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Physics Question on Waves

A wire is stretched between two rigid supports vibrates in its fundamental mode with a frequency of 50 Hz50 \,Hz. The mass of the wire is 30 g30 \,g and its linear density is 4×10−2 kg ms−14 \times 10^{-2}\, kg \,m s^{-1}. The speed of the transverse wave at the string is

A

25 m s−125\, m\, s^{-1}

B

50 m s−150 \, m\, s^{-1}

C

75 m s−175 \, m\, s^{-1}

D

100 m s−1100 \, m\, s^{-1}

Answer

75 m s−175 \, m\, s^{-1}

Explanation

Solution

Here, Mass of the wire, M=30 g=30×10−3 kgM=30\,g =30 \times 10^{-3}\, kg Mass per unit length of the wire, μ=4×10−2 kg m−1\mu=4\times10^{-2} \, kg\, m^{-1} ∴\therefore Length of the wire, L=Mμ=30×10−3 kg4×10−2 kg m−1=0.75 mL=\frac{M}{\mu}=\frac{30\times10^{-3}\,kg}{4\times10^{-2}\,kg\,m^{-1}} =0.75\,m For the fundamental mode, λ=2L=2×0.75 m=1.5 m\lambda=2L=2\times0.75\,m=1.5\,m Speed of the transverse wave, v=υλ=(50 s−1)(1.5 m)v=\upsilon\lambda=\left(50\,s^{-1}\right)\left(1.5\,m\right) =75 m s−1=75\,m\,s^{-1}