The mean kinetic energy of a particle executing SHM over one full oscillation, given  mass of particle = m, amplitude = A, angular frequency =ω  ).

for a simple harmonic oscillator, displacement=x=Asinωt  differentiate with respect to time t, to obtain velocity, velocity=v=Aωcosωt KE=12mv2 KE=12mAωcosωt2 KE=12mω2A2cos2ωt Mean of a function==∫0Tf(t)dt∫0Tdt KEmean=∫0T(KE)dt∫0Tdt KEmean=∫0T12mω2A2cos2ωtdtT KEmean=12mω2A2T∫0Tcos2ωtdt KEmean=12mω2A2T∫0T1+cos2ωt2dt KEmean=12mω2A2Tt2+sin2ωt4ω0T KEmean=12mω2A2TT2 KEmean=14mω2A2=mean kinetic energy of a particle executing SHM over one full oscillation