould be 6.9 x 10-8.

~ The ratatianal inertia af a salid sphere is I = ~MR2; sa as the sun callapses

Li

Ii¡-:;i

2

MR .2¡-:;.-1 1

5

Ri2¡-:;i

= Lf,= Ifr:;Jf,

-~MRf2r:;Jf,-52-= Rf úJ¡.

The angular frequency is inversely proportional to the period of rotation, so

TfRf2 4 .

({6.37Xl06m) )2 . Ti R? = {3.6 x 10 mm) {6.96 x 108m) = 3.0 mm.

ElO-l8 The final angular velocity of the train with respect to the tracks is (Altt = Rv.conservation of angular momentum implies

The

o = MR2úJ + mR2(úJtt + úJ),

or-m v

"' = (m + M)R"

ElO-l9 Thia ia much like a center of masa problem.

o = Ip</>p + Im«/>mp + </>p),

or(Ip + Im)t/>p

-~-.Im

(12.6kg .m2)(25°) = 1.28x 1050,4>mp{2.47x lO-3kg .m2)

That's 354 rotations!

ElO-20 (A)f = (Ii/lf)úJi = [(6.13kg .m2)/(1.97kg .m2)](1.22 rev/s) = 3.80 rev/s.

~ We have two disks which are originally not in contact which then come into contact;there are no external torques. We can write

--11,i + 12,i

1lQl,i + I2Q2,i

= f1,f+t,f,

= 11¡;jl,f + 12¡;j2,fo

The final angular velocities of the two disks will be equal, so the above equation can be simplifiedand rearranged to yield

ElO-22 l.L = lcos{} = mvrcos{} = mvh.

122

ElO-23 (a) (¿}f = (11/12)(¿}j, 11 = (3.66 kg)(0.363m)2 = O.482kg .m2. Then

(¿}f = [(0.482kg .m2)/(2.88kg .m2)](57.7 rad/s) = 9.66 rad/s,

with the same rotational sense as the original wheel.(b) Same answer, since friction is an internal force internal here.

ElO-24 (a) Assume the merry-go-round is a disk. Then conservation of angular momentum yields

I(2mmR2 + mgR2)(¡J + (m rR2)(v/R) = O,

or

ElO-25 Conservation of angular momentum:

2 2 2(mmk + mgR )!4) = mgR (v/R),

so= (44.3 kg)(2.92m/s)/(1.22~) = 0.496 rad/s.

úJ (176kg)(0.916m)2 + (44.3kg)(1.22m)2

ElO-26 Use Eq. 10-22:

Mgr

IUJ

{0.492 kg){9,81 m/s2){3.88 x 10-2m) = 2.04 rad/s = 0.324 rev /s.= {5.12 x 10-4kg .m2){27r28.6 rad/s)ú)p =

~ The relevant precession expression is Eq. 10-22.The rotational inertia will be a sum of the contributions from both the disk and the axle, but

the radius of the axle is probably very small compared to the disk, probably as small as 0.5 cm.Since I is proportional to the radius squared, we expect contributions from the axle to be less than(1/100)2 of the value for the disk. For the disk only we use

Now for (¡),21[ rad

1 rey

1 min

60 sI..; = 975 rev/min = 102 rad/s.

Then L = Ir.,; = 13.8 kg.m2/s.

Back to Eq. 10-22,

Mgr

L

{1.27 kg){9.81 m/s2){0.0610m) = 0.0551 rad/s.= 13.8 kg .m2/s(Up =

The time for one precession is

lrev

!.l)p

27r rad

(0.0551 rad/s)t= = 114 s.

~

123

PI0-l Positive z is out of the page...A A A

(a) 1 = rmvsinOk = (2.91 m)(2.13kg)(4.18m) sin(147°)k = 14.1 kg .m2/8k.

(b) f = rFsinOk = (2.91 m)(1.88N) sin(26°)k = 2.40N .mk.

PI0-2 Regardless of where the origin is located one can orient the coordinate system so that thetwo paths lie in the xy plane and are both parallel to the y axis. The one of the particles travelsalong the path x = vt, y = a, z = b; the momentum of this particle is Pl = mvi. The otherparticle will then travel along a path x = c- vt, y = a + d, z = b; the momentum of this particle isP2 = -mvi. The angular momentum of the first particle is

t = mvbj -mvak,

while that oí the second is~ = -mvbj + mv(a + d)k,

so the total is 11 + t = mvdk.

~ Assume that the cue stick strikes the ball horizontally with a force of constant magnitudeF for a time ~t. Then the magnitude of the change in linear momentum of the ball is given byF ~t = ~p = p, since the initial momentum is zero.

If the force is applied a distance x above the center of the ball, then the magnitude of the torqueabout a horizontal aJds through the center of the ball is 7" = xF. The change in angular momentumof the ball is given by 7"~t = ~l = l, since initially the ball is not rotating.

For the ball to roll without slipping we need v = (J.lR. We can start with this:

v = ¡.)R,

E. = !Rm

F~t--.

mF--.m

I'T~tR

IxFR

I.

Then x = I/mR is the condition for rolling without sliding from the start. For a solid sphere,I = ~mR2, so x = ~R.

PI0-4 The change in momentum of the block is M(V2 -Vl), this is equal to the magnitude theimpulse delivered to the cylinder. According to EI0-8 we can write M(V2 -vl)R = I(¿)f. But in theend the box isn't slipping, so (¿)f = V2/ R. Then

2MV2 -MVl = (I/R )V2,

orV2 = vl/(l + I/MR2).

PI0-5 Assume that the cue stick strikes the ball horizontally with a force of constant magnitudeF for a time At. Then the magnitude of the change in linear momentum of the ball is given byFAt = Ap = p, since the initial momentum is zero. Consequently, FAt = mvo.

If the force is applied a distance h above the center of the ball, then the magnitude of the torqueabout a horizontal axis through the center of the ball is T = hF. The change in angular momentumofthe ball is given by TAt = Al = lo, since initially the ball is not rotating. Consequently, the initialangular momentum of the ball is lo = hmvo = Iú)o.

124

The hall originally slips while moving, hut eventually it rolls. When it has hegun to roll withoutslipping we have v = &. Applying the results from ElO-8,

m(v -vo)R +1(¡.; -¡.;0) = O,

or

then, if v = 9vo/7,

P10-6 (a) Refer to the previous answer. We now want v = 1.1) = O, so

2 2Vm(v -vo)R + 5mR R -hmvo = O,

becomes

-voR -hvo = O,

or h = -R. That'll scratch the felt.

(b) Assuming only a horizontal force then

~ We assume the bowling ball is solid, so the rotational inertia will be I = (2/5)M R2 (seeFigure 9-15).

The normal force on the bowling ball will be N = Mg, where M is the mass ofthe bowling ball.The kinetic friction on the bowling ball is Ff = J1,kN = J1,kMg. The magnitude of the net torque onthe bowling ball while skidding is then T = J1,kM gR.

Originally the angular momentum of the ball is zeroj the final angular momentum will havemagnitude 1 = I ú) = I v I R, where v is the final translational speed of the ball.

( a) The time requires for the ball to stop skidding is the time required to change the angularmomentum to 1, so

(2j5)M R2V j RAlAt = -= .

T .ukMgR 5.ukg

Since we don't know v, we can't solve this for At. But the same time through which the angularmomentum of the ball is increasing the linear momentum of the ball is decreasing, so we also have

Ap Mv -Mvo Vo -v

2v.. -

a 1: = --"'-- = = -"'---F¡ -J.tkMg Ji,k9

Combining,

va -v~t = ¡¡'k9 ,

va -5¡¡,k9~t/2

¡.tkg

125

2vo -5JLkg6.t,2vo

2/Lkg~t =

~t =7JLkg'

2(8.50 mis )-= 1.18s.7(0.210)(9.81 m/s2)

(d) Use the expression for angular momentum and torque,

v = 5Jlkg~t/2 = 5(0.210)(9.81 m/s2)(1.18s)/2 = 6.08m/s.

(b) The acceleration of the ball is F / M = -Jlg .The distance traveled is then given by

1 2X = -at +vot,

21

= --(0.210)(9.81m/s2)(1.18s)2 + (8.50m/s)(1.18s) = 8.6m,2

(c) The angular acceleration i8 T/I = 5JLkg/(2R). Then

1 2(} = -at + (,¡)ot,

2

= ~(0.210)(9.81 m(82) (1.188)2 = 32.6 rad = 5.19 revolutions.4(0.11 m)

PI0-8 (a) 1 = llA)o = (1/2)MR21A)0.(b) The initial speed is vo = &o. The chip decelerates in a time t =

the chip travels with an average speed of vo/2 through a distance of

vo Vo R21A)2y=Vavt= --= -.

2 9 2g

( c) Loosing the chip won 't change the angular velocity of the wheel.

vo/g, and during this time

PlO-9 Since L =J(J) = 27rI/T and L is constant, then I <X T. But I <X R2, so R2 <X Tand

6.T 2R6.R 26.R=~"T R2 R

Then2(30m)

~0.8s.AT = (86400 s) "(6.37 x lO6m)

g7r

PI0-I0 Originally the rotational inertia was

2 2 : 5Ii = -MR = .PoR .

5 15

The average density can be found from Appendix C. Now the rotational inertia is

87r 5 87r 5If = l5(Pl -P2)R1 + l5P2R ,

where Pl is the density of the core, R1 is the radius of the core, and P2 is the density of the mantle.Since the angular momentum is constant we have AT /T = Al/ I. Then

AT Pl -P2 ~ + ~ -1 = 10.3 -4c50 35705 4.50 r --~

R5 Po

~

5,52 63705 T ~ -I = -U.127,T Po

so the day is getting longer .

1?¡::

~ The cockroach initially h8B an angular speed of (Llc,i = -v/r. The rotational inertia ofthe cockroach about the axis of the turntable is I c = mR2. Then conservation of angular momentum

lci+18i = lcf+18f,, , , ,I c(.1c,i + 18(.18,i = I c(.1c,f + 18(.18,f ,

-mR2v/r + 1(.1 = (mR2 + 1)(.1f,

1(.1 -mvR1.I.

PI0-12 (a) The skaters move in a circle of radius R = (2.92m)/2 = 1.46m centered midwaybetween the skaters. The angular velocity of the system will be (,¡}i = v/R = (1.38m/s)/(1.46m) =0.945 rad/s.

,

2MRi2 -(1.46m)2 fO-~4;t) rM/R) = ~-12 rM/R.úJ& = -úJ: --

~

~ (a) Apply Eq. 11-2, W = Fscos4> = (190N)(3.3m)cos(22°) = 5803.(b) The force of gravity is perpendicular to the displacement of the crate, so there is no work

done by the force of gravity.( c ) The normal force is perpendicular to the displacement of the crate, so there is no work done

by the normal force.

Ell-2 (a) The force required is F = ma = (106kg)(1.97m/s2) = 209N. The object moves withan average velocity vav = vo/2 in a time t = vo/a through a distance x = vavt = v8/(2a). So

x = (5l.3m/s)2/[2(l.97m/s2)] = 668 m.

The work done is W = Fx = (-209 N)(668m) = l.40xlO5J.

(b) The force required is

F = ma = (106kg)(4.82m/s2) = 511N.

x = (5l.3m/s)2/[2(4.82m/s2)] = 273m. Theworkdone is W = Fx = (-5llN)(273m) = l.40xlO5J.

Ell-3 (a) W = Fx = (120N)(3.6m) = 430J.(b) W = Fxcos(} = mgxcos(} = (25kg)(9.8m/s2)(3.6m) cos(117°) = 4O0N.(c) W = Fxcos(}, but (} = 90°, so W = 0.

Ell-4 The worker pushes with a force P; this force has components Px = Pcos(} and Py = Psin(},where (} = -32.0°. The normal force of the ground on the crate is N = mg- Py, so the force offriction is f = Jl,kN = Jl,k(mg -Py). The crate moves at constant speed, so Px = f. Then

JLk(mg- Psin ()),

JLkmg

Pcos() =

p = cos (} + Jl,k siDJj .

The work done on the crate is

w =

Ell-5 The components of the weight are WII = mgsin(} and W ~ = mgcos(}. The push p hascomponents PII = Pcos(} and P~ = Psin(}.

The normal force on the trunk is N = W ~ + P.l so the force of friction is f = .uk ( mg cos (} +

Psin(}). The push required to move the trunk up at constant speed is then found by noting that~I = WII + f .

Thenp = mg(tan(J + ILk)

1 -ILk tan (J

(a)-The work done by the applied force is

w = Pxcos(J = {52.3 kg){9.81m/s2)[sin{28.0°) + {0.19) cos{28.0°)]{5.95m) = 60J1- {0.19) tan{28.0°) 21 .

(b) The work done by the force of gravity is

w = mgxcos{(} + 900) = {52.3 kg){9.81 m/s2){5.95m) cos{118°) = -1430J.

12R

Ell-6 O = arcsin(O.902 m/1.62 m) = 33.8° .The components of the weight are WII = mgsinO and W.L = mgcosO.The normal force on the ice is N = W.L so the force of friction is f = JLkmgcosO. The push

required to allow the ice to slide down at constant speed is then found by noting that p = WII -f .Then p = mg( sin O -JLk cos O) .

(a) p = (47.2kg)(9.81m/s2)[sin(33.8°) -(0.110)cos(33.8°)1 = 215N.(b) The work done by the applied force is W = Px = (215N)(-1.62m) = -348J.(c) The work done by the force of gravity is

W = mgxcos(90° -0) = (47.2kg)(9.81 m/s2)(1.62m) cos(56.2°) = 417 J.

~ Equation 11-5 describes how to find the dot product of two vectors from components,

a.b = axbx+ayby+azbz,

= (3)(2) + (3)(1) + (3)(3) = 18.

Equation 11-3 can be used to find the angle between the vectors,

a = yf(3)2 + (3)2 + (3)2 = 5.19,

b = yf(2)2 + (1)2 + (3)2 = 3.74.

Now use Eq. 11-3,a.b

abcos<1>=

= (18)(5.19)(3.74) = 0.927,

and then 1/> = 22.0° .

Ell-8 a. b = (12)(5.8) cos(55°) = 40.

Ell-9 ¡. s= (4.5)(7.3) cos(320° -85°) = -19.

EII-I0 (a) Add the components individually:

r = (5 + 2 + 4)i + ( 4- 2 + 3)j + ( -6- 3 + 2)k = 11i + 5j -7k.

(b) (} = arccos( -7 /Vl12 + 52 + 72) = 120°.(c) (} = arccos(a .b/Vlib), or

(} = (5)(-2) + (4)(~) + (-6)(3) = 124°.

.¡(52 + 42 + 62)(22 + 22 + 32)

~ There are two forces on the woman, the force of gravity directed down and the normalforce of the floor directed up. These will be effectively equal, so N = W = mg. Consequently, the57 kg woman must exert a force of F = (57kg)(9.8m/s2) = 560N to propel herself up the stairs.

From the !jeference frame of the woman the stairs are moving down, and she is exerting a forcedown, so the work done by the woman is given by

W = Fs = (560 N)(4.5 m) = 2500 J,

this work is positive because the force is in the same direction as the displacement.The average power supplied by the woman is given by Eq. 11-7,

p = W/t = (2500 J)/(3.5 s) = 710 W.

129

p = W/t = mgy/t = (100 x 667 N) (152 m)/(55.0s) = 1.84x 105W.Ell-12

p = Fv = (110N)(0.22m/s) = 24 W.Ell-13

F = P/v, but the units are awkward.Ell-14

~ p = Fv = (720 N)(26 m/s) = 19000 W; in horsepower, p = 19000W(1/745.7 hp/W) =

25 hp.

Ell-16 Change to metric units! Then p = 4920W, and the flow rate is Q = 13.9 L/s.density of water is approx.imately 1.00 kg/L , so the mass flow rate is R = 13.9 kg/s.

The

~ (a) Start by converting kilowatt-hours to Joules:

1 kW. h = (1000 W)(3600 s) = 3.6x 106 J.

The gas required to

30 mi'

1 gal,0.026 gal = 0.78 mi.

(b) At 55 mi/h, it will take

lhr

55 mi= 0.014 h = 51 s.0.78 mi

The rate of energy expenditure is then (3.6 x 106 J) / (51 s) = 71000 W.

Ell-18 The linear speed is v = 27r(O.207m)(2.53 rev/s) = 3.29m/s. The frictional force is f =JlkN = (0.32)(180N) = 57.6N. The power developed is p = Fv = (57.6N)(3.29m) = 19ÓW.

Ell-19 The net force required will be (1380kg -1220kg)(9.81m/s2) = 1570N. The work isW = Fy; the power output is p = W/t = (1570N)(54.5 m)/(43.0s) = 1990W, or p = 2.67 hp.

Ell-20 (a) The momentum change of the ejected material in one second is

~p = (70.2kg)(497m/s -184m/s) + (2.92 kg)(497m/s) = 2.34x 104kg .m/s.

The thrust is then F = ~p/ ~t = 2.34 x 104N .(b) The power is p = Fv = (2.34x104N)(184m/s) = 4.31x106W. That's 5780 hp.

130

~ The acceleration on the object as a function oí position is given by

8m'The work done on the object is given by Eq. 11-14,

W = 18 Fzdx = 18 (10 kg).20 m/s2.x dx = 800 J.8m

Ell-22 Work is area between the curve and the line F = O. Then

~ (a) For a spring, F ~ -kx, and ~F = -k~x.

With no force on the spring,

~o) -(-110 N)~x = -c k (6500 N/m) This is the amount less than the 40 mm mark, so the position of the spring with no force on it is 23

mm.

(b) ~x = -10 mm compared to the 100 N picture, so

~F = -k~x = -(6500N/m)(-0.010m) = 65N.

The weight of the last object is 110N -65N = 45N.

~ = -I = -om 7 m

Ell-24 (a) W = ~k(xr -Xi2) = ~(1500N/m)(7.60x 10-3m)2 = 4.33x 10-23.(b) W = ~(1500N/m)[(1.52x 10-2m)2 -(7.60 x 10-3m)2 = 1.30x 10-13.

E 11-25 Start with Eq. 11-20, and let Fx = 0 while Fy = -mg:

W = 1! (Fxdx + Fydy) = -mg1! dy = -mgh.

Ell~26 (a) Fo = mv5/ro = (0.675kg)(10.0m/s)2/(0.500m) = 135N.(b) Angular momentum is conserved, so v = vo(ro/r). The force then varies as F = mv2/r

mv5r5/r3 = Fo(ro/r)3. The work done is

W=JFodr=-(0.500m)-2) = 60.03.

~ The kinetic energy of the electron is

1.60 x 10-19 J'= 6.7 x 10-19 J.4.2 eV

1 eV

Then

= 1.2 x 106m/s.v= ~= f2{6.7xlO-19J)

V m y (9.lxlO-31kg)

131

20 m/s2a= X,

Ell-28 (a) K = ~(110kg)(8.1m/s)2 = 3600J.(b) K = ~(4.2x10-3kg)(950m/s)2 = 1900J.(c) m = 91,400 tons(907.2 kg/ton) = 8.29 x 107kg;

v = 32.0 knots(1.688 ft/s/knot)(0.3048 m/ft) = 16.5m/s.

1K = 2(8.29x lO7kg)(l6.5m/s)2 = l.l3x lOlOJ.

Ell-29 (b) ~K = W = Fx = (1.67x10-27kg)(3.60x1015m/s2)(O.O350m) = 2.10x10-13J. That's2.10x 10-13J/(1.60x 10-19J/eV) = 1.31 x 106eV.

(a) Kf = 2.10x 10-13J + !(1.67x 10-27kg)(2.40x 107m/s2) = 6.91 x 10-13J. Then

Vf = ,¡;¡K¡m = V2(6.91 x 10-13J)/(1.67x10-27kg) = 2.88x 107m/s.

Ell-30 Work is negative ir kinetic energy is decreasing. This happens only in region CD. Thework is zero in region BC. Otherwise it is positive.

~ (a) Find the velocity of the particle by taking the time derivative of the position:

v = ~ = (3.0m/8) -(8.0m/82)t + (3.0m/83)f.

Find v at two times: t = O and t = 48.

v(O)

v(4)

= (3.0m/s) -(8.0m/s2)(O) + (3.0m/s3)(O? = 3.0m/s,

(3.0m/s) -(8.0 m/s2)(4.0s) + (3.0 m/s3)(4.0S)2 = 19.0m/s=

The initial kinetic energy is Ki = ~(2.80kg)(3.0m/s)2 = 13J, while the final kinetic energy isKf = ~(2.80kg)(19.0m/s)2 = 505J.

The work done by the force is given by Eq. 11-24,

w = Kf -Ki = 505J -13J = 492J.

(b) This question is agking for the instantaneous power when t = 3.0s. p = Fv, so first find a;

dv

dt:!: -(8.0 m/s2) + (6.0 m/s3)t.a=

= 3 s this givesThen the power is given by p = mav, and when

p = mav = (2.80kg)(10m/s2)(6m/s) = 168W.

Ell-32 w = ¿lK = -Ki. Then

1W = -2(5.98x 1024kg)(29.8 x 103m/s)2 = 2.66x 1033 J.

Ell-33 (a) K = !(l600kg)(20m/s)2 = 3.2x lO5J.(b) p = W/t = (3.2xlO5J)/(33s) = 9.7xlO3W.(C) p = Fv = mav = (l600kg)(20m/s/33.0s)(20m/s) = l.9xlO4W.

Ell-34 (a) I = 1.40 x 104u .pm2(1.66 x 10-27kg/mboxu) = 2;32 x 10-47kg .m2.(b) K = ~I(¿)2 = ~(2.32x10-47kg. m2)(4.30x 1012 rad/s)2 = 2.14x10-22J. That's 1.34 meV.

132

Ell-35 The translational kinetic energy is Kt = ~mv2, the rotational kinetic energy is KrI I 2 ~ 2K Th2 úJ -3 t. en

2(5.30x 10-26kg)3(1.94x 10-46kg .m2) (500m/s) = 6.75 x 1012 rad/s.úJ~

Y3lV

Ell-36 Kr = ~Iú)2 = ~(5l2kg)(0.976m)2(624 rad/8)2 = 4.75xl07J.(b) t = W/P = (4.75 x l07J)/(8l30W) = 58408, or 97.4 minute8.

~ From Eq. 11-29, Ki = !Ic.Ji2. The object is a hoop, so I = MR2. Then

1 1Ki = 2MR2c.J2 = 2(31.4kg)(1.21m)2(29.6rad/s)2 = 2.01 x 1043.

Finally, the average power required to stop the wheel is

wt

Kf-Ki

tp= =

rBIAJB, or lAJA = 3IAJB.Ell-38 The wheels are connected by a belt, 80 r A(¿}A(a) Ir lA = lB then

IA

IRIA/(.;A -

=~-

1-,3

úJB

úJA

(b) If instead KA KB then

2KA/úJA2

2KB/úJB2

úJB2 1

-9'

1A

18=

:.JA2

Ell-39 (a) LV = 21r/T, so

-47r2 (5.98 x lO24kg)(6.37 x lO6m)21K = -Iw2 =

2= 2.57 x 1029 J

5 (86,4008)2

(b} t = (2.57x 1029J}/(6.17x 1012W} = 4.17x 10168, or 1.3 billion year8.

Ell-40 (a) The velocities relative to the center oí mass are mIVI = m2V2j combine with VI +V2 =910.0 mis and get

(290.0kg)Vl = (150.0kg)(910m/s Vi),

or

Vl = (150kg)(910m/s)/(290kg + 150kg) = 310m/s

and V2 = 600m/s. The rocket case was sent back, so Vc = 7600m/s -310m/s = 7290m/s. Thepayload capsule was sent forward, so Vp = 7600 m/s + 600 m/s = 8200 m/s.

(b) Before;1Ki = 2(290kg + 150 kg)(7600m/s)2 = 1.271 x 10loJ.

After ,

1 1Kf = 2(290 kg)(7290 rn/s)2 + 2(150 kg)(8200 rn/s)2 = 1.275 x 1010 J.

The "extra" energy carne frorn the spring.

133

~ Let the mass of the freight car be M and the initial speed be Vi. Let the mass of thecaboose be m and the final speed of the coupled cars be Vf. The caboose is originally at rest, so theexpression of momentum conservation is

MVi = MVf + mvf = (M + m)vf

The decrease in kinetic energy is given by

1 2 ( 1 2 1 22Mvi -2Mvf + 2mVf

1 ( 2 2)2 M Vi -(M+m)Vf

K;-Kf =

Ell-42 Since the body splits into two parts with equal mass then the velocity gained by one isidentical to the velocity "lost" by the other. The initial kinetic energy is

1Ki = 2(8.0kg)(2.Om/s)2 = 16J.

The final kinetic energy is 16 J greater than this, so

1 1Kf = 32J= 2(4.0kg)(2.Om/s+v)2+2(4.0kg)(2.0m/s-v)2,

= ~(8.0kg)[(2.0m/s)2 + v2],

so 16.0J = (4.0kg)v2. Then v = 2.0m/s; one chunk comes to a rest while the other moves off at aspeed of 4.0 m/s.

Ell-43 The initial velocity of the neutron is vo¡, the final velocity is Vlj. By momentum conser-vation the final momentum of the deuteron is mn(VO¡ -Vlj). Then mdv2 = mn,;;lf+"ij'f.

There is also conservation of kinetic energy:

1 2 1 2 1 22mnVo = 2mnVl + 2mdV2.

Roun(iing the numbers slightly we have md = 2mn, then 4v~ = v3 + v~ is the momentum expressionand v3 = v~ + 2v~ is the energy expression. Combining,

2 2 - 2 2 ( 2 2 )Vo -Vl + Vo + Vl ,

or v~ = v3/3. So the neutron is left with 1/3 of its original kinetic energy.

134

Then the

Ell-44 (a) The third particle must have a momentum

P3 = -(16.7x 10-27kg)(6.22x 106m/s)i + (8.35x10-27kg)(7.85 x 106m/s)j

= ( -1.04i + O.655j) x 10-19kg .m/s.

(b) The kinetic energy can also be written as K = ~mv2 = ~m(p/m)2 = p2/2m.

kinetic energy appearing in this process is

1 1K = 2(16. 7x 10-27kg)(6.22 x 106m/s)2 + 2(8.35 x 10-27kg)(7.85x 106m/s)2

23x lO-12J.+2(11. 7x 10-27kg)(1.23x 10-19kg .m/s}2 =

This is the same 88 7.66 MeV.

~ Change your units! Then

w (4.5 eV)(1.6x10-19 J/eV) = 2.1 x 10-10 N.F = -;- = (3.4 x 10-9 m)

Pll-2 (a) If the acceleration is -g/4 the the net force on the block is -Mg/4, so the tension in

the cord must be T = 3Mg/4.(a) The work done by the cord is W = F .s= (3Mg/4)(-d) = -(3/4)Mgd.

(b) The work done by gravity is W = F .s= ( -Mg)(-d) = Mgd.

Pll-3 (a) There are four cords which are attached to the bottom load L. Each supports a tensionF, so to lift the load 4F = (840 lb) + (20.0 lb), or F = 215 lb.

(b) The work done against gravity is W = F .s = (840 lb) (12.0 ft ) = 10100 ft .lb.(c) To lift the load 12 feet each segment of the cord must be shortened by 12 ftj there are four

segments, so the end of the cord must be pulled through a distance of 48.0 ft.(d) The work done by the applied force is W = F .s= (215lb)(48.0 ft) = 10300 ft .lb.

Pll-4 The incline has a height h where h = W/mg = (680J)/[(75kg)(9.81 m/s2)]. The workrequired to lift the block is the same regardless of the path, so the length of the incline l is l =

W/F = (680J)/(320N). The angle of the incline is then

h F ".".c (320N)O = arcsin -l = arcsin- ""- .-= 25.8°.~mg = ar(;!;lIl (75 kg)(9-:81 m782)

Pll-5 (a) In 12 minutes the horse moves x = (5280 ft/mi)(6.20 mi/h)(0.200 h) = 6550 ft. The

work done by the horse in that time is W = F .i = (42.0 lb) ( 6550 ft ) cos(27 .o°) = 2.45 x 105 ft .lb.

(b) The power output of the horse is

p = ~ 105 ft ~ =- ..¡ ,- ,- 1 hp =0 618 h(720s) .p.= J40 tt .lb/S550 ft .lb/s

Pll-6 In this problem (} = arctan(28.2/39.4) = 35.5°.The weight of the block h8S components WII = mg sin (} and W 1. = mg cos (} .The force of friction

on the block is f = J.tkN = J.tkmgcos(}. The tension in the rope must then be

T = mg(sin (} + J.tk cos (})

in order to move the block up the ramp. The power supplied by the winch will be p = Tv, so

p = (1380kg)(9.81 m/s2)[sin(35.5°) + (0.41) cos(35.5°)](1.34m/s) = 1.66x 104W.

135

~ If the power is constant then the force on the car is given by F = Plv. But the force isrelated to the acceleration by F = ma and to the speed by F = m~ for motion in one dimension.Then

F

dvm-

dtdxdv

m--dt dx

dvmv-

dx

lv mv2dv

-

=

=

=

P-,v

P-,v

~

vP-,v

ix Pdx,

Px.

-

1 3-m v3

=

We can rearrange this final expression to get vas a function of x, v = (3xP/m)1/3.

Pll-8 (a) If the drag is D = bv2, then the force required to move the plane forward at constantspeed is F = D = bv2, so the power required is p = Fv = bv3.

(b) p oc V3, SO if the speed increases to 125% then p increases by a factor of 1.253 = 1.953, orincreases by 95.3%.

Pll-9 (a) p = mgh/t, but m/t is the persons per minute times the average mass, so

p = (100 people/min)(75.0kg)(9.81 m/s2)(8.20 m) = 1.01 x 104W.

(b) In 9.50 s the Escalator has moved (0.620m/s)(9.50s) = 5.89m; so the Escalator has "lifted"the man through a distance of (5.89m)(8.20m/13.3m) = 3.63m. The man did the rest himself.

The work done by the Escalator is then W = (83.5kg)(9.81m/s2)(3.63m) = 2970J.(c)Yes, because the point of contact is moving in a direction with at least some non-zero com-

ponent of the force. The power is

p = (83.5 m/s2)(9.81m/s2)(0.620m/s)(8.20 m/13.3m) = 313W.

( d ) If there is a force of contact between the man and the Escalator then the Escalator is doingwork on the mano

PII-I0 (a) dP/dv = ab -3av2, so Pmax occurs when 3V2 = b, or v = Vb73.(b) F = P/v, so dF/dv = -2v, which means F is a maximum when v = O.(c) No; P = 0, but F = ab.

~ (b) Integrate,

13xo w=

o

F;¡ 3XO O j

Xo O~ -3) ,

F .d8= :.-- :x -Xo)dx = Foxo

or W ,; 3Foxo/2.

136

Pll-12 (a) Simpson's rule gives

1W = 3 [(10N)+ 4(2.4N) + (0.8N)] (1.0m) = 6.8J.

(b) W = JFds = J(A/x2)dx = -A/x, evaluating this between the limits of integration givesW = (9N .m2)(1/lm -1/3m) =6J.

~ The work required to stretch the spring from Xj to Xf is given by

1xf 3 k 4 k 4W = k x dx = ¡Xf -¡Xj .

Xi

Pll-14 (a) The spring extension is 8l = ~ -lo. The force from one spring has magnitude

k8l, but only the x component contributes to the problem, so

F=2k

is the force required to move the point.The work required is the integral, W I; F dx, which is

w = kx2 -2klo.¡¡:¡--+-:;2 + 2kl~

Note that it does reduce to the expected behavior for x » lo.(b) Binomial expansion of square root gives

.¡¡:¡--+-;;; = lo, 1 X2 1 X4 ,1+ ,..

2 l2 8 l4O O

SO the first term in the above expansion cancels with the last term in W; the second term cancelswith the first term in W, leaving

1 X4

W=¡k"l-g-'

Pll-15 Number the springs clockwise from the top of the picture. Then the four forces on eachspring are

= k(lo -..¡X2 + (lo -y)2),

= k(lo -..¡(lo -;)2 +y2),

= k(lo -..¡X2 + (lo + y)2),

= k(lo -..¡(lo + x)Z + y2).

Fj.

F2

Fa

F4

137

We then want to find the work required to stretch from x = l to x = 2l, so

k 4 k) 4

WI-+21 = ¡(2l) -¡(l ,

= 16~l4 -~l44 4 ,

k 4= 15¡1 = 15Wo.

The directions are much harder to work out, but for small x and y we can assume that

F; = k(lo -..¡x2 + (10- y)2)j,

F~ = k(lo -..¡(10- x)2 + y2)i,

F~ = k(lo -..¡x2 + (lo + y)2)j,

F~ = k(lo -..¡(lo + x)2 + y2)i.

Then

Pll-16 (a) Kj = !(1100kg)(12.8m/s)2 = 9.0x 104J. Removing 51 kJ leaves 39 kJ behind, so

or 30 km/h.(b) 39 kJ, 88 W88 found above.

~ Let M be the mass of the man and m be the mass of the boy. Let VM be the originalspeed of the man and Vm be the original speed of the boy. Then

1 2

-mvm2

and1 ) 2 1 22M(VM + 1.0m/s = 2mvm.

Combine these two expressions and solve for v M ,

vt =

o =

The last line can be solved as a quadratic, and v M = (1.0mis) ::1:: (1.41 mis). Now we use the veryfirst equation to find the speed of the boy,

vL

2VM

138

Pll-18 (a) The work done by gravity on the projectile 88 it is raised up to 140 m is W =-mgy = -(0.550kg)(9.81 m/s2)(140m) = -755J. Then the kinetic energy at the highest point is1550J- 755J = 795J. Since the projectile must be moving horizontally at the highest point, thehorizontal velocity is Vx = V2(795J)/(0.550kg) = 53.8m/s.

(b) The magnitude of the velocity at launch is v = V2(1550J)/(0.550kg) = 75.1m/s. ThenVy = V(75.1m/s)2 -(53.8m/s)2 = 52.4m/s.

(c) The kinetic energy at that point is !(0.550kg)[(65.0m/s)2 + (53.8m/sf] = 1960J. Since ith88 extra kinetic energy, it must be below the launch point, and gravity h88 done 410 J of work onit. That means itis y = (410J)/[(0.550kg)(9.81m/s2)] = 76.0m below the launch point.

Pll-19 (a) K = !mv2 = !(8.38 x 1011kg)(3.0 X 104m/s)2 = 3.77 x 102oJ. In terms of TNT,K = 9.0 x 104 megatons.

(b) The diameter will be :r¡8.98 x 104 = 45 km.

PII-20 (a) W 9 = -(0.263kg)(9.81m/s2)(-0.118m) = 0.304J.(b) Ws = -~(252N/m)(-0.118m)2 = -1.75J.( c ) The kinetic energy just before hitting the block would be 1.75 J -0.304 J = 1.45 J. The speed

is then v = V2(1.45J)/(0.263kg) = 3.32m/s.( d) Doubling the speed quadruples the initial kinetic energy to 5.78 J. The compression will then

be given by1

-5.78J = -2(252N/m)y2 -(0.263kg)(9.81m/s2)y,

with solution y = 0.225 m.

~ (a) We can solve this with a trick oí integration.

w = lx F dx,

{X dt {t dxlo maxdt dx = max lo &dt

max lt v x dt = max lt at dt,

1-ma2t22 x .

B88ically, we changed the variable of integration from x to t, and then used the fact the the accel-eration W88 constant so Vx = VOx + axt. The object started at rest so VOx = O, and we are given inthe problem that Vf = atf. Combining,

(b) Instantaneous power will be the derivative of this, so

2

p= ~ =m(*) t.

139

Pll-22 (a) a = (-39.0 rev/s)(27r rad/rev)/(32.0s) = -7.66 rad/s2.

(b) The total rotational inertia of the system about the axis of rotation is

I = (6.40kg)(1.20m)2/12 + 2(1.06kg)(1.20m/2)2 = 1.53kg .m2.

The torque is then T = (1.53kg .m2)(7.66 rad/s2) = 11.7N .m.(c) K = !(1.53kg .m2)(245 rad/s)2 =4.59x 104J.(d) (} = fJ)avt = (39.0 rev/s/2)(32.0s) = 624 rev.

(e) Only the loss in kinetic energy is independent of the behavior óf the frictional torque.

Pll-23 The wheel turn with angular speed úJ = v/r, where r is the radius of the wheel and v thespeed of the car. The total rotational kinetic energy in the four wheels is then

The translational kinetic energy is K t = ~ (1040 kg)V2, so the fraction of the total which is due tothe rotation of the wheels is ---

= 0.0213 or 2.13%.11.3

520 + 11.3

Pll-24 (a) Conservation of angular momentum: I..)f = (6.13kg .m2/1.97kg .m2)(1.22 rev/s) =3.80 rev/s.

(b) K r O( II..)2 O( l2 / I, so

Kf/Ki = Ii/lf = (6.13kg .m2)/(1.97kg .m2) = 3.11.

~ We did the first part of the solution in Ex. 10-21. The initial kinetic energy is (again,ignoring the shaft ) ,

since the second wheel was originally at rest. The final kinetic energy is

since the two wheels moved M one. Then

Kj-Kf

Kj

!11ii:i1,i2 -!(11 + 12)ii:if2

1 1 - 2 ,- 1(.,; 1 .2 ,I

1 -(11 + 12)ii:if2

II¡;jl,i2

1- II

~'

where in the last line we substituted from the results of Ex. 10-21.Using the numbers from Ex. 10-21,

Ki-Kf

Kj79.2%.

140

Pll-26 See the solution to PlO-ll.

while

according to PlO-ll,I(¡) -mvRI +mR2 .úlf =

ThenK -1 (II.JJ-mvR)2f-2 I+mR2 .

Finally,

~K =

Pll-27 See the solution to PlO-l2.(a) Ki = ~Ii(J)i2, so

Ki = ~ (2(51.2kg)(1.46m)2) (0.945 rad/s)2 = 97.5J.

(b) Kr = ! (2(51.2kg)(0.470m)2) (9.12 rad/s)2 = 941 J. Th~~nergy comes from the work they

do while pulling themselves closer together .

Pll-28 K = ~mv2 = ~p2 = ~p. p. Then

Kf -2m

= 2k (pi2 + 2pi .~p + (~p)2) ,

1 .(2pi .~p + (~p)2) .

2mIn all three cases ~p = (3000N)(65.0s) = 1.95x105N .s and Pi = (2500kg)(300m/s) = 7.50x105kg .

m/s.(a) If the thrust is backward (pushing rocket forward),

~ -+2(7.50x105kg.m/s)(1.95x105N.s)+(1.95x105N.s)2= 661107JK -2(2500 kg) + .x .

=

~K

~

(b) If the thru8t i8 forward,

~K = .-2(7.50 x lO5kg .m/8)(l.95x lO5N .8)+S~.~x lO5N: 8)2,-5.09 x lO7J.

2(2500 kg)

( c ) If the thru8t i8 8ideway8 the fir8t term vani8he8,

6.K = +(1:95 x 105N .8)2 = 7.6lxlO6J.2(2500 kg)

141

~ There's nothing to integrate here! Start with the work-energy theorem

1 2 1 2W = K f -K i = 2mVf -2mVi ,

= ~m (Vf2 -Vi2) ,

1= 2m ( Vf -Vi) ( Vf + Vi) ,

where in the last line we factored the difference of two squares. Continuing,

w 12 (mVf -m Vi) (Vf+ Vi) ,

12(Ap)(Vf+Vi),

but ~p = J, the impulse. That finishes this problem.

Pll-30 Let M be the mass of the helicopter. It will take a force Mg to keep the helicopterairborne. This force comes from pushing the air down at a rate ~m/ ~t with a speed of v; soMg = v~m/~t. The blades sweep out a cylinder of cross sectional area A, so ~m/~t = pAv.The force is then Mg = pAv2; the speed that the air must be pushed down is v = ylMg7pA. Theminimum power is then

= 2.49xlO5W.{ii;p = Fv = Mgy PA

Pll-31 (a) Inelastic collision, so Vf = mVi/(m + M).(b) K = ~mv2 = p2/2m, so

~KKj

~m-l/(m+M)=ljm

=~m+M'

Pll-32 Inelastic collision, so

The 1088 in kinetic energy i8

(1.88 kg)(10.3~~K (1.88kg + 4.92kg)(5.21 m/s)2 = 33.7 J.

-22

This change is because of work done on the spring, so

x = v'2(33.7 J)/(1120N/m) = 0.245 m

Pll-33 Pf,B = Pi,A + Pi,B -Pf,A, so

Pf,B [(2.0kg)(15m/s) + (3,Okg)(-10m/s) -(2.0kg)(-6.0m/s)]i

+[(2.0kg)(30m/s) + (3.0kg)(5.0m/s) -(2.0 kg)(30m/s)]j,

(12kg .m/s)i + (15kg .m/s)j.

=

142

Then Vf,B = (4.0m/s)i + (5.0m/s)j. Since K = T(v~ + v~), the change in kinetic energy is

~K = (2.0kg)[(-6.0m/s)2 + (30m/s)2 -(15m/s)2 -(30m/s)2]2

(3.0kg)[(4.0m/s)2 + (5m/s)2 -(-10m/s)2 -(5.0m/s)2]+ -2

-315 J .

Pll-34 For the observer on the train the acceleration of the particle is a, the distance traveled isXt = !at2, so the work done as measured by thetrain is Wt = maxt = !a2t2. The final speed ofthe particle as measured by the train is Vt = at, so the kinetic energy as measured by the train isK = !mv2 = !m(at)2. The particle started from rest, so ~Kt = W t.

For the observer on the ground the acceleration of the particle is a, the distance traveled isXg = !at2 + ut, so the work done as measured by the ground is W 9 = maxg = !a2t2 + maut.The final speed of the particle as measured by the ground is Vg = at + u, so the kinetic energy asmeasured by the ground is

1 1 1 1Kg = 2mV2 = 2m(at + U)2 = 2a2t2 + maut + 2mu2.

But the initial kinetic energy as measured by the ground is ~mu2, so W 9 = LlKg.

Pll-35 (a) Ki = ~mlVl,i2.

(b) After collision Vf = mlVl,i/(ml + m2), SO

21 c

= -ml Vl,i2

m¡V¡,i

m¡ + m2

~ ml

ml + m2

(c) The fraction lost was

The amount lost,

ml +m2 ml +m2(d) Note that Vcm = Vf. The initial kinetic energy of the system is

1 I 2 1 I 2K i = 2ml v l,i + 2m2V 2,i .

The final kinetic energy is zero (they stick together!), so the fraction lost is 1.however, is the same.

Pll-36 Only consider the first two collisions, the one between m and m', and then the one betweenm' and M.

Momentum conservation applied to the first collision means the speed of m' will be betweenv' = mvo/(m+m') (completely inelastic)and v' = 2mvo/(m+m') (completelyelastic). Momentumconservation applied to the second collision means the speed of M will be between V = m'v' /(m' +M)and V = 2m' v' / ( m' + M) .The largest kinetic energy for M will occur when it is moving the fastest,so 2 2 ' , 4 ,v' = mvo and V = m v -m mvo

m+m' m'+M (m+m')(m'+M)

We want to maximize Vas a function of m' , so take the derivative:

dV 4mvo(mM -m'2)-= (m'+M)2(m+m')2.

~

dm'

This vanishes when m' = JTñM .

143

E12-1 (a) Integrate.

U(x) = -{X G~dX = -G~,

Joo x(b) W = U(x) -U(x + d), so

( I I ) G m dW = Gm¡m2 :;; -m = m¡ 2;¡;;-:¡:"ijj"'

E12-2 Ir d « x then x(x + d) ~ X2, SO

w ~ Gmlm2~d.

~ Start with Eq. 12-6.

U(x) -U(Xo) =

Finishing the integration,

Ir we choose Xo

U(x) = U(Xo) + ~ ( e-f:Jx~ -e-f:JX2)

00 and U(XO) = O we would be left with

U(x) = -a -Bx2

2{je

E12-4 ~K = -~U so ~K = mg~y. The power output is then

p = (58%).(lOOOkg/m3)(73,800m3)(9.8lm/s2)(96.3m) = 6.74xlOsW.(608)

E12-5 ~U = -~K, so !kX2 = !mv2, Then

k = ~2 = (2.38kg)(lO.OxlO3/s) = l.lOxlOBN/m.X2 (1.47 m)2

Wow.

E12-6 ~U 9 + ~U s = O, since K = 0 when the man jumps and when the man stops. Then~Us = -mg~y = (220 lb)(40.4 ft) = 8900 ft .lb.

~~

Kf + Uf1 22mVf + mgyf

~Vf2 + g( -r )2

=

Rearranging,Vf = V -29( -r) = 2.15 m/s.-2(9.81 m/s2)( -0.236 m)

144

Ki+Ui,1 2-m Vi + mgyi,2

E12-8 (a) K = ~mv2 = !(2.40kg)(150m/s)2 = 2.70x 104J.(b) Assuming that the ground is zero, U = mgy = (2.40kg)(9.81m/s2)(125m) = 2.94x103J.(c) Kf = Ki + Ui since Uf =O. Then

I

-In (2. 70x 104J) + (2.94 x 103J)2 S2. 70x 104J) + (2.94 x 103J) -158 /Vf -y ~- -m s.

Only (a) and (b) depend on the mass.

(2.40 kg)

E12-9 (a) Since ~y = O, then ~U = 0 and ~K = o. Consequently, at B, v = vo.(b) At C Kc = KA + UA -UC, or

1 2 1 2 h2mvc = 2mvo + mgh -mg2'

or

VB=~=VVO2+9h.(c) At D KD =KA +UA- UD, or

1 2-m VD2

~mvo2 + mgh -mg(O),

orVE = v'VO2 + 2gh.

E12-10 From the slope ofthe graph, k = (0.4N)/(O.O4m) = 10N/m.(a) AK = -AU, so ~mVf2 = ~kXi2, or

I (10N/m)Vf = V ~(O.O550m) = 2.82m/s.

(b) AK= -AU, so ~mVf2 = ~k(Xi2 -Xf2), or

~ (a) The force constant ofthe spring is

k = F/x = mg/x = (7.94kg)(9.81 m/s2)/(0.102m) = 764N/m.

(b) The potential energy stored in the spring is given by Eq. 12-8,

U = ~kX2 = ~(764N/m)(0.286m + O.102m)2 = 57.5J.2 2

(c) Conservation of energy,

=

=

Kf + Uf1 2 1 22mvf + mgyf + 2kxf

1 12(0)2 + mgh + 2k(0)2

Ki+Ui.

1 2 1 2-mv. + m gy . + -kx.2 1 1 2 1 .

1 2 1 22(0) + mg(O) + 2kxi .-

Rearranging,h = ~Xi2 = (764N/m).I 2(7.94kg)(9.81m/s2) (0.388m)2 = 0.738 m.

k

145

E12-12 The annual mass of water is m = (1000kg/m3)(8 x 1012m2)(0.75m). The change inpotential en~rgy each year is then AU == -mgy, where y = -500 m. The power available is then

E12-13 (a) From kinematics, v = ~gt, so K = ~mg2t2 and U = Úo -K = mgh -~mg2t2.(b)U=mgysoK=Uo-U=mg(h-y). ,

E12-14then

The potential energy is the same in both cases. Consequently, mgE~YEmgM~yM, and

y M = (2.05 m - 10m)(9.81m/s2)/(1.67m/s2) + 1.10m = 6.68m.

~ The working is identical to Ex. 12-11,

=

=

=

so

d = k/sinO = (1.92m)/sin(27°) = 4.23m.

E12-16 The vertical position of the pendulum is y = -lcos(}, where (} is measured from thedownward vertical and I is the length of the string. The total mechanical energy of the pendulum is

1E = 2mVb2

In this caseif we set U = O at the bottom of the path and Vb is the speed at the bottom.U = mg(l +y).

(a) K = E- U = ~mvb2 -mgl(l -cose). Then

v = v(8.12m/s)2 -2(9.81 m/s2)(3.82kg)(1- cos58,OO) = 5.54m/s.

(b) U = E -K, but at highest point K = O. Then

(J arccos = 83.1°.

(c) E = ~(1.33kg)(8.12m/s)2 = 43.83.

E12-17 The equilibrium position is when F = ky = mg.~(ky)y = ~mgy. So 2~Us = -~Ug.

Then ~U 9 = -mgy and ~U s =

146

Kf + Uf1 2 1 22mVf + mgyf + 2kxf

K. +U .1 1,1 2 1 22mV¡ + mgy¡ + 2kX¡ ,

The distance up the incline is given by a trig relation,

E12-18 Let the spring get compressed a distance x. If the object fell from a height h = 0.436 m,then conservation of energy gives !kx2 = mg(x + h). Solving for x,

x = f :!:: J (f) 2 + 2fh

only the positive answer is of interest, so

{2.14 kg){9.81 m/s2)( {2.14 kg){9.81 m/s2) ) 2 + 2 {2.14 kg){9.81 m/s2) {0.436 m) = 0.111 m.

:!: {1860N/m) {1860N/m)x = (1860N/m)

~ The horizontal distance traveled by the marble is R = vtf, where tf is the time of flightand v is the speed of the marble when it leaves the gun. We find that speed using energy conservationprinciples applied to the spring just before it is released and just after the marble leaves the gun.

Kj+Uj

1O + 2kx2

= Kf + Uf,

I 2O-m v + .

2=

Kj = O because the marble isn't moving originally, and Uf = O because the spring is no longer

compressed. Substituting R into this,

1 1-kx2 = -m2 2 .~)2.

We have two values for the compression, Xl and X2, and two ranges, R1 and R2. We can put bothpairs into the above equation and get two expressionsj if we divide one expression by the other weget

~ 2(~). = (~)We can easily take the square root of both sides, then

~-&x¡ -R¡ .

R1 was Bobby's try, and was equal to 2.20 -0.27 = 1.93 m. Xl = 1.1 cm was his compression. IfRhoda wants to score, she wants R2 = 2.2 m, then

2.2m 1.1 cm = 1.25 cm.X2 = 1.93m

E12-20 Conservation of energy- U1 + K1 = U2 + K2- but U1 = mgh, K1 = O, and U2 = 0, soK2 = !mv2 = mgh at the bottom of the swing.

The net force on Tarzan at the bottom of the swing is F = mv2 /r , but this net force is equal tothe tension T minus the weight W = mg. Then 2mgh/r = T -mg. Rearranging,

= 241 lb.

This isn 't enaugh ta break the vine, but it is clase.

147

E12-21 Let point 1 be the start position of the first mass, point 2 be the collision point, and point3 be the highest point in the swing after the collision. Then U1 = K2, or ~mlv12 = mlgd, where Vlis the speed of ml just before it collides with m2. Then Vl = yI2gd.

After the collision the speed ofboth objects is, by momentum conservation, V2 = mlvl/(ml +m2).Then, by energy conservation, U3 = K~, or ~(ml +m2)v22 = (ml +m2)gy, where y is the height

to which the combined masses rise.Combining,

V22

2g

:¿m12v12

2(ml +m2)2gd.y= = = mi

mi + m2

E12-22 ~K = -~U, so

1 12mV2 + 2IiJ)2 = mgh,

where I = ! M R2 and iJ) = v / R. Combining,

1 12mv2 + 4Mv2 = mgh,

so

v = ~ = /4(0.0487 kg)(9.81m/s2)(0.540m) = 1.45m/s.y 2;;;: + M V 2(0.0487 kg) + (0.396 kg)

~ There are three contributions to the kinetic energy: rotational kinetic energy of the shell(Ks), rotational kinetic energy of the pulley (Kp), and translational kinetic energy of the block( K b ) .The conservation of energy statement is then

Ksf+K p f+Kbf+Uf ,, , ,1 2 1 2 1 22Is(J.Js +2'p(J.Jp +2mVb +mgy.

;

Ksi+K p i+Kbi+Ui =, , ,

(O) + (0) + (0) + (0) =

Finally, y = -h and

(.;sR= (.;pT = Vb.

Combine all of this together, and our energy conservation statement willlook like this:

which can be fairly easily rearranged into

E12-24 The angular speed of the flywheel and the speed of the car are related by

(1490 rad/s)k=~-1 v -{24.0 mis) = 62.1 radim.

The height oí the slope is h = (1500m) sin(5.000) = 131 m. The rotational inertia oí the Hywheel is

I = ~ (194N) 22 (ij:8i""iñ7B') (0.54 m) = 2.88 kg .m2.

148

(a) Energy is conserved as the car moves down the slope: Ui = Kf, or

1 1 1 1m g h = -mv2 + -I(¿)2 = -mv2 + -Ik2v22 2 2 2 ,

or

~

I 2mgh .1 2(822 kg)(9.81 m/s2)(131 m)v = V m + Ik2 = V (822kg) + (2.88kg .~2)(62.1 rad/m)2 = 13.3m/s,

or 47.9 m/s.(b) The average speed down the slope is 13.3m/s/2= 6.65m/s. The time to get to the bottom

is t = (1500m)/(6.65m/s) = 226s. The angular acceleration ofthe disk is

a = ~ = (13.3'm/s)(62.1 rad/m) = 3.65 rad/s2.t (2268)

(c) p = T(Q = Ia(Q, so

p = (2.88kg .m2)(3.65 rad/s2)(13.3m/s)(62.1 rad/m) = 8680W.

E12-25 (a) Far the salid sphere I = ~mr2; if it ralls withaut slipping w = v/r; canservatian afenergy means K i = U f .Then

1 1 (2 ) ( V ) 22mv2 + 2 Smr2 -;: = mgh.

or

E12-26 Conservation of energy means ~mv2 + ~I(¿)2 = m9h.h = 3v2/49, so

But (.; = v/r and we are told

1 V2-1'--2 r2

3V2

mg49'

1 2-m v2

or

22 (3 1 ) 1 ,I = 2r 4m -2m = 2mr

which cauld be a salid disk ar cylinder .

~ We agsume the cannon ball is solid, so the rotational inertia will be I = (2/5)M R2The normal force on the cannon ball will be N = Mg, where M is the mags of the bowling ball.

The kinetic friction on the cannon ball is Ff = JLkN = JLkMg. The magnitude of the net torque onthe bowling ball while skidding is then T = JLkM gR.

Originally the angular momentum of the cannon ball is zero; the final angular momentum willhave magnitude l = I (¡) = I v / R, where v is the final translational speed of the ball.

The time requires for the cannon ball to stop skidding is the time required to change the angularmomentum to l, so

~l

T

2v

5J1,k9

~t= = .,-"N";=.{2/5)M R2V / R

Since we don't know At, we can't solve this for v. But the same time through which the angularmomentum of the ball is increB8ing the linear momentum of the ball is decreB8ing, so we also have

¡\t= .Ap -Mv-Mvo -vo-v-=-l'J -JLkMg /1kg

Combining,

2v va -v=5Jl,kg Jl,kg

2v = 5(vo -v),

v::;: 5vo/7

2 4r-,--r-,--r-,--1I I I I IL-.J--L- -L-.J--II I I I I I II I I I I I Ir-,-- -,--r-,--1I I I I I IL-.J- L-.J--L-.J--II I I I I I II I I I I I

O 23456

go,-.~

~ -I

3,-.,

'-"2"';;:~1

-2 o

o 1 2 3 4 5 6

x (m)

E12-28

E12-29 (a) F = -~U/~x = -[(-17 J) -(-3J)]/[(4m) -(lm)] = 4.7N.(b) The total energy is !(2.0kg)(-2.0m/s)2 + (-7J), or -3J. The particle is constrained to

move between x = 1 m and x = 14m.(c) When x!= 7m K = (-3J) -(-17 J) = 14J. The speed is v = ..¡2(14J)/(2.0kg) = 3.7m/s.

E12-30 Energy is conserved, so1 2 1 22mvo = 2mv + mgy,

v = .¡;¡=-2;; ,or

which depends only on y.

~ (a) We can find Fx and Fy from the appropriate derivatives of the potential,

{)UFx -= -kx,

axauayFy =-ky.=

The force at point (x, y) is then

F = Fxi + Fyj = -kxi -kyj.

(b) Since the force vector points directly toward the origin there is no angular component, andF9 = O. Then Fr = -kr where r is the distance from the origin.

(c) A spring which is attached to a point; the spring is free to rotate, perhaps?

150

E12-33 We'll jUBt do the paths, showing only non-zero terms.Path 1: W = J~(-k2a)dy) = -k2ab.

Path 2: W = Joa( -k1b) dx) = -k1ab.

Path 3: W = (cos<1> sin<1» JOd( -k1 -k2)rdr = -(kl + k2)ab/2.These three are only equal if k1 = k2.

P12-2 The hall just reaches the top, so K2 = O.

y2(mgL)/m= J29L.

Then K1 = U2 -tJ1 = mgL, so Vi

P12-3 Measure distances along the incline by x, where x = O is measured from the maximallycompressed spring. The vertical position of the mass is given by xsin(}. For the spring k =(268N)/(0.0233m) = 1.15 x 104 N/m. The total energy of the system is

12(l.l5xlO4N/m)(O.O548m)2 = l7.3J.

(a) The block needs to have moved a vertical distance xsin(32.00), where

l7.3J = (3.l8kg)(9.8lm/s2)xsin(32.00),

or x = 1.05m.

(b) When the block hits the top of the spring the gravitational potential energy has changed by

AU = -(3.18kg)(9.81 m/s2)(1.05 m -O.O548m) sin(32.00) = 16.5, J;

hence thespeed is v = .¡2(16.5 J)/(3:18 kg) = 3.22 m/s.

151

P12-4 The potential energy associated with the hanging part is

10Mg 2

2LY -L/4

MgL32 ,

¡O M

U= -gydy=-L/4 L

=-

sothe work required is W = MgL132.

~ (a) Considering points p and Q we have

Kp+Up =

(O) + mg(5R) =

KQ + UQ,

~mv2 + mg(R),21 2-mv,2

v.

4mgR =

There are two forces on the block, the normal force from the track,

mv2

R

m(8gR),

R

N= =8mg,=

and the force of gravity W = mg. They are orthogonal so

Fnet = v(8mg)2 + (mg)2 = V65mg

and the angle from the horizontal by

k: -.:!

8'

-mg

8mgtan (} =

or (} = 7.13° below the horizontal.(b) If the block barely makes it over the top of the track then the speed at the top of the loop

(point S, perhaps?) is just fast enough so that the centripetal force is equal in magnitude to theweight,

mvs2/R=mg.

Assume the block wag releaged from point T. The energy conservation problem is then

KT+UT =

(O) + mgyT =

K s + Us.1 22mvs + mgys,

12(R) + m(2R),

5R/2.

yT S,

P12-6 The wedge slides to the left, the block to the right. Conservation of momentum requiresM Vw + mvb,x = O. The block is constrained to move on the surface of the wedge, so

tan a = Vb,yVb ,

,:¡:-Vw

or

Vb,y = Vb,x tana(l + m/M).

152

Conservation of energy requires

1 2 1 22mVb +2Mvw = mgh.

Combining,

mgh,=

= 2gh,

2M2ghcoS2 a,

2M2ghcos2 a,

2M2ghcoS2 a,

==

=

or

vw=

P12-7 U(x) = -J Fx dx = -Ax2/2 -Bx3/3.(a) U = -( -3.00N/m)(2.26m)2 /2- ( -5.00N/m2)(2.26m)3 /3 = 26.9J.

(b) There are two points to consider:

U1 =

U2 =

K1

-( -3.00N/m)(4.91 m)2 /2- ( -5.00 N/m2)(4.91 m)3/3 = 233J,

-(-3.00N/m)(1.77m)2/2- (-5.00N/m2}(1.77m)3/3 = 13.9J,

12(1.18kg)(4.13m/s)2 = 10.1 J.

P12-8 Assume that Uo = Ko = o. Then conservation of energy requires K = -U; consequently,v = ,fiijFilj .

(a) v = V2(9.81m/s2)(1.20m) = 4.85m/s.(b) v = 2(9.81m/s2)(1.20m -0.45m- 0.45m) = 2.43m/s.

~ Assume that Uo = Ko = o. Then conservation of energy requires K = -U; consequently,

v = ,fiijFii'j .If the ball barely swings around the top of the peg then the speed at the top of theloop is just fast enough so that the centripetal force is equal in magnitude to the weight,

mv2/R = mg.

The energy conservation problem is then

mv2 = 2mg(L- 2(L -d)) = 2mg(2d -L)

mg(L -d) = 2mg(2d- L),

d = 3L/5.

153

( tan2 a(l + m/M)2 + 1 + ~ ) Vb,x2

(sin2 a(M + m)2 + M2 COS2 a + mMcos2 a) Vb,x2

(M2 + mM + mMsin2 a + m2 sin2 a) Vb,x2

((M + m)(M + msin2 a)) Vb,x2

P12-10 The speed at the top and the speed at the bottom are related by

1 2 1 22mVb = 2mVt + 2mgR.

The magnitude of the net force is F = mv2 / R, the tension at the top is

Tt =mvt2/R-mg,

while tension at the bottom isTb = mvb 2/ R + mg,

The difference is

~T = 2mg + m(Vb2 -Vt2)/R = 2mg + 4mg = 6mg.

~ Let the angle (} be meBBured from the horizontal to the point on the hemisphere wherethe boy is located. There are then two components to the force of gravity- a component tangentto the hemisphere, WII = mgcos(}, and a component directed radially toward the center of thehemisphere, W J. = mgsin(}.

While the boy is in contact with the hemisphere the motion is circular so

mv2/R = W.L -N.

When the hoy leaves the surface we have mv2/R = Wl., or mv2 = mgRsin(J.

conservation,Now for energy

K+U - Ko + Uo,

~m(Of + mgR,2

1-mv2 + mgy =2

~gRsin(} + mgy =

R,12Y + y =

y

P12-12 (a) To be in contact at the top requires mVt2 / R = mg. The speed at the bottom wouldbe given by energy conservation .,

1 2 1 22mVb = 2mVt + 2mgR,

SO Vb = ~ is the speed at the bottom that will allow the object to make it around the circle

without loosing contact.(b) The particle willlose contact with the track if mv2 / R ::; mg sin (J .Energy conservation gives

~mv~ = ~mv2 + mgR(l + sin(J)2 2

for points above the half-way point. Then the condition for "sticking" to the track is

or, if Va = O. 775vm,

5(0.775)2- 2 ~ 3 sin(},

or (} = arcsin(I/3).

154

P12-13 The rotational inertia is

Conservation oí energy is

so (JJ = ,/9Ii7(4I¡ .

P12-14 The rotational speed of the sphere is (J) = v/r; the rotational kinetic energy is Kr =!I(J)2 = kmv2.

(a) For the marble to stay on the track mv2 / R = mg at the top of the track. Then the marbleneeds to be released from a point

or h = R/2 + R/5 + 2R = 2.7 R.

(b) Energy conservation gives

1 16mgR = 2mv2 + Smv2 + mgR,

or mv2 / R = 50mg /7. This corresponds to the horizontal force acting on the marble.

P12-15 !mv5 + mgy = o, where y is the distance beneath the rim, or y = -r cos Oo. Then

Vo = v'=2g¡; = \!2grcosOo.

P12-16 (a) For El the atoms will eventually move apart completely.(b) For E2 the moving atom will bounce back and forth between a closest point and a farthest

point.(c) U ~ -1.2x 10-19J.(d) K = El -U ~ 2.2xl0-19J.

(e) Find the slope of the curve, so

which would point toward the larger mass.

~ The function needs to fall off at infinity in both directions; an exponential envelop~would work, but it will need to have an -X2 term to force the potential to zero on both sides. So wepropose something of the form

U(x) = P(x)e-{JX2

where P(x) is a polynomial in x and f3 is a positive constant.We proposed the polynomial because we need a symmetric function which has two zeroes. A

quadratic of the form ax2 -Uo would work, it has two zeroes, a minimum at x = O, and is symmetric.So our trial function is

U(x) = (ax2 -UO) e-í3"'2.

1!'í!'í

This function should have three extrema. Take the derivative, and then we'll set it equal to zero,

dU

dx= 2axe-f30:2-2 (ax2 -UO) {3xe-f3:¡:2,

Setting this equal to zero leaves two possibilities,

x = o,

2a-2(ax2-Uo){3 = 0.

The first equation is trivial, the second is easily rearranged to give

x=:i: ~

y---¡¡a-

These are the points ::I::Xl. We can, if we wanted, try to find Q and {3 from the picture, but youmight notice we have one equation, U(Xl) = U1 and two unknowns. It really isn't very illuminatingto take this problem much farther, but we could.

(b) The force is the derivative of the potential; this expression was found above.( c ) As long as the energy is le88 than the two peaks, then the motion would be oscillatory, trapped

in the well.

P12-18 (a) F = -{)U/{)r, orro 1-+-r2 r

e-r/roF = -Uo

(b) Evaluate the force at the four points:

F(ro)

F(2ro)

F(4ro)

F(10ro)

~~

=

1-2(Uo/ro)e- ,

-(3/4)(Uo/rO)e-2,

-(5/16)(Uo/ro)e-4,

-(11/100) (Uo/ro)e-l0.=

The ratios are then

F('

F(

F(11

==

{3/8)e-l = 0.14,

{5/32)e-3 = 7.8 x 10-3,

{11/200)e-9 = 6.8 x 10-6.

~

156

2ro)

4ro)

Oro)

/FI

IFI

/FI

~ro)

~ro)

~ro)

~ If the projectile had not experienced air drag it would have risen to a height y2, butbecause of air drag 68 kJ of mechanical energy was dissipated so it only rose to a height yl. Ineither case the initial velocity, and hence initial kinetic energy, was the same; and the velocity atthe highest point was zero. Then W = ~U, so the potential energy would have been 68 kJ greater ,and

Lly = LlU/mg = (68xl03J)/(9.4kg)(9:81m/s2) = 740 m

is haw much higher it wauld have gane withaut air frictian.

E13-2 (a) The road incline is (J = arctan(0.08) = 4.57°. The frictional forces are the same; the caris now moving with a vertical upward speed of (15m/s) sin(4.57°) = 1.20m/s. The additional powerrequired to drive up the hill is then ~p = mgvy = (1700kg)(9.81m/s2)(1.20m/s) = 20000W. Thetotal power required is 36000 W.

(b) The car will "coast" if the power generated by rolling downhill is equal to 16000 W, or

Vy = (16000W)/[(1700kg)(9.81m/s2)] = 0.959m/s,

clown. Then the incline is() = arcsin(O.959m/s/15m/s) = 3.67°.

This corresponds to a downward grade of tan(3.67°) = 6.4 %.

E13-3 Apply energy conservation:

1 12mv2 + muy + 2ky2 = O,

sov = J-2(9.81m/s2)(-O.O84m) -(262 N/m)(-O.O84m)2/(1.25kg) = O.41m/s.

E13-4 The car climbs a vertical distance of (225m)sin(10°) = 39.1m in coming to a stop. Thechange in energy of the car is then

1 (16400 N) 2 5~E = -2¡¡¡:s¡;ñ7S')(31.4m/s) + (16400N)(39.1m) = -1.83x10 J.

~ (a) Applying conservation ofenergy to the points where the hall was dropped and whereit entered the oil,

1 2 1 22mvf +mgyf = 2mvi +mgyi,

1 12vf2 +g(O) = 2(0)2 +gyi,

Vf = ~2gyi,

= v2(9.81m/s2)(0.76m) = 3.9m/s.

(b) The change in internal energy of the ball + oil can be found by considering the points wherethe ball was released and where the ball reached the bottom of the container .

~E = Kf+Uf-Kj-Uj,1 2 1 22mVf + mgyf -2m(O) -mgyj,

12(l2.2x lO-3kg)(l.48m/s)2 -(l2.2x lO-3kg)(9.8lm/s2)( -0.55m -O.76m),

-O.l43J

157

E13-6 (a) Ui = (25.3kg)(9.81m/s2)(12.2m) = 3030J.(b) Kf = !(25.3kg)(5.56m/s)2 = 391 J.(c) ~Eint = 3030J -391 J = 2640J.

E13-7 (a) At atmospheric entry the kinetic energy is

K

The gravitational potential energy is

u = (7.9x lO4kg)(9.8m!s2)(l.6x lOsm) = l.2x lO11J.

The total energy is 2.6xlO12J.(b) At touch clown the kinetic energy is

3.8x lO8J.

E13-8 ~E/~t = (68 kg)(9.8m/s2)(59m/s) = 39000J/s.

~ Let m be the mass of the water under consideration. Then the percentage of the potentialenergy "lost" which appears as kinetic energy is

Kf-Ki

Ui-Uf'

Then

Kf-Kj

Uj-Uf= 1 ( 2 2)2m Vf -Vi / (mgyi -mgyf)

Vr -Vi2-2gAy ,

(13m/s)2 -(3.2m/s)2

--2(9.81m/s2)(-15m)'

= 54%.

The rest oí the energy would have been converted to sound and thermal energy.

E13-10 The change in energy is

4. 74x 105 J.1~E = 2(524kg)(62.6m/s)2 -(524 kg)(9.81 m/s2)(292m)

E13-11 Uf = Kj -(34.6J). Then

h = ! {7.81 mLs)2 -

2 {9.81 m/s2)

which means the distance along the incline is (2.28m)/sin(33.00) = 4.19m.

158

E13-12 (a) Kf = Ui -Uf, so

Vf = J2(9.81m/s2)[(862m) -(741 m)] = 48.7m/s.

rhat 's a quick 175 km/h; but the speed at the bottom of the valley is 40% of the speed of sound!

(b)~E=Uf-Ui,SO

~E= (54.4 kg)(9.8lm/s2)[(862m) -(74lm)] = -6.46xlO4J;

which means the internal energy of the snow and skis increased by 6.46 x 104 J .

~ The final potential energy is 15% less than the initial kinetic plus potential energy of the

ball, so

O.85{Ki+U;

But Ui = Uf, SO Ki = O.15Uf/0.85, and then

=Uf.

E13-14 Focus on the potential energy. After the nth bounce the ball will have a potential energyat the top of the bounce of Un = O.9Un-l. Since U <X h, one can write hn = (0.9)nho. Solving for n,

n = log(hn/ho)/ log(O.9) = log(3 ft/6 ft)/ log(O.9) = 6.58,

which must be rounded up to

E13-15 Let m be the mass of the ball and M be the mass of the block.The kinetic energy of the ball just before colliding with the block is given by K1 = UO, so

Vl = V2(9.81 m/s2)(0.687m) = 3.67m/s.Momentum is conserved, so if V2 and V3 are velocities of the ball and block after the collision

then mVl = mV2 + MV3. Kinetic energy is not conserved, instead

Combine the energy and momentum expressions to eliminate V3 :

2mvl

Mv~

which can be formed into a quadratic. The solution for V2 is

2m ~ V2(M2 -mM) VI = (0.600 ~V2 = 2(M+m) 95) m/s.

The corresponding solutions for V3 are then found from the momentum expression to be V3 =0.981 m/s and V3 = 0.219. Since it is unlikely that the ba11 passed through the block we can tossout the second set of answers.

Ef = K f + U f = 3mgh, orE13-16

Vf = ..¡2(9.81 mis2)2(0.18 m) = 2.66 mis.

159

~ We can find the kinetic energy of the center of mass of the woman when her feet leavethe ground by considering energy conservation and her highest point. Then

1 2-m Vi + mgyi2

1-m Vi2

1 22mVf + mgyf ,=

= mgAy,

= {55.0 kg){9.81 m/s2){1.20m -O.90m) 162J.

(a) During the jumping phase her potential energy changed by

~u = mg~y = (55.0 kg)(9.81m/s2)(0.50m) = 270J

while she was moving up. Then

Fext

E13-18 (b) The ice skater needsto lose ~(116kg)(3.24mjs)2 = 609J of internal energy.(a) The average force exerted on the rail is F = (609 J)j(0.340m) = 1790 N.

E13-19 12.6 km/h is equal to 3.50 m/s; the initial kinetic energy of the car is

12(2340 kg)(3.50m/s)2 = l.43xlO4J.

(a) The force exerted on the car is F = (l.43x lO4J)!(0.64m) = 2.24x lO4N.(b) The change increase in interna! energy of the car is

~Eint (2.24 x 104N)(0.640 m -0.083 m) = 1.25 x 104 J.

E13-20 Note that v~ = v~ 2- 2"~ ."cm + Vcm2, Then

K =

L~mnn

2~ -I

L...mnVn

n

V cm+ v cm

Kint - ~ -IL,,¡ mn V n vcm+Kcm.

The middle term vanishes because of the definition of velocities relative to the center of mass.

160

~ Momentum conservation requires mvo = mv+ MV, where the sign indicates the direction.We are assuming one dimensional collisions. Energy conservation requires

1 2 1 2 1 22mvo = 2mv + 2MV + E.

Combining,

J. 2

-mvo2

Mv~

1 2 1 (m m )2-m v + -M -vo- -v +E2 2 M M ,

MV2 +m(VO -V)2 + 2(M/m)E.

=

=

Arrange this as a quadratic in v,

(M + m) V2 -(2mvo) v + (2(M/m)E + mv5 -Mv5) = o.

This will only have real solutions ir the discriminant ( b2 -4ac) is greater than or equal to zero. Then

(2mvo)2 ~ 4 (M + m) (2(M/m)E + mv5 -Mv5)

is the condition for the minimum Va. Solving the equality condition,

4m2v~ = 4(M + m) (2(M/m)E + (m -M)v~) ,

or M2v8 = 2(M+m)(M/m)E. One last rearrangement, and Va = v2(M+m)E/(mM).

~ (a) The initial kinetic energy will equal the potential energy at the highest point plus theamount oí energy which is dissipated because oí air drag.

mgh + fh =

h =

(b) The final kinetic energy when the stone lands will be equal to the initial kinetic energy minustwice the energy dissipated on the way up, so

1 2-m v2

-

V2

v

~

P13-2 The object starts with U = (0.234kg)(9.81 m/s2)(1.05 m) = 2.41 J. It will move back andforth across the flat portion (2.41J)/(0.688J) = 3.50 times, which means it will come to a rest atthe center of the flat part while attempting one last right to left journey.

161

P13-3 (a) The work done on the block block because of friction is

{0.210){2.41 kg){9.81 m/s2){1.83 m) = 9.09 J .

The energy dissipated because of friction is {9.09J)/0.83 = 11.0J.The speed of the block just after the bullet comes toa rest is

3.02 m/s.

(b) The initial speed of the bullet is

M+m,m

Vo =

P13-4 The energy stored In the spring after compression is ~(193N/m)(0.0416m)2 = 0.167 J.Since 117 mJ was dissipated by friction, the kinetic energy of the block before colliding with thespring was 0.284 J. The speed of the block was then

v = y'2(0.284 J)/(1.34 kg) = 0.651 m/s.

P13-5 (a) Using Newton's second law, F = ma, so for circular motion around the proton

mv2

r

e2= F = k-;:2'

The kinetic energy of the electron in an orbit is then

1 1 e2K =; -mv2 = -k-.

2 2 r

The change in kinetic energy is1 2 ( 1 1

)6.K = -ke ---2 r2 rl

(b) The potential energy difference is

1r2 ke2 ( 1 1)~u = --;¡-dr = -ke2 ---

rl r r2 rl

( c ) The total energy change is

'1 1---

.r2 rl

P13-6 (a) The initial energy of the system is (4000 lb)(12ft) = 48,000 ft .lb. The safety deviceremoves (1000 lb )(12ft) = 12,000 ft , lb before the elevator hits the spring, so the elevator has akinetic energy of 36, 000 ft .lb when it hits the spring. The speed of the elevator when it hits thespring is -

= .12(36,000 ft .lb)(32.0 ft/s1 = 240 ft/v V ( 4000 lb) .s.

(b) Assuming the safety clamp remains in effect while the elevator is in contact with the springthen the distance compressed will be found from

36,000 ft .lb = ~(10, 000 Ib/ft)y2- (4000 lb)y + (1000 lb)y.

162

This is a quadratic expression in y which can be simplified to look like

5y2 -3y -36 = O,

which has solutions y = {0.3 :f: 2.7) ft. Only y = 3.00 ft makes sense here.{ c ) The distance through which the elevator will bounce back up is found from

33,000 ft = (4000 lb)y -(1000 lb)y,

where y is measured from the most compressed point of the spring. Then y = 11 ft, or the elevatorbounces back up 8 feet.

(d) The elevator will bounce until it has traveled a total distance so that the safety devicedissipates all of the original energy, or 48 ft.

~ The net force on the top block while it is being pulled is

11.0N -F¡ = 11.0N -(0.35)(2.5kg)(9.81m/s2) = 2.42 N.

This means it is accelerating at (2.42N)/(2.5kg) = O.968m/s2. That acceleration willlast a timet = V2(0.30m)/(0.968m/s2) = O.787s. The speed of the top block after the force stops pulling isthen (0.968m/s2)(0.787s) = O.762m/s. The force on the bottom block is Ff, so the acceleration ofthe bottom block is

(0.35)(2.5kg)(9.81 m/s2)/(10.0kg) = O.858m/s2,

and the speed after the force stops pulling on the top block is (0.858m/s2)(0.787s) = O.675m/s.(a) W = Fs = (11.0N)(0.30m) = 3.3J of energy were delivered to the system, but after the

force stops pulling only

were present as kinetic energy. So 0.296 J is "missing" and would be now present as internal energy.(b) The impulse received by the two block system is then J = (11.0N)(0.787s) = 8.66N.s. This

impulse causes a change in momentum, so the speed of the two block system after the external forcestops pulling and both blocks move as one is (8.66N.g)(12.5kg) = 0.693m/s. The final kinetic energyis

1-(12.5 kg)(0.693m/s)2 = 3.0023;2

this means that 0.0023 are dissipated.

P13-8 Hmm.

163

El4-1

E14-2 Consider the force from the Sun and the force from the Earth. Fs/FE = MSrE2/MErs2,since everything else cancels out in the expression. We want the ratio to be one; we are alsoconstrained because rE + rS = R is the distance from the Sun to the Earth. Then

ME (R -rE)2 = MSrE2,

~R-rE = VM;~rE,

= 2.6 x lO8m.{1.50 x 1011m)/TE = 1 I (1.99 x 1030kg)

+y (5.98 x 10:¿4)

~ The masses of each object are ml = 20.0 kg and m2 = 7.0 kg; the distance between thecenters of the two objects is 15 + 3 = 18 m.

The magnitude of the force from Newton's law of gravitation is then

F = Gmlm2 -(6.67x 10-11N .m2/kg2)(20.0kg)(7.0kg) = 2.9x 10-11N..-

r2 18 m):&

~ The force of gravity on an object near the surface of the earth is given by

GMm .F- (re + y)2 ,

where M is the mass of the Earth, m is the mass of the object, re is the radius of the Earth, andy is the height above the surface of the Earth. Expand the expression since y « re. We'l1 use aTaylor expansion, where F(re + y) ~ F(re) + yaF/arej

GMm -GMm

Since we are interested in the difference between the force at the top and the bottom, we really want

~F = 2y ~ = 2JL~ = 2JLW3 2 ,

re re re re

where in the last part we substituted for the weight, which is the same as the force of gravity,

GMmw = -=--.

2re

Finally,L\F = 2(411 m)/(6.37x 106m)(120 lb) = 0.015 lb.

164

E14-6 9 cx: l/r2, so 91/92 = r~/r~ .Then

r2 = V(9.8lm/s2)/(7.35m/s2)(6.37x lO6m) = 7.36xlO6m.

That's 990 kilometers above the surface of the Earth.

(a) a = GM/r2, orE14-7

(6.67 x 10-11 N. m2 /kg2)(1.99 x 103Okg)l- (10.0 x 103m)2 = 1.33 x 1012m/s2.

(b) v = ..¡¡ax = v'2(1.33x 1012m/s2)(1.2m/s) = 1.79x 106m/s.

17. =

E14-8 (a) 90 = GM/r2, or

(6.67x lO-llN .m2 /kg2)(7.36x lO22kg) = l.62m/s2.go = (l.74x lO6m)2

(b) W m = We(9m/ge) so

w m = (100 N)(1.62 m/s2/9.81 m/s2) = 16.5 N.

(c) Invert 9 = GM/r2;

r = /GM1ii = (6.67x lO-llN .m2 /kg2)(5.98 x lO24kg)/(l.62m/s2) = l.57x lO7m.

That's 2.46 Earth radii, or 1.46 Earth radii above the surface of the Earth.

~ The object fell through y = -10.0mj the time required to fall would then be

t = v=2ii7ii = V-2(-10.0m)/(9.81m/82) = 1.438.

We are interested in the error, that means taking the total derivative of y = -!gt2, and getting

Ióy = -2ógt2 -gtót.

8y = O so -~8gt = g8t, which can be rearranged as

<5t = -~

29

The percentage error in t needs to be 8t/t = 0.1%/2 = 0.05%.(0.05%)(1.43s) = 0.7 ms.

The absolute error is then 8t =

E14-10 Treat mass which is inside a spherical shell as being located at the center of that shell.Ignore any contributions from shells farther away from the center than the point in question.

(a) F = G(M1 + M2)m/a2.(b) F = G(Ml)m/b2.

(c)F=O.

165

~ For a sphere oí uniíorm density and radius R > r ,

M(r)

1;;:33

M

-FR3'

where M is the total mass.The force of gravity on the object of mass m is then

GMm r3-;:2w

GMmrR3 .F= =

9 is the free-fall acceleration of the object, and is the gravitational force divided by the mass, so

GMr GM r GM R-D9 = ~ = W R = w -:n--.

Since R is the distance from the center to the surface, and D is the distance of the object beneaththe surface, then r = R -D is the distance from the center to the object. The first fraction is thefree-fall acceleration on the surface, so

R-D

R

l-QR

=98

E14-12 The work required to move the object is GMsm/r, where r is the gravitational radius.But ir this equals mc2 we can write

mc2 GMsm/r,

GMS/C2.

=

For the sun, r = (6.67 x 10-11 N. m2 /kg2)(1.99 x 103Okg)/(3.00 x 108m/s)2 ; 1.47 x 103m. That's2.1 x 10-6 Rs.

E14-13 The distance from the center is

= (80000) (3.00 x 108m/8)(3.16x 1018) = 7.6x 102om.

The mass of the galaxy is

M = (l.4x lOll)(l.99x lO30kg) = 2.8x lO41kg.

The escape velocity is

v = ,f'iGMF = y2(6.67X lO-llN .m2 /kg2)(2.8x lO41kg)/(7.6x lO20m) = 2.2x lO5m/s.

El4-14 Staying in a circular orbit requires the centripetal force be equal to the gravitational force,so

2 2mVorb Ir = GMmlr ,

or mVorb2 = GMmlr. But -GMmlr is the gravitational potential energy; to escape one requires a

kinetic energy

mvesc2/2 = GMm/r = mVorb2,

which has solution vesc = y'2Vorbo

166

~ (a) Near the surface of the Earth the total energy is

butCMRE2'g=

so the total energy is

GMEm

,

RE

GMEm,

RERE-

RE

This is a positive number, so the rocket will escape.(b) Far from earth there is no gravitational potential energy, so

GMEm

RE

GME1 2--mv-2

R 2 m RE = gmRE

E ,

=

with solution v = V29RE.

E14-16 The rotational acceleration of the sun is related to the galactic acceleration of free fall by

47r2mr/T2 = GNm2/r2,

where N is the number of "sun" sized stars of ma.ss m, r is the size of the galaxy, T is period ofrevolution of the sun. Then

47r2r3 47r2(2.2 x 1020m)3N = = = 5.1xl010.

GmT2 (6.67x 10-11N .m2 /kg2)(2.0x 1030kg)(7.9x 1015S)2

~ Energy conservation is Ki + Ui = Kf+ Uf, but at the highest point Kf = O, so

Uf

GMEm

R1

R

1

R

R

The distance above the Earth's surface is 2.19x 107 m -6.37x 106m = 1.55 x 106 m.

167

El4-18 (a) Free-fall acceleration is 9 = GM/r2. Escape speed is v = ,fiGMr;;. Then v = J2gr =

J2(1.30m/s2)(1.569xl06m) = 2.02xl03m/s.(b) Uf = Ki + Ui. But U/m = -gor~/r, so

12.09 x 106m .

v = 2(l.30m/s2)(l.569xlO6m)2[l/(l.569xlO6m) -l/(2.569xlO6m)] = l260m/s.

(d) M = gr2/G, or

M = (l.30m/s2)(l.569xlO6m)2/(6.67XlO-IIN.m2/kg2) = 4.8xlO22kg.

= 3.34xlO7m/s.

(b) Apply ~K = -~U. Then mv2 = Gm2(1/r2 -l/rl), SO

= 5.49 x 107 m/s.

El4-20 Call the particles 1 and 2. Then conservation of momentum requires the particle to havethe same momentum of the same magnitude, p = mVl = MV2. The momentum of the particles is

given by

Then Vrel = IVll + IV21 is equal to

= mMv2G/d(m+M)Vrel

= m M ..¡2G/d(m + M)

..¡2G(m + M)/d

The maximum speed is mv2 = Gm2 / d, or v = ,fGm1d.El4-21

E14-22 Tl/r~ = Ti/r~, O!

T2 = T1(r2/rl)3/2 = (1.00 y)(1.52)3/2 = 1.87 y.

168

That 's 523 km above the surface.(c) Kf = Uj -Ufo But U/m = -gor~/r, so

~ We can use Eq. 14-23 to find the mass of Mars; all we need to do is rearrange to solvefor M-

M = ~ = 47r2(9.4x106m)3 = 6.5x1023k .GT2 (6.67xl0-11N.m2/kg2)(2.75x104S)2 9

E14-24 Use GM/r2 = 47r2r/T2, so M = 47r2r3/GT2, and

M = 47r2(3.82xl08m)3 = 5.93xl024k .(6.67xlO-llN.m2/kg2)(27.3 X 864008)2 9

El4-25 Tl/rr = Ti /r~, or

T2 = T1(r2/rl)3/2 = (1.00 month)(1/2)3/2 = 0.354 month.

El4-26 Geosynchronous orbit W88 found in Sample Problem 14-8 to be 4.22 x 107 m. The latitudeis given by

(} = arccos(6.37x106m/4.22x107m) = 81.3°.

47r2

{6.67x lO-llN .m2 /kg2){7.36 x lO22kg)(l.74xlO6m)3 = 4.24xlO7s2.

So T = 6.5 x 1038.

(a) The 8peed of the 8atellite i8 the circumference divided by the period, or

v = .27rr ~ 27r(1.74x 106m) = 1.68x103m/8.--

T -(6.5xlO3s)

GMm~'

E14-28 The total energy is -GMm/2a. Then

1 2 GMm-m v --

2 r-

so

V2 = CM ( ~ -.!.

r a

E14-29 ra = a(l + e), so from Ex. 14-28,

Va=VGM(~-~);

rp = a(l -e), so from Ex. 14-28,

= 58.3 km/s.

Dividing one expression by the other,

-!2/(1-e)-1Vp -vay 2/(1 + e) -1

12/0.12- 1= {3.72 km/s)y 27i":"88""=""l

169

El4-30 (a) Convert.

2 33.156x 1078

y

AU

l.496xlOllm,

which is G = 39.49AU3/Ms2 .y2.(b) Here is a hint: 47r2 = 39.48. Kepler's law then looks like

MS2 .y2

AU3,

r3

M'

T2=

~ Kepler's third law states T2 cx: r3, where r is the mean distance from the Sun and T isthe period of revolution. Newton was in a position to find the acceleration of the Moon toward theEarth by assuming the Moon moved in a circular orbit, since ac = v2 / r = 47r2r /T2 .But this meansthat, because of Kepler's law, ac cx: r/T2 cx: 1/r2.

E14-32 ( a) The force of attraction between the two bodies is

F= GMm

(r+R)2"

The centripetal acceleration for the body of mass m is

CM

¡;;-:¡:Rj'i'

CM

T¡.;2

(¿}2 =r3(1 + R/r)2 ,

47r2 3 ( 2

GMr1+R/r).

(b) Note that r = Md/(m+M) and R = md/(m+M). Then R/r = m/M, so the correction is(1 + 5.94 x 1024/1.99 x 1030)2 = 1.000006 for the Earth/Sun 8y8tem and 1.025 for the Earth/Moon8y8tem.

T2

~

El4-33 (a) Use the results of Exercise 14-32. The center of mass is located a distance r =2md/(m + 2m) = 2d/3 from the star of mass m and a distance R = d/3 from the star of mass 2m.The period of revolution is then given by

2 ) 3

-d

3

47r2

G(2m)

47r2

-,

3Gm

T2 = d3,=

(b) Use Lm = mr2(.¡), then

mr2

M:R2

m(2d/3)2

(2m)(d/3)2

Lm

LM=2.= =

(c) Use K = IúJ2/2 = mr2úJ2/2. Then

mr2

MR2

m(2d/3)2

(211i)(d/3)2

KmKM

=2.- -

170

El4-34 Since we don 't know which direction the orbit will be, we will assume that the satelliteon the surface of the Earth starts with zero kinetic energy. Then Ei = Ui.

~U = Uf -Ui to get the satellite up to the specified altitude. ~K = Kf = -Uf/2. We wantto know if ~U -~K is positive (more energy to get it up) or negative (more energy to put it inorbit). Then we are interested in

9600 km, which is 3200 km above theThe "break-even" point is when rf = 3ri/2 = 3(6400 km)/2Earth.

(a) More energy to put it in orbit.(b) Same energy for both.(c) More energy to get it up.

~ (a) The approximate force of gravity on a 2000 kg pickup truck on Eros will be

(b) Use

-c- = 6.9m/s.v = ~ = 1.(6067 x 10-11N om2/kg2)(5.0 x 1015 kg)

V-;:- y (7000 m)

(a) u = -GMm/r. The variation is thenEl4-36

1 1{6.67x lO-llN .m2 /kg2){l.99 x lO30kg){5.98 x lO24kg)~u =

(1.47 x 1011m) {1.52 x 1011m)

78 x 1032 J.

(b) ~K + ~U = ~E = O, so I~KI = 1.78x 1032J.

(c) ~E = 0.

(d) Since ~l = 0 and l = mvr, we have

1-~ra

-2= 3.29x 10 Vp.Vp -Va = Vp =Vp

But Vp ~ vav = 271"(1.5 x 1011m)i(3.16 X 107 s) = 2.98 x 104mis. Then ~v = 981 mis.

El4-37 Draw a triangle. The angle made by Chicago, Earth center, satellite is 47.5° .The distancefrom Earth center to satellite is 4.22xl07 m. The distance from Earth center to Chicago is 6.37xl06m.Applying the cosine law we find the distance from Chicago to the satellite is

(4.22 x 107m)2 + (6.37x 106m)2 -2(4.22x 107m)(6.37x 106m) cos(47.5°) = 3.82x 107m.

Applying the sine law we find the angle rilade by Earth center, Chicago, satellite to be

= 126°.arcsin

That's 36° above the horizontal.

171

( {4.22X107m) .

81]

{3.82 x 107 m) 47.5°)

El4-38 (a) The new orbit is an ellipse with eccentricity given by r = a'(l + e). Then

e= r/a' -1 = (6.64x106m)/(6.52x106m) -1 = 0.0184.

The distance at P' is given by rp¡ = a'(l -e). The potential energy at P' is

Upl = Up~ = 2(-9.76x1010J)~ = -2.03x1011J.1 -e 1 -0.0184

The kinetic energy at P' is then

Kpl = (-9.94xlOlOJ) -(-2.03xlOllJ) = l.O4xlOllJ.

That would mean v = V2(l.O4x lOllJ)/(3250kg) = 8000m/s.

(b) The average speed is

v= . 7820 /(-- .-'I = m s.27r(6.52 x lO6m)

~ (a) The Starshine satellite was approximately 275 km above the surface of the Earth on1 January 2000. We can find the orbital period from Eq. 14-23,

.41T2 .T2 r3,

=CM

471"2

{6.67x lO-llN .m2 /kg2){5.98x lO24kg)(6.65 x 106m)3 = 2.91 x 10782,

so T = 5.39 x 1038.

(b) Equation 14-25 give8 the total energy of the 8y8tem of a 8atellite of mas8 m in a circular orbitof radiu8 r around a 8tationary body of mas8 M » m,

E=-GMm2r

We want the rate of change of this with respect to time, so

dE GMm drdt -2r2 dt

We can estimate the value of dr / dt from the diagram. I'll choose February 1 and December 1 as mytwo reference points.

~r'""'-='""' ~t

-(240 ~(300 km)~ -I km/day

dr

dt t=to (62 days)

The rate of energy 1088 i8 then

dE

dt~~)0-11N.m2/kg2)~24kg)(39kg) -1000m

= -2.0J/s.=2(6.65 x lO6m)2 8.64 x 104 s

~ The object on the top experiences a force down from gravity W1 and a force down fromthe tension in the rope T. The object on the bottom experiences a force down from gravity W2 anda force up from the tension in the rope.

In either case, the magnitude of Wi is

GMmJ.v:. = ~

~ r.

~

172

or

GMm GMmT =

.1 1 'I2 ,2-2

,rl r21GMm r~ -r~-;¡- r2 r2 .

1 2

Now rl ~ r2 ~ R in the denominator, but r2 = rl + I, so r~ -r~ ~ 2RI in the numerator. Then

T~ GMml

R3

Pl4-2 For a planet of uniform density, g = GM/r2 = G(47rpr3/3)/r2

doubled while r is halved we find that g will be unchanged.

= 47rGpr/3. Then ir p is

Pl4-3 (a) F = GMm/r2, a = F/m = GM/r.(b) The acceleration of the Earth toward the center of mass is aE = F/M = Gm/r2. The

relative acceleration is then GM/r + Gm/r = G(m + M)/r. Only if M » m can we assume that a

is independent of m relative to the Earth.

Pl4-4 (a) 9 = GM/r2, 8g = -(2GM/r3)8r. In this case 8r = h and M = 47rpr3/3. Then

8W = m 8g = 87rGpmh/3.

(b) 6.W/W = 6.g/g = 2h/r. Then an error of one part in a million will occur when h is one

part in two million of r, or 3.2 meters.

~ (a) The magnitude of the gravitational force from the Moon on a particle at A is

GMm

¡;:-=Rj2'FA=

173

To simplify assume R « r and then substitute ( r -R) ~ r. The force difference simplifies to

~ 2GMmR

r3FT = GMm~2(r)2 =

( d ) Repeat part ( c ) except we want r + R instead oí r -R. Then

FA-Fc =

To simplify assume R « r and then substitute ( r + R) ~ r. The force difference simplifies to

v = GM ~=-2GMmRrT m 2( )2 3r r r

The negative sign indicates that this "apparent" force points away from the moon, not toward it.(e) Consider the directions: the water is effectively attracted to the moon when closer, but

repelled when farther.

Pl4-6 F net = mrl.l.Js2, where I.l.Js is the rotational speed of the ship. But since the ship is movingrelative to the earth with a speed v, we can write I.l.Js = l.I.J::1::v/r, where the sign is positive ifthe shipis sailing east. Then F net = mr(1.I.J ::I:: v/r)2.

The scale measures a force W which is given by mg -F net , or

W = mg -mr(1.I.J ::I:: v/r)2.

Note that Wo = m(g -TIA)2). Then

w =

~

~

Pl4-7

we have

(a) a = GM/r2 -r(,¡J2. (,¡J is the rotational speed ofthe Earth. Since Frank observes a = 9/2

9/2 = GM/r2 -r(.c)2,

r2 = (2GM -2r3(.c)2)/9,

r = J2(GM~(.c)2)/9

Note thatCM = {6.67x lO-11N .m2 /kg2){5.98 x lO24kg) = 3.99x lO14m3/82

whiler3úJ2 = {6.37x lO6m)3{27r/864008)2 = l.37x lO12m3/82.

174

Consequently, r3(¿}2 can be treated as a perturbation of GM bear the Earth. Solving iteratively,

ro = 2[(3.99x 1014m3/s2) -(6.37x 106m)3(27r/86400s)2]/(9.81 m/s2) = 9.00x 106m,

rl = 2[(3.99xl014m3/s2) -(9.00xl06m)3(27r/86400s)2]/(9.81m/s2) = 8.98xl06m,

which is close enough for me. Then h = 8.98 x 106m -6.37 x 106m = 2610 km.

(b)~E=Ef-Ei=UfI2-Ui. Then

Pl4-8 (a) Equate centripetal force with the force of gravity.

47r2mr

-;¡;;¡-

47r2

~

GMm

T2'

G(4/3)7rr3pr3

rEVGP

=

--.

T =

(b) T = V37r/(6.7xlO-llN .m2/kg2)(3.0xl03kg/m3) = 68008.

Pl4-9 (a) One can find 8g by pretending the Earth is not there, but the material in the hole is.Concentrate on the vertical component of the resulting force of attraction. Then

GMd8g = -.-

r2 r'

where r is the straight line distance from the prospector to the center of the hole and M is the massof material that would fill the hole. A few substitutions later ,

8g = 47rGpR3d3 ( v'iJ'""+X' ) 3

or(0.800)(d2 + 2.25 x lO4m2) = d2,

which has solution d = 300 m. Then

R3 = 3(14.0x10-5m/s2)(300m)241T(6. 7x 10'-11N .m2 /kg2)(2800 kg/m3) ,

so R = 250 m. The top of the cave is then 300 m -250 m = 50 m beneath the surface.(b) All of the formulae stay the same except replace p with the difference between rock and water .

d doesn 't change, but R will now be given by

R3 = 3(14.0~10-5m/s2)(300m)241T(6.7 x 10-11 N. m2 /kg2)(1800 kg/m3) ,

so R = 292 m, and then the cave is located 300 m -292 m = 7 m beneath the surface.

175

Pl4-10 9 = GM/r2, where M is the mass enclosed in within the sphere of radius r.dg = (G/r2)dM -2(GM/r3)dr, so that 9 is locally constant if dM/dr = 2M/r. Expanding,

47rr2Pl = 87rr2p/3,

Pl = 2p/3.

Then

~ The force of gravity on the small sphere of mass m is equal to the force of gravity froma solid lead sphere minus the force which would have been contributed by the smaller lead spherewhich would have filled the hole. So we need to know about the size and mass of the lead which wasremoved to make the hole.

The density of the lead is given byM

p = 1;:R33

The hole has a radius of R/2, so if the density is constant the mass of the hole will be

( M ) 4 ( R ) 3 MMh = pV = FR3 37r 2 = 8

The "hole" is closer to the small sphere; the center of the hole is d -R/2 away. The force of thewhole lead sphere minus the force of the "hole" lead sphere is

GMm G(M/8)m

Pl4-12 (a) Use v = r.,;VR2 -r2, where r.,; = VGME/R3. Then

rT rO dr ro drT = Jo dt = JR d;:7dt = JR -;-'

rO dr= JRr.,;~'

7r

2iJ)

Knowing that

we can find T = 1260 s = 21 min.

(b) Same time, 21 minutes. To do a complete journey would require four times this, or 27r/(JJ.That's 84 minutes!

( c ) The answers are the same.

P14-13 (a) 9 = GM/r2 and M = 1.93x 1024kg + 4.01 x 1024kg + 3.94x 1022kg = 5.98x 1024kg so

9 = (6.67x 10-11N .m2 /kg2)(5.98x 1024kg)/(6.37x 106m)2 = 9.83m/s2.

(b) Now M = 1.93 x 1024kg + 4.01 x 1024kg = 5.94 x 1024kg so

9 = (6.67x 10-11N .m2 /kg2)(5.94x 1024kg)/(6.345x 106m)2 = 9.84m/s2.

(c) For a uniform body, 9 = 47rGpr/3 = GMr/R3, so

9 = (6.67x 10-11N .m2 /kg2)(5.98x 1024kg)(6.345x 106m)/(6.37x 106m)3 = 9.79m/s2.

176

P14-14 (a) Use g = GM/r2, then

9 = {6.67x 10-11N .m2 /kg2){1.93x 1024kg)/{3.490x 106m)2 = 106 m/s2.

The variation with depth is linear if core has uniform density.(b) In the mantle we have 9 = G ( M c + M) / r2 , where M is the amount of the mass of the mantle

which is enclosed in the sphere of radius r. The density of the core is

The density of the mantle is harder to find,

We can pretend that the core is made up of a point mass at the center and the rest has a densityequal to that of the mantle. That point mass would be

Then9 = GMpjr2 + 47rGpmrj3.

Find dg j dr , and set equal to zero. This happens when

2M pfr3 = 471"Pmf3,

or r = 4.93 x lO6m. Then g = 9.29 m/s2. Since this is less than the value at the end points it must

be a minimum.

~ (a) We will use part of the hint, but we will integrate instead of assuming the bit aboutgav; doing it this way will become important for later chapters. Consider a small horizontal slice ofthe column of thickness dr. The weight of the material above the slice exerts a force F(r) on thetop of the slice; there is a force of gravity on the slice given by

GM(r) dm

r2dF=

where M(r) is the mass contained in the sphere oí radius r,

4 3-7rr po3

M(r)

Lastly, the mass oí the slice dm is related to the thickness and cross sectional area by dm = pA dr .Then

dF=3

Integrate both sides of this expression. On the left the limits are O to F center, on the right the limitsare R to O; we need to throw in an extra negative sign because the force increases as r decreases.Then

~rdr

2F = -7rGAp2 R2,

3

Divide both sides by A to get the compressive stress.

177

(b) Put in the numbers!

( c) Rearrange, and then put in numbers;

R = 3(4.0xlO7N/m2) = l.8xlO5m.

27r(6.67x lO-llN .m2/kg2)(3000kg/m3)2

Pl4-16 The two mass increments each exert a vertical and a horizontal force on the particle, butthe horizontal components will cancel. The vertical component is proportional to the sine of theangle, so that

2Gm>..

y

Pl4-11 For any arbitrary point p the cross sectional area which is perpendicular to the axisdA' = r2dn is not equal to the projection dA onto the surface of the sphere. It depends on the anglethat the axis makes with the normal, according to dA' = cos (}dA. Fortunately, the angle made atpoint 1 is identical to the angle made at point 2, so we can write

dn1

dAl/r~

dQ2,

dA2/r~

But the mass of the shell contained in dA is proportional to dA, so

r~dml

Gmdml/r~

= r~dm2,= Gmdm2/r~.

Consequently, the force on an object at point p is balanced by both cones.(b) Evaluate f dn for the top and bottom halves of the sphere. Since every dn on the top is

balanced by one on the bottom, the net force is zero.

Pl4-18

Pl4-19 Assume that the small sphere is always between the two spheres. Then

w = AUl + AU2,

(6.67x 10-llN .m2 /kg2)(0.212kg) [(7.16kg) -(2.53kg)]1I

9.85 x 10-11 J.

178

Pl4-20 Note that ~mVesc2 = -UO. where Uo is the potential energy at the burn-out height.

Energy conservation gives

K Ko + Uo.1 2 1 22mvo -2mvesc ,

/~2 ~ 2V vD -Vesc-.

-

~

v =

~ (a) The force of one star on the other is given by F = Gm2/d2, where d is the distancebetween the stars. The stars revolve around the center of mass, which is halfway between the starsso r = d/2 is the radius of the orbit of the stars. If a is the centripetal acceleration of the stars, theperiod of revolution is then

T= ~= ~= ~.V -¡;- V -p V --a-;;;:-

The numerical value is

T = .1 167r2{1.12 x 1011m)3y {6.67x 10-11N .m2 /kg2){3.22x 1030kg) = 3.21 x 1078 = 1.02 y.

(b) The gravitational potential energy per kilogram midway between the stars is

-2~ = -2 (6.6~ x 10-~1 ~ .m2 /k~~)(~.22 x lO30kg)~~ -3.84 x 109 J /kg.

(1.12 x 1011m)

An object of mass M at the center between the stars would need (3.84 x 109 J /kg)M kinetic energyto escape, this corresponds to a speed of

v = ,fiKTM= V2(3.84xlOgJ/kg)= 8.76xlO4m/s.

Pl4-22 (a) Each differential mass segment on the ring contributes the same amount to the forceon the particle,

dF = ~::,r2 r

where r2 = x2 + R2. Since the differential mass segments are all equal distance, the integration istrivial, and the net force is

GMmxF = (X2 + R2)3/2 ,

(b) The potential energy can be found by integrating with respect to x,

~u = {00 Fdx = {00 ~Mmx d -GMm10 lo (.1:2 +R2)3/2 x- ~.

Then the particle of mass m will pass through the center of the ring with a speed v = .;;¡JS:ü7T;; =

V2GM7R.

179

~ (a) Consider the following diagram.

/ ,/ "/ "'

I

..

\

\ ,The distance r is given by the cosine law to be

r2 = R2 + R2 -2R2 cos()2R2(1- cos()).

The force between two particles is then F = Gm2/r2. Each particle has a symmetric partner, soonly the force component directed toward the center contributes. If we call this the R componentwe have

FR = Fcosa = Fcos(90° -0/2) Fsin(O/2).

Combining,

Fnet

When there are only 9 particles the angles are in steps of 40°j the (}i are then 40°, 80°, 120°, 160°,200°, 240°, 280°, and 320°. With a little patience you will find

= 6.649655,'\:""' ~ij2)

L..t 1 -COS ().

.~~

using these angles. Then F net = 3.32Gm2 / R2.(b) The rotational period of the ring would need to be

~v 3.32Gm .

Pl4-24 The potential energy ofthe system is U = -Gm2 Ir. The kinetic energy is mv2. The totalenergy is E = -Gm2 Id. Then

dr

dt= 2VGm(1/r -l/d),

180

so the time to come together is

[d3dx= ¡vam'

{o dr ~{1 ~T = Jd 2vGm(1/r- l/d) = Y 4Gm Jo Y l=-xl

Pl4-25 (a) E = U/2 for each satellite, so the total mechanical energy is -GMm/r.(b) Now there is no K, so the total mechanical energy is simply U = -2GMm/r. The factor of

2 is because there are two satellites.( c ) The wreckage falls vertically onto the Earth.

Pl4-26 Let ra = a(l + e) and rp = a(l- e). Then ra + re = 2a and ra -rp = 2ae. So the answeris

2(0.0167)(1.50 x 1011m) = 5.01 x 109m,

or 7.20 solar radii.

P14-27

Pl4-28 The net force on an orbiting star is

M-;:2 + m4r2F=Gm

This is the centripetal force, and is equal to 47r2mr/T2. Combining,

47r2

T2

G47=3(4M + m),=

so T = 47rVr3/[G(4M + m)].

P14-29 (a) v = ,¡a¡;¡¡r, so

v = (6.67x lO-llN .m2 /kg2)(5.98x lO24kg)/(7.0l x lO6m) = 7.54x lO3m/s.

(b) T = 27r(7.0lxlO6m)/(7.54xlO3m/s) = 5.84xlO3s.(c) Originally Ea = U /2, or

(6.67x 10-11N .m2 /kg2)(5.98x 1024kg)(220kg) = -6.25x 109J.E = -2(7.01 x 106m)

After 1500 orbits the energy is now -6.25 x 109 J -(1500)(1.40 x 105 J) = -6.46 x 109 J. The new

distance from the Earth is then

(6.67x ¡0-llN .m2 /kg2)(5.98 x ¡024kg)(220kg) = 6. 79x ¡O6m.r = -2( -6.46 x ¡ogJ)

The altitude is now 6790 -6370 = 420 km.(d) F = (1.40x105J)/(27r7.01x106m) = 3.2x10-3N.

(e) No.

181

P14-30 Let the satel1ite S be directly overhead at some time. The magnitudE) of the speed isequal to that of a geosynchronous satel1ite T whose orbit is not inclined, but since there are bothparal1el and perpendicular components to the motion of S it wil1 appear to move north while "losingground" compared to T. Eventually, though, it must pass overhead again in 12 hours. When S is asfar north as it wil1 go (6 hours) it has a velocity which is paral1el to T, but it is located in a regionwhere the required speed to appear fixed is slower. Hence, it wil1 appear to be "gaining ground"against the background stars. Consequently, the motion against the background stars appears to bea figure 8.

P14-31 The net force of gravity on one star because of the other two is

2GM2F= L2

The stars orbit about a point r = L/2cos{300) from any star. The orbita! speed is then found from

cos(300)

M 2 Mv2 2GM2(30 0 )v -cos ,

-.-

L2 r [, /2 cos( 30° )

or v = ,fGM7L.

Pl4-32 A parabolic path wi11 eventually escape; this means that the speed of the comet at anydistance is the escape speed for that distance, or v = ,f2GMF. The angular momentum is constant,and is equal to ~-

l = m VArA = m.'¡2GMrA.

For a parabolic path, r = 2rA/(1 + cosO). Combining with Eq. 14-21 and the equation before thatone we get

dO -~(1 + COSO)2.

"dt -4rA2

The time required is the integral

~ f7r/2 dO ~ (2)T= VGM lo (1+cosO)2 =VGM 3

Note that vrA3jGM is equal to lj27r years. Then thetime for the comet to move is

~ There are three forces on loose matter(of mass mo) sitting on the moon: the force ofgravity toward the moon, Fm = Gmmo/a2, the force of gravity toward the planet, FM = GMmo/(r-a) 2, and the normal force N of the moon pushing the loose matter away from the center of the moon.

The net force on this loose matter is FM + N -Fm, this value is exactlyequal to the centripetalforce necessary to keep the loose matter moving in a uniform circle. The period of revolution of theloose matter is identical to that of the moon,

T = 2 7r ,¡;:¡;JG¡;¡ ,

but since the loose matter is actually revolving at a radial distance r -a the centripetal force is

182

Only if the normal force is zero can the loose matter can lift off, and this will happen when p c =PM -Pm, or

M(r -a)r3

-M m

¡;:--::--aj' -~ ,

Ma2 -m(r -a)2-

Ma2(r -a)3

-3r2a3 + 3ra4 -a4

a2(r -a)2

Mr3a2 -mr3(r -a)2,

~ ( -r5 + 2r4a -r3a2)

==

Let r = ax, then x is dimensionless; let fJ = m/M, thenfJ is dimensionless. The expression then

simplifies to-3x2 + 3x -1 = {3( -X5 + 2x4 -X3).

If we assume than x is very large ( r » a) then only the largest term on each side survives. This means3x2 ~ {3x5, or X = (3/{3)1/3. In that case, r = a(3M/m)1/3. For the Earth's moon rc = 1.lx107m,which is only 4,500 km away from the surface of the Earth. It is somewhat interesting to note thatthe radius r is actually independent of both a and m if the moon has a uniform density!

183

~ The pressure in the syringe is

( 42.3 N) = 4.29 x lO5Pa.p=7r(l.l2 x lO-2m/ 8 )2

E15-2 The total mass oí Huid is

m = (0.5x10-3m3)(2600 kg/m3)+(0.25x10-3m3)(1000 kg/m3)+(0.4x10-3m3)(800 kg/m3) = 1.87 kg.

The weight is (18.7kg)(9.8m/s2) = 18N.

F = AAp, soE15-3

F = {3.43 m){2.08 m){l.OO atm- 0.962 atm){1.01 x 105pa/ atm) = 2.74 x 104N.

E15-4 B~V/V = -~p; V = L3; ~V ~ L2~L/3. Then

= 2.74x lO9Pa.~p = (140 x 109Pa) ~

3(0.85 m)

~ There is an inward force F 1 pushing the lid closed from the pressure of the air outside theboxj there is an outward force F2 pushing the lid open from the pressure of the air inside the box.To lift the lid we need to exert an additional outward force F3 to get a net force of zero.

The magnitude of the inward force is F1 = p outA, where A is the area of the lid and p out isthe pressure outside the box. The magnitude of the outward force F2 is F2 = PinA. We are toldF3 = 108 lb. Combining,

F2

PinA

Pin

===

Fl -F3,

PautA -F3,

Paut -F3/A,

so Pin = (15 lb/in2 (108 lb)/(12 in2) = 6.0 lb/in2.

E15-6 h = Ap/ pg, so

(0.05 atm)(1.01 x 105P~~)h= = 0.52 m.

(1000 kg/m3)([8lli/s2)

90 x 104Pa.~p = (1060kg/m3)(9.81 m/s2)(1.83m) =E15-7

19 x 106Pa. Add this to Poi the total pressure isE15-8 6.p = (1024kg/m3)(9.81m/s2)(118m) =

then 1.29 x 106Pa.

~ The pressure differential assuming we don 't have a sewage pump:

P2 -Pl = -pg(y2 -Yl) = (926kg/m3)(9.8lm/s2)(8.l6m -2.08 m) = 5.52xlO4Pa.

We need to overcome this pressure difference with the pump.

E15-10 {a) p = {1.00 atm)e-5.00/8.55 = 0.557 atm.{b) h = {8.55 km) ln{1.00/0.500) = 5.93 km.

184

E15-11 The mercury will rise a distance a on one side and fall a distance a on the other so thatthe difference in mercury height will be 2a. Since the masses of the "excess" mercury and the waterwill be proportional, we have 2apm = dpw, so

E15-12 (a) The pressure (due to the water alone) at the bottom oí the pool is

p = (62.45 lb/ft3)(8.0 ft) = 500 lb/ft2.

The force on the bottom is

F = (500 Ib/ft2)(80 ft)(30 ft) = 1.2 x 106 lb.

The average pressure on the sirle is half the pressure on the bottom, so

F = (250 lb/ft2)(80 ft)(8.0 ft) = 1.6 x 105 lb.

The average pressure on the end is half the pressure on the bottom, so

F = (250 lb/ft2)(30 ft)(8.0 ft) = 6.0x 104 lb.

(b) No, since that additional pressure acts on both sides.

~ (a) Equation 15-8 can be used to find the height y2 of the atmosphere if the density isconstant. The pressure at the top of the atmosphere would be P2 = O, and the height of the bottomYl would be zero. Then

y2 = {1.01 x 105pa)/ [{1.21 kg/m3){9.81 m/s2)] = 8.51 x 103m.

{b) We have to go back to Eq. 15- 7 for an atmosphere which has a density which varies linearlywith altitude. Linear variation of density means

p=poY'

1--

Ymax )

Substitute this into Eq. 15-7,

P2 -Pl =

dy,

In this case we have Ymax = 2pl / (pg ) , SO the answer is twice that in part ( a) , or 17 km.

E15-14 ~p = (1000kg/m3)(9.8m/s2)(112m) = 1.1 x 106Pa. The force required is then F(1.1 x 106Pa)(1.22m)(0.590m) = 7.9x 105N.

185

E15-15 (a) Choose any infinitesimally small spherical region where equal volumes ofthe two fluidsare in contact. The denser fluid will have the larger mass. We can treat the system as being a sphereof uniform mass with a hemisphere of additional mass being superimposed in the region of higherdensity. The orientation of this hemisphere is the only variable when calculating the potential energy.The center of mass of this hemisphere will be a low as possible only when the surface is horizontal.So all two-fluid interfaces will be horizontal.

(b) If there exists a region where the interface is not horizontal then there will be two differentvalues for ~p = pgh, depending on the path taken. This means that there will be a horizontalpressure gradient, and the fluid will flow along that gradient until the horizontal pressure gradientis equalized.

E15-16 The mags of liquid originally in the first vessel is ml = pAh1; the center of gravity isat hl/2, so the potential energy of the liquid in the first vessel is originally U1 = pgAh~/2. Asimilar expression exists for the liquid in the second vessel. Since the two vessels have the samecross sectional area the final height in both containers will be hf = (h1 + h2)/2. The final potentialenergy of the liquid in each container will be U f = pgA(h1 + h2)2/8. The work done by gravity isthen

U1 + U2 -2Uf,

pg~ [2h~ + 2h~ -(h~ + 2h1h2 + h~)]

4

~(hl -h2)2,4

~ There are three force on the block: gravity (W = mg), a buoyant force Bo = mwg, and atension To. When the container is at rest all three forces balance, so Bo -W -To = o. The tensionin this case is To = (mw -m)g.

When the container accelerates upward we now have B -W -T = ma. Note that neither thetension northe buoyant force stay the same; the buoyant force increases according to B = mw(g+a).The new tension is then

T = mw(g + a) -mg -ma = (mw -m)(g + a) = To(l+ a/g).

E15-18 (a) F1/d'f= F2/d~, so

F2 = (18.6 kN)(3.72 cm)2/(51.3 cm)2 = 97.8 N.

(b) F2h2 = F1h1, so

h2 = (1.65m)(18.6 kN)/(97.8N) = 314m.

E15-19 (a) 35.6 kN; the boat doesn't get heavier or lighter just because it is in different water!(b) Yes.

= -8.51 x 10-2m3.

E15-20 (a) P2 = Pl(V1/V2) = (1000kg/m3)(0.646) = 646kg/m3.(b) P2 = Pl(V1/V2) = (646 kg/m3)(0.918)-1 = 704kg/í:n3.

186

~ The can has a volume of 1200 cm3, so it can displace that much water. This wouldprovide a buoyant force of

B = pVg = (998kg/m3)(1200 x 10-6m3)(9.81 m/s2) = 11.7N.

This force can then support a total mass of (11.7N)/(9.81m/s2) = 1.20kg. If 130 9 belong to thecan, then the can will be able to carry 1.07 kg of lead.

3E15-22 P2 = Pl{V1/V2) = {0.98 g/cm3){2/3)-1 = 1.47 g/cm

E15-23 Let the object have a mass m. The buoyant force of the air on the object is then

Bo = ~mg.Po

There iB alBo a buoyant force on the brasB, equal to

B Pab = -mg.

Pb

The fractional error in the weighing is then

Bo -Bb

mg-(0.00122/cm3) -(0.0012 g/C~3) = 2.0x 10-4

-(3.4 g/cm3) (8.0 g/cm )

E15-24 The volume oí iron is

Vi = (6l3ÜN)/(9.8lm/s2)(787ükg/m3) = 7.94xlü-2m3.

The buoyant force of water is 6130 N -3970 N = 2160 N. This corresponds to a volume of

v w = (2160 N)/(9.81 m/s2)(1000 kg/m3) = 2.20 x 10-1m3.

The volume of air is then 2.20 x 10-1m3 -7.94 x 10-2m3 = 1.41 x 10-1m3.

E15-25 (a) The pressure on the top surface is p = Po + pgL/2. The downward force is

Ft = (Po + pgL/2)L2,

[(1.01 x 105Pa) + (944 kg/m3)(9.81 m/s2)(0.608m)/2] (0.608m)2 = 3.84x 104N,

(b) The pressure on the bottom surface is p = Po + 3pgL/2. The upward force is

Fb = (po + 3pgL/2)L2,

[(1.01 x 105Pa) + 3(944kg/m3)(9.81 m/s2)(0.608m)/2] (0.608m)2 = 4.05x 104N.

(c) The tension in the wire is given by T= W+ Ft -Fb, or

T = { 4450 N) + {3.84 x 104N) -{ 4.05 x 104N) = 2350 N .

(d) The buoyant force is

B = L3 pg = (0.6083)(944 kg/m3)(9.81 m/s2) = 2080 N .

187

E15-26 The fish has the ( average) density of water if

mf

Pw=Vc+Va

ormf

Pw

-VC'va=

We want the fraction V a/(V c + Va), so

Va 1 Vc

-Pw~

~

Vc+Va mf

1- Pw/ Pc = 1- (1.024 g/cm3)/(1.08 g/cm3) = 5.19 x 10-2.

~ There are three force on the dirigible: gravity (W = mgg), a buoyant force B = mag,and a tension T. Since these forces must balance we have T = B -w. The masses are related tothe densities, so we can write

T = (Pa -Pg)Vg = (l.2lkg/m3 -O.796kg/m3)(l.l7xlO6m3)(9.8lm/s2) = 4.75xlO6N.

E15-28 ~m = ~pV, so

~m = [(0.160kg/m3) -(0.0810 kg/m3))(5000m3) = 395kg.

E15-29 The volume of one log is 7r(1.05/2 ft)2(5.80 ft) = 5.02 ft3. The weight of the log is(47.3 lb/ft3)(5.02 ft3) = 237 lb. Each log if completely submerged will displace a weight of wa-ter (62.4 lb/ft3)(5.02 ft3) = 313 lb. So each log can support at most 313 lb- 237 lb = 76 lb. Thethree children have a total weight of 247 lb, so that will require 3.25 logs. Round up to four.

E15-30 (a) The ice will hold up the automobile if

ma+mi

Vi

maAt +Pi.Pw >

~

Then

~ If there were no water vapor pressure above the barometer then the height of the waterwould be yl = p/(pg), where p = Po is the atmospheric pressure. If there is water vapor where thereshould be a vacuum, then p is the difference, and we would have y2 = (Po -Pv ) / (pg ) .The relativeerror is

(Yl -Y2)/Yl= [Po/(pg) -(Po -Pv )/(pg)] / [Po/(pg)] ,

pv/po = (3l69Pa)/(l.OlxlO5Pa) = 3.14%.

p = (1.01 x 105pa)/(9.81 m/s2)(14m) = 740kg/m3.E15-32

E15-33 h = (90)(l.Ol x lO5pa)/(8.60 m/s2)(l.36 x lO4kg/m3) = 78 m.

E15-34 ~U = 2(4.5x lO-2N/m)47r(2.l x lO-2m)2 = 5.0x lO-4J.

188

~ The force required is just the surface tension times the circumference of the circularpatch. Then

F = (0./072N/m)27r(O.l2m) = 5.43x lO-2N.

ilU = 2(2.5xlÜ-2N/m)47r(l.4xlÜ-2m)2 = l.23xlÜ-4J.E15-36

P15-1 (a) One can replace the two hemispheres with an open flat end with two hemispheres witha closed flat end. Then the area of relevance is the area of the flat end, or 7r R2 .The net force fromthe pressure difference is 6.pA = 6.p7rR2; this much force must be applied to pull the hemispheres

apart.(b) F = 7r(O.9)(1.01 x 105Pa)(0.305m)2 = 2.6x 104N.

~ ( a) The resultant force on the wall will be

F = !!Pdxdy,

= 11 -pgy)Wdy,

= pgD2W/2.

(b) The torque will is given by T = F(D-y) (the distance is from the bottom) so ifwe generalize,

T = 11pydxdy,

= !(-pg(D-y))yWdy,

= pgD3W/6.

( c ) Dividing to find the location of the equivalent resultant force,

d = T/F = (pgD3W/6)/(pgD2W/2) = D/3,

this distance being measured from the bottom.

P15-4 p = pgy = pg(3.6 m); the force on the bottom is F = pA = pg(3.6 m)7r(O.60 m)2 = 1.2967rpg.

The volume of liquid is

v = (1.8m) [7r(0.60m) +4.6x10-4m2] = 2.037m3

The weight is W = pg(2.037m3). The ratio is 2.000.

P15-5 The pressure at b is pc(3.2xl04m)+pmY. The pressure at a is pc(3.8xl04m+d)+pm(Y-d).Set these quantities equal to each other:

Pc(3.8 x 104m + d) + Pm(Y -d) = Pc(3.2 x 104m) + PmY,

Pc(6 x 103m + d) = Pmd,

d = Pc(6x 103m)/(pm -Pc),

(2900kg/m3)(6x 103m)/(400kg/m3) = 4.35 x 104m.

189

P15-6 (a) The pressure (difference) at a depth y is Ap = pgy. Since {1= m/V, then

m A V ApAp ~ -Vv = PsB.

Then

(b) ~p/ Ps Psgy/B, so

~p/ p ~ (1000 kg/m3)(9.8m/s2)(4200m)/(2.2x 10gpa) = 1.9%.

~ (a) Use Eq. 15-10, p = (Poi Po)p, then Eq. 15-13 willlook like

(Poi Po)p = (POI po)POe-h/a.

(b) The upward velocity of the rocket as a function of time is given by v = art. The height ofthe rocket above the ground is given by y = !art2. Combining,

Put this into the expression for drag, along with the equation for density variation with altitude;

D = CApv2 = CApoe-Y/a2yaro

Now take the derivative with respect to y,

dD/dy = ( -1/a)CApoe-Y/a(2yar) + CApoe-Y/a(2ar).

This will vanish when y = a, regardless of the acceleration aro

P15-8 (a) Consider a slice of cross section A a depth h beneath the surface. The net force on theHuid above the slice will be

Fnet = ma = phAg,

Since the weight of the Huid above the slice is

W=mg =phAg,

then the upward force on the bottom of the Huid at the slice must be

w + Fnet = phA(g + a),

so the pressure is p = F/A = ph(g + a).

(b) Change a to -a.(e) The pressure is zero (ignores atmospherie eontributions.

P15-9 (a) Consider a portion of the liquid at the surface. The net force on this portion is F =ma = mai. The force of gravity on-. this yort~on is W A= ~mgj. There must then be a buoyantforce on the portion with direction B = F -W = m(ai + gj). The buoyant force makes an angle(} = arctan(a/g) with the vertical. The buoyant force must be perpendicular to the surface of thefluid; there are no pressure-related forces which are parallel to the surface. Consequently, the surfacemust make an angle (} = arctan(a/g) with the horizontal.

(b) It will still vary as pghj the derivation on page 334 is still valid for vertical displacements.

190

P15-10 dp/dr = -pg, but now g = 47rGpr/3. Then

fP 4 frlo dp = -37rGp2 lR rdr,

2p = 37rGp2 (R2 -r2)

~ We can start with Eq. 15-11, except that we'll write our distance interms of r insteadif y. Into this we can substitute our expression for g,

R29 = go2.

r

Substituting, then integrating,

1~

In

~drPo ,

goPoR2 dr

~

=

-

~

If k = 9OPoR2/PO, thenp = poek(l/r-l/R)

P15-12 (a) The net force on a small volrime of the fluid is dF = r(Ll2 dm directed toward thecenter. For radial displacements, then, dF/dr = ~r(Ll2dm/dr or dp/dr = -r(Ll2p.

(b) Integrating outward,

r 1p = Pc + lo fJ'I.J2rdr = Pc + 2pr2(A.!2.

(c) Do part (d) first.( d) It will still vary as pgh; the derivation on page 334 is still valid for vertical displacements.(c) The pressure anywhere in the liquid is then given by

1p = Po + 2pr2iJ)2 -pgy,

where Po is the pressure on the surface, y is measured from the bottom of the paraboloid, and r ismeasured from the center. The surface is defined by p = Po, so

1-pr2iJ)2 -pgy = O,2

or y = r2úJ2/29.

P15-13 The total m88S of the shell is m = Pw7rdo3/3, orit wouldn't barely float. The m88S of ironin the shell is m = pj7r(do3 -dj3)/3, so

d .3 -Pi -Pw d 3l -~ O ,

Pí

so

3 (7870kg/m3)- (1000kg/m3){0.587m) = 0.561 m.di = (7870kg/m3)

191

dp

p

dp

p

dp

p

p

Po

P15-14 The wood will displace a volume ofwater equal to (3.67kg)/(594kg/m3)(0.883) = 5.45x10-3m3 in either case. That corresponds to a mass of (1000 kg/m3)(5.45 x 10-3m3) = 5.45 kg thatcan be supported.

(a) The mass of lead is 5.45kg -3.67kg = 1.78kg.(b) When the lead is submerged beneath the water it displaces water, which affects the "apparent"

mass of the lead. The true weight of the lead is mg, the buoyant force is (Pw / Pl )mg , so the apparentweight is (1- pw/Pl)mg. This means the apparent mass of the submerged lead is (1- pw/Pl)m.This apparent mass is 1.78 kg, so the true mass is

(11400 kg/m3)m = (11400kg/m3) -(¡¡j¡j¡¡kg)(1.78kg) = 1.95kg.

~ We initially have

! -Po4 -Pmercury

When water is poured over the object the simple relation no longer works.Once the water is over the object there are two buoyant forces: one from mercury, F1, and one

from the water, F2. Following a derivation which is similar to Sample Problem 15-3, we have

F1 = Pl V1g and F2 = P2 V2g

where Pl is the density of mercury, V1 the volume of the object which is in the mercury, P2 is thedensity of water, and V2 is the volume of the object which is in the water. We also have

F1 + F2 = PoVog and V1 + V2 = Vo

as expressions for the net force on the object (zero) and the total volume of the object. Combiningthese four expressions,

Pl Vl + P2V2 = PoV o,

or

PI VI + P2 (Vo -VI)

(PI -P2) VI

VI

=PoVo,

(Po-p2)V(

Po-P2

=

=Vo Pl -P2

The left hand side is the fraction that is submerged in the mercury, so we just need to substituteour result for the density of the material from the beginning to solve the problem. The fractionsubmerged after adding water is then

V1 Po -P2=Vo Pl -P2

Pl/4- P2

P15-16 (a) The car floats if it displaces a mass of water equal to the mass of the car. ThenV = (1820kg)/(1000kg/m3) = 1.82m3.

(b) The car has a total volume of 4.87m3+0.750m3+0.810m3 = 6.43m3. It will sink ifthe totalmass inside the car (car + water) is then (6.43m3)(1000kg/m3) = 6430kg. So the mass ofthe waterin the car is 6430kg-1820kg = 4610kg when it sinks. That's a volume of (4610kg)/(1000kg/m3) =4.16m3.

192

P15-17 When the beaker is halffilled with water it has a total mass exactly equal to the maximumamount of water it can displace. The total mass of the beaker is the mass of the beaker plus themass of the water inside the beaker. Then

pw(mg/pg + Vb) = mg + PwVb/2,

where mg/ Pg is the volume oí the glass which makes up the beaker. Rearrange,

(0.390 kg)= 2790kg/m3.

mg -Pg = mg/pw -Vb/2 -

(0.390 kg)/(1000 kg/m3) -(5.00 x 10-4 m3)/2

P15-18 (a) If each atoro is a cube then the cube has a side of length

l = {/(6.64x 10-27 kg)/(145kg/m3) = 3.58x 10-1Om.

Then the atomic surface density is l-2 = (3.58x 10-10m)-2 = 7.8x 1018/m2.(b) The bond surface density is twice the atomic surface density. Show this by drawing a square

array of atoms and then joining each adjacent pair with a bond. You will need twice as many bondsas there are atoms. Then the energy per bond is

(3.5 x lO-4N/m)= 1.4x 10-4 eV.

2(7.8 x 1018 /m2)(1.6 x 10-19 J/eV)

~ Pretend the bubble consists of two hemispheres. The force from surface tension holdingthe hemispheres together is F = 2'YL = 47rr'Y. The "extra" factor of two occurs because eachhemisphere has a circumference which "touches" the boundary that is held together by the surfacetension of the liquid. The pressure difference between the inside and outside is ~p = F / A, where Ais the area of the flat side of one of the hemispheres, so ~p = (47rr'Y)/(7rr2) = 4'Y /r .

P15-20 Use the results of Problem 15-19. To get a numerical answer you need to know the surfacetension; try 'Y = 2.5x 10-2N/m. The initial pressure inside the bubble is Pi = Po + 4'Y/ri. The finalpressure inside the bell jar is P = Pf -4'Y / rf .The initial and final pressure inside the bubble arerelated by piri3 = Pfrf3. Now for numbers:

Pi = {1.00x105Pa) +4{2.5x10-2N/m)/{1.0x10-3m) = 1.001 x 105Pa.

Pf = {1.0 x 10-3m/1.0 x 10-2m)3{1.001 x 105Pa) = 1.001 x 102Pa.

p = {1.001 x 102Pa) -4{2.5x 10-2N/m)/{1.0x 10-2m) = 90.1Pa.

P15-21 The force on the liquid in the space between the rod and the cylinder is F = 'YL =27r'Y(R + r). This force can support a mass of water m = F/g. This mass has a volume V = m/p.The cross sectional area is 7r(R2 -r2), so the height h to which the water rises is

h =27r"((R + r) 2"(

=pg7r(R2 -r2) pg(R -r) ,

2(72.8x lO-3N/m) = 3.71 x 10-3m.(1000 kg/m3)(9.81 m/s2)( 4.0 x 10-3m)

193

P15-22 (a) Refer to Problem 15-19. The initial pressure difference is

4(2.6x lO-2N/m)/(3.20x lO-2m) = 3.25Pa.

(b) The final pressure difference is

4(2.6x lO-2N/m)/(5.80x lO-2m) = l.79Pa.

(c) The work done against the atmosphere is p~V, or

(d) The work done in stretching the bubble surface is 'YAA, or

(2.60x 10-2N/m)47r[(5.80x 10-2m)2 -(3.20 x 10-2m)2] = 7.65x 10-43.

194

E16-1 R = Av = 7r~v/4 and V = Rt, so

t = 4(1600 m3)7r(0.345m)2(2.62m) = 65308

E16-2Then

Alvl = A2v2, Al = 7rd~/4 for the hose, and A2 = N7r4 for the sprinkler, where N = 24.

(0.75 in)2V2 =

~ We'll assume that each river has a rectangular cross section, despite what the pictureimplies. The cross section area of the two streams is then

Al = (8.2m)(3.4m) = 28m2 and A2 = (6.8m)(3.2m) = 22m2.

The volume flow rate in the first stream is

Rl = A1vl = (28m2)(2.3m/s) = 64m3/s,

while the volume flow rate in the second stream is

R2 = A2v2 = (22m2)(2.6m/s)= 57m3/s.

The amount of fluid in the stream/river system is conserved, so

R3 = Rl + R2 = (64m3/s) + (57m3/s) = 121 m3/s.

where R3 is the volume flow rate in the river. Then

D3 = R3/(V3W3) = (121 m3/s)/[(10.7m)(2.9m/s)] =3.9m.

E16-4 The speed of the water is originally zero so both the kinetic and potential energy is zero.When it leaves the pipe at the top it has a kinetic energy of ~{5.30m/s)2 = 14.0J/kg and apotential energy of {9.81m/s2){2.90m) = 28.4J/kg. The water is flowing out at a volume rate ofR = {5.30 m/s)7r{9. 70 x 10-3m)2 = 1.57 x 10-3 m3/s. The mass rate is pR = {1000 kg/m3){1.57 x10-3m3/s) = 1.57kg/s.

The power supplied by the pump is {42.8J/kg){1.57kg/s) = 67.2W.

~ There are 8500 km2 which collects an average oí (0.75) (0.48 m/y), where the 0.75 reflectsthe íact that 1/4 oí the water evaporates, so

R = (8500(103 m)2] (0. 75) (0.48 m/y)

Then the speed of the water in the river is

v = R/A = (97m3 /s)/[(21 m)(4.3m)] = 1.1 m 18.

E16-7 (a) ~p = pg(yl -Y2) + p(V~ -v~)/2. Then

~p = (62.4 lb/ft3)(572 ft) + [(62.4 lb/ft3)/(32 ft/s2)][(1.33 ftjS)2- (31.0 ft/s)2]/2 = 3.48x1041b/ft2.

(b) A2v2 = A1vl, SO

A2 = (7.60 ft2)(1.33 ft/s)/(31.0 ft/s) = 0.326 ft2,

195

E16-8 (a) A2v2 = A1vl, so

V2 = (2.76m/s)[(0.255m)2 -(O.O480m)2j/(0.255m)2 = 2.66m/s.

(b) dp = p(v~ -v~)/2,

~p = (1000 kg/m3)[(2.66m}s)2 -(2.76m/s)2J/2 = -271 Pa

~ (b) We will do part (b) first.

1 yR = (100 m2)(1.6 m/y) = 5.1 x 10-6 m3/s.

365 x 24 x 60 x 608

(b) The speed ofthe flow R through a hole of cross sectional area a will be v = R/a. p = Po+pgh,where h = 2.0 m is the depth of the hole. Bernoulli's equation can be applied to find the speed of

the water as it travels a horizontal stream line out the hole,

1 2

po+2PV =P,

where we drop any terms which are either zero or the same on both sides. Then

v = ..¡2(p -Po)/p = J29h = ..¡2(9.81 m/s2)(2.0m) = 6.3m/s.

Finally, a = (5.1x10-6m3/s)/(6.3m/s) = 8.1 X 10-7m2, or about 0.81 mm2.

E16-10 {a) V2 = {Al/A2)Vl = {4.20 cm2){5.18m/s)/{7.60 cm2) = 2.86m/s.

{b) Use Bernoulli's equation:

1 2 1 2P2 + pgy2 + 2PV2 = Pl + pgyl + 2PVl'

Then

P2 =

2.56 x lO5Pa.

E16-11 (a) The wind speed is (110 km/h)(1000 m/km)/(3600 s/h) = 30.6m/s.difference is then

The pressure

1~p = -(1.2kg/m3)(30.6m/s)2 = 562Pa.

2

(b) The lifting force would be F = (562 Pa)(93m2) = 52000 N.

E16-12 The pressure difference is

1~p = 2(1.23kg/m3)(28.0m/s)2 = 482 N.

The net force is then F (482 N)(4.26m)(5.26m) = lO800N.

196

~ The lower pipe has a radius rl = 2.52 cm, and a cross sectional area of Al = 7rr~ .Thespeed of the fluid flow at this point is Vl. The higher pipe has a radius of r2 = 6.14 cm, a crosssectional area of A2 = 7rr~, and a fluid speed of V2. Then

A - A 2- 2lVl- 2V20rrlvl-r2v2.

Set yl = O for the lower pipe. The problem specifies that the pressures in the two pipes are the

same, so

1 2= Po + 2PV2 + pgy2,

1 2= 2V2 + gy2,

We can combine the results of the equation of continuity with this and get

v~

V21

V~ (1- rt/r~)

V21

= v~ + 2gy2,

= (Vlr~/r~)2 + 2gy2,

= 2gy2,

= 2gy2/ (1- rf/ri) .

Thenv~ = 2(9.81m/s2)(11.5m)/ (1- (0.0252m)4/(0.0614m)4) = 232m2/s2

The volume fiow rate in the bottom (and top) pipe is

R = 1I"r~Vl = 11"(0.0252 m)2(15.2 m/s) = 0.0303 m3/s.

E16-14 (a) As instructed,

1 2Po + 2PVl + pgyl

1 2-Po + 2PV3 + pgy3,

= ~V~ + g(y3 -Yl),

But y3 -yl = -h, so V3 = J29h.(b) h above the hole. Just reverse your streamline!(c) It won't come out as fast and it won't rise as high.

~ Sea level will be defined as y = O, and at that point the Huid is assumed to be at rest.

Then

1 2= Po + 2PV2 + pgy2,

1 2= 2V2 + gy2,

1 2Po + 2PVl + pgyl

where y2 = -200 m. Then

V2 = ,p¡gij; = V-2(9.81m/s2)(-200m) = 63m/s.

197

E16-16 Assume streamlined flow, then

1 2P2 + 2PV2 + pgy2,

1 2-V2

2 .

=

(Pl -p2)/ p + g(yl -Y2) =

Then upon rearranging

V2 = ..¡2 [{2.1){1.01 x 105Pa)/{1000kg/m3) + {9.81 m/s2){53.0m)] = 38.3m/s.

E16-17 (a) Points 1 and 3 are both at atmospheric pressure, and both will move at the samespeed. But since they are at different heights, Bernoulli's equation will be violated.

(b) The flow isn't steady.

E16-18 The atmospheric pressure difference between the two sides will be ~p = ~PaV2. The heightdifference in the U-tube is given by ~p = Pwgh. Then

~ (a) There are three forces on the plug. The force from the pressure of the water, F1 =PIA, the force from the pressure of the air, F2 = P2A, and the force of friction, F3. These threeforces must balance, so F3 = F1 -F2, or F3 = PIA -P2A. But Pl -P2 is the pressure differencebetween the surface and the water 6.15 m below the surface, so

F3 = L\.p A = -pgyA,

-(998 kg/m3)(9.81 m/s2)( -6.15 m)7r(O.O215 m)2,

87.4N

(b) To find the volume of water which flows out in three hours we need to know the volumeflow rate, and for that we need both the cross section area of the hole and the speed of the flow.The speed of the flow can be found by an application of Bernoulli's equation. We'll consider thehorizontal motion only- a point just inside the hole, and a point just outside the hole. These pointsare at the same level, so

1 2Pl + 2PVl + pgyl

I 2= P2 + 2PV2 + pgy2,

I 2-P2 + 2PV2.Pl

Combine this with the resuIts of PascaI's principIe above, and

The volume oí water which flows out in three hours is

v = Rt = (11.0m/8)1r(0.0215m)2(3 x 36008) = 173m3.

E16-20 Apply Eq. 16-12:

Vi = y2(9.81 m/s2) (0.262 m)(810 kg/m3) / (1.03 kg/m3) = 63.6 m/s.

198

~ We'll agsume that the central column of air down the pipe exerts minimal force on thecard when it is deflected to the sides. Then

1 2 1 2Pl + 2PVl + pgyl = P2 + 2PV2 + pgy2,

1 2= P2 + 2PV2.Pl

The resultant upward force on the card is the area of the card times the pressure difference, or

1F = (Pl -P2)A = 2pAv2.

E16-22 If the air blows uniformly over the surf8{:e of the plate then there can be no torque aboutany axis through the center of mass of the plate. Since the weight also doesn 't introduce a torque,then the hinge can 't exert a force on the plate, because any such force would produce an unbalancedtorque. Consequently mg = ApA. Ap = pv2/2, so

'fv = ~ = (l.:i~-:J!~fl~~{~~ 7t~~k)2 = 3O.9m/s.

E16-23 Consider a streamline which passes above the wing and a streamline which passes beneaththe wing. Far from the wing thetwo streamlines are close together, move with zero relative velocity,and are effectively at the same pressure. So we can pretend they are actually one streamline. Then,since the altitude difference between the two points above and below the wing ( on this new, singlestreamline) is so small we can write

The lift force is thenL = ~pA = ~pA(Vt2 -Vu2)

E16-24 (a) From Exercise 16-23,

L= ~(1.17kg/m3)(2)(12.5 m2) {(49.8m/s)2 -(38.2m/s)2] ;;;; 1.49x 104N.

The mass of the plane must be m =L/g = (1.49x 104N)/(9.81 m/s2) = i520 kg.(b) The lift is directed straight up.(c) The lift is directed 15° off thevertical toward the rear of the plane.(d) The lift is directed 15° off the vertical toward the front otthe plane.

~ The larger pipe has a radius rl = 12.7 cm, and a cross sectiona1 area of Al = 1rr~ .Thespeed ofthe fluid flow at this point is Vl. The smaller pipe has a radius of r2 = 5.65 cm, a crosssectional area of A2 = 1rr~ , and a fluid speed of V2. Then

A1vl = A2V2 or r~vl = r~v2.

Now Bernoulli's equation. The two pipes are at the same level, so Yl = y2. Then

1 2 1 2Pl + 2PVl + pgyl = P2 + 2PV2 + pgy2,

1 2 1 2Pl + 2PVl = P2 + 2PV2.

199

Combining this with the results from the equation of continuity,

I 2Pl + 2PVl

v~

v~

( 4)2 rI

vIl -~

2 (P2 -Pl)

p (1- rt/r~) .v~ =

It may laak a mess, but we can salve it ta find v¡

R = Av = 7r(O.127m)2(1.41 mis) = 7.14x lO-3m3is.

That's about 711iters/second.

E16-26 The lines are parallel and equally spaced, so the velocity is everywhere the same. We cantransform to a reference frame where the liquid appears to be at rest, so the Pascal's equation wouldapply, and p + pgy would be a constant. Hence,

1Po =p+pgy+ 2PV2

is the same for all streamlines.

E16-27 (a) The "particles" of fluid in a whirlpool would obey conservation of angular momentum,meaning a particle off mass m would have l = mvr be constant, so the speed of the fluid as a functionof radial distance from the center would be given by k = vr, where k is some constant representingthe angular momentum per mass. Then v = k/r .

(b) Since v = k/r and v = 27rr/T, the period would be T cx: r2.

(c) Kepler's third law says T cx: r3/2.

E16-28 Rc = 2000. Then

~ (a) The volume flux is given; from that we can find the average speed of the fluid in the

pipe.

But

200

Reynold's number from Eq. 16-22 is then

pDv (13600 kgim3)(0.0376 m)(8.03 x 10-4 mis)R= = 265.

{1.55 x 10-3N .s/m2)

This is well below the critical value of 2000.(b) Poiseuille's Law, Eq. 16-20, can be used to find the pressure difference between the ends of

the pipe. But first, note that the mass fiux dm/ dt is equal to the volume rate times the densitywhen the density is constant. Then p dV / dt = dm/ dt, and Poiseuille's Law can be written as

8r¡L dV--

7rR4 dtóp=

P16-1 The volume oí water which needs to fiow out oí the bay is

v = (6l00m)(5200m)(3m) = 9.5x lO7m3

during a 6.25 hour (22500s) period. The average speed through the channel must be

P16-2 (a) The speed of the fluid through either hole is v = .¡2¡¡fi. The mass flux through a holeis Q = pAv, so PIA1 = P2A2. Then Pl/P2 = A2/Al = 2.

(b) R = Av, so Rl/R2 = Al/A2 = 1/2.(c) If R1 = R2 then A1..j2¡¡n¡ = A2J29h2, Then

2 2h2/hl = (Al/A2) = (1/2) = 1/4.

So h2 = h1/4.

~ (a) Apply Torricelli's law (Exercise 16-14): v = V297i:. The speed v is a horizontal velocity,and serves as the initial horizontal velocity of the fluid "projectile" after it leaves the tank. Thereis no initial vertical velocity.

This fluid "projectile" falls through a vertical distance H- h before splashing against the ground.The equation governing the time t for it to fall is

1 2-(H- h) = -2gt ,

Solve this for the time, and t = v'2(H -h)/g. The equation which governs the horizontal distancetraveled during the fall is x = vxt, but Vx = v and we just found t, so

(b) How many values of h willlead to a distance of x? We need to invert the expression, andwe'll start by squaring both sides

X2 = 4k(H- k) = 4kH -4k2,

and then solving the resulting quadratic expression for h,

h = 4!f~i16H2 -16x2 = ~ ( H :!:: ,¡¡¡¡--::-;;-;-)

8

201

For values of x between O and H there are two real solutions, if x = H there is one real solution,

and if x > H there are no real solutions.If h1 is a solution, then we can write hl = (H + ~)/2, where ~ = 2h1 -H could be positive or

negative. Then h2 = (H + ~)/2 is also a solution, and

h2 = (H + 2h1 -2H)/2 = h1 -H

is also a solution.( c ) The farthest distance is x = H, and this happens when h = H /2, as we can see from the

previous section.

P16-4 (a) Apply Torricelli's law (Exercise 16-14): v = V2g(d + h2), assuming that the liquidremains in contact with the walls of the tube until it exits at the bottom.

(b) The speed of the fl.uid in the tube is everywhere the same. Then the pressure difference atvarious points are only functions of height. The fl.uid exits at C, and assuming that it remains incontact with the walls of the tube the pressure difference is given by ~p = p(h1 + d + h2), so thepressure at B is

p = Po -P(hl + d + h2).

(C) The lowest possible pressure at B is zero. A8Sume the flow rate is so slow that Pascal'sprinciple applies. Then the max:imum height is given by O = Po + pgh1, or

h1 = {1.01 x 105pa)/[{9.81 m/s2){1000 kg/m3)] = 10.3 m.

P16-5 (a) The momentum per kilogram of the fluid in the smaller pipe is Vl. The momentum perkilogram of the fluid in the larger pipe is V2. The change in momentum per kilogram is V2 -Vl.There are pa2V2 kilograms per second flowing past any point, so the change in momentum persecond is pa2V2(V2 -Vl). The change in momentum per second is related to the net force accordingto F = ~p/ ~t, so F = Pa2V2(V2 -Vl). But F ~ ~p/a2, SO Pl -P2 ~ Pv2(V2 -Vl).

(b) Applying the streamline equation,

1 2P2 + pgy2 + 2PV2'=

P2 -Pl

~

( c ) This question asks for the 1088 of pressure beyond that which would occur from a gradual1ywidened pipe. Then we want

Ap =1 2 2

)2P(Vl -V2) -Pv2(Vl -V2 ,

1 2 2 22P(vJ -V2) -PV2Vl + PV2'

1 2 2 1 22P(Vl -2V1V2 + V2) = 2P(Vl -V2) .

P16-6 The juice leaves the jug with a speed v = ..¡2"iiij, where y is the height of the juice in thejug. If A is the cross sectional area of the base of the jug and a the cross sectional area of the hole,then the juice flows out the hole with a rate dV/dt = va = a..¡2"iiij, which means the level of jugvaries as dy/dt = -(a/A)..¡2"iiij. Rearrange and integrate,

202

.¡Y) J29at/A.~

~ ~ fif) ( Jh -..jfj) t=

When y = 14h/15 we have t = 12.0 s. Then the part in the parenthesis on the left is 3.539 x 102s.The time to empty completely is then 354 seconds, or 5 minutes and 54 seconds. But we want theremaining time, which is 12 seconds less than this.

~ The greatest possible value for v will be the value over the wing which results in an airpressure of zero. If the air at the leading edge is stagnant (not moving) and has a pressure of Po,then Bernoulli 's equation gives

1 2

po=2PV,

or v = ,j'ip;;Tii = V2(1.01xl05pa)/(1.2kg/m3) = 410m/s. This value is only slightly larger than

P16-8 Bernoulli's equation together with continuity gives

1 2Pl + ?PVl

Pl-P2 -

1 2P2 + 2PV2'

1 ( 2 2)2P V2 -VI'

1(A 2 ,

I 2 2

2P A2VI -VI2 ,

But Pl -P2 =«(J' -(J )gh. Note that we are not assuming (J is negligible compared to (J'. Combining,

J~(P' -p)gh Vl = A2 (A 2 - A 2 ) ." p 1 2

P16-9 (a) Bernoulli's equation together with continuity gives

Pl

Then

= f2(2.12)(1.01x 105Pa) = 4.45m/sVi V (1000 kg/m3)(21.6) ,

and then V2 = (4.75)(4.45m/s) = 21.2m/s.(b) R = 7r(2.60x10-2m)2(4.45m/s) = 9;45 x 10-3 m3/s.

203

the speed oí sound; they are related because sound waves involve the movement oí air particles which"shove" other air particles out oí the way.

P16-10 (a) For Fig. 16-13 the velocity is constant, or v = vi. ds = idx + Jdy. Then

f v. ds = v f dx = O,

because § dx = o.

(b)ForFig.16-16thevelocityisv=(k/r)r. dS=rdr+Ord4>. Then

f v. ds = v f dr = O,

because f dr = O.

Pl6-11 (a) For an element of the fluid of mass dm the net force as it moves around the circle isdF = (v2/r)dm. dm/dV = p and dV = Adr and dF/A = dp. Then dp/dr = pv2/r.

(b) From Bernoulli's equation p + pv2/2 is a constant. Then

dp dv~+Pv~ =O,

or v/r + dv/dr = O, or d(vr) = 0. Consequently vr is a constant.(c) The velocity is v = (k/r)r. dS= rdr + Ord<t>. Then

f v. dS = v f dr = O,

because .f dr = O. This means the fiow is irrotational.

P16-12 F/A = 7Jv/D, so

F/A = (4.0x lO19N .s/m2)(O.O48m/3.l6x lO7S)/(l.9x lO5m) = 3.2x lO5Pa.

~ A fiow will be irrotational if and only if § v. ds = O for all possible paths. It is fairlyeasy to construct a rectangular path which is parallel to the fiow on the top and bottom sides, butperpendicular on the left and right sides. Then only the top and bottom paths contribute to theintegral. v is constant for either path (but not the same) , so the magnitude v will come out of theintegral sign. Since the lengths of the two paths are the same but v is different the two terms don'tcancel, so the fiow is not irrotational.

p 16-14 ( a) The area of a cylinder is A = 27rr L .The velocity gradient if dv / dr .Then the retardingforce on the cylinder is F = -1](27rrL)dv/dr.

(b) The force pushing a cylinder through is F' = A~p = 7rr2 ~p.

(C) Equate, rearrange, and integrate:

Then

~(R2 -r2).41)L

v=

204

~ The volume flux (called R¡ to distinguish it from the radius R) through an annular ringof radius r and width 8r is

tSRf = tSA v = 21Tr tSr V,

where v is a function of r given by Eq. 16-18. The mass flux is the volume flux times the density,so the total mass flux is

dm

dt=

dr,

8r¡L.

P16-16 The pressure difference in the tube is ~p = 4')'/r, where r is the (changing) radius of thebubble. The mass flux through the tube is

4p7rR4'Y

8;¡¡;;:--'

dm

dt

~

R is the radius of the tube. dm = pdV, and dV = 47rr2dr. Then

¡ t R4 '"'1

O 8-;;"Ldt,

pR4'"'1-;¡;¡¡; t ,

1 r2 r3 dr =

rl

4 4rl -r2

~

Then

2(1.80 x 10-5N .s/m2)(0.112 m) [(38.2 x 10-3m)4 -(21.6 x 10-3m)4J = 3630 s.

(0.54 x 10-3m)4(2.50 x 10-2N/m)t=

205

~ For a perfect spring IFI = klxl. x = O.157m when 3.94kg is suspended from it. Therewould be two forces on the object- the force of gravity, W = mg, and the force of the spring, F.These two force must balance, so mg = kx or

(3.94 kg)(9.81 m/s2)k- mg--= .- = 0.246 N /m.x (0.157m)

Now that we know k, the spring constant, we can find the period of oscillations from Eq. 17-8,

E17-2 (a) T = 0.484s.(b) f = l/T = 1/(0.484s) = 2.07s-1.(C) (¿) = 27rf = 13.0 rad/s.(d) k = mw2 = (0.512kg)(13.0 rad/s)2= 86.5N/m.(e) Vm = (¿)Xm = (13.0 rad/s)(0.347m) = 4.51 m/s.(f) Fm = mam = (0.512kg)(13.0 rad/s)2(0.347m) = 30.0N.

E17-3 am = (27rf)2xmo Then

f = V(9.81 m!s2)/(1.20x 10-6~)/(27r) = 455 Hz.

E17-4 (a) "" = (271")/(0.6458) = 9.74 rad/8. k = 7n(,.)2 = (5.22kg)(9.74 rad/8)2 = 495N/m.(b) X m = vm/"" = (0.153m/8)/(9.74rad/8) = 1.57x10-2m.(c) f = 1/(0.6458) = 1.55 Hz.

~ (a) The amplitude is halfofthe distance between the extremes ofthe motion, so A = (2.00mm)/2 = 1.00 mm.

(b) The maximum blade speed is given by Vm = ú)xm. The blade oscillates with a frequency of120 Hz, so ú) = 27rf = 27r(120s-1) = 754 rad/s, and then Vm = (754 rad/s)(0.001m) = 0.754m/s.

(c) Similarly, am = ú)2Xm' am = (754 rad/s)2(0.001m) = 568m/s2.

E17-6 (a) k = mw2 = 2 = l.25xlOSN/m(b) f = /k7m/21T = 830kg/4)/21T = 2.63/s.

E17-7 (a) x = (6.12m)cos[(8.38 rad/s)(1.90s) + 1.92 rad] = 3.27m.(b) v = -(6.12m)(8.38/s)sin[(8.38 rad/s)(1.90s) + 1.92 rad] = 43.4m/s.(c) a = -(6.12m)(8.38/s)2cos[(8.38 rad/s)(1.90s) + 1.92 rad]= -229m/s2.(d) f = (8.38 rad/s)/27r = 1.33/s.

(e)T=I/f=O.750s.

~ If the driye wheel rotates at 193 rey /min then

W = (193 rev/min)(27r rad/rev)(1/60 s/min) = 20.2 rad/s,

then v m =WXm = (20.2 rad/s)(0.3825 m) = 7.73m/s.

206

E17-10 k = (0.325kg)(9.81m/s2)/(1.80x10-2 m) = 177N/m.

= 0.691 s.

E17-11 For the tides (¿} = 27r /(12.5 h). Half the maximum occurs when COS(¿}t = 1/2, or (¿}t = 7r /3.Then t = (12.5 h)/6 = 2.08 h.

E17-12 The two will separate if the (maximum) acceleration exceeds g.(a) Since (AJ = 27r/T = 27r/(1.18s) = 5.32 rad/s the maximum amplitude is

Xm = (9.81m/s2)/(5.32 rad/s)2 = O.347m.

(b) In this case (AJ = ..¡(9.81m/s2)/(O.O512m) = 13.8 rad/s. Then f = (13.8 rad/s)/27r 2.20/8.

E17-13 (a) ax/x = -",2. Then

w = V-(-123m/s)/(0.112m) = 33.1 rad/s,

so f = (33.1 rad/s)/21r = 5.27/s.(b) m = k/'-'l2 = (456N/m)/(33.1 rad/s)2 = O.416kg.(c) x = XmCOS'-'lt; V = -Xm'-'lSin'-'lt. Combining,

X2 + (v/'-'l)2 = Xm2 COS2 '-'lt + Xm2 sin2 '-'lt = Xm2.

Consequently,X m = /(0.112m)2 + (-13.6m/s)2/(33.1 rad/s)2 = 0.426 m.

E17-14 Xl = XmCOS(¿}t, X2 = XmCOS((¿}t + <f¡). The crossing happens when Xl = xm/2, or when(¿}t = 7r/3 (and other values!). The san1e constraint happens for X2, except that it is moving in theother direction. The closest value is (¿}t + <f¡ = 27r /3, or <f¡ = 7r /3.

~ (a) The net force on the three cars is zero before the cable breaks. There are three forceson the cars: the weight, W, a normal force, N, and the upward force from the cable, F. Then

F = WsinfJ = 3mgsinfJ.

The frequency oí oscillation oí the remaining two cars after the bottom car is released is

Numerically, the frequency is

(b) Each car contributes equally to the stretching of the cable, so one car causes the cable tostretch 14.2/3 = 4.73 cm. The amplitude is then 4.73 cm.

207

E17-16 Let the height of one side over the equilibrium position be x. The net restoring forceon the liquid is 2pAxg, where A is the cross sectional area of the tube and 9 is the acceleration offree-fall. This corresponds to a spring constant of k = 2pAg. The mass of the fluid is m = pAL.The period of oscillation is

T = 21T.fi = 1Tfij.

~ (a) There are two forces on the log. The weight, W = mg, and the buoyant force B.

We'll 888ume the log is cylindrical. If x is the length of the log beneath the surface and A thecross sectional area of the log, then V = Ax is the volume of the displaced water. Furthermore,7nw = Pw V is the m888 of the displaced water and B = 7nwg is then the buoyant force on the log.

Combining,B = PwAgx,

where Pw is the density of water. This certainly looks similar to an elastic spring force law, withk = PwAg. We would then expect the motion to be simple harmonic.

(b) The period of the oscillation would be

T~27rfi~27r~,

where m is the total mass of the log and lead. We are told the log is in equilibrium when x = L = 2.56m. This would give us the weight of the log, since W = B is the condition for the log to float. Then

m=~=~=pAL.9 9

From this we can write the period of the motion as

= 3.21 s.T=27r ~

yp:Ag

E17-18 (a) k = 2(1.18J)/(O.O984m)2 = 244N/m.(b) m = 2(1.18J)/(1.22m/s)2 = 1.59kg.(c) f = [(1.22m/s)/(O.O984m)]/(27r) = 1.97/s.

E17-19 (a) Equate the kinetic energy of the object just after it leaves the slingshot with thepotential energy of the stretched slingshot.

mv2 (0.130 kg)(11.2 x 103m/s)2~ = 6.97x lO6N/m.k= =

(1:53 m)2

(b) N = (6.97x 106 N/m)(1.53m)/(220N) = 4.85 x 104 people.

E17-20 (a) E = kxm2/2, U = kx2/2 = k(xm/2)2/2 = E/4. K = E -U = 3E/4. The the energy

is 25% potential and 75% kinetic.(b) If U = E/2 then kx2/2 = kxm2/4, or x = xm/~.

208

~ (a) am = úJ2Xm SO

~ ~

úJ=

X m= 2.06 x 103 rad/s

= I (7,93 x 103 m/s2)y (1.86 x 10-3 m)

The period oí the motion is then211" = 3.05 x 10-38.T=(¿}

(b) The maximum speed of the particle is found by

v m = I.UXm = (2.06 x 103 rad/s)(1.86 x 10-3 m) = 3.83 m/s.

(c) The mechanical energy is given by Eq. 17-15, except that we will focus on when Vx = vm,because then x = O and

E11-22 (a) f = Jk7ffi/27r = i,~~~!1m)/(5.13kg)/27r = 2.21/s.(b) Ui = (988N/m)(0.535m) /2 = 141 J.(c) Ki = (5.13kg)(11.2m s 2 2 = 322J.

( d) Xm = V2E7k = 2(322 J + 141 J) / (988 N /m) = 0.968 m.

E17-23 (a) iAJ = v'(538N/m)/(1.26kg) = 20.7 rad/s.

X m = v'(0.263m)2 + (3.72m/s)2/(20.7 rad/s)2 = 0.319m.

-(-3.72m/s)/[(20.7 rad/s)(0.263 m)]} = 34.3°.(b) 4> = arctan

E17-24 Before doing anything else apply conservation of momentum. If Vo is the speed of thebullet just before hitting the block and Vl is the speed of the bullet/block system just after the twobegin moving as one, then Vl = mvo/(m+M), where m is the mass ofthe bullet and M is the massof the block.

For this system úJ = .¡k/(m + M).(a) The total energy of the oscillation is ~(m+ M)v~, so the amplitude is

11= m voy k¡;ñ:¡:-M)"

The numerical value is

= O.167m.1X m = (O.O50kg)(150mjs) (500 Njm)(O.O50kg + 4.00kg)

(b) The fraction of the energy is

2(m+M)v~ -mv2 -

O

(0.050 kg)( 0.050 kg + 4.00 kg)

m+Mm

= 1.23 x 10-2.m

m+M

m-- ~m+M -

E17-25 L = (9.82 m/s2)(1.00s/27r)2 = 0.249 m.

209

E17-26 T = (1808)/(72.0). Then

227r(72.0)

-¡"i8¡¡;;¡--

9.66 m/soy= {1.53 m)

~ We are interested in the value oí ()m which will make the second term 2% oí the firstterm. We want to solve

which has solution

or (}m = 33°.(b) How large is the third term at this angle?

232 .4 (jm 32 ( 1 .2 (jm) : 9( )2-sm -= --sm -= -0.02

2242 2 22 22 2 4

or 0.0009, which is very small.

E17-28 Since T cx: .¡iTg we have

~ Let the period of the clock in Paris be T1. In a day of length D1 = 24 hours it willundergo n = D/Tl oscillations. In Cayenne the period iB T2. n oscillations should occur in 24 hours,but since the clock runs slow, D2 iB 24 hours + 2.5 minutes elapse. So

T2 = D2jn = (D2jD1)T1 = [(1442.5 min)j(1440.0 min)]Tl = 1.0017Tl.

Since the ratio of the periods is {T2/Tl) = ,¡-(ij;Jg:;j, the 92 in Cayenne is

92 = 91 {Tl/T2)2 = {9.81m/s2)/{1.0017)2 = 9.78m/s2.

E17-30 (a) Take the differential of

2

~

T

47r2 X 105 m

T2

(10m) =g=

so 8g = ( -87r2 X 105 m/T3)8T. Note that T is not the period here, it is the time for 100 oscillations!

The relative error is then ~T8g u

---2--T.9

If óg/g = 0.1% then óT/T = 0.05%.(b) For g ~ 10m/s2 we have

T ~ 27r(100)v(10m)/(10m/82) = 6288.

Then 8T ~ (0.0005) (987 s) ~ 300 InS.

210

E17-31 T = 27rV(17.3m)/(9.81 m/82) = 8.348.

E17-32 The spring will extend until the force from the spring balances the weight, or when Mg =kh. The frequency of this system is then

~ The frequency of oscillation is

where d is the distance from the pivot about which the hoop oscillates and the center of mass of thehoop.

The rotational inertia I is about an axis through the pivot, so we apply the parallel axis theorem.Then

I = Md2 +Icm = Md2 +Mr2.

But d is r, since the pivot point is on the rim of the hoop. So I = 2M ~ , and the frequency is

1 {"j;ii¡.f 1 {9 1 I (9.81 m/s2)f = 2;V -zM(ii = 2;V 2d = 2;V 2(0.653m) = 0.436 Hz.

(b) Note the above expression looks like the simple pendulum equation if we replace 2d with l.Then the equivalent length of the simple pendulum is 2(0.653 m) = 1.31 m.

E17-34 Apply Eq. 17-21:

T2~

47r2= 7.70 xlO-2kg.m2.I=

T = 27r.jf/K = 27rV(O.834kg .m2)/(0.226N .m)= 12.1 s.

E11-36 x is d in Eq. 17-29. Since the hole is driIIed off center we appIy the principIe axis theoremto find the rotational inertia:

Then

T2Mgx=

=

1-ML2+Mx212

1l2(1.00m)2 + X2

(8.33 x 10-2m2) -(1.55 m)x + X2 = o.

This has solutions x = 1.49m and x = 0.0557 m. Use the latter.

211

E17-35 ~ = (0.192N .m)/(0.850 rad) = 0.226 N .m. I = ~(95.2kg)(0.148m)2 = O.834kg .m2.Then

~ For a stick of length L which can pivot about the end, Iof such a stick is located d = L /2 away from the end.

The frequency of oscillation of such a stick is

kML2. The center oí mags

f

~

f =

1!392;Vu'

f =

This means that f is proportional to Ji7L, regardless of the mass or density of the stick. The ratio

of the frequency of two such sticks is then h/ h = .¡t;;7L;, which in our case gives

f2 = hv'L;¡¡;;: = hV(L1)/(2L1/3) = 1.22h.

(, ,\

E17-38 The rotational inertia of the pipe section about the cylindrical axis is

M MIcm = 2 [r~ +r~] = 2 [(0.102m)2 + (0.1084m)2] = (1.11xl0-2m2)M

(a) The total rotational inertia about the pivot axis is

I = 2Icm + M(O.102m)2 + M(O.3188m)2 = (0.134m2)M.

The period of oscillation is

(b) The rotational inertia of the pipe section about a diameter is

M MIcm = 4 [r~ + r~] = 4 [(O.lO2m)2 + (O.lO84m)2] = (5.54x lO-3m2)M

The total rotational inertia about the pivot axis is now

I = M(I.11 x 10-2m2) + M(O.102m)2 + M(5.54x 10-3m2) + M(O.3188m)2 = (0.129m2)M

The period of oscillation is

T = 27r.1 (0.129m2)MV M(9.81 m/s2)(0.2104 m) = 1.57 s.

The percentage difference with part {a) i8 {0.038)/{1.608) = 1.9%.

E17-39

E17-40

E17-41 (a) Since effectively x = y, the path is a diagonalline.(b) The path will be an ellipse which is symmetric about the line x = y.(c) Since cos((A)t + 900) = -sin((A)t), the path is a circle.

212

E17-42 (a)(b) Take two time derivatives and multiply by m,

F = -mALV2 ( i cos (;)t + 9j cos 3(;)t )

(c)U=-JF.dr,sou = ~ mA 2 úJ2 ( COS2 úJt + 9 COS2 3<..Jt )

2

(d) K = ~mv2, so

K = ~mA2w2 (sin2wt+9sin23wt) ;2

And then E = K + U = 5mA2¡.;2.

(e) Yes; the period is 21!"!¡.;.

~ The w which describes the angular velocity in uniform circular motion is effectively thesame w which describes the angular frequency of the corresponding simple harmonic motion. Sincew = Vkliñ, we can find the effective force constant k from knowledge of the Moon 's mass and the

period of revolution.The moon orbits with a period of T, so

27r 27r .= 2.66 x 10-61T (27.3 x 24 x 36008) ~ ~- l'ad/8.

Thi8 can be used to find the value of the effective force constant k from

k = Tn(LI2 = (7.36 x l022kg)(2.66 x lO-6rad/8)2 = 5.2lxlOllN/m.

--.~ I.AJ= -

E17-44

~ln3 =

(a) We want to know when e-bt/2m = 113, or

t = : 2(1.52kg) ln3 = 14.78b (0.227 kg/s)

(b) The ( angular) frequency is

= 2.31 rad/s.

The number of oscillations is then

(14.7s)(2.31 rad/s)/27r = 5.40

~ The first derivative oí Eq. 7-39 is

dx

dtXm( -b/2m)e-bt/2m COS((..1't + <1» + Xme-bt/2m( -(..I') sin((..1't + <1»,

-Xme-bt/2m «(b/2m) cos((..1't + <1» + (..I' sin((..1't + <1»)

=

The second derivative is quite a bit messier;

~ -xm( -b/2m)e-bt/2m ((b/2m) cos((J.}'t + 4» + (J.}' sin((J.}'t + 4»)

-Xme-bt/2m ( (b/2m)( -(J.}') sin((J.}'t + 4» + ((J.}')2 COS((J.}'t + 4») ,

Xme-bt/2m ( ((J.}'b/m) sin((J.}'t + 4» + (b2/4m2 -(J.}'2) cos((J.}'t + 4» )

=dX2

213

Substitute these three expressions into Eq. 17-38. There are, however, some fairly obvious simpli-fications. Every one of the terms above has a factor of xm, and every term above has a factor ofe-bt/2m, SO simultaneously with the substitution we will cancel out those factors. Then Eq. 17-38becomes

m [(w'b/m) sin(w't + 4» + (b2/4m2 -W'2) COS(w't + 4»]-b [(b/2m) cos(w't + 4» + w' sin(w't +4»] + k cos(w't + 4» = {)

Now we collect terms with cosine and terms with sine,

(-.1'b --.I'b) sin(-.1't +<I» + (mb2/4m2 --.1'2 -b2 /2m + k) cos(-.1't + <I» = o.

The coefficient for the sine termis identically zero; furthermore, because the cosine term must thenvanish regardless of the value of t, the coefficient for the sine term must also vanish. Then

( mb2/4m2 -m,{..;'2 -b2/2m + k) = Q1

orb212 k

úJ -m 4m2 .

If this condition is met, then Eq. 17-39 is indeed a solution of Eq. 17-38.

E17-46 (a) Four complete cycles requires a time t4 = 87r/úJ'. The amplitude decays to 3/4 theoriginal value in this time, so 0.75 = e-bt4/2m, or

87rbln( 4/3) = 21n(¿}' -

It is probably reasonable at this time to assume that b/2m is small compared to ú.i so that ú.i' ~ ú.i.We'll do it the hard way anyway. Then

",'2 =

2

~ - (~ )m 2m

k

m=

Numerically, then,

= O.112kg/s.

(b)

E17-47 (a) Use Eqs. 17-43 and 17-44. At resonance 1.1)" = 1.1),80

G = JbV = bIo,;,

and then X m = F m/lJ¡.¡.

(b) V m = (¡)X m = F m/b.

214

E17-48 We need the first two derivatives of

F m ( II )X=- -COS(¡)t-{3

G

The derivatives are easy enough to find,

dx~I

G-úJ") sin(úJ"t -{3),

-=dt

andd2x .dt2 -

We'll substitute this into Eq. 17-42,

m

A cos al cos a2 -A sin al sin a2,

if we let a2 = YJ'it -/3 and choose A and al correctly. The best choice is

Acosal = k-m(YJ")2,A .J.. )1SIn al = (A.II ,

and then taking advantage of the fact that sin2 + COS2 = 1,

A2 = (k- m(v;")2)2 + (/)(.;")2,

which looks like Eq. 17-44! But then we can apply the cosine angle addition formula, and

Acos(al + v;"t -{3) = G cosv;"t.

This expression needs to be true for all time. This means that A = G and al = {3.

andd2x

--dt2

We'll substitute this into Eq. 17-42,

m

215

Then we'll cancel out a.s much a.s we can and collect the sine and cosine terms,

(k -m«(Ll")2) COS«(LlI't -{J) -(búJ")sin«(Ll"t -.B) = GCOS(Ll"t.

We can write the left hand side of this equation in the form

Then we'll cancel out as much as we can and collect the sine and cosine terms,

(k -m(úJ"')2) COS(úJ"'t -/3) -(bú1"') sin(úJ"'t -/3) = G cosúJ"t.

We can write the left hanrl sirle of this equation in the form

Acosal cosa2 -Asinal sina2,

if we let a2 = (.J"'t -/3 and choose A and al correctly. The best choice is

= k -m(w"')2,

= ~II/ ,

A cos al

A sin al

and then taking advantage of the fact that sin2 + COS2 = I,

A2 = (k -m(f.AJ"')2)2 + (bI.J.J"')2,

which looks like Eq. 17-44! But then we can apply the cosine angle addition formula, and

A COS(Ql + ú)"'t -;8)= G COSú)"t.

This expression needs to be true for all time. This means that A = G and al + ",,"'t -.8 = ",,"t andal = .8 and ",,"' = ",,".

E17-50 Actually, Eq. 17-39 is not a solution to Eq. 17-42 by itself, this is a wording mistake inthe exercise. Instead, Eq. 17-39 can be added to any solution of Eq. 17-42 and the result will sti11be a solution.

Let Xn be any solution to Eq. 17-42 (such as Eq. 17-43.) Let Xh be given by Eq. 17-39. Then

x = X n + Xho

Take the first two time derivatives oí this expression.

+b + k (X n + Xh) = F m COSiJ)"t.m

Rearrange and regroup.

Consider the second term on the left. The parenthetical expression is just Eq. 17-38, the dampedharmonic oscillator equation. It is given in the text ( and proved in Ex. 17-45 ) the Xh is a solution,so this term is identically zero. What remains is Eq. 17-42; and we took as a given that Xn was asolution.

(b) The "add-on" solution of Xh represents the transient motion that will die away with time.

216

~ The time between "bumps" is the solution to

vt

~

x,

3600 s

1 hrt = 0.886 s=

The angular frequency is21T

--T 7.09 rad/s

This is the driving frequency, and the problem states that at this frequency the up-down bounceoscillation is at a maximum. This occurs when the driving frequency is approximately equal to thenatural frequency of oscillation. The force constant for the car is k, and this is related to the naturalangular frequency by

=

2 w 2k = rTUJ.J = -úJ ,

9

where W = (2200 + 4 x 180) lb= 2920 lb is the weight of the car and occupants. Then

When the four people get out of the car there is less downward force on the car springs.important relationship is

The

6.F = k6.x.

In this case 6.F = 720 lb, the weight of the four people who got out of the car .6.x is the distancethe car will rise when the people get out. So

AF

k

(720 lb)4590 lb/ft

Ax= = 0.157 ft ~ 2 in.=

E17-52 The derivative is easy enough to find,

dx

dt

Pm-I

G-(¿)") sin((¿)"t -¡3),-

The velocity amplitude is

Fm II

-úJ,G

v m =

Pm~Vm2(w"2 -w2)2 + b2w'12 ,

PmV(m,(JJ" -k/(¡)")2 + b2 .

Note that this is exactly a maximum when iJ.J" = iJ.J.

E17-53 The reduced mags is

m = (1.13kg)(3.24kg)/(1.12kg + 3.24kg) = O.840kg.

The period of oscillation is

T = 27rJ(0.840kg)/(252N/m) = 0.3638

217

E11-54

E17-55 Start by multiplying the kinetic energy expression by (ml + m2)/(ml + m2).

K =

=

2( ) ((m¡v¡ + m2V2)2 + m¡m2(V¡ -V2)2)m¡ + m2

But m¡ V2 + m2V2 = O by conservation of momentum, so

(mlm2)2(m¡ + m2) (V¡ -V2)2 ,

m2(V¡ -V2)2.

K

P17-1 The roass ofone silver atoro is {O.lO8kg)/{6.02xlO23) = l.79xlO-25kg. The effective springconstant is

k = (l.79xlO-25kg)47r2(lO.OxlO12/S)2 = 7.07xlO2N/m.

P17-2 (a) Rearrange Eq. 17-8 except replace m with the total mass, or m+M. Then (M +m)/k =T2/(47r2), or

M = (k/47r2)T2 -m.

(b) When M = O we have

m = [(605.6 N/m)/(47r2)](0.90149 S)2 = 12.467kg.

(c) M = [(605.6 N/m)/(47r2)](2.08832 S)2 -(12.467kg) = 54.432kg.

~ The max.imum static friction is F¡ :::; JLsN. Then

F¡ = JLsN = JLsW = JLsmg

is the maximum available force to accelerate the upper block. So the maximum acceleration is

F¡am = --J.Lsgm

The maximum possible amplitude of the oscillation is then given by

am J1s9xm= . -(,)2 k/(m+M)'

where in the last part we substituted the total mass of the two blocks because both blocks areoscillating. Now we put in numbers, and find

0.119 m.

218

and then add 2ml m2Vl V2 -2ml m2Vl V21

P17-4 (a) Equilibrium occurs when F = O, or b/r3 = a/r2. This happens when r = b/a.(b) dF/dr = 2a/r3 -3b/r4. At r = b/a this becomes

dF/dr = 2a4/b3 -3a4 fb3 = ~a4/b3,

which corresponds to a force constant of a4/b3.(C) T= 27rvmlk = 27rYmb3 /a2, where m is the reduced mass.

P17-5 Each spring helps to restore the block. The net force on the block is then of magnitudeF1 + F2 = k1x + k2x = (k1 + k2)X = kx. We can then write the frequency as

f=~ /I=27r V ~

With a little algebra,

f = 1-~27r y ~'/ 1 k1 .1 k2

y ¡;2 ;;:¡ + ¡;2 ;;:¡ ,

.fii+fi.

P17-6 The tension in the two spring is the same, so k1xl = k2x2, where Xi is the extension of theith spring. The total extension is Xl + X2, SO the effective spring constant of the combination is

F F 1 1 k1k2-====X X1 +X2 xdF + X2/ F 1/k1 + 1/k2 k1 + k2 ,

The period is theD

T- .!.- /I- .!.- I~-211" V:;;;: -211" y (k1 + k2)m

With a little algebra,

f1 .1 klk2

2;y (kl + k2)m ,

1 1 i2;y m/kl + m/k2 ,

1 112;y 1/ú.i~+¡/ú.i~'

1 ~ 2ú.i2 I 2

2; ~'

hf2

v'7ri7i'

219

~ (a) When a spring is stretched the tension is the same everywhere in the spring. Thestretching, however, is distributed over the entire length of the spring, so that the relative amountof stretch is proportional to the length of the spring under consideration. Half a spring, half theextension. But k = -F/x, so half the extension means twice the spring constant.

In short, cutting the spring in half will create two stiffer springs with twice the spring constant,so k = 7.20 N/cm for each spring.

(b) The two spring halves now support a mass M. We can view this as each spring is holdingone-half of the total mass, so in effect

lr--;;-

:2-;YM72

f=

or, solving for M ,

2(720N/m) = 4.43kg.= 47r2(2.S7 S-I )2M= 2k

4;2"12

P17 -8 Treat the spring a being composed of N massless springlets each with a point mass ms/ Nat the end. The spring constant for each springlet will be kN. An expression for the conservationof energy is then

N Nm 2 ms ~ 2 Vk ~ 2

E2v + 2N L..;vn + 2 L..;Xn = .1 1

Since the spring stretches proportionally along the length then we conclude that each springletcompresses the same amount, and then Xn = A/ N sin f.¡)t could describe the change in length of eachspringlet. The energy conservation expression becomes

v = A(,¡) COS (,¡)t. The hard part to sort out is the vn, since the displacements for all springlets to oneside of the nth must be added to get the net displacement. Then

AVn = nN(,¡)cos(,¡)t,

and the energy expression becomes

m m N-+ 8~2 2:N3L..n2

I

Replace the sum with an integral, then

and the energy expression becomes

~ (m+

This will only be constant if

or T = 27r.¡(m + ms/3)/k.

220

P17-9 (a) Apply conservation of energy. When x = Xm v = O, so

1 1 12kxm2 = 2m[v(0)r + 2k[x(0)r,

mXm2 = k[v(O)J2 +[x(O)12,

xm = V[v(0)/tIJ]2 + [X(O)]2.

(b) When t =Ox(O) = XmCOSt/> and v(O) = -wxmsint/>, so

v(O) sint/> = tant/>.=--wx(o) cos~

P17-10

~ Conservation of momentum for the bullet block collision gives m v = ( m + M)Vf or

mVf = ~V.

This Vf will be equal to the maximum oscillation speed Vmo The angular frequency for the oscillationis given by

1.1) = {;;h .

Then the amplitude for the oscillation is

v m

(.;Xm=

P17-12 (a) W = Fs, or mg = kx, so x = mg/k.(b) F = ma, but F = W -Fs = mg -kx, and since ma = md2x/dt2,

f1l!xmd"i2" + kx = mg.

The solution can be verified by direct substitution.(c) Just look at the answer!(d) dE/dt is

P17-13 The initial energy stored in the spring is kxm2/2. When the cylinder pa.sses through theequilibrium point it ha.s a translational velocity Vm and a rotational velocity IJ)r = vm/ R, where Ris the radius of the cylinder. The total kinetic energy at the equilibrium point is

Then the kinetic energy is 2/3 translational and 1/3 rotational. The total energy of the system is

221

(a) Kt = (2/3)(8.40J) = 5.60J.(b) Kr = (1/3)(8.40J) = 2.80J.

( c ) The energy expression is3m'

2

2 1 2Ev + -k x = ,

2

1-I2

which leads to a standard expression for the period with 3M/2 replacing m. Then T = 21T,f3M7'ik.

P17-14 (a) Integrate the potential energy expression over one complete period and then divide bythe time for one period:

1 ¡ xm ~kx2dx =

-22xm -xm

1 1- k 32xm 3 X m ,

1-k X 26 m .

This is one-third the total energy. The average kinetic energy must be two-thirds the total energy,or {1/3)kxm2.

P17-15

'12(0.144m)2 + (0.102m)2 = (2.08 x 10-2m2)M.

The rotational inertia is

I=~MR2+Md2=M2

The period of oscillation is

T=2,,~= 27r (2.08 x 10-2m2)M

M(9.81 m/82)(0.102m) = 0.9068.

P17-16 (a) The rotational inertia of the pendulum about the pivot is

(b) The center of mass location is

(c) The period of oscillation is

T = 211" (0.291 kg .m2)/(0.272kg +0.488kg)(9.81 m/82)(0.496m) = 1.768.

222

~ (a) The rotational inertia oí a stick about an axis through a point which is a distance dfrom the center oí mass is given by the parallel axis theorem,

I = I + md2 = -.::.-mL2+ md2cm "' .

1

The period of oscillation is given by Eq. 17-28,

f L2 + 12d2= 27r V12gd

(b) We want to find the minimum period, so we need to take the derivative of T with respect tod. It 'lllook weird, but

12¿;; -L2dT=7rdd V12gd3(L2 + 12d2) .

This will vanish when 12~ ~ L2, or when d = L/..jf2.

P17-18 The energy stored in the spring is given by kx2/2, the kinetic energy oí the rotating wheelis

~(M R2) (~) 2,

where v is the tangential velocity of the point of attachment of the spring to the wheel. If x =Xm sinú)t, then v = Xmú)COSÚ)t, and the energy will only be constant if

k r2

MW"2-(¿) -

P17-19 The method of solution is identical to the approach for the simple pendulum on page 381except replace the tension with the normal force of the bowl on the particle. The effective pendulumwith have a length R.

P17-20 Let x be the distance from the center of mass to the first pivot point. Then the pertod is

given by

T=21T ~.y-¡;¡-¡¡;-

Salve this far x by expressing the abave equatian as a quadratic:

(~ )471"2x=I+Mx2,

or

Mx2- x+I=OMgT2'\

,~)There are two solutions. One corresponds to the first location, the other the second location. Addingthe two solutions together will yield L; in this case the discriminant of the quadratic will drop out,leaving

,MgT2 -~~ -47r2"

L = XI + X2 = -

Then g = 471"2L/T2,

223

In this problemP17-21

{0.210 mf + {0.760 m + O.210m)2 ) = 2.41kg/ .m2.I = {2.50 kg)2

The center of mass is at the center of the disk.(a) T = 27r (2.41kg/ .m2)/(2.50kg)(9.81m/s2)(0.760m + O.210m) = 2.00s.(b) Replace Mgd with Mgd + K. and 2.00s with 1.50s. Then

47r2(2 41 kg/ .m2)K.= (.) 2 -(2.50kg)(9.81m/s2)(0.760m+O.210m)=18.5N.m/rad.

1.50s

P11-22 The net force on the bob is toward the center of the cÍrcle, and h88 magnitude F net =mv2 / R. This net force comes from the horizontal component of the tension. There is also a verticalcomponent of the tension of magnitude mg. The tension then h88 magnitude

T = J(mg)2 + (mv2/R)2 = mJg2 +v4/R2.

It is this tension which is important in finding the restoring force in Eq. 17-22; in effect we want toreplace 9 with J g2 + v4 / R2 in Eq. 17-24. The frequency will then be

1 I L

f=2;V Jg2+V4/R2.

~ (a) Consider an object of mass m at a point p on the axis of the ring. It experiences agravitational force of attraction to all points on the ring; by symmetry, however, the net force is notdirected toward the circumference of the ring, but instead along the axis of the ring. There is thena factor of cos (} which will be thrown in to the mix.

The di8tance from p to any point on the ring is r = V R2 + Z2, and (} is the angle between theaxis on the line which connects p and any point on the circumference. Con8equently,

cosO = z/r,

and then the net force on the star of mass m at p is

GMm GMmz GMmzF = -;:r¡-- cos(} = r3 = (R2 + Z2)3/2 .

(b) If z « R we can apply the binomial expansion to the denominator, and

1 + (~))-3/2-~ (~)2)

~R-3(R2 + Z2)-3/2 = R-3

Keeping terms only linear in z we have

GMmF = -w-z,

which corresponds to a spring constant k = GMm/ R3. The frequency of oscillation is then

f = ~/(27r) = ,¡¡¡¡;¡¡w /(27r).

(c) Using some numbers from the Milky Way galaxy,

f = (7x 10-11N .m2 /kg2)(2x 1043kg)/(6x 1019m)3/(27r) = 1 x 10-14 Hz.

224

P17-24 (a) The acceleration of the center of mass (point C) is a = F/M. The torque about anaxis through the center of mass is tau = F R/2, since O is R/2 away from the center of mass. Theangular acceleration of the disk is then

a = T/I = (FR/2)/(MR2/2) = F/(MR).

Note that the angular acceleration will tend to rotate the disk anti-clockwise. The tangential com-ponent to the angular acceleration at p is aT = -aR = -F / M; this is exactly the opposite of thelinear acceleration, so p will not (initially) accelerate.

(b) There is no net force at p .

P17-25 The value for k is closest to

k ~ (2000kg/4)(9.8lm/s2)/(O.lOm) = 4.9xlO4N/m.

One complete oscillation requires a time t¡ = 27r/(¿)'. The amplitude decays to 1/2 the originalvalue in this time, so 0.5 = e-btl/2m, or

ln(2) = ~

2m(,¡)"

It is not reasonable at this time to assume that b/2m is small compared to YJ so that YJ' ~ YJ. Then

2'2

úJ =

2 2

~ - (~ )m 2m=

k

m=

Then the value for b is

b= = 1100kg/s.

P17-26 a = ~x/dt2 = -AúJ2cOSúJt. Substituting into the non-linear equation,

-mA(L}2 COS(L}t + kA3 COS3(L}t = FCOS(L}dt.

Now let f.A)d = 3ú.I. ThenkA 3 COS3 (¿}t -mAw2 cos (¿}t = F cos &.Jt

Expanrl the right hanrl sirle aB COS 3(¿}t = 4 COS3 (¿}t -3 cos (¿}t, then

kA3 COS3 (¿}t -mA(¿}2 COS(¿}t = F(4 COS3 (¿}t -3 COS(¿}t)

This will only work if 4F = kA3 and 3F = mA¡,.;2,4mA(,¡)2 = kA3, so A <X (,¡) and then F <X (,¡)3 <X (,¡)d3,

Dividing one condition by the other means

225

P17-27 (a) Divide the top and the bottom by m2. Then

m¡m2

m¡ + m2

mi=(ml/m2) + l'

and in the limit as m2 -00 the value of (ml/m2) -O, so

mlm2lim

m2-+00 m¡ + m2

226

E18-1 (a) f = v!A = (243m!8)!(0.0327m) = 7.43x 103 Hz.

(b) T = 1! f = 1.35x 10-48.

E18-2 (a) f = (12)/(308) = 0.40 Hz.(b) v = (15m)/(5.08) = 3.0m/8.(c) >. = v/f = (3.0m/8)/(0.40 Hz) = 7.5m.

~ (a) The time for a particular point to move from maximum displacement to zero dis-placement is one-quarter of a period; the point must then go to maximum negative displacement,zero displacement, and finally maximum positive displacement to complete a cycle. So the period is

4(178 ms) = 712 ms.(b) The frequency is f = l/T= 1/(712x10-3s) = 1.40 Hz.(c) The wave-speed is v = f>. = (1.40 Hz)(1.38m) = 1.93m/s.

Use Eq. 18-9, except let f = l/T:E18-4

= (0.0213 m) sin [(55.1 rad/m)x -(2420 rad/s)tJ-(385 Hz)ty = (0.0213 m) sin 27rx

(0.114m)

~ The dimensions for tension are [F] = [M][L]/[T]2 where M stands for m8SS, L for length, Tfor time, and F stands for force. The dimensions forlinear m8SS density are [M]/[L]. The dimensions

for velocity are [L]/[T].Inserting this into the expression v = Fa / JLb ,

[L]

ffl

=

There are three equations here. One for time, -1 = -2aj one for length, 1 = a + b; and one formass, O = a- b. We need to satisfy all three equations. The first is fairly quickj a = 1/2. Either of

the other equations can be used to show that b = 1/2.

E18-6 (a) Ym = 2.30mm.(b) f = (588 rad/s)/(27r rad) = 93.6 Hz.(c) v = (588 rad/s)/(1822 rad/m) = O.323m/s.(d) A = (27r rad)/(1822 rad/m) = 3.45mm.(e) Uy = YmW = (2.30mm)(588 rad/s) = 1.35m/s.

E18-7 (a) Ym = 0.060m.(b) .>.. = (27r rad)/(2.07r rad/m) = 1.0m.(c) f = (4.07r rad/s)/(27r rad) = 2.0 Hz.(d) v = (4.07r rad/s)/(2.07r rad/m) = 2.0m/s.(e) Since the second term is positive the wave is moving in the -x direction.(f) Uy = YmW = (0.060m)(4.07r rad/s) = 0.75m/s.

E18-8 v = JF7jL = V(487N)/[(O.O625kg)/(2.15m)] = 129m/s.

227

~ We'll first find the linear mass density by rearranging Eq. 18-19,

FJi,=-

V2

Since this is the same string, we expect that changing the tension will not significantly change the

linear mass density. Then for the two different instances,

F1 F2.=2 V2~

We want to know the new tension, so

= 135 N

E18-10 First v = (317 rad/s)/(23.8 rad/m) = 13.32 m/s. Then

Ji, = F/V2 = (16.3N)/(13.32m/s)2 = O.O919kg/m.

E18-11 (a) Ym = O.O5m.(b) >. = (0.55m) -O.15m) = O.40m.(c)v = JF7Jt = (3.6N)/(O.O25kg/m) = 12m/s.(d) T = ¡If = >.Iv = (0.40m)/(12m/s) = 3.33xl0-2s.(e) Uy = YmúJ = 27rymlT = 27r(O.O5m)/(3.33x 10-2s) = 9.4m/s.(f) k = (27r rad)/(O.40m) = 5.07r rad/m; úJ = kv = (5.07r rad/m)(12m/s) = 607r rad/s.

phase angle can be found from

The

(0.04 m) = (0.05 m) sin( <I> ),

or 4> = 0.93 rad. Then

y = (0.05 m) sin[(5.07r rad/m)x + (607r rad/s)t + (0.93 rad)].

El8-12 (a) The tensions in the two strings are equal, so F = (0.511 kg)(9.81 m/s2)/2 = 2.506 N.

The wave speed in string 1 is

v = JF7j-¡ = V(2.506N)/(3.31x 10-3kg/m) = 27.5m/s,

while the wave speed in string 2 is

v = ~ = v'(2.506N)/(4.87xl0-3kg/m) = 22.7m/s.

(b) We have ,¡F;7¡;; = ,¡F;7¡;;, or Fl/ JLl = F2/ JL2. But Fi = Mig, so Ml/ JLl = M2/ JL2. Using

M=M1+M2,

MI

11.1

MI

11.2

M-M1

/L2

M,

/L2

M

=

~+ =J1,1

1/( 1 MI ---/1.2 /1.1 I /1.2,

{0.511 kg) I= {4.87 x 10-3kg/m)

= 0.207 kg

and M2 = {0.511kg) -{0.207kg) = 0.304kg.

-+-

228

~ We need to know the wave speed before we do anything else. This is found from Eq.

18-19,

Our equations are then Xl = vt = (162 m/s)t, and X2 = v(t+29.6 ms) = (162 m/s)(t+29.6 ms) =

(162m/s)t+4.80m. We can add these two expressions together to solve for the time t at which the

pulses meet,

lO.3m = XI + X2 = (l62m/s)t + (l62m/s)t + 4.80m = (324m/s)t + 4.80 m.

which haB solutionX2 = 7.55m.

= 0.01708.The two pulses meet at XI = (162m/s)(O.O170s) = 2.75m, or

E18-14 (a) [)y/[)r = (A/r)k cos(kr -r.,;t) -(A/r2) sin(kr -r.,;t). Multiply this by r2, and then find

8 28y .(3;r (3; = Ak cos(kr -l.IJt) -Ak2r sm(kr -l.IJt) -Ak cos(kr -l.IJt).

Simplify, and then divide by r2 to get

lo 20Y--r-r2 or or

= -(Ak2 Ir) sin(kr -(LIt).

Now find {}2y/8t2 = -Aú.l2 sin(kr -(,¡)t). But since 1/V2 = k2/(,¡)2, the two sides are equal.

(b) [length]2.

E18-15 The liner mass density is JL = (0.263kg)/(2.72m) = 9.669xlO-2kg/m. The wave speed is

2(85.5W)(0} = ~ I = 1243 radis.V (9.669 x 10-2kgim)(7. 70 x 10-3mr (19.32 mis )

Then f = (1243 rad/s)/27r = 199 Hz.

E18-16 (a) If the medium absorbs no energy then the power flow through any closed surfacewhich contains the source must be constant. Since for a cylindrical surface the area grows as r, thenintensity must fall of as 1/ r .

(b) Intensity is proportional to the amplitude squared, so the amplitude must fall off as 1/ .¡r .

~ The intensity is the average power per unit area (Eq. 18-33); as you get farther from thesource the intensity falls ofI because the perpendicular area increases. At same distance r from thesaurce the total possible area is the area of a spherical shell of radius r, so intensity as a function ofdistance from the saurce would be

Pav

47rr2I=

229

v = y(36.1 N)/(9.669x 10-:¿kg/m) = 19.32m/s.Pav = !jJ,(AJ2ym2v, SO

We are given two intensities: /1 = 1.13W /m2 at a distance rlj 12 = 2.41 W /m2 at a distancer2 = rl -5.30 m. Since the average power of the source is the same in both cases we can equate

these two values as

47rr~ 11

47rr~11

= 47rr~12,= 47r(rl -d)212,

where d = 5.30 m, and then salve far rl. Daing this we find a quadratic expressian which is

r~I1 =

o =

o =

o =

The solutions to this are rl = 16.8 m and rl = 3.15 m; but since the person walked 5.3 m towardthe lamp we will assume they started at least that far away. Then the power output from the lightis

p = 47rr~I1 =47r(16.8m)2(1.13W /m2) = 4.01 x 103W.

E18-18 Energy density is energy per volume, or u = U /V. A wave front of cross sectional area Asweeps out a volume of V = Al when it travels a distance l. The wave front travels that distancel in a time t = l/v. The energy flow per time is the power, or p = U /t. Combine this with thedefinition of intensity, I = p / A, and

I=~A=

uAt

uVAt

uAI

At= = =uv.

~ Refer to Eq. 18-40, where the amplitude of the combined wave is

2ym cos(L\<t>/2),

where y m is the amplitude of the combining waves. Then

cos(~<P/2) = (1.65ym)/(2ym) = 0.825,

which has solution A<1> = 68.8° .

E18-20 Consider only the point x = O. The displacement y at that point is given by

y = Yrnl sin(ú.lt) + Yrn2 sin(ú.lt + 7r /2) = Yrnl sin(ú.lt) + Yrn2 COS(ú.lt).

This can be written asy = Ym(Al sinú)t + A2 cosú)t),

where Ai = Ymi/Ymo But if Y m is judiciously chosen, Al = cos{3 and A2 = sin {3, so that

y = y m sin(c.lt+ /3).

Since we then require A~ + A~ = I, we must have

y m = V(3.20 cm)2 + (4.19 cm)2 = 5.27 cm.

230

E18-21 The easiest approach is to use a phasor representation oí the waves.Write the phasor components as

= ymlcosl/>l!= yml sinl/>l!

-ym2cosl/>2,= ym2sin l/>2!

XI

yI

X2

y2

and then use the cosine law to find the magnitude of the resultant.The phase angle can be found from the arcsine of the opposite over the hypotenuse.

E18-22 (a) The diagrams for all times except t = 15ms should show two distinct pulses, firstmoving closer together, then moving farther apart. The pulses don not flip over when passing eachother. The t = 15 ms diagram, however, should simply be a flat line.

Use a program such as Maple or Mathematica to plot this.E18-23

Use a program such as Maple or Mathematica to plot this.E18-24

~ (a) The wave speed can be found from Eq. 18-19; we need to know the linear massdensity, which is JL = m/L = (0.122kg)/(8.36m) = O.O146kg/m. The wave speed is then given by

[F I (96.7N)v = y ~ = y (O.O146kg/m) = 81.4m/s.

(b) The longest possible standing wave will be twice the length of the string; so). = 2L = 16.7 m.(c) The frequency of the wave is found from Eq. 18-13, v = f)..

E18-26 (a) v = V(152N)I(7.16x 10-3kglm) = 146mls.(b) >.. = (213)(0.894m) = 0.596 m.(c) f = vi>.. = (146 mls)I(O.596m) = 245 Hz.

E18-27 (a) y = -3.9 cm.(b) y = (0.15m) sin[(0.79 rad/m)x + (13 rad/s)t].(c) y = 2(0.15m)sin[(0.79 rad/m)(2.3m)]cos[(13 rad/s)(0.16 s)] = -0.14m.

E18-28 (a) The amplitude is halí oí 0.520 cm, or 2.60mm. The speed is

v = (137 rad/s)/(1.14 rad/cm) = 1.20m/s.

(b) The nodes are (7rrad)/(1.14 rad/cm) = 2.76 cm apart.( c ) The velocity of a particle on the string at position x and time t is the derivative of the wave

equation with respect to time, or

Uy = -(0.520 cm)(137 rad/s) sin[(1.14 rad/cm)(1.47 cm)] sin[(137 rad/s)(1.36s)] = -0.582m/s.

231

~ (a) We are given the wave frequency and the wave-speed, the wavelength is found from

Eq. 18-13,(388m/s)v

>. = 7 = (622 Hz)

The standing wave has four loops, so from Eq. 18-45

= O.624m

is the length of the string.(b) We can just write it clown,

y = (1.90 mm) sin[(27r /0.624m)x] cos[(27r622 S-I )t].

E18-30 (a) In = nv/2L = (1)(250m/s)/2(0.150m) = 833 Hz.(b) >.. = v/1 = (348m/s)/(833 Hz) = O.418m.

v = vfFf¡¡ = ,fFL7rñ. Then In = nv/2L = n.¡FT4mL, soE18-31

h = (1)J(236N)/4(0.107kg)(9.88m)7.47 Hz,

and !2 = 2h = 14.9 Hz while !3 = 3h = 22.4 Hz.

E18-32 (a) v = JF7jl = v'FL?m = ..¡(122N)(1.48m)/(8.62x 10-3kg) = 145m/s.(b) >..1 = 2(1.48m) = 2.96m; >..2 = 1.48m.(c) h = (145m/s)/(2.96m) = 49.0 Hz; f2 = (145m/s)/(1.48m) = 98.0 Hz.

~ Although the tied end of the string forces it to be a node, the fact that the other endis loose means that it should be an anti-node. The discussion of Section 18-10 indicated that thespacing between nodes is always >./2. Since anti-nodes occur between nodes, we can expect that thedistance between a node and the nearest anti-node is >./4.

The longest possible wavelength will have one node at the tied end, an anti-node at the looseend, and no other nodes or anti-nodes. In this case >./4 = 120 cm, or >. = 480 cm.

The next longest wavelength will have a node somewhere in the middle region of the string. Butthis means that there must be an anti-node between this new node and the node at the tied endof the string. Moving from left to right, we then have an anti-node at the loose end, a node, andanti-node, and finally a node at the tied end. There are four points, each separated by >./4, so thewavelength would be given by 3>./4 = 120 cm, or >. = 160 cm.

To progress to the next wavelength we will add another node, and another anti-node. This willadd another two lengths of >./4 that need to be fit onto the string; hence 5>./4 = 120 cm, or >. = 100cm.

In the figure below we have sketched the first three standing waves.

232

E18-34 (a) Note that fn = nh. Then fn+l -fn = h. Since there is no resonant frequencybetween these two then they must differ by 1, and consequently h = (420 Hz) -(315 Hz) = 105 Hz.

(b) v = f>. = (105 Hz)[2(0.756m)] = 159m/s.

(a) >. = v/f and k = 360° />..P18-1

x = (55°)A/(36üO) = 55(353m/s)/360(493 Hz) O.109m.

(b) (J) = 360° f , sof/J = LAJt = (360O)(493 Hz)(1,12 x 10-38) = 199°,

P18-2 (¿) = (27r rad)(548 Hz) = 3440 radls; >. = vi f and then

k = (27r rad)/[(326m/s)/(548 Hz)] = 10.6 rad/m.

Finally, y = (1.12x 10-2m) sin[(10.6 rad/m)x + (3440 rad/s)t].

~ ( a) This problem really isn 't as bad as it might look. The tensile stress S is tension perunit cross sectional area, so

FS = -or F = SA.

A

We already know that linear mass density is JL = m/ L, where L is the length of the wire. Substituting

into Eq. 18-19,

[i ~ r---s---

V=y¡=ym¡Ly;;;¡¡AL).

But AL is the volume of the wire, so the denominator is just the mass density p.(b) The maximum speed of the transverse wave will be

{8 -{(720 x 106 Pa) = 300m/s.y p -Y" (7800 kg/m3)

v

P18-4 {a) f = (,)/211" = {4.08 rad/s)/{211" rad) = 0.649 Hz.(b) >. = v/f = {0.826m/s)/{0.649 Hz) = 1.27m.{c) k = {211" rad)/{1.27m) = 4.95 rad/m, so

y = (5.12 cm) sin[(4.95 rad/mx -(4.08 rad/s)t + 4>],

where <I> is determined by ( 4.95 rad/m)(9.60 x 10-2m) + <I> = (1.16 rad), or <I> = 0.685 rad.(d) F = JLV2 = (0.386kg/m)(0.826m/s)2 = 0.263m/s.

We want to show that dy/dx = Uy/v. The easy way, although not mathematically rigorous:P18-5

dy

dx

P18-6 The maximum value for Uy occurs when the cosine function in Eq. 18-14 returns unity.Consequently, um/Ym = úJ.

233

[E:f!J (a) The linear mass density changes as the rubber band is stretched! In this case,

mJL = L +¿lL .

The tension in the rubber band is given by F = kAL. Substituting this into Eq. 18-19,

v=f¡¡=J~.(b) We want to know the time it will take to travel the length of the rubber band, so

L+AL L+ALv=ort= .

=~

kAL

t vInto this we will substitute our expression for wave speed

t=(L+~L)/~. k~LWe have to possibilities to consider: either ~L « L or ~L » L. In either case we are only

interested in the part of the expression with L + ~L; whichever term is much largerthan the otherwill be the only significant part.

Then if ~L « L we get L + ~L ~ L and

t= (m(L+~L)~ ~V' , V~,

so that t is proportional to 1/JXL".But if AL » L we get L + AL ~ AL and

~~

v~

-rm

-yk'

so that t is constant.

Pl8-8 (a) The tension in the rope at some point is a function of the weight of the cable beneathit. Ifthe bottom ofthe rope is y = O, then the weight beneath some point y is W = y(m/L)g. Thespeed ofthe wave at that point is v = VT/(m/L)= vy(M/L)g/(m/L)= .¡¡¡y.

(b) dy/dt = .¡¡¡y, so

dydt =

iy=2JL79.t =

m'

fL .

lo "m

(c) No.

P18-9 (a) M = jJl,dx, so ¡ L 1M = kxdx = -kL2,

o 2

234

Then k = 2M~(b) v = vFIJL = VF7kZ, then

dt = .¡ki1F dx ,

P18-10 Take a cue from pressure and surface tension. In the rotating non-inertial reference framefor which the hoop appears to be at rest there is an effective force per unit length acting to push oneach part of the loop directly away from the center. This force per unit length has magnitude

V2 V2= (~m/~L)-;: = JL-;:'

~F~L

There must be a tension T in the string to hold the loop tQgether. Imagine the loop to be replacedwith two semicircular loops. Each semicircular loop has a diameter part; the force tending to pulloffthe diameter section is (~F/~L)2r = 2JlV2. There are two connections to the diameter section,so the tension in the string must be half the force on the diameter section, or T = JlV2.

The wave speed is Vw = JT7¡i = v.Note that the wave on the string can travel in either direction relative to an inertial observer.

One wave will appear to be fixed in space; the other will move around the string with twice the

speed of the string.

~ If we assume that Handel wanted his violins to play in tune with the other instrumentsthen all we need to do is find an instrument from Handel 's time that will accurately keep pitch overa period of several hundred years. Most instruments won 't keep pitch for even a few days because oftemperature and humidity changes; some (like the piccolo?) can 't even play in tune for more thana few notes! But if someone found a tuning fork...

Since the length of the string doesn't change, and we are using a string with the same massdensity, the only choice is to change the tension. But f cx: v cx: .JT , so the percentage change in the

tension of the string is

ff2 -fi2

fi2

Tf-Ti

Ti

I= , .~- ~ --, " = 8.46 %.= (440 HZ)2 -(422.5 HZ)2

P18-12

P18-13 (a) The point sources emit spherical wavesj the solution to the appropriate wave equation

is found in Ex. 18-14:Ay~ = -sin(kri -(..;t).ri

If ri is sufficiently large compared to A, and rl ~ r2, then let rl = r- 8r and r2 = r -8r;

A A-+-~rl r2

2A

r

with an error of order (r5r/r)2. So ignore it.Then

yl + y2 ~ ~ [sin(krl -úJt) + sin(kr2 -úJt)] ,r

= ~ sin(kr -úJt) cos ~(rl -r2),

2A kYm = -;:-COS2(rl -!2).

(b) A maximum (minimum) occurs when the operand of the cosine, k(rl -r2)/2 is an integer

multiple of 7r (a half odd-integer multiple of 7r)

235

P18-14 The direct wave travels a distance d from S to D. The wave which refiects off the originallayer travels a distance V d2 + 4H2 between S and D. The wave which refiects off the layer afterit has risen a distance h travels a distance vld2 + 4(H + h)2. Waves will interfere constructively ifthere is a difference of an integer number of wavelengths between the two path lengths. In otherwords originally we have

v ([2 + 4H2 -d = n>..,

and later we have destructive interference so

Vd2 +4(H + h)2- d = (n+ 1/2»...

We don 't know n, but we can subtract the top equation from the bottom and get

Vd2 + 4(H + h)2 -V d2 + 4H2 = AI2

~ The wavelength is

>.. = v/ f = (3.00x 108m/s)/(13.0x 106Hz) = 23.1 m.

The direct wave travels a distance d from S to D. The wave which reflects off the originallayertravels a distance v d2 + 4H2 between S and D. The wave which reflects off the layer one minutelater travels a distance Vd2 + 4(H + h)2. Waves will interfere constructively if there is a differenceof an integer number of wavelengths between the two path lengths. In other words originally wehave

vd2 + 4H2- d = nlA,

and then one minute later we have

vd2 + 4(H + h)2 -d = n2>..

We don't know either ni or n2, but we do know the difference is 6, so we can subtract the topequation from the bottom and get

..¡d2 +4(H + h)2- ..¡d2 + 4H2 = 6>.

Between the second and the third lines we factored out d2 + 4H2; that last line is from the binomialexpansion theorem. We put this into the previous expression, and

.¡d2+4(H+h)2-.¡d2+4H2 = 6>.,/ 4H "\

= 6>.,

4Hh

vál + 4H26>..=

236

Now what were we doing? We were trying to find the speed at which the layer is moving. We knowH, d, and >.; we can then find h,

h = 6(23.1 m)4(510 x 103m) v' (230 x 103mr + 4(510 x 103m)2 = 71.0 m.

The layer is then moving at v = (71.0m)/(60s) = 1.18m/s.

P18-16 The equation of the standing wave is

y = 2ym sin k X cos (J)t.

The transverse speed of a point on the string is the derivative of this, or

Uy = -2ymúJsink x sinúJt,

this has a maximum value when (AJt -7r /2 is a integer multiple of 7r .The maximum value is

Um = 2ym(AJsin kx.

Each mass element on the string dm then has a maximum kinetic energy

{dm/2)um2 = Ym2(AJ2 sin2 k x dm.dKm

Using dm = Jl.dx, and integrating over one loop from k x = O to k x = 7r , we get

Km = Ym2(.,)2JL/2k = 21T2ym2 fJLv.

P18-17 (a) For 100% reflection the amplitudes of the incident and reflected wave are equal, orAi = Ar, which puts a zero in the denominator of the equation for SWR. If there is no reflection,Ar = 0 leaving the expression for SWR to reduce to Ai/Ai = 1.

(b) Pr/Pi = A~/A~. Do the algebra:

Ai +Ar = SWR,

= SWR(Ai -Ar),

= Ai(SWR -1),

= (SWR- l)/(SWR + 1).

Ai-Ar

Ai+Ar

Ar(SWR + I)

Ar/Ai

Square this, and multiply by 100.

P18-18 Mea.sure with a ruler; I get 2Amax = 1.1 cm and 2Amin = 0.5 cm.(a) SWR = (1.1/0.5) = 2.2(b) (2.2 -1)2/(2.2 + 1)2 = 0.14%.

= Asink1(x -Vlt),

= B sin k2(X -V2t),

= Csink1(x + Vlt),

yi

Yt

Yr

where the subscripts i, t, and r refer to the incident, transmitted, and reflected waves respectively.

237

Apply the principle of superposition. Just to the left of the knot the wave has amplitude yi + Yrwhile just to the right of the knot the wave has amplitude yt. These two amplitudes must line upat the knot for a1l times t, or the knot wi1l come undone. Remember the knot is at x = O, so

Yi +Yr

Asin k1 ( -Vlt) + C sink1 ( +Vlt)

-A sin k1vlt + Csink1vlt

= yt!

= Bsin k2(-V2t),

= -Bsin k2v2t

We know that k1vl = k2v2 = (.o), so the three sin functions are all equivalent, and can be canceled.This leaves A = B + C.

(b) We need to match more than the displacement, we need to match the slope just on eitherside of the knot. In that case we need to take the derivative of

yi + Yr = Yt

with respect to x, and then set x o. First we take the derivative,

d-(Yi + Yr) =dx

k1A cos k1 (X -Vlt) + k1 C cos k1 (X + Vlt) =

d

;¡; (Yt) ,

k2Bcosk2(x -V2t),

and then we set x = O and simplify,

klA cos kl ( -Vlt) + kl C cos k1 ( +Vlt)

klAcosklvlt + klCcosklvlt

k2B cos k2( -V2t),

k2B cos k2v2t.

-

=

This last expression simplifies like the one in part (a) to give

k1(A+ C) = k2B

We can combine this with A = B + C to solve for C ,

k1(A + O)

O(k1 + k2)

O

=-

~

k2(A- O),

A(k2 -k1),

A~k1 + k2.

If k2 < k1 C will be negative; this means the reflected wave will be inverted.

P18-20

P18-21 Find the wavelength from

A = 2(0.924m)/4 = 0.462 m.

Find the wavespeed from

v IA = (60.0 Hz)(0.462m) = 27.7m/s.

Find the tension from

F=¡¡,V2 = (0.0442 kg)(27.7m/s)2/(0.924m) 36.7N.

238

P18-22 (a) The frequency of vibration f is the same for both the aluminum and steel wires; theydon not, however, need to vibrate in the same mode. The speed of waves in the aluminum is Vl,that in the steel is V2. The aluminum vibrates in a mode given by nl = 2L1f /Vl, the steel vibratesin a mode given by n2 = 2L2f /V2. Both nl and n2 need be integers, so the ratio must be a rationalfraction. Note that the ratio is independent of f, so that Ll and L2 must be chosen correctly forthis problem to work at all!

This ratio is

(b) There are three nodes in the aluminum and six in the steel. But one of those nodes is shared,and two are on the ends of the wire. The answer is then six.

2?!:1

E19-1 (a) v = fA = (25 Hz)(0.24m) = 6.0m/s.(b) k = (27r rad)/(0.24m) = 26 rad/m; (¿) = (27r rad)(25 Hz) = 160 rad/s. The wave equation is

8 = (3.0x 10-3m) sin[(26 rad/m)x + (160 rad/s)tl

E19-2 (a) [~P]m = 1.48Pa.(b) f = (3347r rad/s)/(27r rad) = 167 Hz.(c) >. = (27r rad)/(1.077r rad/m) = 1.87m.(d) v = (167 Hz)(1.87m) = 312m/s.

~ (a) The wavelength is given by >.. = v/f = (343m/s)/(4.50xlO6Hz = 7.62xlO-5m.

(b) The wavelength is given by >.. = v/f = (l500m/s)/(4.50xlO6Hz = 3.33xlO-4m.

E19-4

"pm".

>.min

Note: There is a typo; the mean free path should have been measured in "JLm" instead of

l.Ox lO-6m; fmax = (343m/s)/(l.OxlO-6m) = 3.4x lO8Hz.

(a) A = (240m/s)/(4.2x lO9Hz) = 5.7x lO-8m.E19-5

E19-6 (a) The speed of sound is

v = (331 m/s)(6.21 x 10-4 mi/m) = 0.206 mi/s.

In five seconds the sound travels (0.206 mi/s)(5.0s) = 1.03 mi, which is 3% too large.(b) Count seconds and divide by 3.

~ Marching at 120 paces per minute means that you move a foot every half a second. Thesoldiers in the back are moving the wrong foot, which means they are moving the correct foot halfa second later than they should. If the speed of sound is 343 m/s, then the column of soldiers mustbe (343m/s)(0.5s) = 172m long.

E19-8 It takes (300m)/(343m/s) = 0.87s for the concert goer to hear the music after it has passedthe microphone. It takes (5.0x106m)/(3.0x108m/s) = 0.017s for the radio listener to hear the musicafter it has passed the microphone. The radio listener hears the music first, 0.85 s before the concertgoer.

E19-9 x/vp = tp and x/vs = ts; subtracting and rearranging,

x = ~t/[l/vs -l/vp] = (1808)/[1/(4.5 km/s) -1/(8.2 km/s)] = 1800 km.

E19-10 Use Eq. 19-8, Sm = [6.p]m/kB, and Eq. 19-14, v = VJirp;;. Then

[6.p]m = kBsm = kv2pOSm = 27rfvposmo

Insert into Eq. 19-18, and= 27r2Pvf2sm2,

~ If the source emits equally in all directions the intensity at a distance r is given by theaverage power divided by the surface area of a sphere of radius r centered on the source.

The power output of the source can then be found from

p = [A = I(47rr2) = (197xl0-6W/m2)47r(42.5m)2 = 4.47W.

240

E19-12

= 3.75 x lO-8m.

Use the results of Exercise 19-10.

(1.13 x 10-6W /m2)Bm = 27r2(1.21 kg/m3)(343 m/s)(3i3 HZ)2

E19-13 U = IAt = (1.60x 10-2W /m2)(4. 70x 10-4m2)(3600s) = 2.71 x 10-2W.

E19-14 Invert Eq. 19-21:11/ 12 = 10(1.00dB)/10 = 1.26.

~ (a) Relative sound level is given by Eq. 19-21,

I I'SLl -SL2 = 10 lag -2 ar -2 = 10(8Ll-8L2)/lO

I2 I2 ,

so if ~SL = 30 then /1/ 12 = 1030/10 = 1000.(b) Intensity is proportional to pressure amplitude squared according to Eq. 19-19j so

~Pm,l/ ~Pm,2 = ,ffJI;; = v'1OOO = 32.

E19-16

is then

We know where her ears hurt, so we know the intensity at that point. The power output

p = 47r(1.3m)2(1.0W/m2) = ~lW.

This is less than the advertised power .

E19-17 Use the results oí Exercise 18-18, I = uv. The intensity is

I = (5200W)/47r(4820m)2 = 1.78x 10-5W /m2,

so the energy density is

u = I/v= (l.78xlO-5W/m2)/(343m/s)= 5.l9xlO-8J/m3.

E19-18 12 = 2/1, since I cx 1/r2 then r~ = 2r~. Then

yf2(D -51.4 m),

yf2(51.4 m),176m.

D =

D(h-l) =

D =

~ The sound level is given by Eq. 19-20,

ISL = lOlog "1;:;

where lo is the threshold intensity of 10-12 W /m2. Intensity is given by Eq. 19-19,

I=~2pv

If we assume the maximum possible pressure amplitude is equal to one atmosphere, then

I = ~ = (1.01 x 105Pa)2 = 1.22 x 107W /m2.

2pv 2(1.21kg/m3)(343m/s)

241

The sound level would then be

I 1.22 x 107W /m2SL = 101ogT = 101og 2 = 191 dB

.Lo (10-12 W /m )

E19-20 Let one person speak with an intensity 11. N people would have an intensity NI1.ratio is N, so by inverting Eq. 19-21,

The

N = 1Ü(15dB)/10 = 31.6,

so 32 people would be required.

E19-21 Let one leafrustle with an intensity 11. N leaves would have an intensity NI1. The ratiois N, so by Eq. 19-21,

SLN = (8.4 dB) + 10 log{2. 71 x 105) = 63 dB.

E19-22 Ignoring the finite time means that we can assume the sound waves travels vertically,which considerably simplifies the algebra.

The intensity ratio can be found by inverting Eq. 19-21,

/1/ 12 = 10(30dB)/10 = 1000.

But intensity is proportional to the inverse distance squared, so 11/12 = (r2/r1)2, or

r2 = (115m),¡-(i"{Xjjj = 3640 m.

~ A minimum will be heard at the detector if the path length difference between thestraight path and the path through the curved tube is half of a wavelength. Both paths involve astraight section from the source to the start of the curved tube, and then from the end of the curvedtube to the detector. Since it is the path difference that matters, we'll only focus on the part ofthe path between the start of the curved tube and the end of the curved tube. The length of thestraight path is one diameter, or 2r. The length of the curved tube is half a circumference, or 7rr .The difference is ( 7r -2)r. This difference is equal to half a wavelength, so

(7r-2)r = >./2,

>.

27r -4

(42.0 cm)

211" -4

= 18.4 cm.r = =

E19-24 The path length difference here is

V(3.75m)2 + (2.12m)2- (3.75 m) = 0.5578 m.

(a) A minimum will occur ifthis is equal to a halfinteger number ofwavelengths, or (n-l/2». =0.5578 m. This will occur when

= (n- 1/2)(615 Hz).(343m/s)f = (n -1/2)¡¡¡:5578m)

(b) A maximum will occur ir this is equal to an integer number or wavelengths, or n>. = 0.5578 m.This will occur when

= n(615 Hz).(343m/s)f = n(O.5578 m)

242

E19-25 The path length difference here is

v(24.4m + 6.10m)2 + (15.2m)2 -v(24.4m)2 + (15.2m)2 = 5.33m.

A maximum will occur ir this is equal to an integer number or wavelengths, or n>. = 5.33 m. Thiswill occur when

f = n(343m/s)/(5.33m) = n(64.4 Hz)

The two lowest frequencies are then 64.4 Hz and 129 Hz.

E19-26 The wavelength is >. = (343m/s)/(300 Hz) = 1.143m. This means that the sound maximawill be half of this, or 0.572 m apart. Directly in the center the path length difference is zero, butsince the waves are out of phase, this will be a minimum. The maxima should be located on eitherside of this, a distance (0.572m)/2 = 0.286m from the center. There will then be maxima locatedeach 0.572 m farther along.

E19-27 (a) h = v/2L and !2 = v/2(L -~L). Then

1-~=~ -!2

or ~L = L(l- l/r).(b) The answers are ~L = (0.80m)(1 -5/6) = O.133m; ~L = (0.80m)(1 -4/5) = O.160m;

~L = (0.80m)(1- 3/4) = O.200m; and ~L= (0.80m)(1 -2/3) = O.267m.

E19-28 The wavelength is twice the distance between the nodes in this case, so >. = 7.68 cm. The

frequency isf = (1520 m/s)/(7.68 x 10-2m) = 1.98 x 104Hz.

~ The well is a tube open at one end and closed at the other; Eq. 19-28 describes theallowed frequencies of the resonant modes. The lowest frequency is when n = 1, so h = v/4L. Weknow h; to find the depth of the we11, L, we need to know the speed of sound.

We should use the information provided, instead of looking up the speed of sound, because maybethe we11 is fi11ed with some kind of strange gas.

Then, from Eq. 19-14,

v = {fi = I {1.41 x 105 Pa)y p y {1.21 kg/m3) = 341 m/s.

The depth of the well is then

L = v/(4f¡) = (341m/s)/[4(7.20 Hz)] = 11.8m.

E19-30 (a) The resonant frequencies of the pipe are given by In = nv/2L, or

In = n(343m/s)/2(0.457m) = n(375 Hz).

The lowest frequency in the specified range is f3 = 1130 Hz; the other allowed frequencies in thespecified range are f 4 = 1500 Hz, and f5 = 1880 Hz.

243

~ The maximum reflected frequencies will be the ones that undergo constructive inter-ference, which means the path length difference will be an integer multiple of a wavelength. Awavefront will strike a terrace wall and part will reflect, the other part will travel on to the nextterrace, and then reflect. Since part of the wave had to travel to the next terrace and back, the pathlength difference will be 2 x 0.914 m = 1.83 m.

If the speed of sound is v = 343 m/s, the lowest frequency wave which undergoes constructiveinterference will be

(3~3m/s)vf = A = (1.83 m) .

Any integer multiple of this frequency will also undergo constructive interference, and will also beheard. The ear and brain, however, will most likely interpret the complex mix of frequencies as asingle tone of frequency 187 Hz.

= 187 Hz

E19-32 Assume there is no frequency between these two that is amplified. Then one of thesefrequencies is fn = nv/2L, and the other is fn+l = (n + 1)v/2L. Subtracting the larger from thesmaller, ~f = v/2L, or

L = v/2~f = (343m/s)/2(138 Hz -135 Hz) = 57.2 m.

E19-33 (a) v = 2Lf = 2(0.22 m) (920 Hz) = 405m/s.(b) F = V2J.t = (405 m/s)2(820x 10-6kg)/(0.220m) = 611N.

E19-34 f ()( v, and v ()( JF, so f ()( JF. Doubling f requires F increase by a factor of 4.

~ The speed of a wave on the string is the same, regardless of where you put your finger ,so f>. is a constant. The string will vibrate (mostly) in the lowest harmonic, so that >. = 2L, whereL is the length of the part of the string that is allowed to vibrate. Then

/2>.2 =

2/2L2 =

L2

So you need to place your finger 5 cm from the end.

E19-36 The open organ pipe has a length

Lo = v/2f¡ = (343m/s)/2(291 Hz) = 0.589 m.

The second harmonic ofthe open pipe has frequency 2!1; this is the first overtone ofthe closed pipe,so the closed pipe has a length

Lc = (3)v/4(2f¡) (3)(343m/s)/4(2)(291 Hz) = 0.442 m.

~ The unknown frequency is either 3 Hz higher or lower than the standard fork. A smallpiece of wax placed on the fork of this unknown frequency tuning fork will result in a lower frequencybecause f cx Jk1iñ. If the beat frequency decreases then the two tuning forks are getting closerin frequency, so the frequency of the first tuning fork must be above the frequency of the standardfork. Hence, 387 Hz.

244

E19-38 If the 8tring i8 too taut then the frequency is too high, or f = (440 + 4)Hz.T = 1/ f = 1/(444 Hz) = 2.25 x 10-38.

Then

E19-39 One of the tuning forks need to have a frequency 1 Hz different from another. Assumethen one is at 501 Hz. The next fork can be played against the first or the second, so it could havea frequency of 503 Hz to pick up the 2 and 3 Hz beats. The next one needs to pick up the 5, 7, and8 Hz beats, and 508 Hz will do the trick. There are other choices.

E19-40 I = vi>. = (5.5 mls)I(2.3m) = 2.39 Hz. Then

I' = I(v + vo)lv = (2.39 Hz)(5.5 mls + 3.3mls)I(5.5mls) = 3.8 Hz.

E19-42 Solve Eq. 19-44 for vs;

v s = (v+vo)!/!' -v = (343m/s+2.63m/s)(1602 Hz)/(1590 Hz) -(343m/s) = 5.24m/s.

E19-43 Vs = (14.7 Rad/8)(0.712m) = 10.5m/8.(a) The low frequency heard i8

f' = (538 Hz)(343m/8)/(343m/8 + 10.5m/8) = 522 Hz.

( a) The high frequency heard i8

f' = (538 Hz)(343m/8)/(343m/8 -10.5m/8) = 555 Hz,

E19-44 Salve Eq. 19-44 far Vs;

V s = V -vi i I' = (343 mis) -(343 mis) (440 Hz)i(444 Hz) = 3.1 mis.

E19-45

E19-46 The approaching car "hears"

E19-47 The departing intruder "hears"

245

The beat ftequency is 28.3 kHz -28.14 kHz = 160 Hz.

E19-48 (a) I' = (1000 Hz)(330m/s)(330 m/s + 10.0m/s) = 971 Hz.(b) I' = (1000 Hz)(330r:n/s)(330 m/s -10.0m/s) = 1030 Hz.(c) 1030 Hz -971 Hz = 59 Hz.

~ (a) The frequency "heard" by the wall is

(343m/s)+(O)I' = f ~ = (438 Hz)- = 464 Hzf -Vs (343m/s) -(19.3m/s)

(b) The wall then reflects a frequency of 464 Hz back to the trumpet player. Sticking with Eq.19-44, the source is now at rest while the observer moving,

= (464 Hz)~-+(19.3m/s) -"= 490 Hz

(343m/s) -(O)f'=f~

v-vs

E19-50 The body part "hears"

V+Vbf'=f v

This sound is reflected back to the detector which then "hears"

I" = I'~ = I.!!---+vbv -Vb v -Vb

Rearranging,I" -I

Vb/V -I" + I

so v ~ 2(lxlO-3m/s)/(l.3xlO-6) ~ l500m/s.

..,., 1 ~f

..'.'2¡'

E19-51 The wall "hears"

I' = I ~ (343m/s)+ (O) = 40.21 kHz= (39.2 kHz). -v -v s (343m/s) -(8.58m/s)

This sound is refl.ected back at the same frequency, so the bat "hears"

I. )~ = (40.21 kHz)S343m/s) + (8.58m/s), = 41.2 kHz.

(343m/s) -(O)f'=f~

v -.,v s

P19-1 (a) tair L/Vair and tm = L/v, so the difference is

6.t = L(l/vair -l/v)

(b) Rearrange the above result, and

L (0.120s)/[1/(343m/s) -1/(6420m/s)] 43.5 m.

P19-2 The 8tone falls for a time tl where y = gt~/2 i8 the depth of the well. Note y i8 po8itive inthi8 equation. The 8ound travel8 back in a time t~where v = y/t2 i8 the 8peed of 8ound in the well.t1 + t2 = 3.008,80

2y = g(3.00s -t2)2 =g[(9.00s2) -(6.00s)y/v + y2/V2],

or, using 9 = 9.81 m/s2 and v = 343 m/s,

y2 -(2.555 x 105 m)y + (1.039 x 107 m2) = 0, .

which has a positive solution y = 40.7 m.

246

12 = I1(rl/r2)2 = (962J1W /m2)(6.11 m/28.5m)2 = 44.2J1W /m2.

(C) We'll do this part first. The pressure amplitude is found from Eq. 19-19,

¡)..Pm = .¡2pvj = 2(1.21 kg/m3)(343 m/s)(962 x 10-6W /m2) = 0.894 Pa.

(b) The displacement amplitude is found from Eq. 19-8,

8m = ¡)..Pm/(kB),

where k = 27rf /v is the wave number. From Eq. 19-14 w know that B = pv2, SO

¡)..Pm (0.894Pa) = 1.64x 10-7m.s m- 27rfpv -27r(2090 Hz)(1.21kg/m3)(343m/s)

P19-4 (a) If the intensities are equal, then A Pm <X .¡-¡;ü, so

J~Pm]water -(998kg/m3)(1482m/~) = 59.9.(1.2 kg/m3)(343 m/s)[¡j.pm]air

{b) If the pressure amplitudes are equal, then I cx:cx: 1! pv , SO

Iwater (1.2 kg!m3)(343m!s) = 2.78xl0-'4,--

(998 kgim3)(1482 mis)Iair

P19-5 The energy is dissipated on a cylindrical surface which grows in area as r, so the intensity isproportional to l/r. The amplitude is proportional to the square root ofthe intensity, so Sm CX l/vr.

P19-6 (a) The first position corresponds to maximum destructive interference, so the waves are halfa wavelength out of phase; the second position corresponds to maximum constructive interference, sothe waves are in phase. Shifting the tube has in effect added half a wavelength to the path throughB. But each segment is added, so

A = (2)(2)(1,65 cm) = 6.60 cm,

and f = (343m/s)/(6.60 cm) = 5200 Hz.(b) Imin CX (Sl-S2)2, Imax CX (Sl +S2)2, then dividing one expression by tbe other and rearranging

we find~= ~+~= V90+VfP=2S2 ~-~ V90-VfP

P19-7 (a) I = P/47rr2 = (31.6W)/47r(194m)2 = 6.68x10-5W/m2.(b) P = IA = (6.68x10-5W/m2)(75.2x10-6m2) = 5.02x10-9W.( c) U = Pt = (5.02 x 10-9W) (25.0 min) ( 60.0 s/min) = 7.53 /l.J .

P19-8 Note that the reverberation time is logarithmically related to the intensity, but linearlyrelated to the sound level. As such, the reverberation time is the amount of time for the sound level

to decrease by~SL = 101og(10-6) = 60 dB.

Thent = (87 dB)(2.6s)/(60 dB) = 3.88

247

~ What the device is doing is taking all of the energy which strikes a large surface area andconcentrating it into a small surface area. It doesn 't succeedj only 12% of the energy is concentrated.We can think, however, in terms of power: 12% of the average power which strikes the parabolicrefiector is transmitted into the tube.

Ifthe sound intensity onthe refiector is 11, then the average power is P1 = I1A1 = 117rr~, whererl is the radius of the refiector. The average power in the tube will be P2 = O.12P1, so the intensityin the tube will be

T -~ -0.12l17rr~ -O 12l 1

.L2- -2 -.12

A2 7rr2 r2

Since the lowest audible sound has an intensity of lo = 10-12 W /m2, we can set l2 = lo as thecondition for "hearing" the whisperer through the apparatus. The minimum sound intensity at theparabolic reflector is

T 2.LO r2I1 = Q"l2 -;:¡.

Now for the whisperers. Intensity falls off a.s 1/d2, where d is the distance from the source. Weare told that when d = 1.0 m the sound level is 20 dB; this sound level ha.s an intensity of

I =101020/10 = 10010

Then at a distance d from the source the intensity must be

11 = 100.l (1 m)2

0-d2

This would be the intensity "picked-up" by the parabolic reflector. Combining this with the conditionfor being able to hear the whisperers through the apparatus, we have

or, upon some rearranging,

= 346 md = (.Jf2m)!:!. = (.Jf2m)~r2 (0.005 m)

P19-10 (a) A di8placement nade; at the center the particle8 have nawhere ta ga.(b) Thi8 8y8tem8 act8 like a pipe which i8 cla8ed at ane end.(c) vVB7p, 8a

T = 4(O.OO9)(6.96x lOSm).¡(l.Ox lOlOkg/m3)/(l.33x lO22Pa) = 228.

P19-11 The cork filing collect at pressure antinodes when standing waves are present, and theantinodes are each half a wavelength apart. Then v = j.>.. = j(2d).

P19-12 (a) f = v/4L = (343 m/s)/4(1.18m) = 72.7 Hz.(b) F = Ji,V2 = Ji,f2>.2, or

F = (9.57xl0-3kg/0.332m)(72.7 Hz)2[2(0.332m)]2 = 67.1 N.

248

~ In this problem the string is observed to resonate at 880 Hz and then again at 1320 Hz,so the two corresponding values of n must differ by 1. We can then write two equations

~

2L

and solve these for v. It is somewhat easier to first salve for n. Rearranging both equations, we get

(880 Hz)

n

v=-

2L

Combining these two equations we get

(880 Hz) ~.=

Now that we know n we can find v,

v = 2(0.300m)~ = 264m/s2

And, finally, we are in a position to find the tension, since

F = JLV2 = (0.652xl0-3kg/m)(264m/s)2 = 45.4 N.

P19-14 (a) There are five choices for the first fork, and four for the second. That gives 20 pairs.But order doesn't matter, so we need divide that by two to get a maximum of 10 possible beat

frequencies.(b) If the forks are ordered to have equal differences (say, 400 Hz, 410 Hz, 420 Hz, 430 Hz, and

440 Hz) then there will actually be only 4 beat frequencies.

P19-15 v = (2.25 x lO8m/s)/ sin(58.00) = 2.65 x lO8m/s.

P19-16 (a) h = (442 Hz)(343m/s)/(343m/s -31.3m/s) = 486 Hz, while

!2 = (442 Hz)(343m/s)/(343m/s + 31.3m/s) = 405 Hz,

so ~f = 81 Hz.(b) h = (442 Hz)(343m/s -31.3 m/s)/(343m/s) = 402 Hz, while

!2 = (442 Hz)(343m/s+ 31.3 m/s)/(343m/s) = 482 Hz,

so A! = 80 Hz.

~ The sonic boom that you hear is not from the sound given off by the plane when it isoverhead, it is from the sound given off before the plane was overhead. So this problem isn 't assimple as distance equals velocity x time. It is very useful to sketch a picture.

249

~

-

~

I

o

We can find the angle (j from the figure, we'll get Eq. 19-45, so

sin{} = !- = S330m/s)v.. I

= 0.833 or (} = 56.4°(396 mis)

Note that v8 is the speed of the source, not the speed of sound!Unfortunately t = 12 s is not the time between when the sonic boom leaves the plane and when

it arrives at the observer. It is the time between when the plane is overhead and when the sonicboom arrives at the observer. That 's why there are so many marks and variables on the figure. Xlis the distance from where the sonic boom which is heard by the observer is emitted to the pointdirectly overhead; X2 is the distance from the point which is directly overhead to the point wherethe plane is when the sonic boom is heard by the observer. We do have X2 = v8(12.0s). This lengthforms one side ofa right triangle HSO, the opposite side of this triangle is the side HO, which isthe height of the plane above the ground, so

h = x2tan(} = (343m/s)(12.0s)tan(56.4°) = 7150m.

P19-18 (a) The target "hears':

f'=fB~'v

This sound is refl.ected back to the detector which then "hears"

f - 1'- v -I .v+v

r- , -8,

v-v u-v'

(b) Rearranging,Ir -!8 ir -isV/v = ~ -.'Ir+18 2 18

where we have assumed that the source frequency and the reflected frequency are almost identical,so that when added Ir + 18 ~ 218.

I' = f ~ = {1030 Hz) {5470 km/h) + {94.6 km/h) = 1050 Hz

v -v s {5470 km/h) -{20.2 km/h)

~

250

(b) The reflected signal has a frequency equal to that of the signal received by the second suboriginally. Applying Eq. 19-44 again,

= 1070 Hz{5470 km/h) + {20.2 km/h)= {1050 Hz)(5470 km/h) -{94.6 km/h)I' = I ~

v-vs

P19-20 In this case v s = 75.2 km/h- 30.5 km/h = 12.4m/s. Then

I' (989 Hz)(1482m/s)(1482 m/s -12.4m/s) 997 Hz.

P19-21 There is no relative motion between the source and observer, so there is no frequency shiftregardless of the wind direction.

P19-22 (a) v s = 34.2m/s and Va = 34.2m/s, so

l' = (525 Hz)(343m/s + 34.2m/s)/(343m/s -34.2m/s) 641 Hz.

(b) v s = 34.2m/s + 15.3m/s = 49.5m/s and Va = 34.2m/s -15.3m/s18.9 mis, so

I' = (525 Hz)(343 mis + 18.9 mis)i(343 mis -49.5 mis) = 647 Hz.

(c) v s = 34.2m/s -15.3m/s = 18.9m/s and Va = 34.2m/s + 15.3m/s = 49.5m/s, so

I' = (525 Hz)(343m/s + 49.5 m/s)/(343m/s -18.9m/s) = 636 Hz.

251

E20-1 (a) t= x/v = (0.20 m)/(0.941)(3.00x 108m/8) = 7.1 x 10-108.(b) y = -gt2/2 = -(9.81 m/82)(7.1 x 10-108)2/2 = 2.5x 10-18m.

E20-2 L = LoV1- U2/C2 = (2.86m)v1- (0.999987)2 = 1.46 cm.

~ L = LoVl- U2/C2= (1.68m)vl- (0.632)2 = 1.30m.

E20-4 Solve ~t = ~to/Vl- U2/C2 for u:

= 2.97x lO8m/s.

~ We can apply ~x = v~t to find the time the particle existed before it decayed. Then

= 3.53 x 10-12 s.

The proper lifetime of the particle is

E20-6 Apply Eq. 20-12:

E20-7 (a) L = Lo..¡l- u2/c2 = (130m)..¡1- (0.740)2 = 87.4m(b) ~t = L/v = (87.4 m)/(0.740)(3.00 x 108m/8) = 3.94x 10-78.

E20-8 ~t = ~to/ ..¡1 -U2 / c2 = (26 ns) ..¡1 -(0.99)2 = 184 ns. Then

L = v~t = (0.99) (3.00 x 108m/8)(184x 10-98) = 55m.

E20-10 Invert Eq. 20-15 to get /3 = V1 -1/'Y2.(a) /3 = V1 -1/(1.01)2 = 0.140.

(b) /3 = V1 -1/(10.0)2 = 0.995.

(c) /3 = V1 -1/(100)2 = 0.99995.

(d) /3 = V1 -1/(1000)2 = 0.9999995.

252

E20-9 (a) Vg = 2v = (7.91 + 7.91) km/s = 15.82 km/s.(b) A relativistic treatment yields Vr = 2v/(1 + v2/c2). The fractional error is

~ The distance traveled by the particle is (6.0 y)c; the time required for the particle totravel this distance is 8.0 years. Then the speed of the particle is

~x

~tv=

(6.0 y)c 3= (&OY) = ¡c.

The speed parameter ,8 is given byv 2c 3

/3=-=1--c c -4.

E20-12 '"Y = 1/ VI -(0.950)2 = 3.20. Then

(3.20)[(1.00 x 105m) -(0.950)(3.00 x 108m/s)(2.00 x 10-48)] = 1.38x 105m,

(3.20) [(2.00 x 10-48) -(1.00x105m)(0.950)/(3.00x108m/8)] = -3.73x10-4s.

x'

t'

==

E20-13 (a) 'Y = 1/ .¡1 -(0.380)2 = 1.081. Then

x' = (1.081)[(3.20x 108m) -(0.380)(3.00x 108m/s)(2.50s)] = 3.78x 107m,

t' = (1.081)[(2.50 s) -(3.20 x 108m)(0.380)/(3.00 x 108m/s)] = 2.26 s.

(b) 'Y = 1/ v'1 -(0.380)2 = 1.081. Then

x'

t'

= (1.081)[(3.20x108m) -(-0.380)(3.00x 108m/8)(2.508)) = 6.54x108m,

= (1.081)[(2.508) -(3.20 x 108m)( -0.380)/(3.00 x 108m/8)) = 3.148.

E20-14

E20-15 (a) v~ = ( -u)/(1 -O) and v~ = cV1 -U2/C2.(b) (V~)2 + (V~)2 = U2 + C2 -U2 = (:2 .

E20-16 v' = {0.787c + O.612c)f[1 + {0.787){0.612)] = O.944c.

~ ( a) The first part is easyj we appear to be moving away from A at the same speed as Aappears to be moving away from us: 0.347 c.

(b) Using the velocity transformation formula, Eq. 20-18,

E20-18 v' = (0.788c -0.413c)/[1 + (0. 788)( -0.413)] = 0.556c.

E20-19 (a) 7 = 1/ VI -(0.8)2 = 5/3.

-3(0.8c)

-5[1 -(O)]

(0) -(0.8c) :1 -(0)

12,v"' = 25c,

-4--5c.

=

,vtJ

=

Vx

/(1- uVy/c2)

v -uy1- UVy/C2 -

Then v' = cV( -4/5)2 + (12/25)2 = 0.933c directed (J = arctan( -12/20) = 31° East of South.

253

(b) 'Y = 1/ .¡1 -(0.8)2 = 5/3.

4

=+5c,

Vx -u1- uvx/& -

v y

1(1- uvx/&)

Iv x =

12

25c.

Iv y =

(O) -( -0.8c)

1 -(O)

-3(0.8c) --5[1 -(0)] -

Then v' = cy(4/5)2 + (12/25)2 = O.933c directed (J = arctan(20/12) = 59° West of North.

E20-20 This exercise should occur in Section 20-9.(a) v = 21r(6.37x106m)c/(ls)(3.00x108m/s) = O.133c.(b) K = {'Y -1)mc2 = (1/ .¡1- (0.133F -1)(511 keV) = 4.58 keV.(c) Kc = mv2/2 = mc2(v2/c2)/2 = (511 keV)(0.133)2/2 = 4.52 keV. The percent error is

(4.52- 4.58)/(4.58) = -1.31%.

E20-21 ~L = L' -Lo so

~L = 2(6.370xlO6m)(l- y -{29.8 x 103m/s)2 /{3.00 x 108m/s)2) = 6.29 x 10-2m.

E20-22 (a) ~L/Lo = 1- L' /Lo so

51 x 10-12.~L {1- VI -{522 m/s)2 /{3.00 x 108m/s)2) =

(b) We want to 8o1ve ~t -~t' = 1 JL8, or

1 Jl,S = ~t(l -1/ ..¡1- (522 m/s)2 /(3.00 x 108m/s)2),

which has solution ~t 6.61 x 1058. That'8 7.64 day8.

~ The length of the ship as measured in the "certain" reference frame is

L = LoVl-V2/C2 = (358m)vl- (0.728)2 = 245 m.

In a time ~t the ship will move a distance Xl = Vl~t while the micrometeorite will move a distanceX2 = V2~t; since they are moving toward each other then the micrometeorite will pass then shipwhen Xl + X2 =L. Then

At = L/(Vl + V2) = (245m)/[(0.728 + 0.817)(3.00x 108m/8)] = 5.29 x 10-18.

This answer is the time measured in the "certain" reference frame. We can use Eq. 20-21 to findthe time as measured on the ship,

~t = ~t' + U~X'/C2 = (5.29x 10-7 s) + (0.728c)(116m)/c2 = 1.23x 10-6s.

V1 -(0.728)2Vl- U2/C2

E20-24 (a) "1 = 1/ ..¡1- (0.622)2 = 1.28.(b) ~t = (183 m)/(0.622)(3.00x 108m/s) = 9.81 x io-7s. On the clock, however,

~t' = ~t/'Y {9.81 x 10-7 s)/{1.28) = 7.66 x 10-7 s.

254

E20-25 (a) ~t = (26.0 ly)/(0.988)(1.00 ly/y) = 26.3 y.(b) The signal takes 26 years to return, so 26 + 26.3 = 52.3 years.(c) ~t' = (26.3 Y)Vl -(0.988)2 = 4.06 y.

E20-26 (a) '1 = (1000 y)(l y) = 1000;

v = cV1 -1/'12 ~ c(l -1/2y) = 0.9999995c

(b) No.

E20-27 {5.6lx lO29 MeV/c2)c/{3.00xlO8m/s) = l.87xlO21 MeV/c.

E20-28 p2 = m2c2 = m2v2/(1 -V2/C2), so 2V2/C2 = 1, or v = V2c.

~ The magnitude of the momentum of a relativistic particle in terms of the magnitude ofthe velocity is given by Eq. 20-23,

m v

p=~.The speed parameter, {3, is what we are looking for, so we need to rearrange the above expressionfor the quantity v/c.

mv/c

.,f1=""V'TC'J'

m,8

~'

~

p/c =

~ =c

mc

pmc

p

={3

VI/ {32 "'- I.=

Rearranging,

mc2VI/ (:]2 -1,=

1:82 -i,=

1

{3'

{3

-

~

(a) For the electron,

{3 = (12;5 MeV /c)c = 0.999.

..¡(0.511 MeV /c2)2c4 + (12.5 MeV /C)2C2

(b) For the proton,

:,a = (12.5 MeV /c)c = 0.0133.

..¡(938 MeV /C2)2C4 + (12.5 MeV /c)2c2

255

E20-30 K = mc2('Y -1), so 'Y = 1 + K/mc2. /3 = V1-1/'Y2.(a) 'Y = 1 + (1.0 keV)/(511 keV) = 1.00196. /3 = 0.0625c.(b) 'Y = 1 + (1.0 MeV)/(0.511 MeV) = 2.96. /3 = 0.941c.(c) 'Y = 1 + (1.0 GeV)/(0.511 MeV) = 1960. /3 = 0.99999987c.

We rearrange this to salve for /3 :::; v / c,

2

It is actually much ea.sier to find 1', since

I-,"1 -..¡1- V2/C2

so K = "Ymc2 -mc2 implies

K+mc2mc21=

(a) For the electron,

= 0.9988,

and

(b) For the proton,

2(938 MeV /C2)C2

fJ=yl-= 0.0145,

(10 MeV) + (938 MeV /c2)c2

and

(b) For the alpha particle,

=0.73,

and

E20-32 'Y = 1/ V1 -(0.99)2 = 7.089.(a) E = 'Ymc2 = (7.089)(938.3 MeV) = 6650 MeV. K = E -mc2 = 5710 MeV.

(938.3 MeV /c2)(0.99c)(7.089) = 6580 MeV /c.(b) E = 'Ymc2 = (7.089)(0.511 MeV) = 3.62 MeV. K = E -mc2 = 3.11 MeV.

(0.511 MeV /c2)(0.99c)(7.089) = 3.59 MeV /c.

p = mv"Y =

p = mv')' =

256

E20-33 ~m/~t = (1.2x1041W)/(3.0x108m/s)2 = 1.33x1024kg/s, which is

~ = (1.33x1024kg/s)(3.16x107s/y) = 21.1~t (1.99 x 103°kg/sun)

E- mc2 = 2mc2, then E = 3mc2, SO 7 = 3, and

v = cv'l -1/72 = cv'l -1/(3)2 = O.943c.

E20-34 (a) If K

(b) If E = 2mc2, then 7 = 2, and

v = cV1 -1/72 = cV1 -1/(2)2 = O.866c.

mc2.--mc2 = mc2 ( (1- .82) -1/2

\11- v2/c2

We want to expand the 1 -.82 part for small .8,

K

(1 -{32) -1/2

Inserting this into the kinetic energy expression,

1 3K = -mc2f32 + -mc2f34 +

2 8

But /3 V/C,80V41 3K = -mv2 + -m + ...

2 8 C2(b) We want to know when the error because of neglecting the second ( and higher) terms is 1 %;

or3 V4 1 2 3 ( V ) 2

0.01 = (-m-)/(-mv ) = --8 C2 2 4 c

This will happen when v/c = V(0.01)4/3) = 0.115.

E20-36 Kc = (lOOOkg)(20m/s)2/2 = 2.0xlOsJ. The relativistic calculation is slightly harder:

Kr = (lOOOkg)(3xlO8m/s)2(l/..¡l-(20m/s)2/(3xlO8m/s)2-l),

Rj (lOOOkg) [~(20m/S)2 + ~(20m/s)4/(3XlO8m/s)2 + ...] ,

= 2.0x lOsJ + 6.7x lO-lOJ.

~ Start with Eq. 20-34 in the form

E2 (pc) 2 + (mc2)2

The rest energy is mc2, and if the total energy is three times this then E = 3mc2, so

(3mc2)2 = (pc)2 + (mc2)2,

8(mc2)2 = (pC)2,

h mc -p.

257

E20-38 The initial kinetic energy is

2mv2

-(1+v2/c2)2'

2v

1+v2/c2

1

Ki=2m

The final kinetic energy is

E20-39 This exercise is much more involved than the previous one!

The initial kinetic energy is

Kj = mc2 2-mcF(~)2/:; ,

mc2(l + v2/c2) 2-mc~2/C2)2 -4V2/C2 ,

m(c2 + V2) m(c2 -V2)1- v2/c2 -1- v2/c2 ,

2mv2

1-v2/c2.

The final kinetic energy is

m¿ 22--2mc,

V;-(V~)2/-:;

Kf =

mc22 I.. 1-?/-?\In -021-2\-2mc2,

y1- (V2/C2)(2 -V2/C2)

mc22 1 2 / 2- 2mc2,

-v c

2~-2~1-v2/c2 1-v2/c2 ,

2mv2

l-v2/c2'

E20-40 For a particle with mass, 'Y = K/mr? + 1. For the electron, 'Y = {0.40)/{0.511) + 1 = 1.78.

For the proton, 'Y = {lO)/{938) + 1 = 1.066.For the photon, pc = E. For a particle with mass, pc = .¡{K + mc2)2 -m2c4. For the electron,

pc = [{0.40 MeV) + {0.511 MeV)]2 -{0.511 MeV)2 = 0.754 MeV.

For the proton,

pc = V[(10 MeV) + (938 MeV)]2 -(938 MeV)2 = 137 MeV.

(a) Only photons move at the speed of light, so it is moving the fastest.(b) The proton, since it has smallest value for 'Y .( c ) The proton has the greatest momentum.(d) The photon has the least.

258

~ Work is change in energy, so

w = mc2/vl- (Vf/C)2- mc2/vl- (Vi/C)2,

(a) Plug in the numbers,

w = {0.511 MeV){1/v1 ~{0.19)2-1/Vl-{0.18)2) = 0.996 keV.

(b) Plug in the numbers,

w = {0:511 MeV)(11 ..¡1-(0.99)2 -11 ..¡1 -(0.98)2) = 1.05 MeV.

E20-42 E = 2¡moc2 = mc2, SO

m = 2lmo = 2(1.30 mg)/ VI -(0.580)2 = 3.19 mg.

E20-43 (a) Energy conservation requires Ek = 2E1r, or mkc2 = 2'"Ym1rC2, Then

'Y = (498 MeV)12(140 MeV) = 1.78

This corresponds to a speed of v = cV1 -11(1.78)2 = O.827c.(b) 'Y = (498 MeV + 325 MeV)I(498 MeV) = 1.65, so v = cV1-11(1.65)2 = O.795c.

(c) The lab frame velocities are then

(0.795) +( -0.827),VI = ~+ (O. 795)( -0.827) c = -0.0934c,

(0.795) + (0.827),V2 = 1 + (O. 795)(0.827) c = 0.979c,

The corresponding kinetic energies are

K1 = (140 MeV)(1/v1- (-0.0934)2-1) = 0.614 MeV

K1 = (140 MeV)(1/ VI -(0.979)2- 1) = 548 MeV

E20-44

P20-1 (a) ')' = 2, so v = V1 -1/(2)2 = O.866c.

(b)')'=2.

P20-2 (a) Classically, v' = (0.620c) + (0.470c) = 1.0gc. Relativistically,

(b) Cla.ssically, v' = (0.620c) + (-0.470c) = 0.150c. Relativistically,

, (0.620c) + ( -0.470c)v = 1 + (0.620)( -0.470) = 0.211c.

259

P20-3 (a) '"Y = 1/ V1 -(0.247)2 = 1.032. Use the equations from Table 20-2.

~t = (1.032)[(0) -(0.247)(30.4x103m)/(3.00x108m/s)] = -2.58x10-5s.

(b) The red flash appears to go first.

P20-4 Once again, the "pico" should have been a J1,."'I' = 1/ VI -(0.60)2 = 1.25. Use the equations from Table 20-2.

At = (1.25)[(4.0xI0-6s) -(0.60)(3.0xI03m)/(3.00xI08m/s)] = -2.5xI0-6s.

~ We can choose our coordinate system so that u is directed along the x axis without anyloss of generality. Then, according to Table 20-2,

~x' = ')'(~x -u~t),

~y' = ~y,

~z' = ~z,

c~t' = ')'(c~t -u~x/c).

Square these expressions,

(~X')2 = '"Y2(~X -u~t)2 = '"Y2 ((~X)2 -2u(~x)(~t) + (~t)2) ,

(~y')2 = (~y)2,

(~Z')2 -(~Z)2,C2(~t')2 = '"Y2(C~t -U~X/C)2 = '"Y2 (C2(~t)2 -2u(~t)(~x) + U2(~X)2/C2) .

We'll add the first three equations and then subtract the fourth. The left hand side is the equal to

(~X')2 + (~y')2 + (~Z')2 -C2(~t')2,

while the right hand side will equal

'"Y2 ((~X)2 + U2(~t)2 -C2(~t)2 -U2/C2(~X)2) + (~y)2 + (~Z)2,

which can be rearranged as

'"Y2 (1- u2/c2) (~X)2 + '"Y2 ( U2 -c2) (~t)2 + (~y)2 + (~Z)2,

'"Y2 (1- U2/C2) (~X)2 + (~y)2 + (~Z)2 -C2'"Y2 (1- u2/c2) (~t)2.

But1"(2 = 1 -U2/C2 ,

so the previous expression will simplify to

(~X)2 + (~y)2 + (~Z)2 -C2(~t)2.

P20-6 (a) Va: = [(0.780c) + (0.240c)]![1 + (0.240)(0.780)] = 0.859c.(b) Va: = [(0) + (0.240c)]![1 + (0)] = 0.240c, while

Vy = (0.780c)vi -(0.240)2fl1 + (0)] = 0.757c.

Then v = V(0.240c)2 + (0.757c)2 = 0.794c.(b) v~ = [(0) -(0.240c)]![1 + (0)] = -0.240c, while

v~ = (0.780c)Vi~ (0.240)2![1 + (O)] =0.757c.

Then v' = V( -0.240c)2 + (0.757c)2 = 0.794c.

260

~ If we look back at the boost equation we might notice that it looks very similar to therule for the tangent of the sum of two angles. It is exactly the same as the rule for the hyperbolic

tangent,tanh al + tanh a2

tanh(~l +~2) =1 h h .

+ tan ~1 tan ~2

This means that each boost of /3 = 0.5 is the same as a "hyperbolic" rotation of ~r where tanh ~r =0.5. We need only add these rotations together until we get to ~f, where tanh~f = 0.999.

~f = 3.800, and ~R = 0.5493. We can fit (3.800)/(0.5493) = 6.92 boosts, but we need an integral

number, so there are seven boosts required. The final speed after these seven boosts will be 0.9991c.

P20-8 (a) Ir ~x' = O, then ~x = u~t, or

u = (730 m)/(4.96x 10-68) = 1.472x 108m/8 = 0.491c.

(b) '"1 = 1/v'1- (0.491)2 = 1.148,

At' = (1.148)¡(4.96 x 10-68) -(0.491) (730 m)/(3 x 108)] = 4.32 x 10-68.

P20-9 Since the maximum value for u is c, then the minimum ~t is

At ?: (730m)/(3.00xlO8m/s) = 2.43xlO-6s.

P20-10 (a) Yes.(b) The speed will be very close to the speed of light, consequently"1 ~ (23,000)/(30) = 766.7.

Thenv = VI -1/,2 ~ 1 -1/2,2 = 1 -1/2(766.7)2 = O.99999915c.

P20-11 (a) 6.t' = (5.00¡¡,s)Vl- (0.6)2 = 4.00¡¡,s.(b) Note: it takes time for the reading on the S' clock to be seen by the S clock. In this case,

6.tl + 6.t2 = 5.00¡¡,s, where 6.tl = x/u and 6.t2 = x/c. Solving for 6.tl,

and~t~ = {3.125 .us)V1 -{0.6)2 = 2.50.us.

P20-12 The only change in the components of 6.r occur parallel to the boost. Then we can choosethe boost to be parallel to 6.r and then

Ar' ='Y[Ar -u(O)] = 'YAr ? Ar,

since "Y ?; 1.

~ (a) Start with Eq. 20-34,

E2 = (pC)2 + (mc2)2,

and substitute into this E = K + mc2 ,

K2 + 2Kmc2 + (mc2)2 = (pc)2 + (mc2)2.

261

We can rearrange this, and then

K2 + 2K mc2 - (pC)2,(pc) 2 -K2

m = 2Kc2

(b) As v/c;;..,. O we have K-+ ~mv2 and P-+ mv, the classicallimits. Then the above expression

becomes

m2v2c2 -~m2v4m """

But v / c -+ O, so this expression reduces to m = m in the classicallimit, which is a good thing.

(c) We get

P20-14 Since E » mc2 the particle is ultra-relativistic and v ~ c. 'Y = (135)/(0.1396) = 967.Then the particle has a lab-life of At'= (967)(35.0 x 10-9s) = 3.385 x 10-5s. The distance traveledis

x = (3.00 x 108m/s)(3.385 x 10-58) = 1.016x 104m,

so the pion decays 110 km above the Earth.

~ (a) A completely inelastic collision means the two particles, each of mass ml, sticktogether after the collision, in e(fectbecoming a new particle of mass m2. We'll use the subscript 1for moving particle of mass ml, the subscript O for the particle which is originally at rest, and thesubscript 2 for the new particle after the collision. We need to conserve momentum,

Pl +Po

"/lmlUl + (O)

= P2,

= 'Y2m2U2,

and we need to conserve total energy,

El + Eo

2 27lmlc +mlc

- E2,

2'"Y2m2C ,=

Divide the momentum equation by the energy equation and then

'YIUl71 + 1 = U2.

But Ul = cyl -l/'Y~ , SO

U2

262

(b) Using the momentum equatio~,

m2

P20-16 (a) K = W = JFdx = J(dp/dt)dx = J(dx/dt)dp = Jvdp.(b) dp = m'Ydv + mv(d'Y/dv)dv. Now use Maple or Mathematica to save time, and get

mdv mv2 dvdp = (1- v2/c2)1/2 + c2(1 -v2/c2)3/2 .

Now integrate:

mv2K dv,=

c2(1- V2/C2)3/2

~.

P20-17 (a) Since E= K + mc2, then

Enew = 2E = 2mc2 + 2K = 2mc2(1 + K/mc2).

(b) Enew = 2(0.938 GeV) + 2(100 GeV) = 202 GeV.(c) K = (100 GeV)/2 -(0.938 GeV) = 49.1 GeV.

P20-18 (a) Assume only one particle is formed. That particle can later decay, but it sets thestandard on energy and momentum conservation. The momentum of this one particle must equalthat of the incident proton, or

p2C2 = [(mc2 + K)2 -m2c4].

The initial energy was K + 2mc2, so the mass of the "one" particle is given by

M2c4 = [(K + 2mc2)2 -p2C2] = 2K mc2 + 4m2c4.

This is a measure of the available energyj the remaining energy is required to conserve momentum.Then

Enew = VM2"¡;4 = 2mc2vl + K/2mc2.

263

P20-19 The initial momentum is m'"Yivio The final momentum is (M -m)'"Yfvfo Manipulating themomentum conservation equation,

=

=

=

=

m'YiVi

1

~

M-m

-;;¡;y;¡¡;-

M-m 1

+

m'Yi/3i

264

~ (a) We'll assume that the new temperature scale is related to the Celsius scale by a lineartransformation; then Ts = mTc + b, where m and b are constants to be determined, Ts is thetemperature measurement in the "new" scale, and Tc is the temperature measurement in Celsiusdegrees.

One of our known points is absolute zero;

Ts = mTc +b,

(O) = m( -273.15°C) + b.

We have two other points, the melting and boiling points for water ,

{Ts )bp

{Ts)mp

= m(100°C) + b,= m(O°C) + b;

we can subtract the top equation from the bottom equation to get

(TS)bp -(Ts)Smp = 100 Com.

We are told this is 180 SO, so m = 1.8 SO /Co. Put this into the first equation and then find b,b = 273.15°Cm = 491.67°S. The conversion is then

Ts = (1.8 SO /CO)Tc+ (491.67°S).

(b) The melting point for water is 491.67°S; the boiling point for water is 180 SO above this, or671.67°S.

E21-2 TF = 9( -273.15 deg O)/5 + 32°F = -459.67°F .

E21-3 (a) We'1l assume that the new temperature scale is related to the Celsius scale by a lineartransformationj then Ts = mTc + b, where m and b are constants to be determined, Ts is thetemperature measurement in the "new" scale, and Tc is the temperature measurement in Celsiusdegrees.

One of our known points is absolute zero;

Ts = mTc +b,

(O) = m( -273.15°C) + b.

We have two other points, the melting and boiling points for water ,

(Ts )hp

(Ts )mp

m(100°C) + b,

m(Ooq)+,b;

==

we can subtract the top equation from the bottom equation to get

(TS)bp -(T8)Smp = 100 Com.

We are told this is 100 Qo, so m = 1.0 Qo /Co. Put this into the first equation and then find b,b = 273.15°C = 273.15°Q. The conversion is then

Ts = Tc + (273.15°8).

(b) The melting point for water is 273.15°Q; the boiling point for water is 100 Qo above this, or373.15°Q.

(c) Kelvin Scale.

265

E21-4 (a) T = (9/5)(6000K -273.15) + 32 = 10000°F.(b) T = (5/9)(98.6°F -32) = 37.0°C.(c) T= (5/9)(-70°F -32) = -57°C.(d) T = (9/5)( -183°C) + 32 = -297°F.

(e) It depends on what you think is hot. My mom thinks 79°F is too warm; that's T =(5/9)(79°F -32) = 26°C.

E21-5 T = (9/5)(310 K -273.15) + 32 = 98.3°F, which is fine.

E21-6 (a) T = 2(5/9)(T -32), so -T/10 = -32, or T = 320°F.(b) 2T = (5/9)(T -32), so 13T/5= -32, orT = -12.3°F,

~ If the temperature (in Kelvin) is directly proportional to the resistance then T = kR,where k is a constant of proportionality. We are given one point, T = 273.16 K when R = 90.35 n,but that is okay; we only have one unknown, k. Then (273.16 K) = k(90.35 n) or k = 3.023 K/n.

If the resistance is measured to be R = 96.28 n, we have a temperature of

T = kR = {3.023 K/0){96.28 n) = 291.1 K.

E21-8 T = (510°C)/(0.028 Y)V , 80 T = (1.82 x 104°C/y)(0.0102 Y) = 186°C.

~ We must first find the equation which relates gain to temperature, and then find the gain.at the specified temperature. If we let G be the gain we can write this linear relationship as

G=mT+b,

where m and b are constants to be determined. We have two known points:

(30.0)

(35.2)

= m(20.0° O) + b"

= m(55.0° O) + b.

Ir we subtract the top equation from the bottom we get 5.2 = m(35.0° C), or m = 1.49C-1. Putthis into either or the first two equations and

(30.0) = (0.149C-1)(20.0° C) + b,

which has a solution b = 27.0Now to find the gain when T = 28.0 °C:

G = mT + b = (0.149C-1)(28.0° C) + (27.0) = 31.2

E21-10 P/Ptr = (373.15 K)/(273.16K) = 1.366.

E21-11 100 cm ~is 1000 torr. PHe = (100 cm Hg)(373K)/(273.16K) = 136.550 cm Hg. Ni-trogen records a temperature whic};l is 0.2K higher, so PN = (100 cm Hg)(373.2K)/(273.16K) =136.623 cm Hg. The difference is 0.073 cm Hg.

E21-12 AL = (23 x 10-6/CO)(33 m)(15CO) = 1.1 x 10-2m~

E21-13 AL = (3.2xlO-6/Co)(20Ü in)(60CO) = 3.8xlü-2in.

266

E21-14 L' =((2.725cm)[l + (23xlü-6¡Co)(l28CO)]= 2.733 c

~ We want to focus on the temper'8.ture change, not the absolute temperature. In thiscase, ~T = Tf -Ti = (42°C) -( -5.0°C) = 47Co.

Then~L = (11 x 10-6 C-1 )(12.0 m)(47 C6;) = 6.2 x 10-3 m.

E21-16 ~A = 2aA~T, so

~A = 2(9xlO-6/Co){2.0m)(3.0m)(30CO) = 3.2x lO-3m2.

~ (a) We'll apply Eq. 21-10. The surface area of a cube is six times the area of one face,which is the edge length squared. So A = 6(0.332 m)2 = 0.661 m2. The temperature change is~T = (75.0°C) -(20.0°C) = 55.0 Co. Then t~e increase in surface area is

~A = 2aA~T = 2(19 x 10-6C-1)(0.661 m2)(55.0 CO) = 1.38 x 10-3 m2

(b) We'll now apply Eq. 21-11. The volume of the cube is the edge length cubed, so

V = (0.332 m)3 = 0.0366 m3.

and then from Eq. 21-11

A V = 2aV AT = 3(19 x 10-6 C-1 )(0.0366 m3)(55.0CO) = 1.15 x 10-4 m3,

is the change in volume of the cube,

E21-18 V' = V(l + 3a~T), so

3v' (530 cm3)[1 + 3(29x 10-6/CO)(-172CO)] = 522 cm:

,6 x 10-4/CO,E21-19 (a) The slope is approximately(b) The slope i~ zero.

E21-20 Ar = ({3/3)rAT, so

(3.2x 10-5 /K)/3](6.37x 106m)(2700 K) = 1.8 x 105m.~r

~ We'll assume that the steel ruler measures length correctly at room temperature. Thenthe 20.05 cm measurement of the rod is correct. But both the rod and the ruler will expand in theoven, so the 20.11 cm measuremept of the rod is not the actuallength of the rod in the oven. Whatis the actuallength of the rod in the oven? We can only answer that after figuring out how the 20.11cm mark on the ruler moves when the ruler expands.

Let L = 20.11 cm correspond to the ruler mark at room temperature. Then

AL = asteelLAT = (11 X lO-6C-1)(20.11 cm)(250CO) = 5.5 x 10-2 cm

is the shift in position of the mark as the ruler is raised to the higher temperature. Then the changein length of the rod is not (20.11 cm) -(20.05 cm) ~ 0.06 cm, because the 20.11 cm mark is shiftedout. We need to add 0.055 cm to this; the rod chánged length by 0.115 cm.

The coefficient of thermal expansion for the rod is

AL (0.115 cm) = 23 x 10-6 C-1,a = ~ =- .", "(20.05 cm)(250CO )L~T

267

E21-22 A = ab, A' = (a+ ~a)(b+ ~b) = ab+ a~b + b~a + ~a~b, so

6.A a~b + b~a + ~a~b,

A(~b/b + ~a/a + ~a~b/ab),

A(a~T+a~T),

2aA~T.

=

~

~ Solve this problem by assuming the solid is in the form of a cube.If the length of one side of a cube is originally Lo, then the volume is originally Vo = L3. After

heating, the volume of the cube will be V = L3, where L = Lo + ~L.Then

v = L3,

(Lo + ~L)3,

(Lo + aLo~T)3,

Lg(l + a~T)3.

As long as the quantity a~T is much less than one we can expand the last line in a binomialexpansion as

V ~ Vo(l + 3a~T +

so,the change in volume is ~V ~ 3aVo~T.

),

E21-24 {a) ~A/A = 2{0.18%) = {0.36%).(b) ~L/L = 0.18%.(c) ~V/V = 3{0.18%) = {0.54%).

{d) Zero.{e) a = (0.0018)/{100CO) = 1.8x10-5/Co.

p' -p = m/V'- m/V = m/(V + AV) -m/V ~ -mAV/V2. ThenE21-25

6.p = -(m/V)(6.V/V) = -p.8AT.

E21-26 Use the results oí Exercise 21-25.(a) AV/V = 3AL/L = 3(0.092%) = 0.276%. The change in density is

Llp/p = -LlV/V = -(0.276%) -0.28

(b) a = /3/3= (0.28%)/3(40CO) = 2.3xlO-5/Co. Must be aluminum.

~ The diameter oí the rod as a function oí temperature is

ds = ds,Q(l + QsLlT),

The diameter of the ring as a function of temperature is

db =$: db,Q(l + ab~T).

268

We are interested in the temperature when the diameters are equal,

ds,o(l + asAT)

asds,oAT -abdb,oAT

= db,O(l + ab~T),= db,O -ds,o,

db,O -ds,o~T

~

asds,o -abdb,O

{2.992 cm) -{3.000 cm)AT = ~~oooc~92-cm1 ,

335 co .

The final temperature is then Tí = (25°) + 335 Co = 360°,

E21-28 (a) ~L = ~L1 + ~L2 = (L1al + L2a2)~T. The effective value for a is then

~LL~T

alL1 + a2L2o:= --

(b) Since L2 = L -L1 we can write

QlLl + Q2(L -Ll)

Ll

= aL,

= L~al -a2 ,

{0.524 m) {13 x 10-6) -{11 x 10-6){19 x 10-6) -{11 x 10-6) = 0.131 m.

The brass length is then 13.1 cm and the steel is 39.3 cm.

E21-29 At lOO°C the glass and mercury each have a volume Va. After cooling, the difference involume changes is given by

~ v = Vo(3ag -,8m)~T.

Since m = pV, the mass of mercury that needs to be added can be found by multiplying though by

the density of mercury. Then

~m = (0.89lkg)[3(9.0xlO-6/CO) -(l.8xlO-4/co)](-l35CO) = O.Ol84kg.

This is the additional amount required, so the total is now 909 g.

E21-30 (a) The rotational inertia is given by I = J r2dm; changing the temperature requiresr -r' = r + ~r = r(l + a~T). Then

so ~I = 2aI~T.

(b) Since L = IúJ, then O = úJ~I +,I~úJ. Rearranging, ~úJ/úJ = -~I/I = -2a~T. Then

~w ~ -2(19x 10-6/co)(230 rev/s)(170CO) ~ -1.5 rev/s."'

269

~ This problem is related to objects which expand when heated, but we never actually needto calculate any temperature changes. We will, however, be interested in the change in rotationalinertia. Rotational inertia is directly proportional to the square of the ( appropriate ) linear dimension,so

If/li = (rf/rJ2,

(a) If the bearings are frictionless then there are no external torques, so the angular momentumis constant.

(b) I[the angular momentum is constant, then

Li

IiúJi

Lf,

If"'fo

==

We are interested in the percent change in the angular velocity, which is

I. / , 2 1 ' 2=~-1=

If

(Llf -(Lli

(Lli

iAJf

iAJj

(~) -1= (~)- -1 = -0.36%.=

(c) The rotational kinetic energy is proportional to Iw2 = (Iw)w = Lw, but L is constant, so

E21-32 (a) The period of a physical pendulum is given by Eq. 17-28. There are two variablesin the equation that depend on length. I, which is proportional to a length squared, and d, whichis proportional to a length. This means that the period have an overall dependence on lengthproportional to vr .Taking the derivative,

1P 1~P ~ dP = --dr ~ -Pa~T.

2 r 2

(b) ~P/P = (O. 7x 10-6Co)(10CO)/2 = 3.5 x 10-6. After 30 days the clock will be slow by

At = {30 x 24 x 60 x 60 s)(3.5 x 10-6) = 9.07 s.

E21-33 Refer to the Exercise 21-32.

~p = (36008){l9x lO-6Co)( -20CO)/2 = 0.688.

E21-34 At 22°C the aluminum cup and glycerin each have a volume Va.difference in volume changes is given by

After heating, the

~ V = Vo(3aa -.Bg)~T.

The amount that spills out is then

~v = (110 cm3)[3(23x10-6/CO) -(5.1 x 10-4/CO)](6CO) = -0.29 cm3.

E21-35 At 20.0°C the gla.ss tube is filled with liquid to a volume Va. After heating, the differencein volume changes is given by

~~V = Vo(3ag -/31)~T.

The cross sectional area oí the tube changes according to

~A = Ao2ag~T.

270

Consequently, the height of the liquid changes according to

= (ha + ~h)(Aa + ~A) -haA,

~ ha~A + Aa~h,

= ~A/Aa + ~h/ha.

~v

~V/VO

Then~h = (1.28m/2)[(1.1 x 10-5/co) (4.2xlÜ-5/CO)](l3CO) = 2.6xlü-4m.

E21-36 (a) 13 = (dV/dT)/V. IfpV = nRT, then pdV = nRdT, so

.8 = (nR/p)/V = nR/pV =l/T.

(b) Kelvins.(c) /3 ~ 1/(300/K) = 3.3x 10-3/K.

E21-37 (a) V = (1 mol)(8.31 J/mol. K)(273 K)/(1.01 x 105Pa) = 2.25 x 10-2m3.(b) (6.02x1023mol-l)/(2.25x104/cm3) = 2.68~1019.

E21-38 n/V = p/kT, so

3n/V = (1.01 X 10-13Pa)/(1.38X 10-23J/K)(295 K) = 25 part/cm;

~ (a) Using Eq. 21-17,

pV

RT

n

(b) Use the same expression again,

nRT

pv=

E21-40 (a) n =pV/RT = (l.OlxlO5pa)(l.l3xlO-3m3)/(8.3lJ/mol.K)(3l5K) = 4.36xlO-2mol.(b) Tí = TiPíVí/PiVi, so

Pi = (14.7+ 24.2) lb/in2 = 38.9 lb/in2.E21-41 Pf =piTfVi/TiVf, SO

(38.9 lb/in2)(299K)(988 in3) ; .2=41.71b/m .Pf =

(270 K)(1020 inv)

The gauge pressure is then ( 41.7 -' 14.7) lb/in2 = 27.0 lb/in2.

E21-42 Sínce p = F/A and F = mg, a reasonable estímate for the mass of the atmosphere ís

m = pA/g = {1.01 x 105Pa)47r{6.37x 106m)2/{9.81 m/s2) = 5.25 x1018kg.

271

E21-43 P = Po + pgh, where h is the depth. Then Pf = 1.01 x 105Pa and

Pi = {1.01 x 105Pa) + {998kg/m3){9.81 m/s2){41.5m) = 5.07x 105Pa.

V f = ViPiTf/PfTi, so

E21-44 The new pressure in the pipe is

Pf =PiVi/Vf= (1.01x105Pa)(2) = 2.02x105Pa.

The water pressure at some depth y is given by P = Po + pgy, SO

1J = .(2.02 x 105Pa) -ll.01 x 105Pa) = 1 n .':t m(998 kg/m3)(9.81 m/s2) ~~.~ Then the water/air interface inside the tube is at a depth of 10.3 m; so h = (10.3m) + (25.0m)/2 =

22.8m.

~ (a) The dimensions of A must be [time]-l, as can be seen with a quick inspection of theequation. We would expect that A would depend on the surface area at the very least; however,that means that it must also depend on some other factor to fix the dimensionality of A.

(b) Rearrange and integrate,

d~T~T

~To)~T

=¡:o'

ln(~T/,

-1tAdt,-At ,

~Toe-At.-

P21-2 First find A.

Then find time to new temperature difference.

t = ln(ó.To/ó.T) = ln[(29CO)/(~12°)] = 97.8min(3.30 x 10-3/min)t

This happens 97.8 -45 = 53 minutes later .

P21-3 If we neglect the expansion of the tube then we can assuIhe the cross sectional area of thetube is constant. Since V = Ah, we can assume that ~ V = A~h. Then since ~ V = ,BVo~T, wecan write ~h = ,Bho~T .

P21-4 For either container we can write piVi = niRTi. We are told that Vi and ni are constants.Then 6.p = AT1 -BT2, where A and B are constants. When T1 = T2 6.p = O, so A = B. WhenT1 = Ttr and T2 = Tb ~ have

(120 m m Hg)= A(373K -273.16K),

so A 1.202 m m Hg/K. Then

T = (90 m m Hg) + (1.202 m m Hg/K)(273.16~= 348K.

(1.202 m m Hg/K)

Actually, we could have assumed A was negative, and then the answer would be 198 K.

272

~ Start with a differential form for Eq. 21-8, dL/dT = aLa, rearrange, and integrate:

{L dL = {T aLa dT,

JLo JToT

Lo f

JTOL-Lo adT,=

LoL =

P21-6 ~L = Q;L~T, so

~T 1 AL--

aL At

= (96xlO-9m/8)(23 XlO-6 iCO)(L8 x lO-2m) = 0.23°C/8.-~~t

~ (a) Consider the work that was done for Ex. 21-27. The length of rod a is

La = La,o(l + Qa6.T),

while the length of rod b isLb = Lb,o(l + ab6.T).

The difference is

La,o(l + aa~T) -Lb,o(l + ab~T),

La,o ~ Lb,O + (La,oaa -Lb,Oab)~T,

La -Lb

~

which will be a constant is La OQa = Lb OQb or, ,

Li,o cx: l/aio

(b) We w~t La,o -Lb,O = 0.30 m so

k/aa -k/ab = 0.30 m,

where k is a constant of proportionality;

k = {0.30 m)/ (l/{llxlO-6/CO) -l/{l9xlO-6/CO)) = 7.84xlO-6m/,Co.

The two lengths are

La = (7.84 x 10-6m/Coj/(11 x 10-6 jCO)= 0.713m

for steel andLb = (7.84xl0-6m/CO)/(19xl0-6/CO) = O.413m

for brass.

P21-8 The fractional increase in length ofthe bar is ~L/Lo = a~T. The right triangle on the lefthas base Lo/2, height x, and hypotenuse(Lo + ~L)/2. Then

x = ~V(LO + ~L)2- L~ =

With numbers,

273

P21-9 We want to evaluate V = Vo(l + J /3 dT); the integral is the area under the graph; the

graph looks like a triangle, so the result is

V = Vo[l + (16CO)(O.OOO2/CO)/2] = (1.0016)Vo.

The density is then

p = PO(Vo/V) = (1000kg/m3)/(1.0016) = O.9984kg/m3.

P21-10 At 0.00°C the glass bulb is filled with mercury to a volume Vo. After heating, the difference

in volume changes is given by~V = Vo((3 -3a)~T.

Since To = 0.0°C, then ~T = T, if it is measured in °C. The amount of mercury in the capillary is~ V, and since the cross sectional area is fixed at A, then the length is L = ~ V / A, or

VL = A((3 -3a)~T.

Let a, b, and c correspond to aluminum, steel, and invar, respectively. Then

a2 + b2 -¿

P21-11

cos c = 2 b -: a ,

We can replace a with ao(l + aa6.T), and write similar expressions for b and c. Since ao = bo = co,

this can be simplified to

(1 + aa~T)2 + (1 + ab~T}2 -:fJ + ac~cosC =

2(1 + aaAT)(l + ab~.l.)

Expand this as a Taylor series in terms of AT, and we find

1 1cosC ~ 2 + 2 (aa + ab -2ac) AT.

Now solve:2 cos(~9=-95°) -1 = 46.4°C.~T = (23 x-10-6JCO) + (11 x 10-6 /CO)--2 (O~7x 10-6/Co)

The final temperature is then 66.4°C.

P21-12 The bottom of the iron bar moves downward according to AL = aLAT. The center ofmass of the iron bar is located in the center; it moves downward half the distance. The mercuryexpands in the glass upwards; subtracting off the distance the iron moves we get

Ah = {3hAT -AL = ({3h -aL)AT.

The center of mass in the mercury is located in the center. If the center of mass of the system is to

remain constant we requiremiAL/2 = mm(Ah -AL)/2;

or, since p = m V = mAy,pjaL = Pm({3h -2aL).

Solving for h,

h = .~ c ~

274

(13.6 x 103kg/m3)(18

~ The volume of the block which is beneath the surface of the mercury displaces a massof mercury equal to the mass of the block. The mass of the block is independent of the temperaturebut the volume of the displaced mercury changes according to

v m = v m,o(l +.BmAT).

This volume is equ8:1 to the depth which the block sinks times the cross sectional area of the block( which does change with temperature) .Then

hshb2 = hs,Qhb,Q2(1 + {3m~T),

where hs is the rlepth to which the block sinks anrl hb,Q = 20 cm is the length of the sirle of theblock. But

hb = hb,o(l + abAT),

sol+f3m~Ths = hs,o -(1 + ab~T)2 .

Since the changes are small we can expand the right hand side using the binomial expansion; keepingterms only in ~T we get

hs ~ hs,o(l + ({:Jm -2ab)~T),

which means the block will sin k a distance hs -hs,o given by

hs,o(/3m -2ab)AT = hs,o [ (1.8 x 10-4/CO) -2(23 x 10-6/CO)J (50 CO) = (6.7 x 10-3)hs,o.

In order to finish we need to know how much of the block was submerged in the first place. Sincethe fraction submerged is equal to the ratio of the densities, we have

h8,0/hb,O = Pb/Pm = (2.7xl03kg/m3)/(1.36xl04kg/m3),

so hs,Q = 3.97 cm, and the change in depth is 0.27 mm.

P21-14 The area of glass expands according to ~Ag = 2agAg~T. The are of Dumet wire expandsaccording to

~Ac + ~Ai = 2(acAc + aiAi)~T.

We need these to be equal, so

QgAg = QcAc + QiAi,

2 - ( 2 2 ) 2Qgrg ~ Qc rc -ri + Qiri ,

( 2 2) - ( 2 2 ) 2

Qg rc + ri -Qc rc -ri + Qiri ,

2:i = Qc -Qg

Qc -Qir~2

P21-15

P21-16 V2 = V1(Pl/P2)(T1/T2), SO

V2 = (3.47m3)[(76 cm Hg)/(36 cm Hg)][(225K)/(295K)] = 5;59m3.

275

~ Call the containers one and two so that Vi = 1.22 L and V2 = 3.18 L. Then the initialnumber oí moles in the two containers are

PiVl PiV2

~.nl,i = RT .and n2i =

1 ,

The total is

n === pi(V1 + V2)/(RTi).

Later the temperatures are changed and then the number of moles of gas in each container is

~

RT1,f~.RT2,f

and n2,f =nl,f =

The total is still n, soV2 pi(Vl + V2)

RT;~(~ =+-.T1,f T2,f)

We can solve this for the final pressure, so long as we remember to convert all temperatures to

Kelvins,pi(VI + V2) ( VI V2

)Pf= -+-Ti T1,f T2,f

or

74 atm.=

P21-18 Originally nA = PAVA/RTA and nB = PBVB/RTB; VB = 4VA. Label the final state ofA as C and the final state of B as D. Aftermixing, nc = pcVA/RTA and nD = PDVB/RTB, butPc = PD and nA +nB = nc +nD. Then

PA/TA + 4PB/TB = Pc (l/TA + 4/TB),

or.(5 x 105Pa)/(300K) + 4(1 x 105Pa)/(400K)

= 2.00 x 105Pa.Pc=1/(300K) + 4/(400 K)

P21-19 If the temperature is uniform then all that is necessary is to substitute Po = nRT /V andp = nRT /V; cancel RT from both sides, and then equate n/V with nv.

P21-20 Use the results of Problem 15-19. The initial pressure inside the bubble is Pi = Po +4'Ylri.The final pressure inside the bell jar is zero, SOPf = 4'Ylrf. The initial and final pressure inside thebubble are related by piri3 = Pfrf3 = 4'Yrf2. Now for numbers:

pj = (1.01 x 105Pa) + 4(2.5 x 10-2N/m)/(2.0 x 10-3m) = 1.0105 x 105Pa.

and

I (1.0105 x 105Pa)(2.0 x 10-3m)3rf = ~ IV 4(2.5x lO=2N/m) = 8.99x lO-2m.

P21-21

P21-22

276

E22-1 (a) n = (2.56 g)/(l97 g/mol) = l.30x lO-2mol.(b) N = (6.02xlO23mol-l)(l.30xlO-2mol) = 7.83xlO21,

E22-2 I(b) n

(a) N = pV/kT = (1.01 x 105pa)(1.00m3)/(1.38x 10-23J/K)(293 K) = 2.50x 1025.

= (2.50x1025)/(6.02~1023mol-l) =41.5 mol. Then

m = (41.5 mol)[75%(28 g/mol) + 25%(32 g/mol)] = 1.20kg.

~ (a) We first need to calculate the molar mass oí ammonia. This is

M = M{N) + 3M{H) = {14.0 g/mol) + 3{1.01 g/mol) = 17.0 g/mol

The number oí moles oí nitrogen present is

n = m/Mr = (315 g)/(17.0 g/mol) = 18.5 mol.

The volume of the tan k is

v = nRT/p = (18.5 mol)(8.31 J/mol. K)(350K)/(1.35 x lO6Pa) = 3.99x lO-2m3.

(b) After the tank is checked the number of moles of g8$ in the tank is

n = pV/(RT) = (8.68x lO5Pa)(3.99x lO-2m3)/[(8.31 J/mol. K)(295K)] = 14.1 mol.

In that case, 4.4 mol must have escaped; that corresponds to a mass of

m = nMr = (4.4 mol)(17.0 g/mol) = 74.8 g.

E22-4 (a) The volume per particle is v/N = kT/P,so

V/N = (1.38x 10-23 J/K)(285K)/(1.01 x 105Pa) = 3.89x 10-26m3.

The edge length is the cube root of this, or 3.39 x 10-9m. The ratio is 11.3.(b) The volume per particle is V/NA, so

V/NA = (18x10-6m3)/(6.02x 1023) = 2.99x10-29m3.

The edge length ls the cube root of this, or 3.10 x 10-1om. The ratio is 1.03.

E22-5 The volume per particle is V / N = kT / p , so

V/N = (1.38x 10~23J/K)(308K)/(1.22)(1.01 x 105Pa) = 3.45x 10-26m3.

The fraction actually occupied by the particle is

47r(0. 710 x 10-10m)3/~ = 4.34 x 10-5.

(3.45 x lO-26m3 )

E22-6 The component of the momentum normal to the wall is

{3.3 x lO-27kg){l;OX lO3mis) cos{55°) = 1.89 x lO-24kg .mis.Py

The pressure exerted on the wall is

F-P=A-

.(1.6 x 1023/8)2(1.89 x 10-24kg ~~18)= 3.0 x 103Pa.

(2.0 x 10-4m2)

277

~ (a) From Eq. 22-9,

Vrms = f!

Then

p = 1.23 x 10-3 atm = 124 Pa

and3100 cm

1mp = 1.32 x 10-5 g/cm3 = 1.32 x 10-2 kg/m3.

lkg

)iOOügFinally,

(b) The molar density of the gas is just n/V; but this can be found quickly from the ideal gaslaw as

~ = p ~ (1240Pa) = J. 71 y ln-l Tnnl/Tn3v :nT -(8.31J/mol.K)(317K) ~ ~..~ " ~~ ...~.f.~ .

(C) We were given the density, which is mass per volume, so we could find the molar mass from

(1.32 x 10-2 kg/m3)= 28.0 g/mol.~= -..p

(4.71 x 10-1 mol/m3)

But what gas is it? It could contain any atom lighter than silicon; trial and error is the way to go.Some of my guesses include C2H4 (ethene), CO (carbon monoxide), and N2. There's no way to tellwhich is correct at this point, in fact, the gas could be a mixt~re of all three.

E22-8 The density is p = m/V = nMr/V, or

p = (0.350 mol)(0.0280 kg/mol)/7r(0.125m/2)2(0.560m) = 1.43kg/m3.

'I'he rms speed is

E22-9 {a) N/V = p/kT = {1.01 x 105pa)/{1.38x 10-23J/K){273K) = 2.68x 1025/m3.{b) Note that Eq. 22-11 is wrong; for the explanation read the last two paragraphs in the first

column on page 502. We need an extra factor of .j2, SO 7rJ2 = V / .j2N >., so ,

.cd = 1/ V21T(2.68 x 1025 /m3)(285 x 10,-9m) = 1.72 x 10-1Om.

E22-10 (a) >. = V¡V2N7rd2, so

(b) Particles effectively follow ballistic trajectories.

278

~ We have v = f>., where >. is the wavelength (which we will set equal to the mean free

path), and v is the speed of sound. The mean free path is, from Eq. 22-13,

>.=~v'27rd2p

so

V27rd2pv

kT

V27r(315 x 10-12m)2(1.02 X 1.01 x 105Pa)(343 mis)f= = 3.88 x lOgHz.= ~~

(1.38x 10-23J/K)(291 K)

E22-12 (a) p = (1.10 x 10-6 m m Hg)(133 Pa/mm Hg) = 1.46 x 10-4Pa. The particle density is

N/V = (1.46x 10-4Pa)/(1.38x 10-23J/K)(295 K) = 3.59x 1016/m3.

(b) The mean free path is

), = 1/ .¡(2) (3.59 x 1016 /m3)7r(2.20 x 10-10m)2 = 130 m.

E22-14 (a) vav = (65 ~m/s)/(10) = 6.5 km/s.(b) Vrms = V(505 km/s)/-(-iO) = 7.1 km/s.

~ (a) The average is

The mean-square value is

~(200s)2 + 2(500m/s)2 + 4(600 m/s)2 = 2.1 x 105 m2/s2.4+2+4

The root-mean-square value is the square root of this, or 458 m/s.(b) I'll be lazy. Nine particles are not moving, and the tenth has a speed of 10 m/s. Then the

average speed is 1 m/s, and the root-mean-square speed is 3.16 m/s. Look, Vrms is larger than vav!(c) Can vrms=vav? Assume that the speeds are not all the same. Transform to a frame of

reference where vav = 0, then some of the individual speeds must be greater than zero, and somewill be less than zero. Squaring these speeds will result in positive, non-zero, numbers; the meansquare will necessarily be greater than zero, so Vrms > 0.

Only if all of the particles have the same speed will Vrrns=Vav.

E22-16 Use Eq. 22-20:

E22-17 Use Eq. 22-20:

279

E22-13 Note that vav cx: VT, while >.. cx: T. Then the collision rate is proportional to l/VT. Then

Eq. 22-14 is in the form N = Av2e-Bv2. Taking the derivative,

dN B 2 B 2= 2Ave- v -2ABv3e- v,

E22-18

dv

and setting this equal to zero,V2 = l/B = 2kT/m.

~ We want to integrate

vav =

vdv,

The easiest way to attack this is first with a change of variables- let x = mv2 /2kT , then kT dx =mvdv. The limits of integration don't change, since .¡-00 = 00. Then

( m )3/2 roo 2kT kTvav = 47r 2;A;T Jo -;;;;-xe-x ~dx,

(2kT ) 1/2 roo= 2 -;;;;:- Jo xe-Qxdx

The factor of a that was introduced in the last line is a Feynman trick; we'll set it equal to one whenwe are finished, so it won 't change the result.

Feynman 's trick looks liked ( { -

-axdx = f ( -x)e-axdx.

e ax ,eada- J e- dx = J 8~e

Applying this to our original problem,

vav =

280

E22-20 We want to integrate

2 1 roo ( 2(V)av = NJo Nv)v dv,

1¡ 00 ( m ) 3/2 2 = -47r N -v2e-mv /2kT V2 dv,

N o 27rkT

( m )3/2 ¡ 00 2 = 47r -v2e-mv /2kT V2 dv.

27rkT o

The eagiest way to attack this is first with a change of variables- let X2 = mv2 /2kT , theny'2)CT7mdx = dv. The limits of integration don't change. Then

E22-21 Apply Eq. 22-20:

Vrms = V3(1.38x 10-23J/K)(287K)/(5:2x 10-17kg) = 1.5x lü-2m/s.

E22-22 Since Vrms (X JT¡m, we have

T= {299 K) {4/2) = 598 K,

or 325°C.

E22-23 (a) The escape speed is found on page 310; v = 11.2 x 103m/s. For hydrogen,

T = (2)(1.67x 10-27kg)(11.2 x 103m/s)2 /3(1.38x 10-23J/K) = 1.0x 104K.

For oxygen,

T = (32)(l.67x lO-27kg)(ll.2x lO3m/s)2 /3(l.38x lO-23J/K) = l.6x lO5K.

(b) The escape speed is found on page 310; v = 2.38 x 103m/s. Por hydrogen,

T = (2)(1.67x 10-27kg)(2.38 x 103m/s)2 /3(1.38x 10-23J/K) = 460K.

For oxygen,

T = (32)(1.67x 10-27kg)(2.38x 103m/s)2 /3(1.38x 10-23 J/K) = 7300K.

(c) There should be more oxygen than hydrogen.

E22-24 (a) vav = (70 km/s)/(22) = 3.18 km/s.

(b) Vrms = V (250 km2/s2)/(22) = 3.37 km/s.

(c) 3.0 km/s.

281

8kT ¡ (X) 4 -x2

d= -xe X~m O

Look up the integral; although you can solve it by first applying a Feynman trick (see solution toExercise 22-21) and then squaring the integral and changing to polar coordinates. I looked it up. Ifound 3~ /8, so

87.3 rad/s.

~ According to the equation directly beneath Fig. 22-8,

(¡) = v4>/L = (212 m/s}(0.0841 rad}/(0.204m}

E22-26 If Vp = Vrms then 2T2 = 3T1, or T2/Tl = 3/2.

E22-27 Read the last paragraph on the first column of page 505. The distribution of speeds isproportional to

2/ B ..

v3e-mv 2kT = v3e- v,

taking the derivative dN / dv and setting equal to zero yields

dN B .. 4 B ..= 3v2e- v -2Bv e- v,

dv

and setting this equal to zero,V2 = 3/2B = 3kT /m.

E22-28 (a) v = V3(8.31 J/mol. K)(4220K)/(0.07261 kg/mol) = 1200m/s.(b) Half of the sum of the diameters, or 273 pm.( c ) The mean free path of the germanium in the argon is

,\ = 1/V2(4.13x 1025 /m3)7r(273x 10-12m)2 = 7.31 x 10-8m.

The collision rate is(l200m/s)/(7.3lxlO-8m) = l.64xlOlO/s.

~ The fraction of particles that interests us is

¡ 0.03kT ~ ~ E1/2e-E/kT dE.

Vii (kT)3/2 0.01kT

Change variables according to E / kT = x, so that dE = kT dx. The integral is then

2 ¡ 0.03 -x1/2e-x dx.

Vii 0.01

Since the value of x is so small compa.red to 1 throughout the ra.nge of integration, we ca.n expa.ndaccording to

e-x ~ 1- x for x « 1.

The integral then simplifies to

2 ¡ 0.03 2 [2~ x1/2(1 -x) dx = --x3/2J7r" 0.01 J7r" 3

] 0.03 2 5/2

-5x 0.01

= 3.09 x 10-¡

E22-30 Write N(E) = N(Ep + E). Then

N(Ep + t:) R! N(Ep) + t: + ...

But the very definition of Ep implies that the first derivative is zero. Then the fraction of [particleswith energies in the range Ep :I:: O.O2kT is

2 1 -( / 1/2 -1/2v'ii (kT)3/2 kT 2) e (O.O2kT),

or 0.04,ff7'ieii = 9.68 x 10-3.

282

~ The volume correction is on page 508; we need first to find d. If we assume that theparticles in water are arranged in a cubic lattice (a bad guess, but we'll use it anyway), then 18grams of water has a volume of 18 x 10-6m3, and

d3 = {18 x10-6 m3){6.02 x 1023) = 3.0 x 10-29m3

is the valume allacated ta each water malecule. In this case d = 3.1 x 10-1om. Then

1 4b = 2(6.02 x 1023)( 37r(3.1 x 10-10m)3) = 3.8 m3/mal.

E22-32 d3 = 3b/27rNA, or

= 2.9 x lO-10m.d = 3 3(32 x 10-6m3/~ol)

211"(6.02 x 1023/mol)

E22-33 a haS units of energy volume per square mole, which is the same as energy per mole timesvolume per mole.

P22-1 Salve (1- x)(1.429) + x(1.250) = 1.293 far x. The result is x = 0.7598.

P22-2

~ The only thing that matters is the total number of moles of gas (2.5) and the number ofmoles of the second gas (0.5). Since 1/5 of the total number of moles of gas is associated with thesecond gag, then 1/5 of the total pressure is associated with the second gas.

P22-4 Use Eq. 22-11 withthe appropriate ~ inserted.

P22-5 (a) Since >. cx 1/J2 , we have

~= r&dn V~

= 1.67.= I (27.5 x lO-8m)

y (9.90x lO-8m)

(b) Since >.. (X l/p, we have

= 49.5 x lO-8m.

(c) Since ). <X T, we have

= 7.87x 10-8 m.

283

P22-6 We can assume the molecule will collide with something. Then

1 = 1°CAe-crdr =A/c,

so A = c. Ir the molecule has a mean free path or A, then

>.=1°Crce-cr dr = l/c,

soA=c=l/A.

~ What is important here is the temperature; since the temperatures are the same then theaverage kinetic energies per particle are the same. Then

1 2 1 22ml(VrmS,I) = 2m2(Vrms,2) .

We are given in the problem that Vav,2 = 2Vrms,l° According to Eqs. 22-18 and 22-20 we have

Vrms = ~ = ~~ = ~va.v.

Combining this with the kinetic energy expression above,

2m¡

m2Vrms,2

Vrms,l

= 4.71.=

~

P22-8 (a) Assume that the speeds are not all the same. Transform to a frame of reference wherevav = O, then some of the individual speeds must be greater than zero, and some will be lessthan zero. Squaring these speeds will result in positive, non-zero, numbers; the mean square willnecessarily be greater than zero, so Vrms > 0.

(b) Only if all of the particles have the same speed will Vrms=Vav.

~ (a) We need to first find the number oí particles by integrating

1°C

1vO

N N(v) dv,

CV2 dv +1°C I

va

2 c 3V dv = 3vo.

Invert, then C = 3N/v3.

(b) The average velocity is found from

1 foo'av = N lo N(v)vdv.

Using our result from above,

vav =

284

As expected, the average speed is less than the maximum speed. We can make a prediction aboutthe root mean square speed; it will be larger than the average speed (see Exercise 22-15 above) butsmaller than the maximum speed.

(c) The root-mean-square velocity is found from

Using our results from above,

~ {VO

Nlo

-; {VO

Vo lo

3N 2

-v

vg

2Vrms v2dv,=

v4dv = ~ ~ -3 2v3 5 -Svo .

Then, taking the square root,

Is .¡375 > 3/4? It had better be.

P22-10

P22-11

P22-12

P22-13

P22-14

~ The mass ofair displaced by 2180 m3 is m = (1.22kg/m3)(2180m3) = 2660kg. The massof the balloon and basket is 249 kg and we want to lift 272 kg; this leaves a remainder of 2140 kg forthe mass ofthe air inside the balloon. This corresponds to (2140kg)/(0.0289kg/mol) = 7.4x104mol.

The temperature of the gas inside the balloon is then

T = (pV)/{nR) = [{1.01 x 105Pa){2180 m3)]/[{7.4x 104mol){8.31 J/mol. K) = 358K.

That's 85°C.

P22-16

P22-17

285

~ We apply Eq. 23-1,

H=kA~~x

The rate at which heat flows out is given as a power per area (mW /m2), so the quantity given isreally H / A. Then the temperature difference is

= 710 K~T = !:!: ~xA k = (O

The heat flow is out, so that the temperature is higher at the base oí the crust. The temperaturethere is then

710 + 10 = 720 °C.

E23-2 We apply Eq. 23-1

E23-3 (a) AT/Ax = (136 CO)/(0.249m) = 546CO/m.(b) H = kAAT/Ax = (401W /m. K)(1.80m2)(546CO /m) = 3.94x 105W.(c) TH = (-12°C + 136CO) = 124°C. Then

T = (124°C) -(546 CO /m)(O.ll m) = 63.9°C.

E23-4 (a) H= (0.040W/m.K)(1.8m2)(32CO)/(0.012m) = 190W.(b) Since k has increased by a factor of (0.60)/(0.04) = 15 then H Bhould alBo increase by a

factor of 15.

~ There are three possible arrangements: a sheet of type 1 with a sheet of type 1; a sheetof type 2 with a sheet of type 2; and a sheet of type 1 with a sheet of type 2. We can look back onSample Problem 23-1 to see how to start the problem; the heat flow will be

for substances of different typesj and

A~T j L

(Ljkl) + (Ljk1)H1l =

for a double layer if substance 1. There is a similar expression for a double layer of substance 2.For configuration ( a) we then have

while for configuration (b) we have

klk2

kl + k2.((1/k1) + (1/k2))-1 = ((kl + k2)!(k2k2))-l =

286

54 w /m2) (33,000 m)

(2.5 W /m. K)

Then which is larger2k1k2 ?

~.

(k1 + k2)/2 or

If k1 » k2 then the expression become

kl/2 and 2k2,

so the first expression is larger, and therefore configuration (b) has the lower heat flow. Notice thatwe get the same result if k1 « k2!

E23-6 There's a typo in the exercise.H = A,1.T / R; since the heat flows through one slab and then through the other, we can write

(T1 -T",)/ R1 = (T", -T2)/ R2. Rearranging,

T", = (T1R2 + T2R1)/(Rl + R2).

E23-7 Use the results of Exercise 23-6. At the interface between ice and water Tx = 0°C. ThenR1T2 + R2T1 = O, or klTl/ L1 + k2T2/ L2 = 0. Not only that, L1 + L2 = L, so

k1T1L2 + (L -L2)k2T2 =0,

so

(1.42 m)(1.67W /m. K)( -5.20°C)L2 = = 1.15m.(1.67W /m .K)(-5.20°C) -(0.502W /m .K)(3.98°C)

E23-8 ~T is the same in both cases. So is k. The top configuration has Ht = kA~T/(2L). Thebottom configuration has Hb = k(2A)~T/L. The ratio of Hb/Ht = 4, so heat flows through thebottom configuration at 4 times the rate ofthe top. For the top configuration Ht = (10J)/(2 min) =5J/min. Then Hb = 20J/min. It will take

t = (30J)/(20J/min) = 1.5 min.

~ (a) This exercise has a distraction: it asks about the heat fiow through the window, butwhat you need to find first is the heat fiow through the air near the window. We are given thetemperature gradient both inside and outside the window. Inside,

AT (20°C) -(5°C) = 190 CO /m;

~

~x (0.08 m)

a similar expression exists for outside.

From Eq. 23-1 we find the heat flow through the air;

H = kA.AT = (O-O2R w /m .K)(n R m )2(1 an ~o /m' -1 ~ 'XT---~x \ .., 1\~.~ ...1 \~~~ ~ I...J -~.v ...

The value that we 8J:rived at is the rate that heat flows through the air across an 8J:ea the size ofthe window on either side of the window. This heat flow had to occur through the window as well,so

H = 1.8W

answers the window question.(b) Now that we know the rate that heat flows through the window, we are in a position to find

the temperature difference across the window. Rearranging Eq. 32-1,

~T = H~x (1.8W)(O.OO5m) = () ()')~~o-=kA (1.0 W /m. K)(0.6 m)2 -~.~~~ '"' ,

so we were well justified in our approximation that the temperature drop across the glass is verysmall.

287

E23-10 (a) W = +214J, done on means positive.(b) Q = -293J, extracted from means negative.(c) AEint = Q + W = ( -293J) + (+214J) = -79.0J.

(a) AEint along any path between these two points isE23-11

~Eint = Q + W = (50J) + (-20J) = 30J.

Then along ibf W = (30J) -(36J) = -6J.(b) Q = (-30J) -(+13J) = -43J.(c) Eint.f = Eint.i + ~Eint = (10J) + (30J) = 40J.(d) ~Eintib = (22J) -(10J) = 12J; while ~Eintbf = (40J) -(22J) = 18J. There is no work

done on the path bf, so

Qbf = AEintbf -Wbf = (18J) -(O) = 18J,

and Qib = Qibf -Qbf = (36J) -(18J) = 18J.

Q = mL = (0.10)(2.1 x 108 kg)(333x 103J/kg) = 7.0x 1012J.E23-12

~ We don 't need to know the outside temperature because the amount of heat energyrequired is explicitly stated: 5.22 GJ. We just need to know how much water is required to transfer

this amount of heat energy. Use Eq. 23-11, and then

Q (5.22 x 109 J) ~ = 4.45 x 104 kg.-

m=-

(4iOOJ/kg. K)(50.0°C -22.0°C)c~T

This is the mass of the water, we want to know the volume, so we'll use the density, and then

E23-14

E23-15 Q = mL, so m = (50.4 x 103J)/(333x 103J/kg) = 0.151kg is the mount which freezes.

Then (0.258 kg) -(0.151 kg) = 0.107 kg is the amount which does not freeze.

(a) w = mg~y; if IQI = IWI, thenE23-16

mg6.y

mc

~T=

~ There are three "things" in this problem: the copper bowl (b), the water (w), and thecopper cylinder (c). The total internal energy changes must add up to zero, so

AEint,b + AEint,w + AEint,c = O.

As in Sample Problem 23-3, no work is done on any object, so

Qb + Qw + Qc = O.

288

The heat transfers for these three objects are

Qb

Qw

Qc

= mbCb(Tf,b -Ti,b),

= mwCw(Tf,w -Ti,w) + Lvm2,

= mccc(Tf,c -Ti,c).

For the most part, this looks exactly like the presentation in Sample Problem 23-3; but there is anextra term in the second line. This term refiects the extra heat required to vaporize m2 = 4.70 9 ofwater at loo°C into steam loo°C.

Some of the initial temperatures are specified in the exercise: Ti b = Ti w = 21.0°C and T f b =, , ,Tf,w = Tf,c = loo°C.

( a) The heat transferred to the water, then, is

Qw = (0.223 kg)(4190J/kg.K) ((100°C) -(21.0°C)) ,

+(2.26 x 106 J /kg) ( 4.70 x 10-3kg) ,= 8.44 x 104 J .

Thi8 an8Wer differ8 from the back ofthe book. I think that they (or was it me) u8ed the latent heatof fu8ion when they 8hould have U8ed the latent heat of vaporization!

(b) The heat tran8fered to the bowl, then, i8

Qw = (0.146 kg)(387 J/kg .K) ((100°0) -(21.0°C)) = 4.46 x 103 J.

( c) The heat transfered from the cylinder was transfered into the water and bowl, so

Qc = -Qb -Qw = -{4.46 x lO3J) -{8.44 x lO4J) = -8.89 x lO4J.

The initial temperature of the cylinder is then given by

Qc

mcCcTi c = Tí c -

, ,

E23-18 The temperature of the silver must be raised to the melting point and then the heatedsilver needs to be melted. The heat required is

Q = mL + mcLlT = (O.l30kg)[(lO5x lO3J/kg) + (236J/kg .K)(l235K -289 K)] = 4.27x lO4J.

E23-19 (a) Use Q = mc6.T, m = pV, and t = Q/P. Then

JmaCa + Pw v wCw],6.T

p

~)(900J/kg.K) + (998kg/m3)(0.64x 1~(4190J/kg.K)](100~2~q2= 117 s.

(2400 W)

(b) Use Q = mL, m = pV, and t = Q/P. Then

PwVwLw

pt=

is the additional time required.

289

E23-20 The heat given off by the steam will be

Qs = msLv + msCw(50CO).

The hear taken in by the ice will be

Qi = miLf + miCw(50 Co;

Equating,

Lf + Cw(50 CO)ms =

~ The linear dimensions of the ring and sphere change with the temperature change ac-cording to

~dr = ardr(Tf,r -Ti,r),

~ds = asds(Tf,s -Ti,s).

When the ring and sphere are at the same (final) temperature the ring and the sphere have thesame diameter. This means that

dr + 6.dr = ds + 6.ds

when Tf,s = Tf,r. We'll solve these expansion equations first, and then go back to the heat equations.

dr + 6.dr = ds + 6.ds,

dr (1 + ar(Tf,r -Ti,r)) = ds (1 + as(Tf,s -Ti,s)) ,

which can be rearranged to give

ardrTf,r -asdsTf,s = ds (1- asTi,s) -dr (1- arTi,r) ,

but since the final temperatures are the same,

Tf = .ds (1- asTi,s) -dr (1- arTi,r)

ardr -asds

Putting in the numbers,

Tf =

= 34.1oC.

No work is done, so we only have the issue of heat flow, then

Qr + Qs = o.

Where "r" refers to the copper ring and "s" refers to the aluminum sphere. The heat equations are

Qr = mrCr(Tf -Ti,r),

Qs = msCs(Tf -Ti,s).

290

E23-22 The problem is compounded because we don't know if the final state is only water, onlyice, or a mixture of the two.

Consider first the water. Cooling it to 0°C would require the removal of

Qw = (0.200kg)(4190J/kg .K)(O°C -25°C) = -2.095x 104J.

Consider now the ice. Warming the ice to would require the addition of

Qi = (0.100kg)(2220J/kg .K)(O°C + 15°C) = 3.33x 103 J.

The heat absorbed by the warming ice isn 't enough to cool the water to freezing. However, the icecan melt; and if it does it will require the addition of

Qim = (0.100kg)(333x 103J/kg) = 3.33x 104J.

This is far more than will be liberated by the cooling water, so the final temperature is o°C, andconsists of a mixture of ice and water.

(b) Consider now the ice. Warming the ice to would require the addition of

Qi = (0.050kg)(2220J/kg .K)(O°C + 15°C) = 1.665 x 103 J.

The heat absorbed by the warming ice isn 't enough to cool the water to freezing. However, the icecan melt; and if it does it will require the addition of

Qim = (0.050kg)(333x103J/kg) = 1.665x104J.

This is still not enough to cool the water to freezing. Hence, we need to solve

Qi + Qim + miCw(T -o°C) + mwCw(T -25°C) = 0,

which has solution

T = (4190J/kg. K)(0.200kg)(25°C) -(1.665x 103J) + (1.665x 104J)= 2.5°C.

(4190J/kg. K)(0.250kg)

E23-23 (a) c = (320J)/(0.0371 kg)(42.0°C -26.1°C) = 542 J/kg .K.(b) n = m/M = (37.1 g)/(51.4 g/mol) = 0.722 mol.(c) c = (542J/kg .K)(51.4x 10-3 kg/mol) = 27.9J/mol. K.

E23-24 (1) W = -p6.V = (15Pa)(4m3) = -60J for the horizontal path; no work is done duringthe vertical pathj the net work done on the gas is -60 J .

(2) It is easiest to consider work as the (negative of) the area under the curvej then W =-(15Pa + 5Pa)(4m3)/2 = -40J.

(3) No work is done during the vertical path; W = -p6.V = (5Pa)(4m3) = -20J for thehorizontal pathj the net work done on the gas is -20J.

~ Net work done on the gas is given by Eq. 23-15,

W=-IPdV.But integrals are just the area under the curve; and that 's the easy way to solve this problem. In thecase of closed paths, it becomes the area inside the curve, with a clockwise sense giving a positivevalue for the integral.

The magnitude of the area is the same for either path, since it is a rectangle divided in half bya square. The area of the rectangle is

(15x103Pa)(6m3) = 90x103J,

so the area of path 1 (counterclockwise) is -45 kJ; this means the work done on the gas is -(-45 kJ)or 45 kJ. The work done on the gas for path 2 is the negative of this because the path is clockwise.

291

E23-26 During the isothermal expansion,

V2 -Pl Vlln P.!P2

Wl = -nRT InVi

During cooling at constant pressure,

W2 = -P2¡lV = -P2(Vl -V2) = -P2Vl(1-Pl!P2) = Vl(Pl-P2).

The work done is the sum, or

E23-27 During the isothermal expansion,

V2V2w = -nRTln .

V¡

= -PI VIln VI'

so

= l.l4xlO3J.

E23-28 (a) pV'Y is a constant, so

P2 = P1(V1/V2)'Y = (1.00 atm)((11)/(0.5 1))1.32

T2 = T1(P2/P1)(V2/V1), SO

= 2.50 atm;

-=341K.T2 = (273 K) .(2.50 at~ (0.5 I)

(1.00 atm) ~

(b) V3 = V2(P2/Pl)(T3/T2), SO

=0.40

( c ) During the adiabatic process,

(1.01 x 105Pa/atm)(1 x 10-3m3 /1)[( 2.5 atm )( 0.5 1) -(1.0 atm)(11)] = 78.9 J.

W12 = .

(1.32) -1

During the cooling process,W23 = -p~V = -(1.01 x 105pa/atm)(2.50 atm)(l x 10-3m3/1)[(0.4 I) -(0.5 J)] = 25.2J.

The net work done is W123 = 78.9J + 25.2J = 104.1 J.

292

E23-30 Air is mostly diatomic (N2 and O2), so use T = 1.4.(a) pV'Y is a constant, so

T2 = T1(P2/Pl)(V2/V1), SO

or 96°C.(b) The work required for delivering 1 liter of compressed air is

W12 = .(1.01 x 105Pa/atm)(lx10-3m3/1)[(2.3 atm)(1.0 I) -(1.0 atm)(1.0 1/0.552)] = 123J.

(1.40) -1

The number of liters per second that can be delivered is then

6.V/6.t = (230W)/(123J/l) = 1.87

E23-31 Eint,rot = nRT = (1 mol)(8.31 J/mol. K)(298K) = 2480J.

E23-32 Eint,rot = ~nRT = (1.5)(1 mol)(8.31 J/mol. K)(523K) = 6520J.

~ (a) Invert Eq. 32-20,

Tf = Ti~ = (250 K) (l450x lO3Pa)(l.36m3) -PiVi (l22xlO3Pa)(lO.7m3) -378 K,

~

which is the same as 105°C.( c ) Ideal gas law, again:

n = [P~]/[RT] = [(1450 x 103pa)(1.36m3)]/[(8.31 J/mal. K)(378K)] = 628 mal.

(d) From Eq. 23-24,

before the compression and

2.96 x 106 J

after the compression.( e) The ratio of the rms speeds will be proportional to the square root of the ratio of the internal

energies,V(1.96x 106J)/(2.96x 106J) = 0.813;

we can do this because the number of particles is the same before and after, hence the ratio of theenergies per particle is the same as the ratio of the total energies.

293

E23-34 We can assume neon is an ideal gas. Then ~T = 2~Eint/3nR, or

E23-35 At constant pressure, doubling the volume is the same as doubling the temperature. Then

7Q = nCp~T = {1.35 mol)2{8.31J/mol. K) {568 K -284 K) = 1.12x104J.

E23-36 (a) n = m/M = (12 g)/(28 g/mol) = 0.429 mol.(b) This is a constant volume process, so

Q = nCvAT = {0.429 mol)~{8.31 J/mol. K){125°C -25°C) = 891J.

Q = nCp~T = {4.34 mol){29. J/mol. K)(62.4K) = 7880J.

(b) From Eq. 23-28,

5 5Eint = 2nR~T = 2(4.34 mol)(8.31J/mol. K)(62.4K) = 5630J.

(c) From Eq. 23-23,

3 5Ktrans = 2nR~T = 2(4.34 mol)(8.31 J/mol. K)(62.4K) = 3380J.

E23-38 Cv = ~(8.31 J/mol. K)/(4.00 g/mol) = 3120J/kg .K.

E23-39 Each species will experience the same temperature change, so

Q = Ql + Q2 + Q3,= nlC1~T + n2C2~T + n3C3~T,

Dividing this by n = n¡ + n2 + n3 and ~T will return the specific heat capacity oí the mixture, so

G -n¡ G¡ + n2G2 + n3G3

n¡ + n2 + n3

E23-40 W AB = O, since it is aconstant volume process, consequently, W = W AB+W ABC = -15J.But around a closed path Q = -W, so Q = 15 J. Then

QCA = Q -QAB -QBC = (15J) -(20J) -(OJ) = -5J.

Note that this heat is removed from the system!

294

~ According to Eq. 23-25 (which is specific to ideal gases),

3~Eint = 2nR~T,

and for an isothermal process ~T = O, so for an ideal gas ~Eint = 0. Consequently, Q + W = 0 foran ideal gas which undergoes an isothermal process.

But we know W for an isotherm, Eq. 23-18 shows

Then finally

Vi

E23-42 Q is greatest for constant pressure processes and least for adiabatic. W is greatest (inmagnitude, it is negative for increasing volume processes) for constant pressure processes and leastfor adiabatic. AEint is greatest for constant pressure (for which it is positive), and least for adiabatic(for which is is negative).

E23-43 (a) For a monatomic gas, 'Y = 1.667. Fast process are often adiabatic, so

T2 = T1(Vl/V2)'Y-l = (292 K)[(1)(1/10)]1.667-1 = 1360 K.

(b) For a diatomic gas, "Y = 1.4. Fast process are often adiabatic, so

T2 = T1(V1/V2)'Y-l = (292K)[(1)(1/10)]l04-1 = 733 K.

E23-44 This problem cannot be solved without making some assumptions about the type of pro-cess occurring on the two curved portions.

~ Ir the pressure and volume are both doubled along a straight line then the process canbe described by

p=!?!-VV1

The final point involves the doubling of both the pressure and the volume, so according to the idealgas law, pV = nRT, the final temperature T2 will be four times the initial temperature T1.

Now for the exercises.(a) The work done on the gas is

¡ 2 ¡ 2 ( 2 2

W = -pdV = -f!;..VdV = -f!;.. ~ -~ )I I VI V I 2 2

295

We want to express our answer in terms oí TIo First we take advantage oí the íact that V2 = 2VI,then

E23-46 The work done is the area enclosed by the path. If the pressure is measured in units of10MPa, then the shape is a semi-circle, and the area is

W = (1T/2)(1.5)2(10MPa)(lx10-3m3) = 3.53x104J.

The heat is given by Q = -W = -3.53 x 104 J .

E23-48 Constant Volume(a) Q = 3(3.15 mol)(8.31 J/mol. K)(52.0K) = 4080J.(b) W = o.( c ) ~Ermint = 3(3.15 mol) (8.31 J /mol .K) (52.0 K) = 4080 J .

Constant Pressure(a) Q = 4(3.15 mol)(8.31J/mol. K)(52.0K) = 5450J.(b) W = -P~V = -nR~T = -(3.15 mol)(8.31 J/mol. K)(52.0K)= -1360J.(c) ~Ermint = 3(3.15 mol)(8.31 J/mol. K)(52.0K) = 4080J.

Adiabatic(a) Q = 0.(b) W = (PfV f -Pi Vi)/('Y -1) = nR~T/('Y -1) = 3(3.15 mol)(8.31 J/mol. K)(52.0K) = 4080J.(c) ~Ermint = 3(3.15 mol)(8.31 J/mol. K)(52.0K) = 4080J.

P23-1 (a) The temperaturedifference is

(5 Co /9 FO)(72°F --20°F) = 51.1 Co.

The rate of heat loss is

H = (1.0W /m. K)(1.4m2)(51.1 CO)/(3.0x 10-3m) = 2.4x 104W.

(b) Start by finding the R values.

Rg = (3.0x 10-3m)/(1.0W /m. K) = 3.0x 10-3m2 .K/W,

Ra = (7.5x 10-2m)/(O.O26W /m. K) = 2.88m2 .K/W.

Then use Eq. 23-5,

Get double pane windowsl

296

P23-2 (a) H = (428W /m. K)(4. 76x 10-4m2)(100 CO)/(1.17m) = 17.4 W.(b) ~m/~t = H/L = (17.4 W)/(333x103J/kg) = 5.23x10-5kg/s, which is the same as 188 g/h.

~ Follow the example in Sample Problem 23-2. We start with Eq. 23-1:

H -kA~ dr ,

H2 dT

= k( 47rr ) "dr" ,

j T2= kdT,

Tl

= k(T1 -T2),

411"k(Tl -T2),

411"k(Tl -T2)rlr2 .

r2 -rl

=

H

P23-4 (a) H = (54xlO-3W/m2)47r(6.37xlO6m)2 = 2.8xlO13W.

(b) Using the results of Problem 23-3,

AT = (2.8x lOI3W)(6.37x lO6m -3.47x lO6m)

47r(4.2W /m. K)(6.37x lO6m)(3.47 x lO6m)

Since T2 = 0°C, we expect TI = 7.0xlO4co.

7.0x lO4co.

-kAdTjdx, then Hdx = -aTdT. H is a constant, so integrate both sideP23-5 Since Haccording to

JHdx = -JaTdT,

HL =

H

P23-6 Assume the water is all at 0°C. The heat flow through the ice is then Hrate of ice formation is ~m/ ~t = H/ L. But ~m = pA~x, so

kAAT/x; the

6.x

6.t

H

PAL=

k~T

-P-j;;;'

=

il. 7W /m. K)(10CO)= 1.11 x 10-6m/s.

(920 kg/m3)(333x lO3J/kg)(0.05 m)

That's the same as 0.40 cm/h.

~ (a) Start with the heat equation:

Qt + Qi + Qw = O,

297

,1

where Qt is the heat from the tea, Qi is the heat from the ice when it melts, and Qw is the heatfrom the water (which used to be ice). Then

mtCt(Tf -Tt,i) + miLf + mwCw(Tf -T w,i) = O,

which, since we have assumed all of the ice melts and the masses are all equal, can be solved for T fas

CtTt,i + CwT w,i -LfTi =

Ct+Cw(4190J/kg. K)(90°C)+ (4190J/kg .K)(O°C) -(333 x 103J/kg)

(4190J/kg .K) + (4190J/kg .K)5.3°C.

(b) Once again, assume all of the ice melted. Then we can do the same steps, and we get

TI ,--- CtTt.i + CwT w,i =--LE

~

Ct+Cw

,(~190J/kg .K)(70°C) + (4190J/kg .K)(O°C) -(333~103J/kg)

(4190J/kg. K) + (4190J/kg .K)-4.7°C.

So we must have guessed wrong when we assumed that all of the ice melted. The heat equationthen simplifies to

TntCt(Tf -Tt,J + miLf = O,

and then

TntCt(Tt,i -Tf)mi =

Lf

(0.520 kg)(41~OJ/kg .K)(90°C -00)

(333 x lO3J/kg)

0.458 kg.

P23-8 c = Q/m~T = H/(~m/~t)~T. But ~m/~t = p~V/~t. Combining,

= 2.4x lO3J/kg .K.(AV/At)pAT (8.2 x lO-6m3/s)(0.85 x lO3kg/m3)(l5 CO)

P23-9 (a) n = NA/M, so

(b) Eav = ~kT, so

P23-10 Qw + Qt = O, so

CtATt + mwCw(Tr -TU = O,

or.(0.3 kg)(4190J/kg .m)(44.4°C) + (46.1 J/K)J44.4°C -15.0°C)

Ti= = 45.5°C.(0.3kg)(4190J/kg. m)

298

~ We can use Eq. 23-10, but we will need to approximate c first. If we assume that theline is straight then we use c = mT + b. I approximate m from

m = .(14J /mol. K) -(3 J /mol .K) = 3.67 x 10-2 J /mol.

(50ijK) -(200 K)

Then find b from those same data points,

b = (3 J /mol .K) -(3.67 x 10-2 J /mol) (200 K) = -4.34 J /mol .K.

Then from Eq. 23-10,

Q

= {0.45mal) ( {3.67 x 10-2 J /mal) ( (500 K)2 -{200 K)2)

, 2+ {-4.34J/mol. K){500K -200K))

l.l5xlO3J.

P23-12 8Q = nC8T, so

Q

goKn [(0.318 J/mol. K2)T2/2 -(0.00109 J/mol. K3)T3/3 -(0.628 J/mol. K)T]50K ,

n(645.8 J/mol).

Finally,Q {645.8 J/mol){316 g)/{107.87 g/mol) = 189J.

P23-13 TV"Y-l is a constant, so

T2 = (292 K)(1/1.28)(1.40)-1 = 265 K

P23-14 w= -JpdV, so

J [ nRT an2]-v-nb-V2 dV,w =

~I-nRTln(V -nb) -

Vf-nb

Vi-nb2-nRT In -aff

299

hTf -n cdT,

Ti

{Tf= nJTi (mT+b)dT,

[m 2 ] Tf = n 2T +bT Ti ,

= n (~(Tf2 -Ti2) + b(Tf -Ti) ) ,

~ We can use Eq. 23-10, but we will need to approximate c first. If we assume that theline is straight then we use c = mT + b. I approximate m from

m = .(14J /mol. K) -(3 J /mol .K) = 3.67 x 10-2 J /mol.

(50ijK) -(200 K)

Then find b from those same data points,

b = (3 J /mol .K) -(3.67 x 10-2 J /mol) (200 K) = -4.34 J /mol .K.

Then from Eq. 23-10,

Q

= {0.45mal) ( {3.67 x 10-2 J /mal) ( (500 K)2 -{200 K)2)

, 2+ {-4.34J/mol. K){500K -200K))

l.l5xlO3J.

P23-12 8Q = nC8T, so

Q

goKn [(0.318 J/mol. K2)T2/2 -(0.00109 J/mol. K3)T3/3 -(0.628 J/mol. K)T]50K ,

n(645.8 J/mol).

Finally,Q {645.8 J/mol){316 g)/{107.87 g/mol) = 189J.

P23-13 TV"Y-l is a constant, so

T2 = (292 K)(1/1.28)(1.40)-1 = 265 K

P23-14 w= -JpdV, so

J [ nRT an2]-v-nb-V2 dV,w =

~I-nRTln(V -nb) -

Vf-nb

Vi-nb2-nRT In -aff

299

hTf -n cdT,

Ti

{Tf= nJTi (mT+b)dT,

[m 2 ] Tf = n 2T +bT Ti ,

= n (~(Tf2 -Ti2) + b(Tf -Ti) ) ,

P23-17 The speed of sound in the iodine gas is

v = fA = (1000 Hz)(2 x 0.0677m) = 135m/s.

w = -Q = mL = (333xlO3J/kg)(O.l22kg) = 4.06xlO4J.P23-18

P23-19 (a)Process ABQ = ~(1.0 mol)(8.31 J/mol. K)(300K) = 3740J.

W=O.~Eint = Q + W = 3740J.

Process BC

Q=O.W = (PfV f -Pi Vi)/('Y -1) = nR~T/('Y -1) = (1.0 mol)(8.31 J/mol. K)( -145K)/(1.67 -1) =

-1800 J~Eint = Q + W = -1800J.

Process ABQ = ~(1.0 mol)(8.31 J/mol. K)( -155K) = -3220J.W = -P~V = -nR~T = -(1.0 mol)(8.31J/mol. K)(-155K) = 1290J.~Eint = Q + W = 1930J.

CycleQ = 520J; W = -510J (rounding error!); ~Eint = 10J (rounding error!)

P23-20

P23-21 Pf = (16.0 atm)(50/250)1.40 = 1.68 atm. The work done by the gas during the expansion

is

w =

96.0J.

This process happens 4000 times per minute, but the actual time to complete the process is half ofthe cycle, or 1/8000 of a minute. Then p = (96J)(8000)/(60s) = 12.8x 103W.

301

.

~ For isothermal processes the entropy expression is almost trivial, ~S = Q/T, where if Qis positive (heat flow into system) the entropy increases.

Then Q = T~S = (405K)(46.2J/K) = 1.87xl04J.

E24-2 Entropy is a state variable and is path independent, so(a) ~Sab,2 = ~Sab,l = +0.60J/K,(b) ~Sba,2 = -~Sab,2 = -0.60J/K,

E24-3 (a) Heat onlyenters along the top path, so

Qin = T6.S = (400 K)(0.6J/K -O.lJ/K) = 200J.

(b) Heat leaves only bottom path, so

Qout = T~S = (250 K)(0.l 3/K -0.63/K) = -1253.

Since Q + W = O for a cyclic path,

w = -Q = -[(200J) + (-125J)] = -75J.

E24-4

(b) For isothermal process, Q = -W, then

~s = Q/T= (l.69xlO4J)/(4l0K) = 4l.2J/K.

( c ) Entropy change is zero for reversible adiabatic processes.

~ (a) We want to find the heat absorbed, so

Q = mcAT = (1.22 kg)(387 J/mol. K) ((105°C) -(25.0°C)) = 3.77x 104J.

(b) We want to find the entropy change, so, according to Eq. 24-1,

~s

~

The entropy change oí the copper block is then

E24-6 ~s = Q/T = mL/T, so

~s = (0.001 kg)( -333 x 103 J/kg)/(263K) = -1.27J/K

302

E24-7 Use the first equation on page 551

E24-8 ~S = Q/T c -Q/Th.(a) ~S = (260J)(1/100K -1/400K) = 1.95J/K.(b) ~S = (260J)(1/200K -1/400K) = 0.65J/K.(c) ~S = (260J)(1/300K -1/400K) = 0.217 J/K.(d) ~S = (260J)(1/360K -1/400K) = 0.0722J/K.

~ (a) If the rod is in a steady state we wouldn't expect the entropy of the rod to change.Heat energy is flowing out of the hot reservoir into the rod, but this process happens at a fixedtemperature, so the entropy change in the hot reservoir is

2!!

TH

( -1200 J)

( 403 K)ilSH = = -2.98J/K.

~

The heat energy flows into the cold reservoir, so

2.!!

TH

(1200 J)

(297K)L\Sc = = 4.04J/K.=

The total change in entropy of the system is the sum of these two terms

~s = ~SH + ~SC = 1.06J/K.

(b) Since the rod is in a steady state, nothing is changing, not even the entropy.

E24-10 (a) Qc + Ql = O, so

mccc(T- T c) + mlC¡(T -Ti) = O,

which can be solved for T to give

~~!~)S387 J/kg .K) (400 K) + (0.10kg)(129J/kg .K) (200 K)T= = 320 K.(0.05 kg)(387 Jjkg .K) + (0.10 kg)(129Jjkg .K)

(b) Zero.(c) ~S = mclnTf/Tj, so

~ The total mass of ice and water is 2.04 kg. If eventually the ice and water have the samemass, then the final state will have 1.02 kg of each. This means that 1.78 kg -1.02 kg = 0.76 kg ofwater changed into ice.

(a) The change of water at o°C to ice at o°C is isothermal, so the entropy change is

(0.76 kg)(333 x 103J/kg)~s = .2

r=

-mLT -927 J/K..

(273 K)

(b) The entropy change is now +927 J/K.

303

E24-12 (a) Qa + Qw = O, 80

maca(T- T a) + mwCw(T -T w) = O,

= -24.4J/K.

(c) For the water,

= 27.5J/K.

(d) AS = (27.5J/K) + (-24.4J/K) = 3.1J/K.

E24-13 (a) e = 1- (36.2 J/52.4J) = 0.309.(b) W = Qh -Qc = (52.4J) -(36.2J) = 16.2J.

E24-14 (a) Qh = (8.18 kJ)/(0.25) = 32.7 kJ, Qc = Qh -W = (32.7 kJ) -(8.18 kJ) = 24.5 kJ.(b) Qh = (8.18 kJ)/(0.31) = 26.4 kJ, Qc = Qh -W = (26.4 kJ) -(8.18 kJ) = 18.2 kJ.

~ One hour's worth oí coal, when burned, will provide energy equal to

{382 x 103kg) {28.0 x 106 J /kg) = 1.07 x 1013 J .

In this hour , however, the plant only generates

(755 x lO6W)(3600s) = 2.72xlO12J.

The efficiency is thene = (2.72xl012J)/(1.07xl013J) = 25.4%.

E24-16 We use the convention that all quantities are positive, regardless of direction. W A = 5W B;Qi,A = 3Qi,B; and Qo,A = 2Qo,B. But W A = Qi,A -Qo,A, so

5WB = 3Qi,B -2Qo,B,

or, applying WB = Qi,B -Qo,B,

= 3Qi,B -2(Qi,B -WB),= Qi,B,= 1/3 = eB.

5WB

3WB

WB/Qi,B

Then

304

E24-22 (a) The area of the cycle is ~V ~p = PoVo, so the work done by the gas is

w = (l.Olxl05pa)(0.0225m3) = 2270J.

(b) Let the temperature at a be Tao Then

Tb = Ta(Vb/Va)(Pb/Pa) 2Ta.

Let the temperature at c be Tc' Then

Tc = Ta(Vc/Va)(Pc/Pa) = 4Ta.

Consequently, ~Tab = Ta and ~Tbc = 2Tao Putting this information into the constant volume andconstant pressure heat expressions,

and

Qbc

so that Qac = YPo Yo, or

13Qac = 2{1.01 x 105Pa){0.0225 m3)

{c) e = {2270J)/{14800J) = 0.153.

{d) ec = 1- {Ta/4Ta) = 0.75.

l.48x lO4J.

~ According to Eq. 24-15,

Now we salve the question out oí order .(b) The work required to run the freezer is

IWI = IQLI/K = (185 kJ)/(5.70) = 32.5 kJ.

(a) The freezer will discharge heat into the room equal to

IQLI + IWI = (185 kJ) + (32.5 kJ) = 218 kJ.

E24-24 (a) K = IQLlllwl = (5683)/(1533) = 3.71.(b) IQHI = IQLI + IWI = (5683) + (1533) = 7213.

E24-25 K = TL/(TH -TL); IWI = IQLI/K = IQLI(TH/TL -1).(a) IWI = (10.0J)(300K/280K -1) = O.714J.(b) IWI = (10.0J)(300K/200K -1) = 5.00J.(c) IWI = (10.0J)(300K/100K -1) = 20.0J.(d) IWI = (10.0J)(300K/50K -1) = 50.0J.

E24-26

306

~ We wi11 start with the 8Bsumption that the air conditioner Ís a Carnot refrigerator .K = TL/(TH -TL); IWI = IQLI/K = IQL1(TH/TL -1). For fun, 1'11 convert temperature to the

abso1ute Rankine sca1e! Then

IQLI = (1.0J)/(555°R/5300R -1) 21J.

E24-28 The best coefficient of performance is

Kc = (276 K)/(308K -276 K) = 8.62.

The inventor claims they have a machine with

K = (20 kW -1.9kW)/(1.9 kW) = 9.53.

Can 't be done.

E24-29 {a) e = 1- {258 K/322K) = 0.199. IWI = {568 J){0.199) = 113J.{b) K = {258K)/{322K -258K) = 4.03. IWI = {1230J)/{4.03) = 305J.

E24-30 The temperatures are distractors!

IWI = IQHI-IQLI = IQHI- KIWI,

soJWI = IQHI/(l + K) = (7.6 MJ)/(l + 3.8) = 1.58 MJ.

Then p = (1.58 MJ)/(3600s) = 440W.

E24-31 K = (260K)/(298K -260 K) = 6.8.

E24-32 K = (0.85)(270K)/(299K-270K) = 7.91. In 15 minutes the motor can do (210W)(900s) =

1.89x 105J of work. Then

IQLI = KIWI = (7.9l)(l.89xlO5J) = l.50xlO6J.

~ The Ca;rnot engine has an efficiency

E = I -T2T;=

IWI

TQJ"

The Carnot refrigerator has a coefficient of performance

IQ41

TWr'

T4 =K = T;-::"T¡

Then we combine the last two expressions, and

IQ31-IWIIWI

~

IWI-1.=~-T3 -T4 -

307

Finally, combine them all,T4 -

;¡;-::-T¡- IQ1I Tl -T2

Now, we rearrange,

~IQli

=

E24-34 (a) Integrate:

ln N! ~ lN ln x dx = N ln N -N + 1 ~ N ln N -N .

(b) 91, 752, and about 615,000. You will need to use the Stirling approximation extended to adouble inequality to do the lBSt two:

.[i;nn+l/2e-n+l/(12n+l) < n! < .[i;nn+l/2e-n+l/(12n) .

~ (a) For this problem we don't care how the particles are arranged inside a section, weonly care how they are divided up between the two sides.

Consequently, there is only one way to arrange the particles: you put them all on one side, andyou have no other choices. So the multiplicity in this case is one, or Wl = 1.

(b) Once the particles are allowed to mix we have more work in computing the multiplicity.Using Eq. 24-19, we have

N! N!W2 = (N /2)!(N /2)! = ""("{Ñ7"iji)"

(c) The entropy oí a state oí multiplicity w is given by Eq. 24-20,

s = klnw

For part (a), with a multiplicity of 1, 81 = O. Now for part (b),

N!

¡¡N¡2¡ij'

82 = kln = klnN! -2kln(N/2)!

and we need to expand each of those terms with Stirling's approximation.Combining,

82 = = k (NlnN -N) -2k ((N/2) ln(N/2) -(N/2)) ,

= kNlnN- kN -kNlnN + kNln2 + kN,

= kNln2

Finally, ~8 = 82 -81 = kN ln 2.(d) The answer should be the same; it is a free expansion problem in both cases!

308

~ We want to evaluate

~s = {Tf~JT; T ,

{Tf nAT3 dTJT; T ,

l Tf nAT2 dT ,

Ti

~ (Tf3 -Ti3) .

Into this last expression, which is true for many substances at sufficiently low temperatures, wesubstitute the given numbers.

-( 4.8 mal) (3. 15 x 10-5 J /mal. K4){ (10K)3 -(5.0 K)3) = 4.41 x 10-'}; J /K:

AS- 3

P24-2

P24-3 (a) Work is only done along path ab, where Wab = -p6.V = -3Po6.Vo. So Wabc= -3poVo.(b) 6.E Jbc = ~nR6.Tbc, with a little algebra,

3 3 3~Eintbc = 2(nRTc -nRTb) = 2(PcVc -PbVb) = 2(8 -4)poVo = 6poVo.

6.Sbc = ~nRln(Tc/Tb), with a little algebra,

3 3ASbc = 2nRln(pc/Pb) = 2nRln2.

(c) Both are zero for a cyclic process.

P24-4 (a) For an isothermal process,

P2 = Pl (Vl/V2) = Pl/3.

For an adiabatic process,

P3 = P1 (V1/V2)'Y = P1 (1/3)1.4 = O.215p1.

For a constant volume process,

T3 = T2(P3/P2) = Tl(O.215/0.333) = O.646T1,

(b) The easiest ones first: ~Eint12 = O, W23 = 0, Q31 = 0, ~S31 = 0. The next easier ones:

5 5~Eint23 = 2nR~T23 = 2nR(O.646Tl -T1) = -0.885Pl Vl,

Q23 = ~Eint23 -W23 = -0.885pl Vl,

~Eint31 = -~Eint23 -~Eint12 = O.885pl V1,

W31 = ~Eint31 -Q31 = O.885pl V1.

309

Finally, some harder ones:

W12 -nRT11n(V2/V1) = -P1 V11n(3) = -1.10p1 V1,

Q12 = ~Eint12 -W12 = 1.10p1 V1

And now, the hardest:~Sl2 = Ql2/Tl = 1.10nR,

~S23 = -~Sl2 -~S3l = -1.10nR.

P24-5 Note that TA = TB = Tc/4 = TD.

Process I: ABC(a) QAB = -WAB = nRToln(VB/VA) =PoVoln2. QBC = ~nR(Tc-TB) = ~(PCVC-PBVB) =

~(4poVo -PoVo) = 4.5poVo.

(b) WAB= -nRToln(VB/VA) = -PoVoln2. WBC = O.(c) Eint = ~nR(Tc -TA) = ~(PcVc -PAVA) = ~(4poVo -PoVo) = 4.5poVo.

(d) ~SAB = nRln(VB/VA) = nRln2; ~SBC = ~nRln(Tc/TB) = ~nRln4 = 3nRln2. Then

~SAC = 4nR.

Process II: ADC(a) QAD = -WAD = nRToln(VD/VA) = -PoVoln2. QDC = ~nR(Tc -TD) = ~(pcVC -

PDVD) = ~(4poVo -PoVo) = 10poVo.

(b) WAB = -nRToln(VD/VA) =PoVoln2. WDC = -p~V = -Po(2Vo -Vo/2) = -~poVo.

(c) Eint = ~nR(Tc -TA) = ~(PcVc -PAVA) = ~(4PoVo -PoVo) = 4.5poVo.

(d) ~SAD = nRln(VD/VA) = -nRln2; ~SDC = ~nRln(Tc/TD) = ~nRln4 = 5nRln2. Then

~SAC = 4nR.

P24-6 The heat required to melt the ice is

Q = m(Cw~T23 + L + ci~T12),

(O.O126kg)[(4190J/kg. K)(15CO) + (333 x 103J/kg) + (2220J/kg .K)(10CO)],

5270J.

The change in entropy of the ice is

~Si = m[Cw ln(T3/T2) + L/T2 + Ci ln(T2/Tl)],

(O.O126kg)[(4190J/kg.K) ln(288/273) + (333 x 103J/kg)/(273K),

+(2220J/kg.K) ln(273/263)],

19.24J/K

The change in entropy ofthe lake is ~Sl = (-5270 J)/(288K) = 18.29J/K. The change in entropyof the system is 0.95 J /kg.

~ (a) This is a problem where the total internal energy of the two objects doesn't change,but since no work is done during the process, we can start with the simpler expression Ql + Q2 = O.The heat transfers by the two objects are

Ql

Q2

= mlCl(Ti -T1,i),

= m2C2(T2 -T2,i).

Note that we don't call the final temperature Tf here, because we are not assuming that the twoobjects are at equilibrium.

310

We combine these three equations,

m2c2(T2 -T2,i) = -m1c1(T1 -T1,i),

m2c2T2 = m2c2T2,i + m1c1(T1,i -T1),

m1c1T2 = T2 i + -(T1 i -T1), m2C2 ,

As object 1 "cools clown" , object 2 "heats up" , as expectecl.(b) The entropy change of one object is given by

~S= rTf~=mcln~,JT; T Ti

and the total entropy change for the system will be the sum of the changes for each object, so

TI T2

Ti,2

~s = mlclln T + m2c21ni,l

Into the this last equation we need to substitute the expression for T2 in as a function of T1. There'sno new physics in doing this, just a mess of algebra.

(c) We want to evaluate d(AS)/dT1. To save on algebra we will work with the last expression,remembering that T2 is a function, not a variable. Then

dT2

dTl=

mlcl

We could consider writing T2 out in all of its glory, but what would it gain us? Nothing. There isactually considerably more physics in the expression as written, because...

(d) ...we get a maximum for ~S when d(~S)/dTl = O, and this can only occur when T1 = T2according to the expression.

P24-8 Tb = {10.4 x 1.01 x 105Pa){1.22 m3)/{2 mol){8.31 J/mol. K) = 7.71 x 104K. Maybe not sorealistic? T a can be found after finding

Pc = Pb(~/Vc)'Y = (10.4 x 1.01 x 105Pa)(1.22/9.13)1.67 = 3.64x 104Pa,

ThenTa = Tb(pa/Pb) = (7.7lx lO4K)(3.64x lO4/l.05X lO6) = 2.67xlO3K.

311

Similarly,Tc = Ta(Vc/Va) = (2.67x 103K)(9.13/1.22) = 2.00x 104K.

(a) Heat is added during process ab onlyj

3 3Qab = 2nR(Tb -Ta) = 2(2 mol)(8.31 J/mol. K)(7.71 x 104K -2.67x 103K) = 1.85x 106J.

(b) Heat is removed during process ca only;

Qca = ~nR(Ta -Tc) = ~ (2 mol)(8.31 J /mol .K)(2.67 x 103K -2.00 x 104K) = -0.721 x 106 J .

(c) W = IQabl-IQcal = (1.85x106J) -(0.721 x 106J) = 1.13x106J.(d) e = W/Qab = (1.13x106)/(1.85x106) = 0.611.

P24-9 The pV diagram for this process is Figure 23-21, except the cycle goes clockwise.(a) Heat is input during the constant volume heating and the isothermal expansion.

heating,During

3 3Ql = 2nR~T = 2(1 mol)(8.31 J/mol. K) (600 K -300 K) = 3740J;

During isothermal expansion,

Q2 = -W2 = nRTln(V f/Vi) = (1 mol)(8.31 J/mol. K)(600K) ln(2) = 3460J;

so Qin = 7200J.(b) Work is only done during the second and third processes; we've a;lready solved the second,

W2 = -3460J;

W3 = -P~V = PaVc -PaVa = nR(Tc -Ta) = (1 mol)(8.31 J/mol. K)(600K -300K) = 2490J;

So w = -970J.(c) e = IWI/IQinl = (970 J)/(7200J) = 0.13.

P24-10 (a) Tb = Ta(pb/Pa) = 3Ta;

Tc = Tb(Vb/Vc)'Y-l = 3Ta(1/4)0.4 = 1.72Ta;

Pc = Pb(Vb/Vc)'Y ~3pa(1/4)1.4 = O.430pa;

Td = Ta(Va/Vd)'Y-l = Ta(1/4)0.4 = O.574Ta;

Pd = Pa(Va/Vd)'Y =Pa(1/4)1.4 = O.144pa.

(b) Heat in occurs during process ab, so Qi = ~nR~Tab = 5nRTa; Heat out occurs during

process cd, so Qo = ~nR~Tcd = 2.87nRTa. Then

e F 1- (2.87nRTa/5nRTa) = 0.426.

P24-11 (c) (VE/VA) = (PA/PE) = (0/0.5) = 2. The work done on the gas during the isothermal

compressionis

w = -nRTln(VB/VA) = -(1 mol)(8.31J/mol. K)(300K) ln(2) = -1730J.

Since ó.Eint = O along an isotherm Qh = 1730 J .The cycle has an efficiency of e = 1- (100/300) = 2/3. Then for the cycle,

w = eQh = (2/3)(1730J) = 1150J.

312