A flexible steel cable of total length `L` and mass per unit length `mu` hangs vertically from a support at one end. (a) Show that the speed of a transverse wave down the cable is `v =sqrtg(L - x)`, where `x` is measured from the support. (b) How long will it takes for a wave to travel down the cable?

Answer

B

Explanation

(a) `T_(x) =(m_(L-x)) g =mu(L - x) g`
where, `mu = "mass"` per unit length`
`:. v_(x) =sqrt((T_(x))/(mu)) =sqrt(g(L - x))`
(b) `v_(x) = -(dx)/(dt) = sqrt g(L - x)`
`:. int_(0)^(t) dt = -int_(x = L)^(x = 0) sqrt (g(L - x) dx)`
Solving we get, `t = 2sqrt((L)/(g))`