A wire is stretched between two rigid supports vibrates in its fundamental mode with a frequency of $50\,\, Hz$ . The mass of the wire is $30\,\, g$ and its linear density is $4\, \times \, 10^{-2}\,\, kg/m$ . The speed of the transverse wave at the string is ...... $ms^{-1}$
Medium
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Here, mass of the wire, $M=30 \mathrm{\,g}=30 \times 10^{-3} \mathrm{\,kg}$
Mass per unit length of the wire.
$\mu=4 \times 10^{-2} \mathrm{\,kg} \mathrm{m}^{-1}$
$\therefore $ length of the wire, $L=\frac{M}{\mu}$
$=\frac{30 \times 10^{-3} \mathrm{\,kg}}{4 \times 10^{-2} \mathrm{\,kgm}^{-1}}=0.75 \mathrm{\,m}$
For the fundamental mode $\frac{\lambda}{2}=L$
$\Rightarrow \lambda=2 \mathrm{L}=2 \times 0.75=1.5 \mathrm{\,m}$
Speed of the transverse wave.
$\mathrm{v}=\mathrm{n} \lambda=\left(50 \mathrm{\,s}^{-1}\right)(1.5 \mathrm{\,m})=75 \mathrm{\,ms}^{-1}$
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