PHY 3221 Classical Mechanics, Homework #7, due 8:30 am, 03/26/12
1 A particle of mass, m, is attached to a spring of spring constant, k, and is subject to a velocitydependent damping force with damping coefficient, b.
a) Introducing new variables if you wish (such as ω0 and β), write out the most general solution to
the equation of motion, assuming m ∗ k >> b2 /4.
b) If the system is driven by a force of f0 cos(ωt), find the coefficients B1 and B2 for a solution of
the form:
x = B1 cos(ωt) + B2 sin(ωt).
c) Rewrite this solution in the form:
x = A cos(ωt − δ),
by finding the appropriate A and δ.
d) Find the driving frequency ω for which A is a maximum?
e) For this driving frequency, what is the value of δ?
f) Consider the condition A2 (ω) = A2max /2, solve it for ω at the these “half power” points, and
calculate the half power width of the resonance.
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2 A clamped mechanical device (such as a metal ruler) oscillates in vacuum at a frequency (assume
there is no damping in this case) of 10.0 Hz.
a) In water the oscillation frequency drops to 9.6 Hz. Find the characteristic decay time for the
damped motion.
b) In molasses, the characteristic decay time is 0.125 s. Find the frequency of oscillation of the
damped oscillator when it is being damped in molasses.
c) If the system in molasses is driven by an electro-mechanical coupling (such as a guitar pick-up
in reverse), what is the frequency at which the resonant response is maximum? Careful, this is a
tricky question.
d) In corn syrup, the frequency of maximum resonance is reduced to 3 and 1/3 Hz (i.e., 3.3̇ Hz).
What is the characteristic decay time of the oscillator in corn syrup?
e) What is the oscillation frequency of the damped oscillator in this case?
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3 Consider a periodic function of period 2 s, which is given by y(t) = −t for −1 s < t < 1 s.
a) Find the complete Fourier series representation for y(t).
b) In the domain −1.5 s < t < 1.5 s, plot y, the sum of the first three terms in its Fourier representation, and the sum of the first six terms in its Fourier representation. Compare them.
c) Suppose that an oscillator has an undamped frequency of 1.5 Hz, and is subject to a damping
force which results in a characteristic decay time of 1 s. Find the response, x(t), of the damped
driven oscillator as a Fourier series in t.
d) What is the amplitude of this response at the oscillation frequency of the undamped oscillator?
Does the response also have an amplitude at the oscillation frequency of the damped oscillator?
Explain.
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