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Wave pulse can travel along a tense string like a violin spring. Aseries of experiments showed that the wave velocity $\mathrm{V}$ of a pulse depends on the following quantities, the tension T of the string, the cross-section area A of the string and then as per unit volume $\rho$ of the string. Obtain an expression for $V$ in terms of the $\mathrm{T}, \mathrm{A}$ and $\rho$ using dimensional analysis.
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Verified Answer
The correct answer is:
$\mathrm{V}=\mathrm{k} \sqrt{\frac{\mathrm{T}}{\mathrm{A} \rho}}$
Let $\mathrm{V}=\mathrm{kT}^{\mathrm{a}} \mathrm{A}^{\mathrm{b}} \rho^{\mathrm{c}}$,
$\mathrm{k}=$ dimensional constant
Writing dimension on both we side
$$
\begin{aligned}
\left[\mathrm{LT}^{-1}\right] &=\left[\mathrm{MLT}^{-2}\right]^{a}\left[\mathrm{~L}^{2}\right]^{\mathrm{b}}\left[\mathrm{ML}^{-3}\right]^{\mathrm{c}} \\
&=\left[\mathrm{M}^{\mathrm{a}+\mathrm{c}} \mathrm{L}^{\mathrm{a}+2 \mathrm{~b}-3 \mathrm{c}} \mathrm{T}^{-2 \mathrm{a}}\right]
\end{aligned}
$$
Comparing power on both sides we have
$$
\begin{array}{l}
a+c=0, a+2 b-3 c=1, \quad-2 a=-1 \\
\therefore \quad a=\frac{1}{2}, c=-\frac{1}{2} \Rightarrow b=-\frac{1}{2} \therefore V=k \sqrt{\frac{T}{A \rho}}
\end{array}
$$
$\mathrm{k}=$ dimensional constant
Writing dimension on both we side
$$
\begin{aligned}
\left[\mathrm{LT}^{-1}\right] &=\left[\mathrm{MLT}^{-2}\right]^{a}\left[\mathrm{~L}^{2}\right]^{\mathrm{b}}\left[\mathrm{ML}^{-3}\right]^{\mathrm{c}} \\
&=\left[\mathrm{M}^{\mathrm{a}+\mathrm{c}} \mathrm{L}^{\mathrm{a}+2 \mathrm{~b}-3 \mathrm{c}} \mathrm{T}^{-2 \mathrm{a}}\right]
\end{aligned}
$$
Comparing power on both sides we have
$$
\begin{array}{l}
a+c=0, a+2 b-3 c=1, \quad-2 a=-1 \\
\therefore \quad a=\frac{1}{2}, c=-\frac{1}{2} \Rightarrow b=-\frac{1}{2} \therefore V=k \sqrt{\frac{T}{A \rho}}
\end{array}
$$
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