19 Jan 08
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Printed Name:
Please Circle your Section!
Jau
10 am Marino
Garcia-Garcia
10 am Garcia-Garcia
Ianni
10 am Ianni
Rothman
10 am Rothman
Yazdani
10 am Yazdani
Yost
10 am Yost
Problem
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Score
/50
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/300
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.
2 Course textbooks
2 Course Notes
2 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
FURTHER INSTRUCTIONS
The exam contains SIX problems with multiple parts, worth varying numbers of
points, which total to 300 points.
Do not panic or be discouraged if you cannot do every part of 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!
Read each problem carefully. You must show your work —the grade you get
depends on how well the grader can understand your solution even when you write
down the correct answer. Always write down analytic answers first and only then
calculate numerical values if requested. Check that the dimensions of your
answers are correct.
Please Box Your Answers .
To refresh your memory, here is the calculator policy from the Blackboard website.
Calculator Policy:
Calculators are tools used by all physicists. Many of
these devices have advanced features that are very useful. However, not all students
have calculators with these features, and our rules are designed for fairness. On tests
and the final: No advanced features may be used at any time. Calculator use
is restricted to arithmetic and evaluation of trig functions, exponentials, logs, square
roots, etc. You are not allowed to use, for example,
• calculator memory for storage of physics formulas or constants
• automatic equations solving
• graphing features
• functions such as quadratic formula
You are also not allowed to use of any of these features for checking answers. Violation
of these rules is a violation of the Honor Code.
DO ALL THE WORK YOU WANT TO HAVE GRADED IN THIS
EXAMINATION BOOKLET! YOU WILL NOT BE ALLOWED TO
HAND IN ANYTHING ELSE.
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 0, Page 3
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Physics 103 Last In-Class Exam, 18 Dec 07
Problem 0, Page 4
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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 N/m. It is compressed
a distance x0 = 15 cm, 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 θ = 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?
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.
iv) Is the momentum conserved while the block is sailing through the air? Comment
on both the horizontal and vertical components of the momentum.
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 1, Page 6
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?
vt :
v2 :
d:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 1, Page 7
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 µk = 0.20. Find the
work done on the block by the spring (Wspr ) and by friction (Wf ). What is the work
Wg,1 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?
Wspr :
Wg,1 :
Wf :
Wg,2 :
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 1, Page 8
d) (13 pts) What is the distance d in this case (with friction on the inclined plane)?
d:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 2, Page 9
Problem 2. Collisions and Kinematics
a) (20 pts) A particle of mass m = 10.0 g moves with a velocity v0 = 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 ∆E lost in the collision.
vf :
∆E:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 2, Page 10
b) A particle of mass m1 =100 g moves along the x-axis with velocity v0 = 2.0î m/s.
The particle hits a second particle of mass m2 = m1 , which is at rest. The collision
is elastic. After the collision m1 has velocity v1 and m2 has velocity v2 .
i) (10 pts) Prove that v1 ·v2 = 0 in this case. (Hint: use conservation of momentum
and energy, and the definition of the dot product.)
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 2, Page 11
ii) (20 pts) After the collision v1 makes an angle θ1 =45◦ with respect to v0 . Determine the angle θ2 that v2 makes to v0 , and find the magnitudes v1 and
v2 .
θ2 :
v1 :
v2 :
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 = 10R. 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.
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 3, Page 13
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?
c) (4 pts) Suppose at t = 0, the spacecraft is at rest at x = 4R. What is the net force
F on the spacecraft?
F =
Direction:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 3, Page 14
d) Suppose instead that at t = 0, the spacecraft is at rest at x = 5R. The captain
fires a huge cannonball of mass mc = m/10 in the −y direction. After the cannon is
fired, the spacecraft has net mass (9/10)m and moves off in the +y direction.
i) (8 pts) Is angular momentum conserved before and after the cannonball is fired?
If vc is the speed of the cannonball after the cannon is fired, what is the magnitude Lc of the angular momentum of the cannonball about the z-axis?
yes or no:
Lc :
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 3, Page 15
ii) (20 pts) With must vc be in order to permit the spacecraft to escape the gravitational pull of the binary star system? As always, show all the steps of your
work clearly for full credit.
vc :
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 4, Page 16
Θ
Problem 4. Banked Curves
An engineer designing the bank on a curved road has to take friction into account.
Suppose 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 while maintaining some speed v. The coefficient of friction between the wheels
and the surface of the road is µs .
a) (15 pts) Draw a freebody diagram of the car. Label every force clearly. You do
not need to resolve the forces on your diagram, but make it clear where the angle θ
comes in.
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 4, Page 17
b) (20 pts) For what speed v is the angle θ a perfect bank? (On a perfect bank, the
car is able to go around the curve, but the road exerts no sidewise frictional force on
the car.) Express your answer for v in terms of θ, g, and R.
v:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 4, Page 18
c) (15 pts) At what speed ṽ will the car just begin to slide up a bank of angle θ?
Express ṽ in terms of θ, g, R, and µs .
ṽ:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 5.
Problem 5, Page 19
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 =0. 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 T0 in the
string when the mass is at y = 0. Express your answer in terms of variables given in
the problem and g as needed.
T0 :
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 5, Page 20
b) (10 pts) Find the period P of oscillations of the mass m, assuming that the amplitude 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.)
P , algebraic:
P , numerical:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 5, Page 21
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.
direction:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 5, Page 22
d) (20 pts) Determine the period P̃ 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 =1.0 kg and m= 0.10 kg. The moment of inertia for the disk
is I = (1/2)M R2 .
P̃ , algebraic:
P̃ , numerical:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 6, Page 23
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, î
and ĵ. Mathematically, this spiral wave is described by
R(z, t) = R sin(kz − ωt)î + R sin(kz − ωt + φ)ĵ.
Here, R is the wave amplitude, and φ = π/2. The values k and ω are constants.
a) (5 pts) Based on the R(z, t) given above, in what direction is the wave traveling
(+k̂ or −k̂)?
direction:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 6, Page 24
b) (5 pts) Consider the small chunk of string in the xy 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 φ = +π/2, what is the direction of the circular motion
(clockwise or counterclockwise)?
direction:
c) (15 pts) Consider that small chunk to have length δz, with δz small enough that
you can treat the chunk as a point mass. What is its kinetic energy, K? What is its
angular momentum L about the origin? Express your answers in terms p
of k, ω, T ,
R, and δz as needed.. (Hint: Remember that the velocity of the wave is T /µ, and
note that the mass of the chunk is m = µδz. Don’t leave µ in your answers.)
K:
L=
Direction:
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, ω, T , R,
and δz as needed.
F =
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, ω, T , R, and δz.
U:
Physics 103 Last In-Class Exam, 18 Dec 07
Problem 6, Page 26
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