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I
PHYSICS 103 FINAL EXAM
Instructions: When you are told to begin, check that this examination booklet
contains all the numbered pages from 5 through 25.
Princeton University Undergraduate Honor Committee
This examination is administered under the Princeton University Honor Code. Students
should sit one seat apart from each other, if possible, and refrain from talking to other
students during the exam. All suspected violations of the Honor Code must be reported to
the Honor Committee Chair at honor@princeton.edu.
The checked items below are permitted for use on this examination. Any item that is not
checked may not be used and should not be in your working space. Assume items not on
this list are not allowed for use on this examination. Please place items you will not need
out of view in your bag or under your working space at this time. University policy does not
allow the use of electronic devices such as cell phones, PDAs, laptops, MP3 players, iPods,
etc. during examinations. Students may not wear headphones during an examination.
o
Course textbooks
o
Course Notes
o
Other printed materials
NOTHING IS ALLOWED EXCEPT THIS EXAMINATION, A PEN OR PENCIL, AND
YOUR CALCULATOR. THE CALCULATOR POLICY (SEE NEXT PAGE) IS IN
PLACE AS ALWAYS.
Students may only leave the examination room for a very brief period without the explicit
permission of the instructor. The exam may not be taken outside of the examination room.
This is a timed examination. You will have 3 hours to complete this exam. During the
examination, the Professor or a preceptor will be outside the exam room.
Rewrite and sign the pledge: I pledge my honor that I have not violated the Honor Code
during this examination.
Signature
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 1, Page 5
Problem 1. Conservation Laws
The spring in the figure has a spring constant k = 1000.0 NJm. It is compressed
a distance Xo = 15 em, then launches a block of mass m = 0.20 kg. The horizontal
surfaces and the inclined plane are frictionless. The second horizontal surface is a
height h = 2.0 m above the first, and the angle of the inclined plane is () = 45°.
a) (12 pts) For each question below, first answer yes or no, then write 1-2 sentences
explaining your reasoning.
i) Is the energy of the block conserved after it leaves the spring?
Y£5,
ii) Is the momentum conserved after the block is released by the spring, while the
block is on the lower horizontal surface?
iii) Is the momentum conserved while the block is on the inclined plane? Comment
on both the horizontal and vertical components of the momentum.
No.
(\.L1"vL
" he I- e
J
lke
do,.
iVlCLhfJ
¥"
e
i· 1,
r . .,
pia. v,.::,
LiE
Dv; \,."
r
.J-;
I'-ie;they
J
l·",.
d
fA
VI,,'-
<>
\'V ' " \;;.r
1'1'
(>.1--,,(;;
cv'-"Iuvd.
"'_K
iv) Is the momentum conserved while the block is sailing through the air? Comment
on both the horizontal and vertical components of the momentum.
No
.J'~e:Lkl:;Y1.
T \",e. r e
i :;
0.­
(.vvec.3 it t), 'f
y
he.
is
Fj(
I
~-Y'~
CA/VV'-J
L
C,,,,,,- Uy v e J
\."\0
o.
Lhe
.
\le.,..
PIC •
f;ciA., \
kvw~ltl:'
r
Problem 1, Page 6
Pbysics 103 Last In-Class Exam, 18 Dec 07
b) (15 pts) What is the block speed Vt at the top of the incline? What is the block
speed V2 just before landing? What distance d (see figure) on the upper horizontal
surface does the block travel before landing?
r\(~I
~'5e rlle..~
C'Vt
1- ~)(: :::
1
'wi.
dh
L V\ev- 'rJ
+
Jw:V-t
'
I
=?
IS]
f
1 r,vi ~
Ij
2<",,)
ivco
~ iJ.(" VI. ~
8.H 'l. = k z
Problem 1, Page 7
Physics 103 Last In-Class Exam, 18 Dec 07
c) (10 pts) The frictionless surface on the incline is replaced by one such that the
coeffcient of kinetic friction between the block and the surface is fLk 0.20. Find the
work done on the block by the spring (Wapr) and by friction (WI)' What is the work
Wg,l done by the gravitational force while the block is on the inclined plane? What
is the work Wg ,2 done by the gravitational force while the block is flying?
0"'(
2-
h-
'W" - -
f
};, €
S(VI.U:
tk.:
llR J
-.-
.fJk
vV{.(3!d
w u.r --'!
( ,11""", '.
I
Wspr:
I1
W g ,l: -
I
W"L
I
~""'G
I/-:.
~v~
t' ~.
C'"'\A.!
\"'-I.(!JvVVltl
W.
0"-
i.eVv.- L J
'J""
i
W u.r
?
(j 't.
t .+-Vll
\:)1.-,/)"
I
VI(:!.-w
I
+
v"2h
v,
::-il
:=0
\t"( ~ :: ,)
0?-
l L IS- J
1
k
--
qi.
WI:
II
II
...,...
-
W g ,2:
J
I
II O·7f
)
0 J
II Problem 1/ Page 8
Physics 103 Last In-Class Exam, 18 Dec 07
d) (13 pts) What is the distance d in this case (with friction on the inclined plane)?
.At (?-w
~: c .k
'to~r t
(j"V'\ •
-
2
h
(I+~k) II
d:
bbq
~
II
Problem 2, Page 9
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 2. Collisions and Kinematics
a) (20 pts) A particle of mass m 10.0 g moves with a velocity Vo = 60 m/s along
the x-axis. The particle hits a body of mass M = 2.0 kg and sticks to it. Determine
the velocity vf of the body plus particle after the collision. Determine the amount of
energy t1E lost in the collision.
Uo:;; Go
4j'
'Yh -.::: O\O\O~
U!t /~
::>
fI)
/1::: 2ol2t
VVL()o
6,0\ y.,
WfMjs
;:=:
eM.). UJ) ~
I
~ O~ ~O UIJ/S
2- ,ol
(2T
I[
M:
19
~
[I
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 2, Page 10
b)
of mass ml=100 g moves along the x . ' t h '
ThA particle
.
~
. e partIcle
hits a second particle 0 f mass m
WI h .velocIty Vo = 2.0i m/s .
m-axIS h'
'
IS e as IC, After the collision m h i '
2
1, W IC IS at rest, The collision
I t
1 as ve OCIty Vi and m2 has velocity V2.
i) and
(10 pts)
Prove
that
0 inofthis
'
energy,
and
theVi'V2
definition
th case, (Hldnt:
use conservation of momentum
e d0 t pro uct.)
;j \j e br--d}; fr::='o{:
fA- o"f!je)lt a It?
0'1] 5£ f
u«ku:v:
(J )
:::=
12.
11. \)2
Vi J I
f
I..L
.1
z.
z..,"i{Z.Vl
(2)
G-e.aue.j,~_ ~i g
AjaM I Ui~qw.-
T~)
tit e Ef"i
a
=Jts:c
\J 0 J \J ( ) \h
I
<Z:{(£e rL>;et-IO'1
7
~
~~
(jilt:
'" "'-
\)2
I
. 2
-f. lJ"(. ,
,+- -b-IQ~ wAt ~~(/~
IJ I ..l qz.,
nlv
-)1M
\j \.J.
\J-c,
c€ <f +1<e~ ,
si4:s
\Jz
\J'"=
::::;;i)
~
--"
\)0,
f Ji'f"S \; I' Vz.. =0,
"T f .
l.W,rt
~)
/\
I~~,..;,
j
Problem 2, Page 11
Physics 103 Last In-Class Exam, 18 Dec 07 ii) (20 pts) After the collision VI makes an angle 01=45° with respect to Vo- De­
termine the angle O2 that V2 makes to vo, and find the magnitudes VI and
fh:
C(So
I
VI:
II
I
~ lM( ~
V2:
Ii
I
J2
~(S
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 3, Page 12
Problem 3. Binary Star System
Two stars each have mass M and radius R. One has its center on the origin of an xyz coordinate system, and the center of the other is at X2 = lOR. A spacecraft of mass m (with m « M) moves along the x-axis under the force of gravity. (Its engines have been destroyed.) a) (12 pts) Below, sketch a graph of the space capsule's potential energy as it moves on the x-axis. Don't forget to label the axes. Indicate the value of the potential energy at any extrema, and the positions of any extrema) and how the potential energy varies with distance from each star when it is very near the star. .-
r-.: .
:>')"
t?a?
G
.
( '\ - - G-)).11t
lA s -kr/ V .~
Potet1b-I~Q
()
I
.) -
L'1:"­
crt....
o
-G-- ~ut- GI-{ lt1 ,
'\o~- K1
\',(,1
"IS
!)
\Oh
_-L---'#----S--;-~---~-----t~'~ rx
e{lQ~1 t(&J~~ ,;
U
i
i
1MHI~4;lt1m
Dt{
GJJ..CU
th s-b t-s)
K-
SCi
Tfcoces ~
G-UC/A _- 10 C).-{}ft
CfK) - cr ~
)
Problem 3, Page 13
Physics 103 Last In-Class Exam, 18 Dec 07
b) (6 pts) Suppose the spacecraft is at rest exactly midway between the stars. Is this
a point of equilibrium? If so, is it a stable or unstable equilibrium point?
M\&W"'d ~-t- cV-~V1 tluz skssQiL!;
~
H
lJb:,;u~
Sk,C
kfGU'S£
Ci' \'€
<2i(L{ ~
1<:: '!Vt~~;()t) )e
~1~ clv,~.pJ
crtt-[oh-&",(i
fl1t 0 ~
~ :J\C{'u,iuH~J ~~ces c~
ceuJ) <ifrOSt-t;? )
j=' =0,
C>l&
J
1'\iQ
? Ji
I~
C)~
iLlo~ s~6 io ~
r(Q\hj"~ QCO"';:)
C(
( ( (.)
<;:b
~e(U'Lts-e '\<r:
+0
)!5
+Jm 0{
4.-,
J+
Sfar.e cr~
'
, '
I~
-S-b~~;
'I'S; ~U's-)
c/O.'>P)
I; b,; CllV1.
c) (4 pts) Suppose at t = 0, the spacecraft is at rest at x = 4R. What is the net force
F on the spacecraft?
~
~=::
\S
+~
Direction:
'-
J,
Problem 3, Page 14
Physics 103 Last In-Class Exam, 18 Dec 07 d) Suppose instead that at t = 0, the spacecraft is at rest at x = 5R. The captain
fires a huge cannonball of mass me milO in the -y direction. Mter the cannon is
fired, the spacecraft has net mass (9/1O)m and moves off in the +y direction.
i) (8 pts) Is angular momentum conserved before and after the cannonball is fired?
If Ve is the speed of the cannonball after the cannon is fired, what is the mag­
nitude Le of the angular momentum of the cannonball about the z-axis?
II .-f:
LIIAms:-
rAIl {i11.el,biAA
.)~
'\
CZ5irsel<rrJ2
LeC<RlIt~ -(L 'FTee
ere tL -'f'=- -;l'f d1.-tL C"'71~ ~ 'f/» ,;,.(e -tGz
.+"c€
04·-tk C""h~ JV/-~ sf?:9=d )(j> ~ So t~
'I
J
1.{.~..{-
i i.vJ p,.J>se i (
\Lv1 d?~-I~
z-em.
~~D L es-o.
n"
iLuq> )ret.-
t-l.e. at] W/e{
Problem 3, Page 15
Physics 103 Last In-Class Exam, 18 Dec 07
ii) itational
(20 pts) With
be in torder to permIt
. the spacecraft to escape the gray
pull ofmust
the Vc
bm'ar
y s ar system? As I
h
­
, a ways, s ow all the steps of your
work clearly for full credit.
S
-­
~
)
\\jI c. -::::;
,'JG4
g\
S; f2- ~
1
Physics 103 Fall 2007 Final Exam Problem 4 Solution
(This solution is from Weekly Problem Set 3)
An engineer designing the bank on a curved road has to take friction into account. Sup­
pose the road is curved into a circle at a certain point. The radius of the circle is R
(measured to the position of the car). Assume the driver wants to negotiate the curve
with speed v. The coefficient of friction between the wheels and the surface of the road
is Ms.
Figure 1: Car on bank.
a) Draw a free-body diagram showing the forces on the car:
Figure 2: Car sliding up.
(The centripetal force, Fc is a resultant, and it was not necessary to show it on the
diagram. The direction of the frictional force Fs can be up or down depending on the
speed of the car. The angle 0 is also the angle between Fs and Fc, and the angle between
the normal force N and the vertical.)
b) For what speed v is the angle 0 a "perfect bank?" (On a perfect bank, the car goes
around the curve and the road exerts no sidewise frictional force on the car.) Express
your answer for v in terms of 0, g, and R.
Figure 3: Perfect bank.
2
In this case there are only two forces on the car, N and mg. The resultant cen­
tripetal force is not pointed down along the bank but radially inward. Therefore
N -I mg cos B. We have Fc = mv 2 / R = N sin 0 and N cos B = mg. Dividing the
first equation by the second to eliminate N gives tan 0 = v 2 / Rg, or v J Rg tan (j.
(Explanation was expected.)
c) At what speed v will the car just begin to slide up a bank of angle ()? Express v in
terms of (), g, R, and /1s.
To slide up the car must overcome the force of static friction pointing down. There
are two components to the forces involved. From figure 2 we have
Fc = N sin 0 + Fs cos 0
and N cosO = mg
Remembering that Fs
+ Fs sinO. PsN and Fc = mv 2 / R always,
R
mg
N sin 0 + PsN cos 0
(1)
N cos 0 - PsN sin O.
(2) sin 0 + Ps cos 0
,
cos 0 - Ps sin 0 (3)
Dividing Eq. (1) by Eq. (2) gives
v2
gR
-=
or V=
9
R ( tan 0 + Ps )
1 - Ps tan 0
t
Physics 103 Last In-Class Exam, 18 Dec 07
k
Problem 5, Page 19
1'
.'
y
x
Problem 5.
Oscillations and Pulleys
In the system shown in the figure, a spring with spring constant k is attached
to a massless string which passes over a pulley and is attached to a mass m. The
pulley is a homogeneous disk of mass mp and radius R. (It is supported by a rigid
rod extending out from the table, but the details of that do not matter.) The disk
rotates without friction around its axis perpendicular to the page. The string does
not slip on the pulley.
a) (10 pts) First, take mp=O. The mass is at rest in its equilibrium position when it
is at y = 0 and the spring is elongated by d = 0.100 m. Find the tension To in the
string when the mass is at y = O. Express your answer in terms of variables given in
the problem and 9 as needed.
y
6
t I"
CI
t
t'<a- bOdy
rmo...
0
d a. ~'tArvr7
I'
1 'L
o
(molt)
10 - rrn~
.-­
(YYl <a
II
To:
rrn ~
II
rrn
Problem 5, Page 20
Physics 103 Last In-Class Exam, 18 Dec 07
b) (10 pts) Find the period P of oscillations of the mass m, assuming that the am­
plitude A of the oscillations satisfies A < d so that the string never goes slack. After
finding an algebraic expression for the period, also provide a numerical answer in
seconds. (Hint: yes, you have all the information you need to do so.)
y
Newtd'o'''~,
tM
fA.
k (01 -- y) -- (Ynd
rrn~:::I
\XI i th
6t
ky
k .-
(rn ea
d
T :::: ~7){~
K
P, algebraic:
~JT{ ~
P, numerical:
I
0.63 s
I
·
.
Problem 5, Page 21
Physics 103 Last In-Class Exam, 18 Dec 07
c) (10 pts) What is the direction of the net torque on the pulley when the mass m is
below the equilibrium position and moving downward? You must defend your answer
with 1 or 2 sentences and a sketch for full credit.
\T \= R (T~ - Ti )
8
~QAbtv:wm
ace,
(; P\xi <n)l J
pv~
'JR.ows
t-0\ fAA€-
direction:
)'S
oW- of p~
I
Problem 5, Page 22
Physics 103 Last In-Class Exam, 18 Dec 07
d) (20 pts) Determine the period P of oscillations of the mass m in its motion along
the y axis. As before, the spring is elongated by d = 0.100 m in the equilibrium
configuration, and you should consider only oscillations with amplitudes smaller than
d. In this case, after finding an algebraic expression, use the following values to find
a numerical result: mp=l.O kg and m= 0.10 kg. The moment of inertia for the disk
is I = (1/2)MR 2 •
k
y
o.~~ ¥
-C ==-
i.norm ~. e;~~u.'O)
RC-s - T1 )
fot fPJ~
(rr1
:
{O\ .sp~ro~:
~-T4
=.
:::.
-r
;(h A;
0(
I ~.
R
rvn ~ =- T1 ~ 1M ~
~ =. k (d- y)
I ~
::. ~-KY-(tr,~_~S
R~
r~
(rm + f('rr; p) y+- k Y =- ()
thi ~ ~ +
(% +
::::
1-
~
f~ ~ i1 + ~- ~~ -r I" y ~ -
071
pr~ +- k Y
;=-
0
-0
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 6, Page 23
y
y
x
,,
,
,~
I
spiral wave
,,
R
I
/
/
-~
,,
\
X
I
,,
,
~-
...... ,
,
I
I
~
view along z-direction
Problem 6. Spiral String Wave
A string 'with tension T is found in the lab. Instead of vibrating the string,
someone spirals it and produces a spiral 'wave, as sketched in the figure. A spiral
wave on a string, traveling along the z axis, can be thought of as a superposition of
two independent sinusoidal waves undulating along the two orthogonal directions, i
and j. ~Iathematically, this spiral wave is described by
R(z. t)
Rsin(kz - wt)i + Rsin(kz
Here, R is the ,vave amplitude, and 0
=
wt + ¢)j.
The values k and ware constants.
a) (5 pts) Based on the R(z, t) given above, in what direction is the wave traveling
(+k or -k)?
Problem 6, Page 24
Physics 103 La.st In-Class Exam, 18 Dec 07
b) (5 pts) Consider the sma.ll chunk of string in the ;l:Y plane (that is, with z = 0).
As the figure on the right indicates, this chunk is executing circular motion as the
wave travels through. Since ¢ = +rr/2, what is the direction of the circular motion
(clockwise or counterclockwise)?
II
direction: ~feV'
CtcJ..w : ~
\i
'I
oz,
oz
c) (15 pts) Consider that small chunk to have length
with
small enough that
you can treat the chunk as a point nUlSS. \Nhat is its kinetic energy, J{? vVhat is its
angular momentum L about the origin? Express your answers in terms of k, uJ, T,
R, and
as needed .. (Hint: Remember that the velocity of the wave is /T/Ill and
note that the ma.ss of the chunk is m = lL6 z. Don't leave /1 in your answers.)
oz
k.E.:
~
e.=~ =
'" tV?
-r-J'l'
(W~. R') !;' Physics 103 Last In-Class Exam, 18 Dec 07
Problem 6, Page 25
d) (15 pts) Considering that the chunk of string is executing circular motion, what is
the net force F on that chunk of string? Express your answer in terms of k, w, T, R,
and
as needed.
oz
-
W
cl 'C
't
Direction:
e) (10 pts) You should have found in the previous part that the force F on the chunk
of string is a function of R. Find the potential energy U of the chunk, taking the
zero point to be at the origin. (Hint: you might need to evaluate an integral along a
radius.) Express U in terms of k, w, T, R, and Oz.
~
\)
:::
f
t
J..t . T k. . R. it.
\:)
U .::::
- I
z..
J ~. -,fr,
2.
(Z 1­