First slide

Simple hormonic motion

Question

The period of small oscillations of a particle of mass m in a potential energy field given by $U = U_{0} (1 - \cos (α x))$ is

Difficult

A

$2 π \sqrt{\frac{m}{U_{0}}}$
B

$π \sqrt{\frac{m}{α U_{0}}}$
C

$2 π \sqrt{\frac{m}{α U_{0}}}$
D

$\frac{2 π}{α} \sqrt{\frac{m}{U_{0}}}$

Solution

$p o t e n t i a l e n e r g y U = U_{0} (1 - \cos (α x)) U = U_{0} 2 \sin^{2} (\frac{α x}{2}) U = U_{0} 2 {(\frac{α x}{2})}^{2} = U_{0} (\frac{α^{2} x^{2}}{2}) f o r s m a l l a m p l i t u d e o f o s c i l l a t i o n U = \frac{1}{2} m ω^{2} x^{2} = U_{0} (\frac{α^{2} x^{2}}{2}) ω = \sqrt{\frac{U_{0}}{m}} α T = \frac{2 π}{α} \sqrt{\frac{m}{U_{0}}}$

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