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A steel wire has a length of $12 \mathrm{~m}$ and a mass of $2.10 \mathrm{~kg}$. What will be the speed of a transverse wave on this wire when a tension of $2.06 \times 10^4 \mathrm{~N}$ is applied?
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Verified Answer
As given that
length of the wire, $l=12 \mathrm{~m}$
Total mass of wire $M=2.10 \mathrm{~kg}$
Tension, $(T)=2.06 \times 10^4 \mathrm{~N}$
Mass per unit length $m=\frac{M}{l}=\left(\frac{2.10}{12}\right)$
Speed of transverse wave, $v=\sqrt{\frac{T}{m}}$
[where $m=$ mass per unit length]
$$
\begin{aligned}
&=\sqrt{\frac{2.06 \times 10^4}{\left(\frac{2.10}{12}\right)}}=\sqrt{\frac{2.06 \times 12 \times 10^4}{2.10}} \\
&=\sqrt{\frac{1236 \times 10^4}{105}}=343 \mathrm{~m} / \mathrm{s} \\
v &=343.0 \mathrm{~m} / \mathrm{s}
\end{aligned}
$$
length of the wire, $l=12 \mathrm{~m}$
Total mass of wire $M=2.10 \mathrm{~kg}$
Tension, $(T)=2.06 \times 10^4 \mathrm{~N}$
Mass per unit length $m=\frac{M}{l}=\left(\frac{2.10}{12}\right)$
Speed of transverse wave, $v=\sqrt{\frac{T}{m}}$
[where $m=$ mass per unit length]
$$
\begin{aligned}
&=\sqrt{\frac{2.06 \times 10^4}{\left(\frac{2.10}{12}\right)}}=\sqrt{\frac{2.06 \times 12 \times 10^4}{2.10}} \\
&=\sqrt{\frac{1236 \times 10^4}{105}}=343 \mathrm{~m} / \mathrm{s} \\
v &=343.0 \mathrm{~m} / \mathrm{s}
\end{aligned}
$$
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