2

1. Antilock braking systems (ABS) automatically release brakes when wheels areiocked. Once the wheels recommence rolling, the ABS reapplies the brakes. Thesestop a vehicle under slippery conditions in a more rapid and controlled fashionbecause: l2(af+he@:eti<xr is larger than that of kinetic friction.(u)ffiie*f+ietionisIargerthanthatofroIIingfriction.

a'reb rolling motion a car's wheels remain at rest relative to the road, and the!ffiffi/cient of Jatic friction is larger than that of kinetic friction.

(d)Jff-strdmgrn€tion, a car's wheels remain at rest relative to the road, and thecoefficierrt of static friction is larger than that of kinetic friction.(e)@aCaI'SwheeIsremainatrestre1ativetothecar,andthe

coefficient of kinetic friction is larger than that of static friction.

v^tWn plwls ,oNt^J,#{t,orr d, d s(tp o^ {an,"l

haN 1, tor.;tbl€{ S*u}t'c {r,dton, r'v+i ktntlt,:-{rur1,r^

AAS bnrults {r 'J +" !nrt'.rvrfe^{v1 {, =fs tr ( vnr,tx. f, JaS c(tkctspoSs"tblr bl, /slAr.

2. A simple harmonic oscillator has a displacement as a function of time given byr(t) : Acos(fr). When half of the oscillator's energy is potential energy, what is

its displacement from equilibrium? l2(")":*A /: L,trl(

* vttzt< e d,t sy(ur{ur*wh K = C

: E-- *r^',W*wl x Lvl+t,yr A= E /Z =

*"t *-

A,

(b) r:

(-G)):(d) ":(e) r: * an'.T

AIr-2

AIL------=

JzTI

B

AI

4

U= Llyrzru,r i F

:- y{*(^'

4{7'1

/ ,l Lx.J X- i.

3

3. A simple pendulum swings at small angles. If the amplitude and length ofthe pendulum are both quadrupled, the maximum spegd of the pendulum's bob is:

(you may assume that the amplitude remains sufficiently small that it still undergoessimple harmonic motion)

(a) quadrupled

f@)oubted\-il(c) unchanged(d) halved(e) quartered

l2

il=

Vw^, ; UJA :A

",. V*l* ryA LIA TT'tul-

l\/ L =/ A

= f l/y,*x

J c*Lle ,tr

_il-L

cir/

L

4

For 5 mark multiple choice questions, please show all work for part marks.Correct answers will get full marks, incorrect choices will get part marks for any

correct work. including crossing off incorrect answers, provided the correct answer

(or answers for question 5) is (are) not crossed off.

4. A compressed spring fires a ball of mass zn directly upwards. If the mass ofthe ball and the initial compression of the spring are both doubled, (neglect airresistance) then, 15

q!ftne ball will reach twice the height as before.(b) The ball will reach the same height as before.

(c) The ball will reach half the height previousiy reached.

(d) The ball will reach eight times the height previously reached.

(e) The ball wiil reach one eighth of the previous height.

5. If at some position in space, a particle in a conservative potential has no force

acting on it, (Choose all that apply).

-JtThe particJe must'be atrest. f*+ =D ,',upli?; ,a*ii. velJr:,,(V . wri af .ntJ.i.

( (!)zThe particle must not be accelerating. ',/ J ' ':k)

The-par$ei+hasonly kinetic energy' s'c,vttc(t ttt+^ l,r, ^( "4!1 ptrcttitt,-l /b"v,r,J

[3i frffmf,*T.,"fJ*H:il]i enersv "---7- r"". )"'i'< /

$.'v rr"ttpff55r..{ '3pvtvt6' t ti = L?,' K,' =

fil vwat l,vr,1l,l ,t L,,rlj il1 = rAIh, K+ =

f :,hSr yy;air)h IDL u3l"'.L rf rfJ h'-r#{ ' t&'= Jl,

"t- kr'n" **J

l5

(") (b) (.) (d)

*tFL oo/s- %a7

":

*=O W=iJ gy'=4f *=-?JgrqD" &=(0"

cass=s ,,so=l /, ,:;:=, !,=r,lr='_FIG. 1:

6. Calculate-the work done by a constant force of magnitude F : 2N applied over

a displacement of magnitude Ar --2m for all four of the diagrams of Fig. 1.

W = [ O

^, (os| = (f ,"s 0)

^y {0, aowst. fx.< .J

= (+ t) (c"s o)

iffr"" j$ caronthe I

3. Turn the assembled tower so that it is facing in the

direction shown in figure 3. Complete the circuit as shown.

NOTE: lt takes two cars to make it through this Super Set.The standard car (red motor) will go throuoh the tower,then tap the Stop & Go car (blue motor), and the Stop &Go car will go through the straight-away and loops.

small baselarge base plates

&"u*HS9*

I 14"

I straight g

, track iusti before

L'G'j,g*

180'curue

FIG. 2:

7. Darda (TM) cars are spring loaded toy cars that, when released from rest,can complete complicated roller coaster-like motions around tracks to quite largeheights. Wi,thout calculation) express in words how you might estimate the energystored in the spring of the "standard" Darda (TM) car which is able to completethe tower track shown in F ig. 2 if started at the indicated position from rest. (Hint:how m,uch energA i,s lost prr,or to arri,uing at the highest poi,nt, and at the highestpoi,nt what type or types of energy are present?) l5

fp,ti,'i fo-lru1h;i etarrTT ii4',y;rttt, W,u't W tit'(..)t,tnl ,{rr:

- ,i,l{ t{,ovt,:( i in nMt{tcttt it Ll r^.rn"ry ^

rl"*ttt'a 7'v1t{)tv1

* *t{ i^,;r rk. ,Ja,l4" L}J t4cr,t.{ova:itr'9'}t\'l 7"'t;' {{;ractttn)* o^J rotlrw kt,a{,t tinLN-Jy ,# ,*l*tl t cf' {''+v

{ Pt':b'^b}' S*atul"l )

-A{tr, l,.ru*". lo (za.s,rlr.v- llryJ *.,1 $of ){ila(ttsI {rap Nyw ,t"i t(:,y ra'fu lhdli^rrt r-t *ffrae k. .. ht,rwr;.,-[. ln,,r ,( +r^t('i'.,^ J,,,iy

J; b 7tv> r:t,l,$ r'.J it"tc,ts ivi7i,( - At iylr*p,:,nt .:*ly t'a rr- Jv^ ('t"v rs trAi ,b^tt il1ttt 1..,r.

l',a ir jf ,'.prLrJ L€LLl{rp't*-J ,r/r"t*rrr*;,,rt \o yl^.rJ

c,ry^ r,'ill $l,l{ dtit*e i*ilAe ftitt*f(o.r_, K.{, c;i."

P*,L iu r3 [*.

7

8. (a) Some babies like to be bounced to calm them down. A baby's mother (82

kg) sits on a "birthing" ball and bounces a 4kg baby with her with frequency 1

Hz, and finds that the baby goes to sleep. The baby's father (95 kg) tries the same

technique, but fincls that the baby won't go to sieep. He reaiizes that this is becausehe can't bounce on the ball with the same frequency. Assuming this to be simpleharmonic motion, what is the spring constant of the ball (assume that it remainsunchanged by the added weight)?

llk( c, f"trtf{ Jtt ^ {fr/}t^1,

/rLk= (lv{) n^ = N

n/1-

(r* tr@)'(u^

p'J = 2n{ jl4

fi7r4+uk).=

rr-tl_.Vnu

" s l, ftt,,, " :>

(b) What adjustment to theput his child to sleep?

fr, !v,- l"llw' t*qutw

spring constant

k, ( nrt

be made

'fsli r

father to

3'rc8

for the

Lt'L, )=

needs to

l3

q)-( un^

I^^ &S{ tvtc trfctt ( k bj 5 t 3 t5%

(c) How could such an adjustment be made to the ball, which is filled with air?

l2

&)J wod at,,- It ikCN0-51 pvrsyLtrr /h lfu lntl.

(noyta tr,rl p rc,;cu|'\^ lutcu?rS bt' ll kr

Nilr(

lI k"^^1vw3

{.^t(r {*)l( r hitr,^y

1^",; ht l1

abSr.t ?

IM6( (

,vtnrf hollow sphere,solid sphere, &or solid disk15

mass m

? massless,

frictionlesspulley

It^

voiitvry

rll*Nrtn 5J ), C.S-pn

\q

mass mr

FIG. 3:

9. One end of a massless string is attached to a massless axle that passes throughthe centre of mass of an object (mass m) that has a circular cross-section (radiusE) and rolls without slipping up the incline. The string passes over a massless,

frictionless pulley and a block (also of mass m) is suspended from the opposite endof the string as shown in Fig. 3. In this problem you are to find the speed of therolling object after the hanging mass has fallen by 1.0 m (the system is reieased fromrest). Find the rolling object's speed assuming that it is : (a) a solid sphere, (b) uhollow sphere, and (c) a solid cylinder. l\o

, {f," (7* c) * = {/rhj- Vtt

(^) 5,3it'cl sylvv<

!;Cb) lnoiiow €yl,'u.v o A --

!; i.I1,

e :- zlri-cL n^l S

7(3Ynl t\ (z-

V * 1.1,3

tl7 lv\ge trvbdl J'n o+ €rLt^lAAd

AktAtx;QhuvXtni firrttt" + AV'= *^u

-

+ Av' -*?h (t^-{*)

fe iir u' 't r;brtt (.l ,t

-+ A(.. Al(r^r A(v,(

= *'*" + !rut"-L= CtwQz

r cwa1l L

' AK= I.i,,n v,- ( {*<- )2\/l

I

AU =- -r 6ar1 hra

.: *rv' * fn^v'( W c)

& *vn1Lr {w?k=a

D( \r

k\l(

"k+ c

q(^t,

LtC

(, ) ralrl cytunl€ ( L:

f'o*

cJr.

Ceuntnt" ,t 'tv"ut f

rLt-kvta{

{ c "tt^

'tf rvttfl;

;f sot,J spv*tw drsK

,i ;i4 oi l,,tlP( frutof s*t.e,{ cyitn'l('r '

1o*

X

tP-t'rr7 =o,g(n^ru{

/FrG 4: tn s'ftwtrt 'hwv v, ld-|ft ^ffert

,^( a r1A (ot i,./ tr h .

10. A disk that will appear to roll uphill is made from spruce (pro,.,.":450#)and steel (prt""r : ZSSOS) by drilling a 3 cm diameter hole, centred 8 cm above thecentre of a 30 cm diameter spruce cylinder, as shown in Fig. 4. Into this hole isinserted a piece of steel of identical dimensions. The disk has a 20 cm width.

(i) Calculated the mass of the spruce removed from this disk, and the mass of thesteel that replaces it. l4fltass;{ ;o{ d_syx_wtr A.ts(a

^,; fnf,u /,2 d f ; l{cry Lo = Z0 r.;i,w

PArp' {.StZ kJ

WL6$ 5 rt' Sy v-,t,4. rt wc& l\ M( . fry"(arw a> l, fc,,^

t/^^s( bl sted c'1tt^/,err

(ii) The location of the centre of mass of the weighted disk is closest to:

tN 1 gt ft"|. c yil n d tv r ng vrle,N

& y,;'kI r., f i .

l5(u) ":1cm.y:0cmalT): o cm. y: 1 cm\y_lc)

":0cm.y:Bcm(d) "-0cm,y:2cm(e) x:0cm,y:4cm * (oo'4,r

c6.)(f r {r i)ks'{,ie,6lr,,1i,

t

(:,::o,)

'()k,)io,,

f+i(, ,r; k1 )G*^) t ,| (({.stz- c(

Mr. 0.c{3( irJ

f/,sl= ftt,fd'^ = l,ll0

:. 1!^,( i.tez*

10

FIG. 5:

11. As shown in Fig. 5, a solid, uniform, frictionless cylindrical reel of mass

M - 3.00 kg and radius B : 0.400 m is used to draw water from a we1l. A bucketof mass T(L : 2.00 kg is attached to a massless cord that is wrapped around thecylinder. (a) If the bucket is released from rest at the top of the well, find thetension 7 in the cord and the acceleration a of the bucket (the cord does not slipon the reel). (b) If the bucket falls for 3.00 s before hitting the water, how far does

it fall? lL\

(^) ?BD bw*tt:',^b @ u*o Q

(i= ct - tA o.^-Uz ilw

(A= V ^ = 5,<f*l(,t-- /v1*f * .tY-'l.-w

( rr) c a',n rtc,",J c,ca<,(Lu ^f

,,ru

TvXvc ,>fi w€(

,: lYat ^fr,tT

t hrr

(\rk,^t 0. > C

T=Kf=]""(r.= *n4Rz

tuslq:.T ,

{ttol k<

r')

trL*J

r( "(t"#) = V

^5-- 6r:**(,+)'

dr=

lr\' 1''{vr a 19

11

r

-l^r* S' r\"

:- f < L^'4'* &FIG. 6

12. As shown in Fig. 6, a small block of mass m is placed inside an inverted cone

that is rotating around a verticai axis with period 7. The walls of the cone are

frictionless and make an angle 0 with the vertical. What is the value of 7 if theblock is to remain at a height h above the apex of the cone? Give your answerfor T in terms of m, h, g, and 0. (Your final answer may not require all of these

variables.)

FsD cf w

lq

Srn (9

r il.tal Ju t w

Aco90 = Wl rLJ)-

: WtLcoiS; y\Ar tt5 iv.0

t r^)L

,-(A = d

l"

N=ryT

." T-

lrc

+a;e

_(4r^&

rw3 4e(,r)

L.

T 3 e"F Iany

lngvl,r,J J t"r ' ,n :

14 st,a 0 *w,,, =c0

: 14" = w"fr<'

q

--k**orr-&

L1

"'- . (t

4r".^ 6= hlr'^^0 c'J

?