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

Potential energy, `U=kx^(3)`,
Force, `F=(-dU)/(dx)=-3kx^(3)`
Max. force `=F_(max)=-3ka^(2)=-momega^(2)a`
or `omega^(2)=(3ka^(2))/(ma)=(3ka)/(m)`
or `(4pi^(2))/(T^(2))=(3ka)/(m) or T^(2) prop (1)/(a) or T prop (1)/(sqrt(a))`