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Question

A particle of mass m is executing oscillations 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 1a
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B
Independent of a
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C
Proportional to a
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D
Proportional to a3/2
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Solution

The correct option is A Proportional to 1a
Dimensions of time period (T] = (M0L0T1]Time period T can depend on the three variablesm, k and a.(m] = (M1L0T0](a] = (M0L1T0]We are given U = kx3.(U] = (M1L2T2] and (x] = (M0L1T0]Therefore,(k] = (M1L1T2]Let T (m]a(k]b(a]c i.e T (M]a+b(L]b+c(T]2b (M]0(L]0(T]1a+b =0 b+c=0 2b = 1Solving the equations, we get c = 12Therefore, T 1a

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