Find analytic expressions for the arbitrary constants A and phi

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.

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

Express the amplitude A and phase phi (separated by a comma) in terms of C and S.

Answer 1

♦ 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);