Consider one dimensional motion of a particle of mass m . It has potential energy U = a + b x 2 , where a and b are positive constants. At origin ( x = 0 ) it has initial velocity V 0 . It performs simple harmonic oscillations. The frequency of the simple harmonic motion depends on

Step-by-step solution

Given, Potential energy, U = a + b x 2 Force, F = - d U d x = - d . ( a + b x 2 ) d x = - d d x a + d d x ( b x 2 ) ⇒ F = 0 - b . d d x ( x 2 ) ⇒ F = 0 - b . 2 x = - 2 b x ⇒ a = F m = - 2 b m x .......(i) Compare equation. (i) with the standard relation of displacement in case of simple harmonic motion. i.e., a = F 2 m = - ω 2 x i.e., ω 2 = 2 b m ω = 2 b m i.e., the frequency of the simple harmonic motion depends on b and m alone.