A particle of mass (m) is executing oscillations about the origin on the (x) axis. Its potential energy is `V(x) = k|x|^3` where (k) is a positive constant. If the amplitude of oscillation is a, then its time period (T) is.

Options

A. proportional to `1//sqrt(a)`
B. independent of (a)
C. proportional to `sqrt(a)`
D. proportional to `a^(3//2)`.

Answer

A

Explanation

`U (x) = k|x|^3`
:. `[k] = ([U])/([x^3]) = (M L^2 T^-2)/(L^3) = ML ^-1 T^-2`
now time period may depend on ` T prop (mass)^x (amplitude)^y (k)^z`
:. `[M^0 L^0T] = [M]^x [L]^y [M L^-1 T^-2]^z`
=`[M^x + z L^y -z T^(-2z)]`
Equating the powers, we get
`-2z = 1 or -1//2`
`y - z = 0 or y = z = - 1//2`
Hence `T prop (amplitude)^(-1//2) prop a^(1//2)`.