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A steel wire has a length of $90 \mathrm{~cm}$ which is under a constant tension of $100 \mathrm{~N}$. The speed of the transverse waves that can be produced in the wire will be (take, the mass of the steel wire to be $6 \times 10^{-3} \mathrm{~kg}$ )
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Verified Answer
The correct answer is:
$50 \sqrt{6} \mathrm{~m} / \mathrm{s}$
Given, tension is the wire, $T=100 \mathrm{~N}$
Mass of steel wire, $m=6 \times 10^{-3} \mathrm{~kg}$
Iength, $l=90 \mathrm{~cm}=0.9 \mathrm{~m}$
Mass per unit length of steel wire,
$$
\begin{aligned}
\mu &=\frac{m}{l} \\
&=\frac{6 \times 10^{-3}}{0.9} \\
&=\frac{2}{3} \times 10^{-2} \mathrm{~kg}-\mathrm{m}^{-1}
\end{aligned}
$$
Speed of transverse wave,
$$
\begin{aligned}
v &=\sqrt{\frac{T}{\mu}}=\sqrt{\frac{100}{\left(\begin{array}{c}
2 \\
3
\end{array}\right) \times 10^{-2}}} \\
&=\frac{300}{\sqrt{6}}=50 \sqrt{6} \mathrm{~m} / \mathrm{s}
\end{aligned}
$$
Mass of steel wire, $m=6 \times 10^{-3} \mathrm{~kg}$
Iength, $l=90 \mathrm{~cm}=0.9 \mathrm{~m}$
Mass per unit length of steel wire,
$$
\begin{aligned}
\mu &=\frac{m}{l} \\
&=\frac{6 \times 10^{-3}}{0.9} \\
&=\frac{2}{3} \times 10^{-2} \mathrm{~kg}-\mathrm{m}^{-1}
\end{aligned}
$$
Speed of transverse wave,
$$
\begin{aligned}
v &=\sqrt{\frac{T}{\mu}}=\sqrt{\frac{100}{\left(\begin{array}{c}
2 \\
3
\end{array}\right) \times 10^{-2}}} \\
&=\frac{300}{\sqrt{6}}=50 \sqrt{6} \mathrm{~m} / \mathrm{s}
\end{aligned}
$$
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