First slide

Simple harmonic motion

Question

A particle of mass m is executing oscillations about the origin on the X-axis with amplitude A. Its potential energy is given as U(x)= $β$ x⁴, where $β$ is a positive constant. The x-coordinate of the particle, where the potential energy is one-third of the kinetic energy, is

Difficult

A

$\pm \frac{A}{2}$
B

$\pm \frac{A}{\sqrt{2}}$
C

$\pm \frac{A}{3}$
D

$\pm \frac{A}{\sqrt{3}}$

Solution

$\begin{array}{l} U = β x^{4} (Given) \\ ∴ U_{max} = β \cdot A^{4} \\ K (x) = U_{max} - U (x) = β A^{4} - β x^{4} = β (A^{4} - x^{4}) \end{array}$
$\begin{array}{l} U (x) = \frac{1}{3} K (x) (Given) \\ ∴ β x^{4} = \frac{1}{3} β (A^{4} - x^{4}) \\ \Rightarrow x = \pm \frac{A}{\sqrt{2}} \end{array}$

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