T3 (Fs2012) - 1

Fall 2012 Physics 23 Print LAST Name: RJ Bieniek

Rec Sec Letter TEST 3 (4 pages) and First Name: Worked Solution

Points for a question are indicated in parentheses. Your solution to a

question with OSE in front of it MUST begin with an Official Starting

Equation, with the math subsequently flowing from it for full credit. If

you need more space to finish a question, write and circle “BPP” at the end of

the space provided and complete your work on the Back of Previous Page.

For Questions on this page, write the letter which you believe to be the best answer in the underlined space

provided to the left of the question number. On subsequent pages, draw a box around your answer to each

question. The expression for the final result must be in system parameters and simplified as far as possible. All

information and algebraic quantities that you use to solve the problem must appear in the figure. Neglect air

resistance. Calculators and notes cannot be used during the test. If you have any questions, ask the proctor.

D 1)(5) A ball of radius R rolls without slipping down an inclined plane. The moment of inertia about its

center of mass is I. At a particular moment it has angular speed ω and center of mass speed Vcm. Select a

correct statement about the ball at this moment:

A) The magnitude of the force of friction on the ball is zero.

B) The kinetic energy of the ball is Iω2/2

C) ω = VcmR.

D) None of the preceding statements are true.

A 2)(5) A uniform disk, a hoop, and a solid uniform ball all have the same mass and radius. They are

rolling without slipping at the same speed on a horizontal surface. They then start going up the same inclined

plane. Which one reaches the greatest vertical height?

A) hoop B) disk C) ball D) all reach the same height

B 3-4)(10) A string is tightly wrapped around the circumference of a solid uniform sphere

of mass M and radius R. The string’s other end is held in place and the sphere is released from

rest. What is the square of the speed of the ball’s center of mass after it has fallen a distance d?

A) 6Mgd/5 B) 10gd/7 C) Mgd D) 4gd/3

C 5)(5) A buoy in the ocean bobs up and down, completing 2 full oscillations in 4 seconds. The angular

frequency of its oscillation is (in radians/sec):

A) 4π B) ¼ C) π D) ½

ABCD 6)(5) Prof. Bieniek has no memory of making a mistake in class after Test 2. Do you think that it is fair

to him that you should still be given an easy question?

A) Why not B) I agree that Prof. Bieniek has no memory

C) No thank you D) Four

Test Total = 180 / 180

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R

M

7. A beam of mass M and length L can rotate on a frictionless axis at

one end (the black dot at point A). A massless rope, attached to the

other end, is wound tightly around a unifonn solid disk of mass '/2A1 and radius L that can rotate on a frictionless horizontal shaft B.

a)(SO) aSE: Using torque methods, derive an expression for the angular acceleration of the disk, in tenns of relevant system parameters,

at the moment shown when the beam is at angle e from the vertical, at

which time the rope is perpendicular to the end of the beam. Assume

that the rope does not slip on the cylinder. c\ -:::. 6\') \< b""- \o"'2.o...~'1\..-0(\ --x -=.

~

Ct+ c\lS\c<'5 '\ IYV\..

'7 ,

1 \u~ ldo..~c= ZLd~

3: b J b l: -=-~L~-l[1l-~ Nt') LL ] ~ ~ -= L_ t:i 0 + y D

Icl c S ~ /. ~~i: (1- M. L"L) to!. ~ T -t r -t ~A

\0 2: Tb

l ~b:C / is b ~

.J<f 1\'\ L-(O<:t:: -= -+1~ i~ M L q Ie ~ I:: Ii) -+ [F.:~( \:y(m 8)J -4-I (\1 L ~

:~ ::- -\- T

~ ~ M L r:f t ~ ...:T + ~ (\~~'5\'t') t:r...1- N\L - ~ 1 ~ "' lX.:c - - \ -+ 2m(J~\YI~

\l...­-

1 5 D / 50 for this page

TJ (Fs20l2) - 2

0

8. A unifonn rod of mass 3M and length 4L is

hanging at rest from a frictionless pivot P that is

located at distance L from the upper end. The rod

is simultaneously struck by two putty balls,

traveling as shown. The balls hit exactly at the ends of the rod. The lower ball has mass M and

speed V, while the upper ball has mass 2M and

speed 2V. Both balls stick to the rod AFTER the

collision.

CD BEFORE AFTER

•2M

2V • tL:: b~ p

w?f:t~ . D~L 4L~

3M I Li9U-"'.:\u~

<:..6('---'-"~

CD c..\M)~

M V e-.

a)(50) OSK Treating the balls as point particles, derive an expression, in tenns of relevant system parameters,

for the angular velocity (magnitude and direction) of the rod after the collision.

Jf ~

~ cet.AS ~ k\jk...>.( Q'( I"A O\' t="" e..c~CL \s t,-€....-n:.l,12A?+ udt

fi'l ;: ~t:

11,;2.- \ )2;~ + L, r- L -c

?

&IT\, v, \.\1 t i':- "l,V,- \,j +I,..rcit~ ~ \tot L0h

+ lM.\j") (:3 L) - (21\\) (2 'l "'J L ~ [I, ~-t ~ l")C)'-+ (2M)L~] Lo.-+-z­

~ fV\ \{ L ~ [ (1-" c~~ L3 ""') ti') -+ 9 !'<I L' n tv\ L~ (}.j-' "

"2- [f:L (j"'"') (Ll (J2. + ~ M L~ -+ 1\N\ L'21 U:')-f2

- [ \ £. -] r It \"f\ e-+ \4 '(Y\ C-Jlu.(.c~ C\~LL) + \It N\L cO-\-'-c ~ L<-r-I1\

( \"f t: tI') w~,

M.G6-=- w ~ ~ ~ ~

~ '{'e.c.\-u...-:m. -::.. c.t(jc\:..w l~

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9. A block attached to a massless spring is oscillating with simple hannonic motion on a level frictionless surface. It takes time T to complete one full oscillation. The figure shows the situation at time t = 0 when the spring is at its equilibrium position and the block is moving with speed V to the right, which is the positive direction.

a)(40) OSE: Derive an expression, in tenTIS of relevant system parameters, for the speed of the block whenever the spring is compressed to one-half of its maximum compression from equilibrium.

~--t---7~

'xm:"+A V,..-t:::CO

~L~Y\~~ "X rC'l ~ A c~ (lU tJ c.p)

\Ix l-t)::: -~: "'- - r~U_D s\'V\Lwt+ ~ ') 'J \;, /VUL'K 1m..\...J,1M-- Go..'\ "Y- :::. 0 V0­

l'­

.: \Ix (-te -.: 0) -= V ~x --=- \j -:. Aw

- ~ --XL--:.-x ttL) -:::c, A C:~ LlD-t( t \f»~ ~

- -l "'- Cos @ 6 v",(tc.l::: Vc--x.""'- -Aws\Y\Luytc..i-Lfl)

~

G ,', \j~= \ \fc~ 1--== Aw \ S\'i'd~fu\

EL"~,\ -:: E.. (1-0

)

~lv),VI- ~'l&'1M 'v <- + Y-~~ (~ ~) 2. ~ f ~'1\. + t k( D)L- (c

A\'f~ ~ ~k-A'- -:0 N\ '-J"- ? -:0, V V\- (-~)<. \ =\{~

'J' "L< + '<t \4 ( ~'j \j~ ~ t'l '-- ( ~c ~ vrilVc'~ v'- ~1J2~ ~ \JL_) 8¥ vJ 1

b)(IO) OSE: Derive an expression, in tenns of relevant system parameters, for the phase constant <p of the

block's motion . _ ') ~') ~ 'SlY, "t d "",,~\"'"

~({) - AC0S ('--~t+f j '-, 5\'r\l~Z) ~~\'>-O o -:: 'X (ot- ~ b);:: A C-0~-{j C<f'J ~~, s~ L-~) ~ - \ < ~

C6'S('f)~O ~"> 'i'~+'f "y'-~ \''{'f~~' '\} -X it) ~ if =- A&,'\ Lwt +'i')1 Lu )

-+ \j -:::. 'U (-t""- 0) ~ - Aw)\'(\ ~ ? <j,''-\Cf =- ~f\ <0x

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