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 $\text{U}\left(x\right)=α \, x^{4}$ , where $\alpha $ is a positive constant. The x-coordinate of mass where potential energy is one-third the kinetic energy of the particle 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

Answer: (B)

The energy of oscillation is $\textit{E}=\alpha \textit{A}^{4}$
$\therefore $ the kinetic energy of mass at x = x is
$\text{K}=\text{E}-\text{U}=\alpha \left(\left(\textit{A}\right)^{4} - \left(\textit{x}\right)^{4}\right)$
Given, K = 3U
$\therefore \alpha \left(\left(\textit{A}\right)^{4} - \left(\textit{x}\right)^{4}\right)=3\alpha \left(\textit{x}\right)^{4}$
Or $\textit{x}=\pm\frac{\textit{A}}{\sqrt{2}}$