In describing the heat capacities of solids in Section 18.4, we stated that the potential energy U = 1/2kx2 of a harmonic oscillator averaged over one period of the motion is equal to the kinetic energy K = 1/2mU2 averaged over one period.
Prove this result using Eq. (13.13) and (13.15) for the position and velocity of a simple harmonic oscillator. For simplicity, assume that the initial position and velocity make the phase angle.Ф equal to zero. (Hint: Use the trigonometric identities cos2 (θ) = 1 + cos (2θ) l/2 and sin2 (θ) = [1 - cos (2 θ )l/2. What is the average value of cos (2wt) over one period?)