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Alpa Shenoy
Asked: 2022-11-08T23:01:45+05:30 In:

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`.

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`.
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  1. Correct Answer – B::C
    `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)`

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