Question

A ring of radius $r$ made of wire of density $\rho$ is rotated about a stationary vertical axis passing through its centre and perpendicular to the plane of the ring as shown in the figure. Determine the angular velocity (in rad/s) of ring at which the ring breaks. The wire breaks at tensile stress $\sigma$. Ignore gravity. Take $\sigma /\rho =4$ and $r=1m$.

Answer 1

$2T\sin \frac{△\theta }{2}=dm{\omega }^{2}r$
$2T\left(\frac{△\theta }{2}\right)=\rho \times A\times r^2{\omega }^2$
$\sigma =\frac{T}{A}=\rho r^2{\omega }^2$
$\therefore \omega =\frac{1}{r}\sqrt{\frac{\sigma }{\rho }}=2rad/s$.