a) Variation with time of energy in simple harmonic motion

If the variation with time of displacement is as shown, then the energies should be drawn as shown.

recalling Energy formula

KE = ½ m v²

PE = ½ k x²

in terms of time t,

x = x₀ sin(ωt)

differentiating with t gives

v = v₀ cos (ωt)

therefore, KE = ½ m v²= ½ m (v₀ cos (ωt))²= ½ m (x₀²ω²)cos (ωt))²

similarly

PE = ½ k x²= ½ (mω² )(x₀ sin (ωt))²= ½ m (x₀²ω² )sin (ωt))²

therefore total energy is a constant value in the absence of energy loss due to drag (resistance)

TE = KE + PE = ½ m (x₀²ω² )[cos²(ωt) + sin²(ωt))] = ½ m (x₀²ω²)

this is how the x vs t looks together of the energy vs t graphs

the table shows some of the common values

Model: