A uniform rope of mass 0.1 kg and length 2.45 m hangs from a ceiling, (a) Find the speed oftransverse wave in the rope at a point 0.5 m distant from the lower end, (b) Calculate the time taken by a transverse wave to travel the full length of the rope. `(g = 9.8 m//s^(2))`

Explanation

(a) As the string has mass and it is suspended vertically, tension in it will be different at different points. For a point a distance x from the free end, tension will be due to the weight of the string below it. So if M is the mass of string of lengthL, the massof length x for the string will be `(M/L)x`.
`therefore T = [M/Lx]g`

So, `v = sqrt(T/m) = sqrt((Mgx)/(L(M//L))) = sqrt(gx)`......(i)
Here, `x = 0.5 m`
So, `v = sqrt(0.5 xx 9.8) = 2.21 m//s`
(b) From part (a) it is clear that the tension and so the velocity of the wave is different at different points. So if at point x the wave travels as distance dx in time dt,
`v = (dx)/(dt)` or `sqrt(gx) = (dx)/(dt)` [from Eqn.(i)]
or `intdt = int(dt)/sqrt(gx)`, i.e., `t = 1/sqrt(g) int_(0)^(L) x^(-1//2) dx` i.e., `t = 2 sqrt((L/g))`...... (ii)
Hence, `L = 2.45 m` so `t = 2 sqrt(2.45//9.8) = 1` sec