First slide

Simple harmonic motion

Question

A particle of mass m is moving in a potential well, for which the potential energy is given by $U (x) = U_{0} (1 - \sin b x)$ where $U_{0}$ and a are constants. Then for small oscillations

Moderate

A

time period, $T = 2 π \sqrt{\frac{m}{b U_{0}}}$
B

speed of the particle is maximum at x = b
C

amplitude of oscillations is $\frac{π}{b}$
D

time period, $T = 2 π \sqrt{\frac{m}{b^{2} U_{0}}}$

Solution

$\begin{array}{l} U (x) = U_{0} (1 - \sin bx) \\ F = - \frac{dU}{dx} = - {aU}_{0} \sin bx \\ For small oscillations, \sin b x \approx b x \end{array}$
$\Rightarrow m a' = - a^{2} U_{0} x \Rightarrow a' = - \frac{a^{2} U_{0}}{m} x$ . Since, $a' α - x$ , oscillations are simple harmonic.
Hence, $T = 2 π \sqrt{\frac{m}{b^{2} U_{0}}}$ .
Speed is maximum when a’ = 0. It happens at x = 0.
$\begin{array}{l} \frac{dv}{dt} = - a^{2} U_{0} x \\ \frac{dv}{dx} \frac{dx}{dt} = - a^{2} U_{0} x \\ vdv = - a^{2} U_{0} xdx \\ \int_{\max}^{0} vdv = - a^{2} U_{0} \int_{0}^{A} xdx \end{array}$
Since, $v_{\max}$ is unknown, A cannot be found.

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