Question

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

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

Answer 1

Answer: (B)

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

The force can be given by $F=-\frac{dU}{dx}$
$F=-U_{0}dsindx$
Since dx is small, $F=-U_0{d}^{2}x$
or, $a=-\frac{U_0{d}^2}{M}x$
Comparing with above equation, $\omega =\sqrt{\frac{U_0{d}^2}{M}}$
Therefore, $T=2\pi \sqrt{\frac{M}{U_0{d}^2}}$
Option B is correct.