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. `E_(1)+E_(2)`
B. `sqrt(E_(1)E_(2))`
C. `sqrt(E_(1)^(2)+E_(2)^(2))`
D. `E_(1)+E_(2)+2sqrt(E_(1)E_(2))`

Answer

D

Explanation

`E_(1)=(1)/(2)momegat^(2)x^(2), E_(2)=(1)/(2)momega^(2)y^(2)`
`:. sqrt(E_(1)E_(2))=[(1)/(2)m omega^(2)x^(2)xx(1)/(2)m omega^(2)y^(2)]^(1//2`
`=(1)/(2)momega^(2)xy`
`E=(1)/(2)momega^(2)(x+y)^(2)=(1)/(2)momega^(2)(x^(2)+y^(2)+2xy)`
`(1)/(2)momega^(2)x^(2)+(1)/(2)momega^(2)y^(2)=2xx(1)/(2)momega^(2)xy`
`=E_(1)+E_(2)+2sqrt(E_(1)E_(2))`