Question

A uniform rope of length $L$ and mass $m$ is held at one end and whirled in a horizontal circle with angular velocity $\omega$. Ignore gravity. The time required for a transverse wave to travel from one end of the rope to the other is $\frac{\pi }{\sqrt{x}\omega }$. Find $x$.

Answer 1

Making FBD of a portion x from free end, we get, $T=\frac{m}{L}x{\omega }^2(L-\frac{x}{2})$

Since $v=\frac{dx}{dt}=\sqrt{\frac{T}{\mu }}=\sqrt{x{\omega }^2(L-\frac{x}{2})}$,

Integrating this equation form 0 to L, we get $T=\frac{\pi }{\sqrt{2}\omega }$.