A circular loop of string is rotating about its axis on a...

Consider a small element of the string subtending an angle $d\theta$ at the center of the circular loop.

Let the radius of the loop be $R$ and the mass per unit length of the string be $\mu$.

The mass of this small element is $dm = \mu R d\theta$.

The element is rotating in a circle of radius $R$ with a constant tangential speed $v$. The centripetal force required for this circular motion is provided by the inward component of the tension $T$ at the two ends of the element.

The net inward force is $2T \sin\left(\dfrac{d\theta}{2}\right) \approx T d\theta$ since $d\theta$ is very small.

Equating this to the required centripetal force:

$T d\theta = \dfrac{(dm) v^2}{R}$

$T d\theta = \dfrac{(\mu R d\theta) v^2}{R}$

$T = \mu v^2$

The speed of a transverse wave propagating along a string under tension $T$ is given by:

$c = \sqrt{\dfrac{T}{\mu}}$

Substituting $T = \mu v^2$, we get:

$c = \sqrt{\dfrac{\mu v^2}{\mu}} = v$

Thus, the speed of the disturbance relative to the string is $v$.