Question

A ring of radius $R$ is rotated about its axis with angular speed $w$. Find tension produced in it.

Answer 1

Consider a small element $dx$ of the ring and writing the equilibrium equation,

Mass per unit length of ring$=\frac{M}{2\pi R}$

Mass of the $dx$ element$=dm=dx\frac{M}{2\pi R}$

For Equlibrium,

$2T\sin \theta =dm\times \frac{v^2}{R}$

Since,$\theta =$small $\sin \theta =\theta ,$

$\begin{array}{c}2T\theta =dx\times \frac{\frac{M}{v^2\left(2\pi R\right)}}{R}\\ 2T\left(\frac{dx}{R}\right)=dx\times \frac{\frac{M}{v^2\left(2\pi R\right)}}{R}\end{array}$

Tension in the ring=$T=\frac{M{v}^2}{4\pi R}$