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 in the string ???
(HCV PG 325 q- 25)...
detald ans shall be appreciated....
1ITS IS EITHER 0 OR 2V DEPENDING ON THE DIRECTIONS OF ROTATION OF THE DISTURBANCE W.R.T THE ANGULAR VELOCITY OF OF THE ROPE. DERIVATION IS THE SAME AS FINDING VELOCITY OF WAVE IN A TAUT STRING.
1
Considering a semicircular portion
mass per unit length = M2Î R [ where R is Radius ]
considering an element mass dm
dm = M2ΠR Rdθ = M2Πdθ
centrifugal force = dmv2R
net force
2T-\int_{0}^{\pi/2 }{2.\frac{v^{2}}{R}sin\theta \frac{m}{2\pi }d\theta }=0
\Rightarrow T = \frac{mv^{2}}{2\pi R}
V = \sqrt{\frac{T}{m}}
= \sqrt{\frac{mv^{2}}{2\pi R}X\frac{2\pi R}{M}}=v