Question

To understand how the two standard ways to write the general solution to a harmonic oscillator are related.There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:x(t)=Acos(ωt+ϕ) andx(t)=Ccos(ωt)+Ssin(ωt).Either of these equations is a general solution of a second-order differential equation (F⃗ =ma⃗ ); 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.)A)Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B) in terms of the constants A and ϕ in Equation 1 (found in Part A), which are now considered as given parameters.Give your answers for the coefficients of cos(ωt) and sin(ωt), separated by a comma. Express your answers in terms of A and ϕ.

Answer 1

Answer:

a)   C = A cos φ , S = A sin φ  

Explanation:

a) The two equations given are equivalent ,

we know the formula of  

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

let say

          a  = ωt     and      b = φ

          x(t) = A (cos ωt cosφ- sin ωt sin φ )

the second equation is

          x(t) = C cos ωt + S  sin ωt

they both are equivalent

          C cos ωt + S sin ωt = A cos φ cosωt - a sin φ sin ωt

The coefficients of the equation should be equal

        C = A cos φ

         S = A sin φ