A uniform rope of mass 0.1 kg and length 2.45 m hangs from a ceiling. (a) Find the speed of transverse 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 length L, the mass of length x for the string will be `((M)/(L))x`.
`:.T=[(M)/(L)x]g`

So, `v= sqrt((T)/(m)) = sqrt((Mgx)/(L(M//L)))=sqrt(g x)..........(i)`
Here `x=0.5m`
So, `v=sqrt(0.5xx9.8) =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 pointx the wave travels as distance dx in time dt,
`v=(dx)/(dt) or sqrt(g x) =(dx)/(dt)` [from Eqn. (i)]
or `int dt =int (dx)/(sqrt(gx)), "i.e., "t=(1)/(sqrt(g)) int_(0)^(L)x^(-1//2)dx" i.e., "t=2sqrt((L//g))........(ii)`
Hence L = 2.45m so `t=2sqrt(2.45//9.8)=1sec`