Question

A body of mass M is situated in a potential field $U\left(x\right)=U_0(1-cos\alpha x)$ where $U_0$ and $\alpha$ are constants. The time period of small oscillations is

• A. $2\pi \sqrt{\frac{M}{U_0{\alpha }^2}}$
• B. $2\pi \sqrt{\frac{U_0}{M{\alpha }^2}}$
• C. $2\pi \sqrt{M{U}_0{\alpha }^2}$
• D. $2\pi \sqrt{\frac{U_0{\alpha }^2}{M}}$

Answer 1

Answer: (B)

$2\pi \sqrt{\frac{U_0}{M{\alpha }^2}}$

Given : $U=U_0(1-cos\alpha x)$

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

$F=-\frac{d}{dx}\left[U_0\right(1-cos\alpha x\left)\right]=-U_0\alpha sin\alpha x$

As $\alpha x$ is small, so sin $\alpha x\sim \alpha x$

$\therefore F=-U_0{\alpha }^{2}x$ ...(i)

$F\propto x$ and $-ve$ shows that F is directed towards the mean position, hence the body executes SHM.

For SHM, $F=-kx$

Comparing (i) and (ii), we get, $k=U_0{\alpha }^2$

Time period of oscillation,

$T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{m}{U_0{\alpha }^2}}$