A heavy but uniform rope of lenth 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 teh celling ? (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

A, B, C, D

Explanation

`(a) m rarr` mass per unit length of string. Consider an element at distance 'x' form lower end. Here wt acting down ward = (mx)g = Tension in the string of upper part velocity of transverse vibration =
`upsilon = sqrt((T)/(m))`
`= sqrt((mgx)/(m)) = sqrt(gx)`
(b) For small displacemetn, `dt, (dx)/(sqrt((gx)))`
Total time, `T = = int_0^L (dx)/(sqrt(gx)) = sqrt((4L)/(g))`
(c ) Suppose after time 't' from start the pulse meet the particle at distance 'y' from lower end y
`t = int_0^y (dx)/(sqrt(gx))`
`=sqrt((4y)/(g))`
:. Distance travelled by the particle in this time is (L - y)..
`:. S = ut +(1)/(2)gy^2`
`rArr L- y = ((1)/(2))gxx{(sqrt((4y)/(g))^2}`
`rArr L - y = 2y`
`rArr 3y = L`
`rArr y = (L)/(3)`
So the particle meet at distance `(L)/(3)` from lower end.