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

Show that for a particle executing simple harmonic motion the average

1.	Show that for a particle executing simple harmonic motion the average value of kinetic energy is equal to the average value of potential energy.
Answer» Solution :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 trignometryidentity `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)