Question

A bead of mass $m$ can slide on a frictionless wire as shown in figure. Because of the given shape of the wire, near $P$, the bottom point, it can be approximated as parabola. $\mathrm{Near}P$, the potential energy of the bead is given $U=c{x}^2$ where $c$ is a constant and $x$ is measured from $P$. The bead, if displaced slightly from point $P$ will oscillate about $P$. The period of oscillation is

• A. $2\pi \sqrt{c/m}$
• B. $2\pi \sqrt{m/2c}$
• C. $2\pi \sqrt{m/c}$
• D. $2\pi \sqrt{2c/m}$

Answer 1

Answer: (B)

$U=C{x}^2$
$F=-\frac{dU}{dx}\Rightarrow ma=-2cx$
$\Rightarrow a=-\frac{2c}{m}x$
$\Rightarrow \omega =\sqrt{\frac{2c}{m}}$
$\Rightarrow T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{2c}}$