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

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Correct answer: A. 1 / a

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 θ d x=a θ d θ ∫_0^ π / 2 a θ d θ a^3-(a θ)^3 = (2 k)/(m) (T)/(4)

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