Problem 1:
A rope has a mass of 2 kg and a length of 10 m. It is stretched with a tension of 50 N and fixed at both ends. What is the frequency of the first harmonic on this rope?
Problem 2:
Consider a damped harmonic oscillator. Let us define T1 as the time between adjacent zero crossings, 2T1 as its “period”, and w1 = 2p/(2T1) as its “angular frequency”. If the amplitude of the damped oscillator decreases to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [ 1 - (8p2n2)-1] times the frequency of the corresponding undamped oscillator.
Problem 3:
When the system shown in the diagram is in equilibrium, the right spring is stretched by x1. The coefficient of static friction between the blocks is µs. There is no friction between the bottom block and the supporting surface. The force constants of the springs are k and 3k (see the diagram). The blocks have equal mass m. Find the maximum amplitude of the oscillations of the system shown in the diagram that does not allow the top block to slide on the bottom.
Problem 4:
A train has a whistle, which emits a 400 Hz sound. You are stationary and you hear the whistle, but the pitch is 440 Hz. How fast is train moving towards or away from you?
Problem 5:
Phonons are quantized lattice vibrations, and many aspects
of these excitations can be understood in terms of simple mode counting.
(a) Estimate the number of phonon modes in 1 cm3 of a crystalline
material with an inter-atomic spacing of 2Ǻ.
(b) Assuming that in thermal
equilibrium each phonon mode has kBT of energy, give a numerical
estimate of the heat capacity DE/DT of this 1 cm3 of material, in [J/K].
Problem 6:
Use Lagrange's equations to find the normal modes and normal frequencies for linear vibrations of the CO2 molecule shown below.
Problem 7:
A mass on a spring moves in one dimension and is subject to a velocity-dependent force. The net force is F = -kx – bv. Assuming b is small, what fraction of energy is dissipated in each cycle?