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 $U\left(x\right)=\alpha x^4$ where $\alpha$ is a positive constant. The x-coordinate of the particle where its potential energy is one third the corresponding kinetic energy is:

• A. $\pm \frac{A}{2}$
• B. $\pm \frac{A}{\sqrt{2}}$
• C. $\pm \frac{A}{3}$
• D. $\pm \frac{A}{\sqrt{3}}$

Answer 1

The correct option is B $\pm \frac{A}{\sqrt{2}}$

Potential energy $a{x}^4$

Given potential energy , $U=\frac{1}{3}$ kinetic energy(k)

$k=3U$

$k=3A{x}^4$

At maximum amplitude A,

Total energy =$a{A}^4$

$U+k=a{A}^4$

$a{x}^4+3a{x}^4=a{A}^4$

$4a{x}^4=a{A}^4$

$x^4=\frac{A^4}{4}$

$x=\pm \frac{A}{\sqrt{2}}$