See attached file for full problem description.
1. Consider the four equivalent ways to represent simple harmonic motion in one dimension:
To make sure you understand all of these, show that they are equivalent by proving the following implications: I-->II--> III--> IV. For each form, given an expression for the constants (C1, C2, etc.) in terms of the constants of the previous form.
2) a) solve for the coefficients B1 and B2 of the form (II) in terms of the initial position x0 and velocity v0 at t = 0.
b) if the oscillator's mass is m = 0.5 kg and the force constnat is k = 50 N/m, what is the angular vlocity w? If x0 = 3 m and v0 = 50 m/s, what are B1 and B2? Sketch x(t) for a couple of cycles.
c) What are the earliest times at which x = 0 and at which x dot = 0?
3) consider a simple harmonic oscillator with period t. Let (f) denote the average value of any variable f(t), averaged over one complete cycle.
Prove that <T> = <U> = 1/2 E, where E is the total energy of the oscillator.
4) The potential energy of a one dimensional mass m at a distance r from the origin is
find the equilibrium position. Let x be the distance from equilibrium and show that, for small x, the PE has the form U = const + 1/2kx^2. What is the angular frequency of small oscillations?
5) The general solution for a two dimentional isotropic oscillator is given by:
Show that by changing the origin of time you can cast this in the simpler form:
The solution investigates the simple harmonic oscillator and the potenital energy of the oscillator. The solution is detailed and well presented.