Question

The potential energy of a particle of mass $1kg$ in motion along the $x$-axis is given by $U=4(1-\cos 2x)J$. Here $x$ is in meter. The period of small oscillations (in sec) is _______.

• A. $2\pi$
• B. $\pi$
• C. $\frac{\pi }{2}$
• D. $\sqrt{2\pi }$

Answer 1

The correct option is C $\frac{\pi }{2}$

$M=1kg$

$U=4(1-\cos 2x)$

$F=-\frac{du}{dx}$

$F=-4[0-(-\sin 2x\left)\right(2\left)\right]$

$F=-8\sin 2xN$

$a=-8\sin 2xm/{\sec }^2[\because m=1kg]$

$\Rightarrow a\simeq -8\left(2x\right)$

$a=-16x$ [For small $x,\sin x\approx x$]

$\Rightarrow {\omega }^2=4^2$

$\omega =4$ rad /ac

$T=\frac{2\pi }{w}=\frac{\pi }{2}\sec$