A particle executes SHM.
(a) What fraction of total energy is kinetic and what fraction is potential when displacement is one half of the amplitude?
(b) At what value of displacement are the kinetic and potential energies 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))`