Find the displacement of a simple harmonic oscillator at which its PE is half of the maximum energy of the oscillator.

Explanation

Let us assume that the required displacement be x.
:. Potential energy of the simple harmonic oscillatro `=(1)/(2)kx^(2)`
where, k= force constant `=momega^(2)`
`:.PE=(1)/(2)momega^(2)x^(2)" "....(i)`
Maximum energy of the oscillator
`TE=(1)/(2)momega^(2)A^(2)" " [:'x_(max)=A].....(ii)`
where, A = amplitude of motion
Given, `PE=(1)/(2)TE`
`rArr(1)/(2)momega^(2)x^(2)=(1)/(2)[(1)/(2)momega^(2)A^(2)]`
`rArrx^(2)=(A^2)/(2)`
or `x=sqrt(A^(2)/(2))=+-(A)/(sqrt2)`
Sign `+-` either side of mean position.