Is Anybody Smart Enough To Solve This Problem?

Question

1) Find C and S in terms of the initial position and velocity of the oscillator. Give your answers in terms of x0, v0, and omega. Separate your answers with a comma.

Problem Background Information:

A common problem in physics is to match the particular initial conditions - generally given as an initial position x0 and velocity v at - once you have obtained the general solution. You have dealt with this problem in kinematics, where the formula:

x(t) = x0 + v0(t) + (1/2)at^2

has two arbitrary constants (technically constants of integration that arise when finding the position given that the acceleration is a constant). The constants in this case are the initial position and velocity, so "fitting" the general solution to the initial conditions is very simple.

For simple harmonic motion, it is more difficult to fit the initial conditions, which we take to be

x0, the position of the oscillator at t = 0, and

v0, the velocity of the oscillator at t = 0.

Answer 1

The oscillator is at position x0 when t=0, so you can plug 0 in for t to get the equation:

x0 = C cos[0] + S [sin[0] = C

so C is just x0.  Moreover, velocity is the derivative of position, so the velocity of the system is given by:

v = d/dx(C cos[om*t] + S sin[om*t]) = -C*om sin[om*t]+S*om cos[om*t]

where "om" is shorthand for omega.  Again plugging in t=0 to find v0:

v0 = -C*om sin[0] + S*om cos[0] = S*om

Therefore, S = v0/om.