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= 1 m.`

Explanation

Due to rotation, each part of the ring experinces an outward force (centrifugal force). Because of this force, the ring will ruputre.
Let us consider small part of the ring, which subtendd an angle `theta` at the centre.
Mass fo the element `(dm) = delta dV = delta (AR theta)`
`therefore 2T sin theta//2 = (dm) omega^(2) R`
When `theta` is small `sin theta//2 rarr theta//2`
`therefore 2T xx theta//2 = (delta AR theta) omega^(2) R rArr T = delta A omega^(2) R^(2)`
The stress at any selection any section of the ring
`f = (T)/(A) = (delta A omega^(2) R^(2))/(A) = delta omega^(2) R^(2)`.
Rupture takes place when `f = sigma`
`therfore sigma = delta omega^(2) R^(2) rArr = sqrt((sigma)/(delta R^(2)))` and
`n = (1)/(2pi) sqrt((sigma)/(delta R^(2))) = (1)/(2pi R) sqrt((sigma)/(delta))`