A uniform circular hoop of string is rotating clockwise in the absence of gravity. The tangential speed is `v_(0)`. Find the speed of the wave travelling on this string.

Explanation

Let `T` be the tension in the string. Consider a small circular element `AB` of the string of length,
`Deltal = R (2theta)` (R =radius of hoop)
The components of tension `T cos theta` are equal and opposite and thus cancel out. The cpmponents towards centre `C` (i.e. `T sin theta`) provides the necessary force to elemment `AB`.
`:. 2T sin theta =(mv_(0)^(2))/(R)` ...(i)
Here, `m = muDeltal = 2muRtheta` `(mu=("mass")/("length"))`
As `theta` is small, `sin theta ~~ theta`
Substituting in Eq. (i), we get
`2T theta = (2muRtheta v_(0)^(2))/(R)`
or `(T)/(mu) = v_(0)^(2)`
or `sqrt((T)/(mu)) = v_(0)` ...(ii)
Speed of wavw travelling on this string,
`v =sqrt((T)/(mu)) = v_(0)` [from Eq. `(ii)`]
i.e. the velocity of the transverse wave along the hoop of string is the same as the velocity of rotation of the hoop, viz. `v_(0)`.