Energy of SHM, total energy constant

Simple harmonic motion (SHM) is periodic motion of a particle in which the restoring force (or acceleration) is proportional to the displacement from the equilibrium position and is directed towards it: F = −kx.

Equation and solution

m d²x/dt² = −kx gives x = a sin(ωt + φ), with ω = √(k/m), a = amplitude. Then v = dx/dt = aω cos(ωt + φ).

Potential energy

Work done against the restoring force to displace by x:

PE = ∫₀ˣ kx dx = ½ k x² = ½ m ω² a² sin²(ωt + φ)

Kinetic energy

KE = ½ m v² = ½ m ω² a² cos²(ωt + φ) = ½ m ω² (a² − x²)

Total energy

E = KE + PE = ½ m ω² a² [cos²(ωt + φ) + sin²(ωt + φ)]
E = ½ m ω² a² = ½ k a²

This does not depend on time t.

Variation with time

• KE and PE both vary with time between 0 and ½mω²a², with frequency 2ω (twice that of the motion).
• At the mean position (x = 0): PE = 0, KE = ½mω²a² (maximum).
• At extreme positions (x = ±a): KE = 0, PE = ½mω²a² (maximum).
• When KE falls, PE rises by exactly the same amount.

 E |-------------------------  total E
   |  \    /\    /\
   |   \  /  \  /     KE and PE are
   |    \/    \/      out of phase
   |__________________ t

So KE and PE change with time, but their sum ½ka² stays constant; no energy is lost in ideal SHM.