Question

Two identical simple pendulums each of length ${}^{\prime }{l}^{\prime }$ are connected by a weightless spring as shown. In equilibrium, the pendulum are vertical and the spring is horizontal and under formed. Find the time period of small oscillations of the linked pendulums, when they are deflected from their equilibrium positions through equal displacements in the same vertical plane in the opposite directions and released.

Answer 1

Displace both masses by a distance x as shown.

$sin\theta =\frac{x}{l}\theta =\frac{x}{l}$

as $\theta$ is very small

$x=l\theta$

Net restoring force acting on the particle

$Tcos\theta$=$mg$

force by string in horizontal direction.

$[Tcos\theta -sin\theta ]=mgsin\theta =mg\theta$

$F_{net}=mg\theta +2kx{F}_{net}=\frac{mgx}{L}+2kxa=[\frac{g}{L}+\frac{2k}{m}]xw=\sqrt{\frac{g}{L}+\frac{2k}{m}}T=2\pi \sqrt{\frac{1}{\frac{g}{L}+\frac{2k}{m}}}$