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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

1. Problem 3-11 in the text. For the last part, remember to use fact that the oscillator is lightly dampled, i.e. beta << omegao [20 points]

2. Consider the harmonic oscillator driven by a force F(t) = Fo cos(omega t). For the steady state solution, compute Wd, the work done in one cycle of oscillation on the oscillator by the damping force, Fd = -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 + Wd = 0. Since -Wd is the work done by the oscillator on the source of the damping, -Wd 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 + Wd = 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]

3. Problem 3-31 in the text. Also, if we define the complex number cn = cne-iphin, show how cn is simply related to the usual Fourier coefficients an and bn. Derive a single expression for cn, 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]

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