A non-uniform wire of length `l` and mass `M` has a variable linear mass density given by `mu = kx`, where `x` is distance from one end of wire and `k` is a constant. Find the time taken by a pulse starting at one end to reach the other end when the tension in the wire is `T`.

Answer

B, C

Explanation

`M = int_(0)^(L)kx dx = (kL^(2))/(2)`
`:. k = (2M)/(L^(2)`
`v_(x) = sqrt((T)/(mu_(x))) = sqrt((T)/(kx))= sqrt((T)/((2M//L^(2)) x))`
`=(Lsqrt((T)/(2M)))x^(-1//2) = (dx)/(dt)`
`:. int_(0)^(t) dt = (1)/(L) sqrt((2M)/(T))int_(0)^(L)x^(1//2) dx`
or `t=((2)/(3L)sqrt((2M)/(T)))L^(3//2)`
`=(2)/(3) sqrt((2ML)/(T)`