A body is executing simple harmonic motion. At a displacement `x` its potential energy is `E_(1)` and at a displacement `y` its potential energy is `E_(2)` The potential energy `E` at displacement `(x+y)` is

Options

A. `sqrt(E) = sqrt(E_(1)) + sqrt(E_(2))`
B. `E = E_(1) + E_(2)`
C. `E = E_(1) = E_(2)`
D. `sqrt(E) = sqrt(E_(1))-sqrt(E_(2))`

Answer

A

Explanation

PE at a displacement x is
`E_(1) = (1)/(2) Kx^(2)`
`therefore" "x = sqrt((2E_(1))/(K))" "...(i)`
and `E_(2)=(1)/(2)Ky^(2)`
`therefore" "y = sqrt((2E_(2))/(K))" "...(ii)`
and `E = (1)/(2)K(x+y)^(2)`
`therefore" "x + y = sqrt((2E)/(K))" "...(iii)`
`therefore" "sqrt((2E_(1))/(K))+sqrt((2E_(2))/(K))+sqrt((2E)/(K))`
`therefore" "sqrt(E_(1))+sqrt(E_(2))=sqrt(E)`