Chapter 7, Problem 7.13P

The equation of motion for a damped harmonic oscillator is s(t) = Ae^(−kt) sin(ωt + δ),where A, k, ω, δ are constants. (This represents, for example, the position of springrelative to its rest position if it is restricted from freely oscillating as it normally would).(a) Find the velocity of the oscillator at any time t.(b) At what time(s) is the oscillator stopped?

In the above question let us consider the position of mass when the spring is relaxed as x = 0, and the left to right direction as the positive direction of the x-axis.Provide x as a function of time t for the oscillating mass, if at the moment we start the stopwatch (t = 0), the mass is:( i ) at the mean position,( ii ) at the maximum stretched position, and( iii ) at the maximum compressed position.

A frictionless table has a mass m on it. The table has a hole in it, where mass m is connected to a string that passes through the hole, attached to a hanging mass M.(a) What is the Lagrangian for this system? Using this Lagrangian, find the equations of motion (a = function of position) for each of the generalized coordinates used.(b) Under what conditions does m make a circular motion?(c) What is the frequency of small oscillations (in r) about this circular motion? Further, assume the oscillations are small: use a Taylor Expansion at the point we choose to use the small angle approximation.)