Q: A particle of mass m is executing oscillation about the origin on the x-axis, its potential energy is U(x) = k |x|^{3}, where k is a positive constant. If the amplitude of oscillation is a , then its time period T is

(a) Proportional to 1/√a

(b) Independent of a

(c) Proportional to √a

(d) Proportional to a^{3/2}

Ans: (a)

Sol: $ k = \frac{U}{x^3} = \frac{ML^2T^{-2}}{L^3}= [M L^{-1}T^{-1}]$

Time period may depend upon;

$ T \propto (mass)^x (amplitude)^y (k)^z$

$ T \propto (mass)^x (amplitude)^y (k)^z $

$ [M^0 L^0 T] = [M]^x [L]^y [M L^{-1}T^{-1}]^z$

On equating Powers & solving we get;

y = -1/2

$ T \propto (a)^{-1/2} $