Question 1 (8 points)

Figure shows a three-pa,rticle system, with masses ml, m2, o,\d ms.

(7) If m2 is graduatly decreased, does the center of mass of the system (circle the rightanswer)

mga

a*'

(2) If the only forces acting in the system a.re internal, the net force acting.on the center ofmass of the system is

(b) Can not be deternined from the information given.

ml

(a) shlft towards rn2,

(c) remain statibnary.

*ttu-.-&,/ +6l.*r"^

Problem L (LZ points)

A rocket is fired vertically upward with an initial speed of. s00 mls. when it reaches analtitude of 1000 rn, and its speed has slowed to 480 mf s, it explodes producing three equal-mass ftagments. Just a,fter the explosion, one fragment continues to move upward with aspeed of 450 m/s. The second fragment has a speed of 240 ml s and is moving east righta"fter the explosion.

(1) What is the velocity of the third ftagment just after the explosion?

?. -P.'d - "-

$,2+oe *M , t4"3 s 3 +M

3

(2) what is the maximum a.ltitude (ftom the ground) the center of mass of the system willreach?

i " Q M o - -

t -J) i4.Qg6 -

O +

4so5* g o ' * T413 * t= 77o yg

tL Ce*/et, 4 /r.,4r4 aonl'rr^'"a 4o t*<, ^ d ed 4f;

4 A* 'A%c "'to. 45=o,

& - - XfV1: -E+ * SooV - + $ = o 4 +'?r4r *

V e . - * + " * % ? f * o L/r^,= -;

6,.fn 'eoos.'

J x x = @ ' = v - l s s . t n *

r tcp-- y,E r '

9;3

Question 2 (8 points)

A block slides across a frictionless floor and then collides with a second block that wasinitially stationa.ry. The figure shows four possible graphs for the kinetic energies of theblocks both before and after the collision.

(1) Determine whidr of the graphs represents a possible result of the collision and whichdoes not. Circle the graphs you consider possible

graph (c) $aph (d) €-

4f

q{'Jt €ccrorQ'

Cnp'rbLA:; Elo u!,r.V n"' lgo5r.loL

(2) Determine which of the graphs, if any, represents an elastic collision

6or--s lC**r'' €t!'

Nonegraph (b) saph (c) graph (d)

0"b['[

K;ot;t\T'

f f i , l i * * ' \ { 'e i w- '

Problem 2 (17 points)

A block of mass M lies at rest upon a frictionless floor with its right face attached to asprirg of spring constant ft which is at its rest length as iliustrated in the figure. A bulletof mass rn traveling at a speed V strikes the lefi face of the biock and comes to a comoletestoo within the block.

(1) Write down an expression for the linear momentum of the bulled-block svstem beforethe bullet strikes the block.

:E -- a^ir d

(2) obtain an expression for the velocity of the composite system (comprising the block andthe bullet) right after the impact as a function of m,,M andV.

d* 'q,* ")-€-s, 5

=, F"&e + q

(3)Calculate the kinetic energy of the system right a,fter the impact as a function of m, M

J

= N -@*u)

a n d V .

K= i(y"n) {i'=}@") ffir,= io^tfrl

@+rr)

(4) After the collision, the combined block/bullet moves to the right until momentarilystopped by the spring. Calculate the maximum compression of the spring as a function ofm.M.k andV.

A€o*- o e: CF-A--G"Ai{"" = / 6:,

Question 3 (8 points)

The figure shows two massive pulleys connected by a non-slippingpulley, A, has radius 11; the larger one, C, has radius rcr.

(1) If ,4 has a non-zero angula.r acceleration, ad, then C's a.ngular

(u) ot. Irr,_= ̂ t

(b) a,a(rslra). Lr;fu -rdoGrl

massless belt. The smaller

acceieration is:

Qr=%o4 44q4 = c/sR* a *a- d*ft(d) none of the above.

(2) Assume that the system is initially at rest. F\rther assume that the only external torquein the system is e><erted on the pulley A, a.nd that the system accelerates as described above.The tensiori of the top part of the belt is:

(a) not equal to the tension on the lower paxt of the belt.

(b) depends on the sign of oa. /

.1-@\-:__*__-___-

(d) none of the above a.re true.

y- (6+c.rr"* '^t J"'\'-'l)

Problem 3 (LT points)

A pulley (of unknown mass M, but lnown rotational inertia I : 0.025 kgm2 and radiusE: 0.15 m) has a non-siipping massless cord vround around it. The cord is tied to a blockof mass rn: 0.39 kg, as shown in the figure. The system is initially at rest.

(1) Find the kinetic energy of the system a,fter the block has fallen by 2 m. Use energymethods.

tDr**d &0, A6^.r.=c = N)+ trK + AK= -\o

oW A(= Xg 'K" = K,l (xn'<- a[ 'td''=o )

4**- 4 , -a<J. e0 A".Qst72(1&+4.)Q*1= 7'653-

(2) Find the anguia,r speed of the pulley and the speed of the block at that time.

Kb",^ - * Ttt + f lr,r ,'ra el'o qr= ats

n nuSA= 7 (-t+nrz') usz

+.6S t, ! (o.ozri"'*@l3Pt) (arsJ-a) '^rt

t4* : t"nt *

::.> LrJ=

i:=-"'-'t V I

Question 4 (8 points)

Two identical disks A and B are placed at an attitude h on inciines Ia and Is which havethe same angles. There is friction in between A and Ia while there is none in between Band 1s. The disks a,re released from rest. Circle the correct answer.

(1) When the disks reach the bottom disk's A total kinetic energy isAF--o'+ au+sKao

smaller than @

ra.rgerthan A K, )-ft 6#

that of disk B.

(2) When the disk B reaches the bottom its rotational kinetic energy is zero:

FALSE. '-to co0l- Jo.r^ +{""rr u'st t--

(3) When the disks reach the bottom the velocity of the center of mass of disk A is

equal to la.rger than

that of disk B.

$*.{"=-", dfl^cnrr\r- tt- st&a ='$".?'-

r*l{^."uF nc{,*-''a a ^,o nat d'r"-l' |6an! AA--.-----\t

4or fta.a,^e6 ,'r .Cer'd4.*/4^-!- {

N -{-,^t|'dlt-tn' + Ct+tsft'}'!- e"l

* V t a - b O

[ o o * % .

$r t s K . ( \ r ' L t 4 : . . '

Problem 4 (L7 points)

A constant horizohtal force Fo* is apptied to a uniform solid cylinder by a fishing line ofnegligible mass. The mass of the cylinder is M and its radius is B. The cvlinder rollssmoothly on the horizontal surface.

&*Fishing line T= {eL

2-

. . C l - *

A (4.I'$o- -..ts1o.o.,

'r.^rid( -rt 11'

oLce.r. ale'Eb-

<r

F--------r

(1) What is the acceleration of the center of mass of the cylinder? (Express your answer interms of f'o* and M ).

LL{*'. 0t. d* 6 _ rt q rt,,.fl'l"a l* &- rcJer{>' fr.* - T *^ 1 .- e<€ +e.E. -

coJlq- : € €, + €i-tC- -€6-sf ,,t3o.

(2) What is the magnitude of the force of friction? (Express your answer 1" f.i-ffif

q = 7 1 a - Q t- - re.

,g€-- no.- E- ---

]4e2

' 9!'reo.F t

= tb-

s= !

3