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Physics 103 – Fall 2009
Fourth Hour Examination
Friday, December 18, 2009
50 minutes
Instructions: When you are told to begin, check that this examination booklet contains
all the numbered pages from 2 through 7.
Do not be discouraged if you cannot do all three problems, or all parts of a given
problem. If a part of a problem depends on a previous answer you have not obtained,
assume an answer 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 we can understand your solution even when you write down the correct answer.
Always write down analytic answers first and only then calculate numerical values.
Include correct units where appropriate. For the purposes of this exam g = 9.8m/s2.
Please Box Your Answers.
THE ONLY MATERIALS ALLOWED DURING THE EXAM ARE
THE EXAMINATION BOOKLET, PEN OR PENCIL, AND YOUR
CALCULATOR. DO ALL THE WORK YOU WANT TO HAVE
GRADED IN THIS EXAMINATION BOOKLET! YOU MAY USE
THE BACK OF EACH SHEET, BUT YOU WILL NOT BE
ALLOWED TO HAND IN ANYTHING ELSE.
If you need to use the restroom briefly you may do so, but your exam booklet cannot leave
the room. This is a timed examination. You will have 50 minutes to complete this exam.
Two preceptors will be outside the room to answer questions during the exam.
_________________________________________________
_________________________________________________
REWRITE IN FULL AND SIGN THE PLEDGE IN THE SPACE ABOVE
“I pledge my honor that I have not violated the Honor Code during this examination.”
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U
SIGNATURE
PHY103 – Exam 4
Problem 1, Page 2
Problem 1. Standing Waves. (35 pts)
String
Tube
L
D
A string of length L  25 cm is fixed at both ends and located near the open end of an
open-closed tube of length D (note: the diagram is not necessarily to scale). The tension
in the string is 50N, and the speed of sound in air is vsound  343 m/s. When the string is
plucked so that it vibrates at its fundamental frequency of 440Hz, the air in the tube
resonates in its second vibrational mode (i.e., the next higher frequency after the
fundamental).
a) (6 pts) On the diagram, draw the standing wave patterns of displacement for the
fundamental mode of the string, and the second vibrational mode of the air in the tube.
b) (5 pts) What is the wavelength of the fundamental frequency of the string?
c) (5 pts) What is the speed of waves on the string?
PHY103 – Exam 4
Problem 1, Page 3
Problem 1 (cont)
d) (5pts) What is the mass of the string in grams?
e) (7 pts) What is the length D of the tube?
f) (7 pts) If the tube were open at both ends, which vibrational mode would be excited if the
string were excited in its second vibrational mode?
PHY103 – Exam 4
Problem 2, Page 4
Problem 2. (30 pts) Damped Oscillations: Emma and Isabella Return!
Emma and Isabella, our future physicists, are on their way to Grandma’s house in the
country. They have been reading up on damped oscillations and can’t wait to test out
their knowledge on Grandma’s classic car, a 1975 Cadillac Eldorado with shock
absorbers so loose that when you push down on the hood the car bounces up and down
many times before coming to rest. The car is effectively a lightly-damped oscillator
with a mass of m  1958 kg, spring constant k, and damping constant b.
Upon arrival the girls pull out their meter stick and stopwatch to make some
measurements. Their data are plotted in the graph above, which shows the vertical
displacement y t  of the car in cm relative to its equilibrium position as a function of
time t . For convenience, we also give the exact values for the amplitude at the four
times indicated on the graph.
a) (8 pts) What is the car’s frequency of oscillation 0 ?
b) (8 pts) What is the car’s spring constant k?
PHY103 – Exam 4
Problem 2, Page 5
Problem 2 (cont)
c) (8 pts) What is the car’s damping constant b? Write down the analytic formula for
y t  as a function of t, b, and m, and then give the numerical result for b.
d) (6 pts) Grandma takes the girls for a ride on a bumpy country road near her house. If
Grandma is driving at a constant speed v over bumps spaced 15 m apart, what speed
should she avoid so that the car does not resonate? Assume the mass of the car with
passengers is 2200 kg.
PHY103 – Exam 4
Problem 3, Page 6
Problem 3. Torsional Pendulum. (35 pts)
M
R

k
m
A torsional pendulum consists of a uniform solid disk of mass M and radius R pivoted
around an axis perpendicular to the page and passing through its center of mass. A small
mass m is attached to the edge of the disk and connected to a massless spring with force
constant k , which itself is attached to a rigid immovable wall. The entire apparatus is in
equilibrium when the mass m is directly below the pivot (   0 ). If the mass is rotated
from equilibrium through a small angle  and released, it will execute simple harmonic
motion. The moment of inertia of the disk (without the mass m ) is 12 MR 2 , and you may
assume that the spring remains horizontal throughout this problem.
a) (5 pts) What is the moment of inertia of the disk including the mass m ?
b) (5 pts) Draw a free-body diagram for the disk with the mass m attached when it is in
the position shown in the drawing.
PHY103 – Exam 4
Problem 3, Page 7
Problem 3 (cont)
c) (5 pts) Assuming the spring remains horizontal, but not making any assumption about
 , what is the net torque on the disk in the position shown in the drawing?
d) (15 pts) Now assume the small-angle approximation for  and find the frequency 
of small oscillations around the equilibrium position?
e) (5 pts) What is the angular velocity
oscillation frequency  , and time t ?
d
dt
of the disk in terms of the initial amplitude  0 ,