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 1√aDimensions 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] = (M1L2T−2] and (x] = (M0L1T0]Therefore,(k] = (M1L−1T−2]Let T ∝ (m]a(k]b(a]c i.e T ∝ (M]a+b(L]−b+c(T]−2b ∝ (M]0(L]0(T]1⇒a+b =0 −b+c=0 −2b = 1Solving the equations, we get c = −12Therefore, T ∝ 1√a

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