A particle of mass m is executing oscillation about the origin on 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 proportional to

Step-by-step solution

Given, V=k x^3 Total energy (E)= Maximum potential energy =k a^3 From conservation of energy aligned 1 2 m ( d x d t )^2+k x^2 & =k a^3 1 2 m ( d x d t )^2 & =k (a^3-x^3 ) d x d t & = 2 k m (a^3-x^3 ) d x a^3-x^3 & = 2 k m d t aligned or Integrating ₀^a d x a^3-x^3 = 2 k m ₀^ T / 4 d t or ₀^a d x a^3-x^3 = 2 k m T 4 Let x=a d x=a d ₀^ / 2 a d a^3-(a )^3 = 2 k m T 4