In case of simple harmonic motion -
(a) What fraction of total energy is kinetic and what fration is potential when displacement is one hall of the amplitude.
(b) At what displacement the kinetic and potential energies are equal.

Explanation

In `S.H.M.` : Kinetic Energy `K = 1/2 k(A^(2) - x^(2))`
Potential Energy `U = 1/2kx^(2)`
Total Energy `(TE) = 1/2 KA^(2)`
(a) Fraction of Kinetic Energy `f_(K.F.) = (K)/(T.E.) = (A^(2) - x^(2))/(A^(2))`
Fraction of Potential Energy `f_(P.E.) = (U)/(T.E) = (x^(2))/(A^(2))`
at `x = (A)/(2) , f_(K) = (A^(2) - A^(2)//4)/(A^(2)) = 3/4` and `f_(u) = (A^(2)//4)/(A^(2)) = 1/4`
(b) `K = U rArr 1/2 k(A^(2) - x^(2)) = 1/2 kx^(2) rArr 2x^(2) = A^(2) rArr x = +-(A)/(sqrt(2))`