A particle of mass $m$ is located in a one dimensional potential field where potential energy is given by : $V(x) = A(1 -cos\, px)$ ,

where $A$ and $p$ are constant.

The period of small oscillations of the particle is

Question

A particle of mass $m$ is located in a one dimensional potential field where potential energy is given by : $V(x) = A(1 -cos\, px)$ ,

where $A$ and $p$ are constant.

The period of small oscillations of the particle is

Diffcult

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Solution

$\mathrm{V}=\mathrm{A}(1-\cos \mathrm{px})$

$\mathrm{F}=-\frac{\mathrm{d} \mathrm{V}}{\mathrm{dx}}=-\mathrm{Ap} \sin \mathrm{px}$

for small displacement

$\sin \mathrm{px}=\mathrm{px}$

$\therefore \mathrm{F}=-\mathrm{Ap}^{2} \mathrm{x}$

$\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{K}}}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{Ap}^{2}}}$

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