Question

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

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

Answer 1

Answer: (C)

Here, $U=4(1-\cos 2 x) \,J$
$\therefore F=-\frac{dU}{dx}=-8 \,sin \,2 x$
Acceleration, $a=\frac{F}{m}=-8\, sin \,2 x$
$(\because m=1 \,kg )$
For small oscillations, $\sin 2 x \approx 2 x$
$ \therefore a=-16 x\,\,\,\,\,\,\dots(i)$
Since $a \propto-x$, the oscillations are SHM in nature. In SHM, $a=-\omega^{2} x\,\,\,\,\,\,\dots(ii)$
Comparing (i) and (ii), we get
$\omega^{2}=16$ or $\omega=\sqrt{16}$
$\therefore $ Time period, $T=\frac{2 \pi}{\omega}=\frac{2 \pi}{\sqrt{16}}=\frac{\pi}{2} s$