Question

A particle of mass $m$ in a unidirectional potential field have potential energy $U(x)=\alpha+2 \beta x^{2}$, where $\alpha$ and $\beta$ are positive constants. Find its time period of oscillation.

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

Answer 1

Answer: (C)

$U(x)=\alpha+2 \beta x^{2}$
$F=-\frac{d U(x)}{d x}$
$F=-4 \beta x$
$T=2 \pi \sqrt{\frac{m}{k}}$
$T=2 \pi \sqrt{\frac{m}{4 \beta}} \cos [k=\beta]$
$T=\sqrt[\pi]{\frac{m}{\beta}}$