PHY 3221: Mechanics I
Fall Term 2009
Final Exam, December 16, 2009
• This is a closed book exam lasting 90 minutes.
• Since calculators are not allowed on this
√ test, if the problem asks for a numerical answer,
answering 2 + 2 is as good as 4, and 2 is as good as 1.4142....
• There are six problems worth a total of 21 points, out of a maximum of 20 points. Each
problem carries equal weight. The problems appear on the second and third page of this
test. Begin each problem on a fresh sheet of paper. Use only one side of the paper. Avoid
microscopic handwriting.
• Put your name, the problem number and the page number in the upper right-hand corner
of each sheet.
• To receive partial credit you must explain what you are doing. Carefully labelled figures
are important! Randomly scrawled equations are not helpful.
• Draw a box around important results (or at least results which you think might be important).
• Good luck!
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Problem 1. [3.5 pts] In Cartesian coordinates, some force is given by F~ = k(y, x, 0), where
k is a constant. Is this a conservative force? If the answer is “yes”, find the corresponding
potential energy U(x, y, z).
Answer: Yes. U(x, y, z) = −kxy + const.
Problem 2. [3.5 pts] Refer to Fig. 1. A clumsy skateboarder lets her skateboard roll down
a frictionless “quarter pipe” ramp, which has the profile of a cylinder of radius R. Assume that
the skateboard started rolling from rest at point A, which is a distance H above the ground,
and landed at point B, which is a distance D from the end of the ramp. Find D in terms of
H, R and g.
A
R
~g
H
B
D
Figure 1: An illustration for the skateboarder problem.
q
Answer: D = 2 R(H − R).
Problem 3. [3.5 pts] A particle starting at rest is attracted to a force center according to
the relation F = −mk 2 /x3 . Show that the time required for the particle to reach the force
center from a distance L is L2 /k.
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Problem 4. [3.5 pts.] The natural frequency of a critically damped oscillator is ω0 . Initially,
at time t = 0, the oscillator is displaced a distance L from its equilibrium position and then
released. Find the velocity v of the oscillator at a later time t.
Answer: ẋ(t) = −ω02 Lte−ω0 t .
Problem 5. [3.5 pts.] Refer to Fig. 2. A hole has been drilled straight through the center
of a spherical planet of mass M and radius R. Assume uniform mass density and neglect
rotational effects and friction.
(a) A stone of mass m has been dropped in the hole at point A. Find the time it takes to
arrive at point B. Hint: show that the stone’s motion in the hole is simple harmonic and find
its frequency.
(b) Now suppose the stone is launched into a circular orbit just above the surface of the planet.
How long does it take to reach point B in this case?
A
R
M
B
Figure 2: An illustration for the supertunnel problem.
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Answer: a) TAB =
πR 2
√
.
GM
b) the same answer.
Problem 6. [3.5 pts.] A point mass m is located a distance D from the nearest end of a
thin rod of mass M and length L, along the axis of the rod. Find the gravitational force exerted
on the point mass by the rod.
Answer: F =
GmM
.
D(D+L)
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