Chapter 15
[HOMEWORK 4]
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Problem 1 [20 pts] The position of a particle is given by x = (7 cm) cos(6πt), where t is in
seconds. What are (a) the frequency, (b) the period, and (c) the amplitude of the particle's
motion? (d) What is the first time after t = 0 that the particle is at its equilibrium position? In
what direction is it moving at that time? (e) Sketch x as a function of t. (f) Find the distance
traveled in the first, second, third, and fourth second after t = 0.
Solution:
Chapter 15
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Problem 2 [20 pts] Figure shows a uniform disk of radius R = 0.8 m and 6-kg mass with a
small hole a distance d from the disk's center that can serve as a pivot point. (a) What should
be the distance d so that the period of this physical pendulum is 2.5 s? (b) What should be
the distance d so that this physical pendulum will have the shortest possible period? What is
this period?
Solution:
Chapter 15
[HOMEWORK 4]
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Chapter 15
[HOMEWORK 4]
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Chapter 15
[HOMEWORK 4]
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Problem 3 [20 pts] A simple pendulum of length L is attached to a massive cart that slides
without friction down a plane inclined at angle θ with the horizontal as shown in Figure.
Find the period of oscillation of the pendulum on the sliding cart.
Solution:
Chapter 15
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Problem 4 [20 pts] Figure shows a pendulum of length L with a bob of mass M. The bob is
attached to a spring of spring constant k as shown. When the bob is directly below the
pendulum support, the spring is at its equilibrium length. (a) Derive an expression for the
period of this oscillating system for small amplitude vibrations. Please, derive first using the
dynamical method, then the energy method. Show that they give the same result.
(b) Suppose that M = 1 kg and L is such that in the absence of the spring the period is 2.0 s.
What is the spring constant k if the period of the oscillating system is 1.0 s?
Chapter 15
[HOMEWORK 4]
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Chapter 15
[HOMEWORK 4]
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Chapter 15
[HOMEWORK 4]
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Problem 5 [20 pts] Find the resonance frequency for each of the three systems shown in
figure.
Solution:
Chapter 15
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Optional Problems
Please note that you do not have to do the optional problems. However, I expect ambitious
students to do these problems, too.
Optional Problem 1: Figure shows a uniform solid half-cylinder of mass M and radius R
resting on a horizontal surface. If one side of this cylinder is pushed down slightly and then
released, the object will oscillate about its equilibrium position. Determine the period of this
oscillation.
.
Solution:
Chapter 15
[HOMEWORK 4]
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Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
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Chapter 15
[HOMEWORK 4]
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Optional Problem 2: The Morse potential, which is often used to model interatomic forces,
can be written in the form U (r )  D(1  e  ( r r0 ) )2 , where r is the distance between the two
atomic nuclei. (a) Using a spreadsheet program or graphing calculator, make a graph of the
Morse potential using D = 5 eV, β= 0.2 nm- 1 , and r0 = 0 . 75 nm. (b) Determine the
equilibrium separation and "spring constant" for small displacements from equilibrium for
the Morse potential. (c) Determine a formula for the oscillation frequency for a homonuclear
diatomic molecule (that is, two of the same atoms), where the atoms have mass m .
Chapter 15
[HOMEWORK 4]
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