A particle of mass m is executing oscillation about the origin on X- axis Its potential energy is V(x)=kIxI Where K is a positive constant If the amplitude oscillation is a, then its time period T is proportional

Options

A. Proportinal to `a^(3//2)`
B. Independent of a
C. Proportional to `(1)/(sqrt(a))`
D. Proportional to `sqrt(a)`

Answer

C

Explanation

The `P.E. U=K(x)^(3) therefore F=(dU)/(dx)= - K 3x^(2)`
and in S.H.M. `x=a "sin" omega t and (d^(2)x)/(dt^(2))= - omega^(2)x`
`F=(md^(2)x)/(dt^(2)) =- momega^(2)x`
`therefore -m omega^(2)x =- K3x^(2)`
`therefore omega =sqrt((3Kx)/(m))`
`therefore T=(2pi)/(omega)=2pisqrt((m)/(3Kx))=2pisqrt((m)/(3K(a "sin" omega t))`
`therefore T prop (1)/(sqrt(a))`