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  • 32. A body of mass m is situated in a potential field U(x) = U_0(1-cos αx ) when U_0 and α are constants. Find the time period of small oscillations.

A body of mass m is situated in a potential field U(x) = U_{0}(1- \cos \alpha x ) when U_{0} and \alpha are constants. Find the time period of small oscillations.

Answers (1)

Here, dW = F.dx

So, if W = U, dU = F.dx

Or F = -\frac{dUx}{dx}         ….. (since, restoring force here is opposite to displacement)

F = -\frac{d}{dx} [U_{0} (1 - \cos \alpha x) = - \frac{d}{dx} [U_{0} + U_{0} \cos \alpha x]

F = -\alpha U_{0} \sin \alpha x

Now, \alpha x is small for SHM, So sin \alpha x will become \alpha x   …… (i)

Thus, F = -\alpha U_{0} \alpha x

 Now, since, U_{0} & \alpha are constants,

F \propto -x

Thus, the motion will be SHM.

From (ii)

k = \alpha ^{2}U_{0}

m \omega ^{2}= \alpha ^{2}U_{0}

Thus, \omega ^{2} = \alpha ^{2} .\frac{U_{0}}{m}

\left ( \frac{2\pi}{T} \right )^{2} = \alpha ^{2} .\frac{U_{0}}{m}

T = \frac{2\pi }{\alpha }\sqrt{\frac{m}{U_{0}}}

Thus, considering (i) time period is valid for the small-angle \alpha x.

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