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?

Question

Philoid · Accepted Answer

Suppose, m is the mass per unit length of the rope and there is an arbitrary element x unit away from the lower end.

So, downward weight for that element is,

,

This must be equal to the active tension at the top part.

Now, we know that, for transverse wave,

[v=velocity of wave, T=tension and m=mass density]

(A)

(B)

Suppose, it will meet the pulse after y distance.

To get the in between time, we integrate,

Getting the travelled distance,

By computing with the formula, we get,

(C)