Examination Cover Sheet Princeton University Undergraduate Honor Committee January 30, 2008

Examination Cover Sheet
Princeton University Undergraduate Honor Committee
January 30, 2008
Course Name & Number: PHY 101
Professors: Galbiati, McDonald, Nappi, Pretorius
Date: January 23, 2007
Time: 9 AM
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 items marked with YES 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.
•
•
•
•
Course textbooks: NO
Course Notes: NO
Other books/printed materials: NO
Formula Sheet: YES, only the one available at the end of the exam
booklet
• Comments on use of printed aids: NO
• Calculator: YES, but only for standard functions. Cannot use graphing
functions or advanced functions, like equation solving
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. During the examination, the Professor or a preceptor will be
available at the following location: outside the exam room.
This exam is a timed examination. You will have 3 hours and 0 minutes to
complete this exam.
Write your name in capital letters in the appropriate space in the next page. Also,
dont forget to write and sign the Honor Code pledge in the appropriate line on the
next page:
“I pledge my honor that I have not violated the Honor Code during this examination”
1
2
Your Name:
PHYSICS 101 FINAL EXAM
January 23, 2006
1
3
5
3 hours
Please Circle your section
9 am
Nappi
2 10 am McDonald
10 am
Galbiati
4 11 am McDonald
12:30 pm Pretorius
Problem
1
2
3
4
5
6
7
8
9
Total
Score
/13
/10
/14
/18
/20
/18
/10
/8
/14
/125
Instructions: When you are told to begin, check that this examination booklet
contains all the numbered pages from 2 through 27. The exam contains 9 problems.
Read each problem carefully. You must show your work. The grade you get depends
on your solution even when you write down the correct answer. BOX your final
answer. Do not panic or be discouraged if you cannot do every problem; there are
both easy and hard parts in this exam. If a part of a problem depends on
a previous answer you have not obtained, assume it and proceed. Keep
moving and finish as much as you can!
Possibly useful constants and equations are on the last page, which you
may want to tear off and keep handy.
Rewrite and sign the Honor Pledge: I pledge my honor that I have not violated the
Honor Code during this examination.
Signature
3
Problem 1
Circle the true statements. There may be none or more than one true statement
per group.
1. (3 points)
(a) In any perfectly inelastic collision, all the kinetic energy of the bodies
involved is lost.
(b) Mechanical energy is conserved in an elastic collision.
(c) After a perfectly inelastic collision, the bodies involved move with a
common velocity.
2. (3 points)
(a) If the angular velocity of an object is zero at some instant, the net torque
on the object must be zero at that instant.
(b) If the net torque on an object is zero at some instant, the angular velocity
must be zero at that instant.
(c) The moment of inertia of an extended solid object depends on the location
of the axis of rotation.
4
3. (3 points) A horizontal pipe narrows from a diameter of 10 cm to a diameter
of 5 cm. For a non-viscous liquid flowing without turbulence from the larger
diameter to the smaller,
(a) the velocity and the pressure both increase
(b) the velocity increases and the pressure decreases
(c) the velocity decreases and the pressure increases
(d) the velocity and the pressure both decrease
4. (4 points)
(a) work can never be completely turned into heat
(b) heat can never be completely turned into work
(c) all heat engines have the same efficiency
(d) all reversible engines operating between two thermal baths at the same
temperatures have the same efficiency
(e) it is impossible to transfer heat from a cold to a hot reservoir
(f) all Carnot engines are reversible
5
Problem 2
A pendulum bob consists of a string of length L = 1.0 m with a small bob on the
end. The mass of the bob is M = 10 kg. The bob is released from rest at the position
sketched in the drawing, i.e. with the string horizontal. At the lowest point of its
trajectory, the string catches on a thin peg a distance R above the lowest point.
1. (2 points) Calculate the velocity v of the bob at the lowest point of its trajectory.
6
2. (2 points) Calculate the centripetal acceleration ac of the bob at the lowest point
of its trajectory, just before it hits the peg.
3. (6 points) Calculate the largest value of R such that the bob swings around the
peg for one full circle keeping the string taut at all times.
7
Problem 3
A guitar string made of steel (density ρ = 7.85 × 103 kg/m3 ) has a diameter
d = 0.9652 mm.
1. (2 points) What is the linear density µ = m/L of the string?
2. (4 points) What is the tension FE in a string of length 66.24 cm that has a
fundamental mode of 82.41 Hz (an “E” note)?
8
3. (4 points) Different notes can be played on the same string by pressing down at
different positions on the fretboard while plucking the string, as this shortens
the part of the string that vibrates. Suppose you want to play an “A” note
with a fundamental frequency of 110.0 Hz on the same string of the previous
question. How much shorter does the vibrating part of the string need to be?
4. (4 points) Suppose you plucked the string without pressing down on the
fretboard (so the length of the vibrating part is still 66.24 cm) such that only
the third harmonic of the fundamental frequency is heard. Where along the
string will the nodes of vibration be?
9
Problem 4
The tidal forces between the earth and the moon cause the moon to exert a torque
on the earth, which affects its angular frequency. Data: Mearth =5.98 × 1024 kg, and
assume the earth to be a uniform density sphere of radius Rearth =6.36 × 106 m.
1. (3 points) What is the angular frequency ωf of the earth’s rotation around its
axis today?
2. (3 points) The effect of the torque on the earth causes the earth’s rotation to slow
down. If the angular frequency of rotation one century ago was 1.67×10−12 rad/s
more than it is today, what was the average angular acceleration of the earth
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over the last century?
3. (4 points) What was the magnitude of the average torque exerted on the earth
over this period?
4. (4 points) If this torque had been applied to the earth since its formation
4.5 billion years ago, how long was an earth “day” (the time of one revolution)
11
then?
12
5. (4 points) What was the change in the earth’s angular momentum over the 4.5
billion years?
13
Problem 5
A cycle of a gasoline engine can be approximated by an Otto cycle, which consists
of two reversible adiabatic and two isochoric processes as shown in the figure below.
In this problem, an ideal Otto engine works with 500 moles of a monoatomic gas.
The following data are provided: VA = 3.53 m3 , VB = 1.55 m3 , TA = 301 K and
TC = 615 K.
P
c
B
VB
D
A
VA
V
1. (10 points) Compute V, P and T at the other vertices, and fill the entries in the
table below. For each numerical answer, clearly show your calculations.
14
A
V [m3 ]
P [Pa]
T [K]
B
C
D
15
2. (10 points) Fill now the entries in the table below by computing ∆U , Q, W and
∆S along each part of the cycle. The change of entropy from D to A is given
to you and is -1030 J/K. Show your formulas and calculations clearly.
A→B
∆U [J]
Q [J]
W [J]
∆S [J/K]
B→C
C→D
D→A
16
Problem 6
A rectangular slab of ice of base area A = 5.30 m2 and height H = 3.20 m floats on
water. Keep in mind that the densities of water and ice are: ρwater = 1.00×103 kg/m3
and ρice = 0.917 × 103 kg/m3 .
1. (2 points) What is the depth of immersion h of the slab?
2. (2 points) A polar bear climbs on the ice. Now the depth of immersion of the
ice slab is h0 =3.10 m. What is the mass M of the bear?
17
3. (4 points) Still holding on the ice, the bear (ρbear = 0.872 × 103 kg/m3 ) partially
slides into the water, so that 30 percent of the volume of her body is submerged.
What is the depth of immersion h00 of the block now?
4. (4 points) The bear dives off the ice. What is the pressure exerted by the water
on the bear at a depth d = 10.0 m?
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5. (4 points) What is the upward push (apparent weight) on the bear at that same
depth?
6. (2 points) The bear emerges to get a deep breath of air and promptly submerges
again. The upward push (apparent weight) is now 1333 N, directed upward. By
how much has the volume of her lungs expanded? (You may neglect the weight
of air.)
19
Problem 7
A rectangular block of steel has dimensions 1.0 × 2.0 × 3.0 cm3 , and Young’s modulus
Y = 2.0 × 1011 N/m2 . The mass of this block can be neglected in this problem.
1. (3 points) Suppose the block is welded to a metal ceiling such that the 2.0-cm
length of the block is vertical, and a mass M = 100 kg is welded to the bottom
face of the block. How much does the block stretch vertically?
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2. (4 points) Suppose the mass M is struck from below with a hammer so that it
oscillates vertically. According to Young’s law, the block behaves like a spring,
where the displacement from the equilibrium position is now ∆L. What is the
frequency of oscillation?
3. (3 points) The block could also be welded so that either 1.0-cm length or its
3.0-cm length were vertical, leading to three different frequencies of vertical
oscillation. What is the ratio of the largest to the smallest of these frequencies?
21
Problem 8
A glider of mass M = 250 g is connected by a string that runs over a pulley to
mass m = 75 g. The glider is connected to the wall by two identical springs of
constant k = 15 N/m.
M
m
1. (4 points) What is the period of oscillation of the glider, assuming that it slides
without friction on an air track? The mass of the springs can be neglected in
this problem.
22
2. (4 points) Two gliders of mass M = 250 g are connected via three springs of
constant k = 15 N/m on an air track, as shown in the figure in the bottom.
What is the period of oscillation of the gliders when they move with the same
velocities, and when the move with opposite velocities?
M
M
23
Problem 9
A thermodynamic model of a bit of computer memory is a pair of boxes each of volume
V = 10−12 m3 that are in contact with a thermal bath at temperature T = 25 ◦ C.
There is exactly one monoatomic gas molecule present. If it is in the left box, we say
the bit is a 0, while if the molecule is in the right box we say the bit is a 1.
The single-molecule “gas” obeys the ideal gas law if the pressure is taken to be the
average effect of many collisions of the molecule with the wall of the box. You can
treat it as a regular gas where the number of moles is n = 1/NA .
Thermodynamics of Memory Loss. If the partition is removed (and not
reinserted), the usefulness of the memory is irreversibly lost. This operation results
in the free expansion of the gas from volume V to 2V . Although no heat flows
during a free expansion, we cannot say that there is zero entropy change, because free
expansion is not a quasi-equilibrium process during which the laws of thermodynamics
apply.
A reversible process during which these laws apply and which produces the same final
state,and same entropy change, of the gas as the free expansion is to suppose the
partition were a piston that is retracted (sideways) slowly and isothermally until the
volume doubles.
The only formulas needed for this problem are ∆S = Q/T , P V = nRT , ∆U =
nC∆T = Q − W , where Cmono = 3R/2, R = 8.31 J/◦ K, W = nRT ln(Vf /Vi ) during
an isothermal process. Factoid: k = R/NA = 1.38 × 10−23 J/◦ K = Boltzmann’s
constant.
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1. (4 points) How much work is done by the single molecule during this isothermal
expansion?
2. (2 points) How much heat flows into the gas during the isothermal expansion?
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3. (2 points) What is the entropy change ∆Sgas of the gas/memory, and ∆Sbath of
the outside thermal bath during the isothermal expansion?
4. (2 points) Now consider the case of memory loss due to removal of the partition.
What is the entropy change ∆Sgas of the gas/memory, and ∆Sbath of the outside
thermal bath during the subsequent (irreversible) free expansion?
26
5. (2 points) Thermodynamics of Memory Erasure. We can partially recover
from the “loss of memory” considered in part 4 if, after the free expansion,
the righthand wall/piston compresses the volume 2V back down to V , the
partition is reinserted, and the righthand wall/piston is retracted creating an
empty volume V on the right. The memory is now in the well-defined state 0,
so we have erased the memory (which is more useful than merely “losing” the
memory).
0
0
What is the entropy change ∆Sgas
of the gas/memory, and ∆Sbath
of the outside
thermal bath during the isothermal compression that resets the memory to 0?
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6. (2 points) What is the total entropy change of the system of gas + bath during
the erasure of the memory consisting of the free expansion (part 4) followed by
the isothermal compression (part 5)?
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POSSIBLY USEFUL CONSTANTS AND EQUATIONS
You may want to tear this out to keep at your side
x = x0 + v0 t + 12 at2
F = µFN
s = rθ
p = mv
ω = ω0 + αt
KE = 21 Iω 2
W = F s cos θ
2
ac = vr
τ = F ` sin θ
Ifull cylinder = 21 mr2
Ifull sphere = 52 mr2
F = Y A ∆L
L0
P V = nRT
Q = mL
V
Wisotherm = nRT ln Vfi
Q
= kA
∆T
t
L
Q = σT 4 At
v = v0 + at
F = −kx
v = rω
F ∆t = ∆p
P E = 12 kx2
KE = 12 mv 2
L = Iω
ω 2 =p
ω02 + 2α∆θ
ω = k/m√
2πr3/2 = T GM
Ihollow cylinder = mr2
∆L = αL0 ∆T
P1 V1γ = P2 V2γ
P V = nkT
Q = cm∆T
γ =q
CP /CV
v=
γkT
v2 = v02 + 2ax
F = −G Mr2m
a = rα
P
xcm = M1tot i xi mi
P E = mgh
Wnc = ∆KE + ∆P E
P
τ = Iα
∆θ =pω0 t + 12 αt2
ω = g/l
P
I = mi ri2
Q = πR4 ∆P/8ηL
Q = Av
P + ρgh + 21 ρv 2 = const
∆S = ∆Q
T
∆U = Q − W
U = 32 nRT
q
F
v = m/L
√
v = Yρ
β = (10 dB) log II0
f0 = fs / 1 ∓ vvs
sin θ = Dλ
p m
v = Bad /ρ v
1± 0
f0 = fs 1∓ vvs
v
v
fn = n 2L
sin θ = 1.22 λd
Monatomic:
Diatomic:
CV = 3R/2
CV = 5R/2
CP = 5R/2
CP = 7R/2
R = 8.315 J/K/mol
u = 1.66 × 10−27 kg
σ = 5.67 × 10−8 W/m2 /K4
NA = 6.022 × 1023 mol−1
k = 1.38 × 10−23 J/K
Mearth = 5.98 × 1024 kg
Mmoon = 7.35 × 1022 kg
0◦ C = 273.15 K
Rearth = 6.36 × 106 m
Rmoon = 1.74 × 106 m
1 kcal = 4186 J
GNewton = 6.67 × 10−11 Nm2 /kg2
v = λf
f0 = fs (1 ± vv0 )
v
fn = n 4L
vsound = 343 m/s