Problem Set 5

Due Friday, October 12, 1pm, in metal homework locker B&L 1st floor

Problem 3-11 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 3-31 in the text. Also, if we define the complex number c_n = c_ne^-iphi_n, 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]

Last update: Tuesday, August 14, 2007 at 5:35:56 PM.