A circular ring of radius `R` and mass `m` made of a uniform wire of cross sectional area `A` is rotated about a stationary vertical axis passing throgh its centre and perpendicular to the plane of the ring. If the breaking stress of the material of the ring is `sigma_(b)`, then determine the maximum angular speed `omega_("max")` at which the ring may be rotated without failure.

Explanation

To determine the maximum angular speed \(\omega_{\text{max}}\) at which a circular ring of radius \(R\) and mass \(m\) can be rotated without failure, we can follow these steps:

### Step 1: Understand the Forces Acting on the Ring
When the ring rotates, it experiences a centripetal force that is provided by the tension in the wire. The tension in the wire must counteract the centripetal force required to keep the mass of the ring moving in a circular path.

### Step 2: Calculate the Centripetal Force
For a small segment of the ring, the mass \(dm\) can be expressed in terms of the total mass \(m\) and the angle \(d\theta\) as follows:
\[
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