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A steel wire has a length of $12 \mathrm{~m}$ and a mass of $2.1$ kg. What should be the tension in the wire so that the speed of a transverse wave on the wire equals the speed of sound in dry air at $20^{\circ} \mathrm{C}=343 \mathrm{~m} / \mathrm{s}$.
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Verified Answer
Given, $l=12 \mathrm{~m}, M=2.1 \mathrm{~kg}, T=? v=343 \mathrm{~m} / \mathrm{s}$;
$$
\begin{aligned}
\text { Mass/per unit length }(\mathrm{m}) &=\frac{M}{l}=\frac{2.1}{12} \\
&=0.175 \mathrm{~kg} / \mathrm{m}
\end{aligned}
$$
$$
\begin{aligned}
\because \quad v &=\sqrt{\frac{T}{m}} \\
\Rightarrow T &=v^2 \times m=(343)^2 \times 0.175 \\
&=2.06 \times 10^4 \mathrm{~N}
\end{aligned}
$$
$$
\begin{aligned}
\text { Mass/per unit length }(\mathrm{m}) &=\frac{M}{l}=\frac{2.1}{12} \\
&=0.175 \mathrm{~kg} / \mathrm{m}
\end{aligned}
$$
$$
\begin{aligned}
\because \quad v &=\sqrt{\frac{T}{m}} \\
\Rightarrow T &=v^2 \times m=(343)^2 \times 0.175 \\
&=2.06 \times 10^4 \mathrm{~N}
\end{aligned}
$$
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