Question

A particle of mass $m$ is located in a one dimensional potential field where potential energy is given by : $V\left(\right.x\left.\right)=A\left(\right.1-\cos px\left.\right),$ Where $A$ and $p$ are constant. The period of small oscillations of the particle is :-

• A. $2\pi \sqrt{\frac{m}{\left(\right. A p \left.\right)}}$
• B. $2\pi \sqrt{\frac{m}{\left(A p^{2}\right)}}$
• C. $2\pi \sqrt{\frac{m}{\left(\right. A \left.\right)}}$
• D. $\frac{1}{2 \pi }\sqrt{\frac{A p}{m}}$

Answer 1

Answer: (B)

$V=A\left(\right.1-\cos px\left.\right)$
$F=-\frac{d V}{d x}=-Ap \sin px$
For small displacement
$\sin px=px$
$\therefore F=-Ap^{2}x$
$T=2\pi \sqrt{\frac{m}{K}}=2\pi \sqrt{\frac{m}{A p^{2}}}$