Show that for a particle in linear system, the average K.E. over a period of oscillation equals the average potential energy ever the same period.

Explanation

`=(1)/(T)int_(0)^(T)E_(k)dt` and average P.E. per cycle `=(1)/(T)int_(0)^(T)E_(p)dt`
`" where "E_(k)=(1)/(2)m(a^(2)omega^(2)cos^(2)omegat)`
`and Ep=(1)/(2)m(a^(2)omega^(2)sin^(2)omegat)`
Since `(1)/(T)int_(0)^(T) sin^(2)omegatdt=(1)/(T)int_(0)^(T)cos^(2)omega t dt=(1)/(2)`
We note that average KE over one cycle `=(1)/(2)[(1)/(2)ma^(2)omega^(2)]=(1)/(4)ma^(2)omega^(2)`
Hence the average PE = average KE over one cycle.