Question

A circular loop of string rotates about its axis on a frictionless horizontal place at a uniform rate so that the tangential speed of any particle of the string is ν. If a small transverse disturbance is produced at a point of the loop, with what speed (relative to the string) will this disturbance travel on the string?

Answer 1

Speed of the Disturbance is V

Explanation:

m = mass per unit length of the string  

R = Radius of the loop  

$\omega$ = Angular Velocity, V = Linear Velocity of the String  

Consider one half of the string as shown in figure.  

The half loop experiences a centrifugal force at every point, away from the center, which is balanced by tension 2T.  

Consider an element of angular part $d\theta$ at angle $\phi.$  

• Consider another element symmetric to this centrifugal force experienced by the element  

= $mRd(\theta)\omega2R$.

(…Length of element = $Rd\theta, mass = mRd\theta)$

• Resolving into rectangular components net force on the two symmetric elements,

DF = $2mR^2d\theta \omega^2 sin\theta$ [Horizontal components cancel each other]

So, total F = $\int_{0}^{\pi/2}2mR^2\omega^2sin\thetad\theta = 2mR^2\omega^2[-cos\theta]$

$2mR^2\omega^2$

Again, $2T = 2mR^2\omega^2$

T = $mR^2\omega^2$

Velocity of transverse vibration V = $\sqrt T/m = \omega R = V$

So, The speed of the disturbance will be V.