Question

Example 14 A wave pulse starts propagating in the +x
direction along a non-uniform wire of length 10 m with mass
per unit length given by m=mo + ax and under a tension of
100 N. Find the time taken by the pulse to travel the lighter
end (x = 0) to the heavier end. (m, = 10-2 kg/m and a =
9 x 10-kg/m²).​

Answer 1

The velocity of the wave propagating along the wire is given by,

$\sf{\longrightarrow v=\sqrt{\dfrac{T}{m}}}$

or,

$\sf{\longrightarrow \dfrac{dx}{dt}=\sqrt{\dfrac{T}{m_0+ax}}}$

$\sf{\longrightarrow dt=\sqrt{\dfrac{m_0+ax}{T}}\ dx}$

Integrating,

$\displaystyle\sf{\longrightarrow\int\limits_0^tdt=\int\limits_0^{10}\sqrt{\dfrac{m_0+ax}{T}}\ dx}$

$\displaystyle\sf{\longrightarrow t=\dfrac{1}{a\sqrt T}\cdot\dfrac{2}{3}\left[(m_0+ax)^{\frac{3}{2}}\right]_0^{10}}$

Here,

• $\sf{T=10\ N}$

• $\sf{m_0=10^{-2}\ kg\,m^{-1}}$

• $\sf{a=9\times10^{-3}\ kg\,m^{-2}}$

Then,

$\displaystyle\sf{\longrightarrow t=\dfrac{1}{9\times10^{-3}\sqrt{100}}\cdot\dfrac{2}{3}\left[\left(10^{-2}+(9\times10^{-3})x\right)^{\frac{3}{2}}\right]_0^{10}}$

$\displaystyle\sf{\longrightarrow t=\dfrac{200}{27}\left(10^{-\frac{3}{2}}-10^{-3}\right)}$

$\displaystyle\sf{\longrightarrow\underline{\underline{t=0.2268\ s}}}$

Answer 2

$\huge{\underline{\underline{\mathrm{\red{AnswEr}}}}}$

The velocity of the wave propagating along the wire is given by,

$\sf{\longrightarrow v=\sqrt{\dfrac{T}{m}}}$

or,

$\sf{\longrightarrow \dfrac{dx}{dt}=\sqrt{\dfrac{T}{m_0+ax}}}$

$\sf{\longrightarrow dt=\sqrt{\dfrac{m_0+ax}{T}}\ dx}$

Integrating,

$\displaystyle\sf{\longrightarrow\int\limits_0^tdt=\int\limits_0^{10}\sqrt{\dfrac{m_0+ax}{T}}\ dx}$

$\displaystyle\sf{\longrightarrow t=\dfrac{1}{a\sqrt T}\cdot\dfrac{2}{3}\left[(m_0+ax)^{\frac{3}{2}}\right]_0^{10}}$

Here,

$\sf{T=10\ N}$

$\sf{m_0=10^{-2}\ kg\,m^{-1}}$

$\sf{a=9\times10^{-3}\ kg\,m^{-2}}$

Then,

$\displaystyle\sf{\longrightarrow t=\dfrac{1}{9\times10^{-3}\sqrt{100}}\cdot\dfrac{2}{3}\left[\left(10^{-2}+(9\times10^{-3})x\right)^{\frac{3}{2}}\right]_0^{10}}$

$\displaystyle\sf{\longrightarrow t=\dfrac{200}{27}\left(10^{-\frac{3}{2}}-10^{-3}\right)}$

$\displaystyle\sf{\longrightarrow\underline{\underline{t=0.2268\ s}}}$