Show that for a particle executing simple harmonic motion the average value of kinetic energy is equal to the average value of potential energy.

Explanation

Total energy
`T.E=(1)/(2)m omega^(2)y^(2)+(1)/(2)m omega^(2)(A^(2)-y^(2))`
But `y=a sin omega t`
`T.E. =(1)/(2) m omega^(2)A^(2) sin^(2)omega t+(1)/(2)m omega^(2)A^(2) cos ^(2) omega t`
`=(1)/(2)m omega^(2)A^(2)(sin^(2) omegatt+ cos^(2) omegat)`
From trignometry identity
`Sin^(2)omegat+cos^(2) omega t=1`
`:. T.E.=(1)/(2)m omega^(2)A^(2)(1)`
`=(1)/(2)m omega^(2)A^(2)`
`:. {:("Average"),("potential energy"):}}= {{:("Average"),("Kinetic energy"):}`
`=(1)/(2)` (total energy)