Question

A heavy but uniform rope of length $L$ is suspended from a ceiling. (a) Write the velocity of a transverse wave travelling on the string as a function of the distance from the lower end. (b) If the rope is given a sudden sideways jerk at the bottom, how long will it take for the pulse to reach the ceiling? (c) A particle is dropped from the ceiling at the instant the bottom end is given the jerk. Where will the particle meet the pulse?

Answer 1

(a) $m\to$ mass per unit of length of string

consider an element at distance ‘x’ from lower end.

Here wt acting down ward $=\left(mx\right)g=$ Tension in the string of upper part

Velocity of transverse vibration $=v=\sqrt{T/m}=\sqrt{\left(mgx/m\right)}=\sqrt{\left(gx\right)}$

(b) For small displacement $dx,dt=dx/\sqrt{\left(gx\right)}$

Total time $T={\int }_0^{L}dx/\sqrt{gx}=\sqrt{\left(4L/g\right)}$

(c) Suppose after time ‘t’ from start the pulse meet the particle at distance y from lower end.

$t={\int }_0^{y}dx/\sqrt{gx}=\sqrt{\left(4y/g\right)}$

$\therefore$ Distance travelled by the particle in this time is $(L–y)$

$\therefore S-ut+1/2g{t}^2$

$\Rightarrow L-y\left(1/2\right)g\times \left\{\sqrt{(4y/g{)}^2}\right\}\{u=0\}$

$\Rightarrow L-y=2y\Rightarrow 3y=L$

$\Rightarrow y=L/3$. So, the particle meet at distance $L/3$ from lower end.