Physics 11
Fall 2012
NAME:
Discussion Session 10
Simple Harmonic Oscillator Worksheet
Simple harmonic motion occurs everywhere in physics and is essential to understanding
such diverse topics as mass-spring systems, a pendulum, electric circuits, the index of refraction, the fundamental laws of quantum mechanics, and many, many other topics. In fact, as
we have seen, the motion of any system for small displacements near a stable equilibrium
point is approximately given by simple harmonic motion. Therefore, a deep understanding
of simple harmonic motion yields a deep understanding of a large variety of topics.
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Conceptual Questions
1. What happens to the frequency of a simple harmonic oscillator when the spring constant is doubled? When the mass is doubled? How does the frequency of a simple
harmonic oscillator depend on the amplitude of its motion?
2. How would the frequency of a horizontal mass-spring system change if it was taken to
the Moon? What about a vertical mass-spring system? What about a pendulum?
3. One pendulum consists of a solid rod of mass m and length L, and another consists of
a compact ball of the same mass m on the end of a massless string of the same length
L. Which has the greater period? Why?
4. The length of the string or wire supporting a pendulum bob increases slightly when the
temperature of the string or wire increases. How does this affect a clock operated by a
simple pendulum? If a pendulum clock keeps perfect time at the base of a mountain,
will it also keep perfect time when it is moved to the top of the mountain? Explain.
5. A pendulum bob is made from a sphere
lled with water. What would happen to the frequency of vibration of this pendulum if
there were a hole in the sphere that allowed the water to leak out slowly? What if the
bob was attached to a spring, instead of a pendulum?
6. What is resonance? What are some examples of a system in resonance? Opera singers
have been known to break glasses with their voices. How?
7. Light consists of electromagnetic waves oscillating at a certain frequency (the frequency
determines the color of light). When this light interacts with bound electrical charges
(like electrons in an atom), the light wiggles the charges, and they act like harmonic
oscillators. What frequency are the charges wiggling at? When charges oscillate with a
certain frequency, they reemit light at the same frequency. With this in mind, explain
why we can see through materials like glass or water (the light dies away quickly in
electrical conductors, like metals).
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A Mass-Spring System and Simple Harmonic Motion
1. The position of a particle is given by x (t) = (7.0 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?
2. A straight tunnel is dug through Earth, as
shown in the figure. Assume that the walls
of the tunnel are frictionless.
(a) The gravitational force exerted by
Earth on a particle of mass m at
a distance r from the center of
Earth when r < RE is Fr =
3
− (GmME /RE
) r, where ME is the
mass of Earth and RE is its radius.
Show that the net force on a particle of mass m a distance x from
the middle of the tunnel is given by
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Fx = − (GmME /RE
) x and that the
motion of the particle is therefore
simple harmonic motion.
(b) Show that the period of the motion is
independent of the length ofpthe tunnel and is given by T = 2π RE /g.
(c) Find its numerical value in minutes.
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A Pendulum
1. Show that the total energy of a simple pendulum undergoing small oscillations of
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amplitude φ0 (in radians) is E = 12 mgLφ20 . Hint: use the approximation cos φ ≈ 1 − φ2
for small φ.
2. You have designed a cat door that consists of a square piece of plywood that is 1.0 in
thick and 6.0 in on a side, and is hinged at its top. To make sure the cat has enough
time to get through it safely, the door should have a natural period of at least 1.0 s.
Will your design work? If not, explain qualitatively what you would need to do to
make it meet your requirements.
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