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Physics 235: Classical Mechanics
Prof. S. Teitel stte@pas.rochester.edu  Fall 2001
Problem Set 5
Due Friday, October 12, 1pm, in metal homework locker B&L 1st floor
 Problem 311 in the text. For the last part, remember
to use fact that the oscillator is lightly dampled, i.e.
beta << omega_{o} [20 points]
 Consider the harmonic oscillator driven by a force
F(t) = F_{o} cos(omega t).
For the steady state solution, compute W_{d},
the work done in one cycle of oscillation on the
oscillator by the damping force,
F_{d} = 2 m beta dx/dt.
If W is the work done in one cycle of oscillation on the
oscillator by the external driving
force F(t), show that
W + W_{d} = 0. Since
W_{d} is the work done by the
oscillator on the source of the damping,
W_{d} is the energy lost by the oscillator
in one cycle of oscillation to the degrees of freedom
that give rise to the damping. The result
W + W_{d} = 0
therefore just says that the total energy absorbed
by the oscillator equals the total energy lost by
the oscillator, so that the average energy of the
oscillator in one cycle remains constant in the
steady state. [10 points]
 Problem 331 in the text. Also, if we define the
complex number
c_{n} = c_{n}e^{iphin},
show how c_{n} is simply related to the
usual Fourier coefficients a_{n} and b_{n}.
Derive a single expression for c_{n},
of a form similar to Eq.(3.102b) in the text, however
involving an integration over a complex exponential instead
of cosine and sine functions. [10 points]
