Problem 1:
A particle that hangs from a spring oscillates with an angular frequency of 2
rad/s. The spring is suspended from the ceiling of an elevator car and hangs
motionless (relative to the car) as the car descends at a constant speed of 1.5
m/s. The car then suddenly stops. Neglect the mass of the spring.
(a) With what amplitude does the particle oscillate?
(b) What is the equation of motion for the particle? (Choose the upward
direction to be positive.)
Problem 2:
Three small identical coins of mass m each are connected by two light non-conducting strings of length d each. Each coin carries an unknown charge q. The coins are placed on a horizontal frictionless non-conducting surface as shown (the angle between the strings is very close to 180°). After the coins are released, they are observed to vibrate with period T. Find the charge q on each of the coins in terms of m, d, and T.
Problem 3:
A string of length 2l is suspended at points A and B located on a horizontal line. The distance between A and B is 2d (d < l). A small, heavy bead can slide on the string without friction. Find the period of the small-amplitude oscillations of the bead in the vertical plane containing the suspension points. The acceleration due to gravity is g.
Problem 4:
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 5:
Use Lagrange's equations to find the normal modes and normal frequencies for linear vibrations of the CO2 molecule shown below.
Problem 6:
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 7:
A helicopter accelerates upward with a cable of mass 8 kg and length 17 m attached to a mass of 150 kg hanging vertically below it. If a transverse pulse initiated by wobbling of the mass takes 0.238 seconds to travel the length of the cable, what is the upward acceleration of the helicopter?