A circular loop of string rotates about its axis on a frictionless horizontal plane at a uniform rate so that the tangential speed of any particle of the string is v. 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 ?

Explanation

m=mass per unit length of the string.
R=Radius of the loop
w=angular velocity
V=Linear velocity of the string

consider one half of the string as shown in ure.
The hal loop experiences centrifugl force at every point, away from centre, which ils balanced by tension 2T.
Consider an elemennt of angular part `dtheta` at angle `theta.
consider another element symmetric to this.
Cenrifugal force experience by the element `=(mRdtheta)w^2R`
`(......Length of element =Rdtheta, mass =mRdtheta)
Resolving in to rectangular components, net force on the two symmetirc elements,
dF=2mR^2dthetaw^2sintheta`
[Horizontal components cncels each other]
so, total `F=int_0^(pi/2)2mR^2omega^2sithetadtheta`
`=2mR^2omega^2[-costheta]`
`=2mR^2w^2`
`Again 2T=2mr^2w^2`
`rarr T=mR^2w^2`
velocity of transverse vibration
`V=sqrt((T/m))`
`=sqrt((mR^2omega^I2)=omegaR=V`