A wave pulse starts propagating in +x-direction along a non-uniform wire of length 10 m with mass per unit length given `m = m_(0) + ax` and under a tension of 100 N. Find the time taken by the pulse to travel from the lighter end (x=0) to the heavier end. `(m_(0) = 10^(-2) kg//m` and `alpha = 9 xx 10^(-3) kg//m^(2))`

Explanation

Velocity of transverse wave in a string,
`v = (dx)/(dt) = sqrt(T/m) = sqrt(T/(m_(0) + alphax))`
Integrating within proper limits, `int_(0)^(L) sqrt(m_(0) + alpha x) dx = sqrt(T) int_(0)^(t) dt`
`[(2(m_(0) + alphax)^(3//2))/(3 alpha)]_(0)^(L) = sqrt(T) t`
or `therefore t =2/(3alphasqrt(T)) [(m_(0) + alphaL)^(3//2) - m_(0)^(3//2)] = 2/(3 xx 9 xx 10^(-3) xx 10) [(10^(-2) + 9 xx 10^(-3) xx 10)^(3//2) -(10^(-2))^(3//2)] = 2.335` sec