Physics 3221
Fall Term 2004
Test 2, November 5, 2004
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This is an open notes test lasting 50 minutes.
•
There are two problems. Problem 2 is divided into 4 subsections. The
points for each part are marked.
•
Begin each problem on a fresh sheet of paper. Use only one side of a
sheet of paper.
•
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
labeled figures are important. Randomly scrawled equations aren't helpful.
•
Try to draw a box around important results.
There are 3 pages including this page. Do not forget to look at all parts of the
problems.
Page 1 of 3
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Problem 1.
Two pendulums are suspended from the same point. They have masses m and 3m
attached to massless, extensionless strings, each of length l (see figure). The mass m
is displaced to a height h above the mass 3m (as shown in the figure) and then
released. Mass m collides with mass 3m and they stick together. Find the maximum
height h1 that the combined mass reaches in terms of h.
(8 points)
[Remember that the collision is inelastic.]
l
l
l
l
m
h1
h
3m
PROBLEM 2 ON NEXT PAGE
Page 2 of 3
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Problem 2.
A massive spring (massive does not mean gigantic but just something which has
mass).
We usually consider massless springs when we solve oscillation problems. What if a
spring has mass too? A uniform spring has a mass m, equilibrium length l and a
spring constant k. This spring has a block of mass M attached to it as shown. There is
no friction between the mass M and the surface. This system undergoes simple
harmonic oscillations.
a) Find the maximum value of potential energy of this system in terms of the
spring constant k and the amplitude of oscillation A.
(2 points)
[Hint: Remember that the potential energy only depends on the spring
constant and how much the spring is stretched and not on the masses.]
b) The block of mass M has the maximum velocity vmax when it passes
through the equilibrium position. At that moment the different parts of the
spring are also moving but at different velocities. The fixed point of the
spring (at x=0) is obviously stationary while the point attached to mass M
(x=l) is moving at vmax. Assume that the velocity of each point of the
spring goes linearly from 0 to vmax as x goes from 0 to l. Write down the
equation for the velocity of a point on the spring as a function of x in terms
(3 points)
of vmax and l.
c) Since the spring has a mass m, it contributes to the total kinetic energy
along with the mass M. Using the equation from part (b) find the
maximum value of the total kinetic energy of the spring and block in terms
of m, M and the maximum velocity of the mass M, vmax.
(4 points)
[Hint: Some integral calculus is needed]
d) Find the effective spring constant and mass factor of this system by
comparing the maximum potential energy and maximum kinetic energy
for this system to that of the massless spring case. Hence find the angular
frequency (ω0) of oscillation for this system.
(3 points)
x=0
x
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