First slide

Simple harmonic motion

Question

A particle of mass m is moving in a potential well, for which the potential energy is given by $U (x) = U_{0} (1 - \cos a x)$ where $U_{0}$ and a are constants. Then for small oscillations, time period is

Moderate

A

$2 π \sqrt{\frac{m}{a U_{0}}}$
B

$2 π \sqrt{\frac{m}{a^{2} U_{0}}}$
C

$2 π \sqrt{\frac{m}{3 a^{2} U_{0}}}$
D

$2 π \sqrt{\frac{2 m}{3 a^{2} U_{0}}}$

Solution

$\begin{array}{l} U (x) = U_{0} (1 - cosax) \\ F = - \frac{dU}{dx} = - {aU}_{0} \sin ax \\ For small oscillations, \sin a x \approx a x \\ \Rightarrow {ma}^{'} = - a^{2} U_{0} x \Rightarrow a^{'} = - \frac{a^{2} U_{0}}{m} x . Since, a^{'} α - x, oscillations are simple harmonic. \\ Comparing with the relation a = - ω^{2} x with a^{'} = - \frac{a^{2} U_{0}}{m} x \\ We get ω^{2} = \frac{a^{2} U_{0}}{m} . Here ω = \frac{2 π}{T} \\ Therefore \frac{2 π}{T} = \sqrt{\frac{a^{2} U_{0}}{m}} \\ Hence, T = 2 π \sqrt{\frac{m}{a^{2} U_{0}}} . \end{array}$

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