Find analytic expressions for the arbitrary constants C and S in Two General Simple Harmonic Motion Solutions

Question

There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:

1. x(t) = A*cos(omega*t phi)

and

2. x(t) = C*cos(omega*t) S*sin(omega*t)

Either of these equations is a general solution of a second-order differential equation F = m*a; hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.)

----------------------

Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B: C and S) in terms of the constants A and phi in Equation 1 (found in Part A: A and phi), which are now considered as given parameters.

Give your answers for the coefficients of cos(omega*t) and sin(omega*t), separated by a comma. Express your answers in terms of A and phi.

Answer 1

Using the trig identity:

cos(a+b) = cos(a) cos(b) - sin(a) sin(b),

and solving for the appropriate values gives the answer:

C= A*cos(phi), S= -A*sin(phi)

**You don't need the last part that guy above me gave you. You would only need that if the question asked you to find analytic expressions for the arbitrary constants A and phi in terms of the constants C and S, which is the answer he gave you in the last part that he wrote.

Answer 2

x= A*(cos(w*t)*cos(phi) – sin(w*t)*sin(phi)) =

= A*cos(phi)* cos(w*t) – A* sin(phi)* sin(w*t);

thus C= A*cos(phi), S=- A* sin(phi);

or; phi=-atan(S/C), A=√(C^2 +S^2);