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 the small oscillations)

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$t h e t i m e p e r i o d o f s m a l l o s c i l l a t i o n s i s T = 2 π \sqrt{\frac{m}{a U_{o}}}$

$t h e s p e e d o f t h e p a r t i c l e i s m a x i m u m a t x = 0$

$t h e a m p l i t u d e o f o s c i l l a t i o n s i s \frac{π}{2 a}$

$t h e t i m e p e r i o d o f s m a l l o s c i l l a t i o n s i s T = 2 π \sqrt{\frac{m}{a^{2} U_{o}}}$

answer is B, C, D.

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Detailed Solution

$U = U_{0} (1 - \cos a x)$

$∴ F = - \frac{d U}{d x} = - a U_{0} \sin a x$

For small value of $x, \sin a x ≃ a x$

$so F = - a U_{0} \times a x = a^{2} U_{0} (- x) a c c = (a^{2} U_{0} / m) (- x)$

On comparing with, $a = - ω^{2} x,$ we get

$ω = \sqrt{\frac{a^{2} U_{0}}{m}} and T = 2 π \sqrt{\frac{m}{a^{2} U_{0}}} . At x = \frac{π}{2 a} . U = U_{0} [1 - \cos (a \times \frac{π}{2 a})] = U_{0} S o a m p l i t u d e o f o s c i l l a t i o n, A = \frac{π}{2 a} . E q u i l i b r i u m p o s i t i o n o f t h e p a r t i c l e i s a t x = 0 a n d s p e e d i s m a x i m u m a t t h i s p o s i t i o n .$