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=9xx10^(-3)kg//m^(2))`

Explanation

Velocity of transverse wave in a string,
`v=(dx)/(dt) = sqrt((T)/(m))= sqrt((T)/(m_(0)+alpha x))`
`:.sqrt(m_(0)+alpha x) dx= sqrt(T)dt`
Integrating within proper limits, `int_(0)^(L) sqrt(m_(0)alpha x)dx= sqrt(T) int_(0)^(t)dt`
`[(2(m_(0)+ax)^(3//2))/(3alpha)]_(0)^(L)= sqrt(T)t`
or `(2)/(3alpha) [(m_(0)+alphaL)^(3//2)-m_(0)^(3//2)]= sqrt(T)t`
`:.t=(2)/(3alpha sqrt(T))[(m_(0)+alphaL)^(3//2)-m_(0)^(3//2)]=(2)/(3xx9xx10^(-3)xx10) [(10^(-2) +9xx10^(-3)xx10)^(3//2) -(10^(-2))^(3//2)]`
`=(2)/(27xx10^(-2)) [ (10^(-2) +9xx10^(-2))^(3//2) -10^(-3)]=2.335sec`