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.

Show that for a particle in linear system, the average K.E. over a per

1.	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.
Answer» Solution :`=(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.