A point particle is acted upon by a restoring force - k x 3 . The time period of oscillation is T when the amplitude is A . The time period for an amplitude 2   A will be:

Step-by-step solution

Given F = - k x 3 - d U d x = - k x 3 ⇒ U = 1 4 k x 4 ∴ Energy of oscillations will be E = 1 2 m v 2 + U = 1 2   m d x d t 2 + 1 4 k x 4 … 1 If we pull d x d t = 0 in above equation, we will get amplitude as A = 4 E k 4 … 2 Also on rearranging equation ( 1 ) , we get d t = ± d x m 2 E 1 - k 4 E x 4 - 1 / 2 Now, use A = 4 E k 4 , to reduce above equation as d t = ± d x 2   m k A - 2 1 - x A 4 - 1 / 2 The time period can be found by integrating above equation. T = 4 ∫