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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?

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Solution

## (a) m→ mass per unit of length of stringconsider an element at distance ‘x’ from lower end.Here wt acting down ward =(mx)g= Tension in the string of upper partVelocity of transverse vibration =v=√T/m=√(mgx/m)=√(gx)(b) For small displacement dx,dt=dx/√(gx)Total time T=∫L0dx/√gx=√(4L/g)(c) Suppose after time ‘t’ from start the pulse meet the particle at distance y from lower end.t=∫y0dx/√gx=√(4y/g)∴ Distance travelled by the particle in this time is (L–y)∴S−ut+1/2gt2⇒L−y(1/2)g×{√(4y/g)2}{u=0}⇒L−y=2y⇒3y=L⇒y=L/3. So, the particle meet at distance L/3 from lower end.

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