Question

A wave pulse starts propagating in the positive $x-$ direction along a non-uniform wire of length $10\text{m}$ with a mass per unit length given by $m=m_0+\alpha x$ and under a tension of $100\text{N}$. Find the time taken by the pulse in seconds to travel from the lighter end $x=0$ to the heavier end. ($m_0={10}^{-2}\text{kg/m}$ and $\alpha =9\times {10}^{-3}{\text{kg/m}}^2$). Answer upto 2 decimal places.

• A. 0.21
• B. 0.22
• C. 0.23
• D. 0.24

Answer 1

Answer: (A), (B), (C), (D)

The speed of transverse wave is given by
$v=\sqrt{\frac{F}{\mu }}$
Here $F=100\text{N}$ and $\mu =m=m_0+\alpha x$
$v=\sqrt{\frac{100}{(m_0+\alpha x)}}$ or $\frac{dx}{dt}=10\frac{1}{(m_0+\alpha x{)}^{1/2}}$
or $dt=\frac{1}{10}(m_0+\alpha x{)}^{1/2}dx$
$\Rightarrow \underset{0}{\overset{t}{\int }}dt=\frac{1}{10}\underset{0}{\overset{10}{\int }}(m_0+\alpha x{)}^{1/2}dx$
$\Rightarrow t=\frac{1}{10}{\left[\frac{(m_0+\alpha x{)}^{3/2}}{\left(3/2\right)\alpha }\right]}_0^{10}$
$=\frac{1}{15\alpha }[{(m_0+\alpha \times 10)}^{3/2}-m_0^{3/2}]$
$=\frac{1}{15\times 9\times {10}^{-3}}[{({10}^{-2}+9\times {10}^{-2})}^{3/2}-({10}^{-2}{)}^{3/2}]$
$=0.23\text{s}$

Accepted answers: $0.21,0.22,0.23,0.24$