Potential ENERGY (PE):

Restoring FORCE F = mass * acceleration = m * (- ω2y) = -ky where k = spring factor of SHM = m ω2

Work done for the additional displacement dy against the restoring force dW = -F dy = - (-ky) dy = k dy

Total work done in displacing the particle from the mean position to the position of displacement y = W = ∫ ky dy within the limits 0 to y = ½ ky2

Potential energy = PE = ½ ky2 = ½ m ω2y2 = ½ m ω2a2sin2ωt

Kinetic Energy (KE):

Kinetic Energy, KE = ½ mv2 = ½ m (a ω cos ωt)2

                                                   = ½ m a2 ω2 cos2 ωt

= ½ m a2 ω2 (1 - sin2 ωt)

= ½ m a2 ω2 (1 – y2 / a2 )

= ½ m ω2 (a2 – y2 )

Total Energy:

Total energy of a particle at an instant t = PE + KE = ½ m ω2y2 + ½ m ω2 (a2 – y2 ) = ½ m ω2 a2

Where m. a, ω are all constants. Hence, total energy of the system remains constant at all times

As ω = 2∏f, Total energy T = ½ m (2∏f) 2 a2 = 2m ∏2f2 a2 which SHOWS the total energy is proportional to square of amplitude, a and frequency,