First slide

Simple hormonic motion

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. 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

Difficult

A

$2 π \sqrt{\frac{c}{m}}$
B

$2 π \sqrt{\frac{m}{2 c}}$
C

$2 π \sqrt{\frac{m}{c}}$
D

$2 π \sqrt{\frac{2 c}{m}}$

Solution

U = ${Cx}^{2}$
$F = - \frac{dU}{dx} \Rightarrow ma = - 2 cx \Rightarrow a = - \frac{2 c}{m} x$
$\Rightarrow ω = \sqrt{\frac{2 c}{m}} \Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{2 c}}$

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