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Question

A sinusoidal transverse wave travels on a string of length 4 m and mass 4 g. The wave speed is 20 m/s and the wavelength is 0.2 m. If the wave is to have an average power of 50 W, then

A
tension induced in the string is 0.4 N.
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B
frequency of wave will be 100 Hz.
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C
linear mass density of the given string must be 103 kg/m.
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D
amplitude of the wave will be 11.25 cm
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Solution

The correct option is D amplitude of the wave will be 11.25 cm
Given data,
Length (l)=4 m and wave speed (v)=20 m/s
Mass of string (m)=4×103 kg
Wavelength (λ)=0.2 m
Linear mass density μ=ml=4×1034=103 kg/m
Let T be the tension in the string.
From formula, v=Tμ(1)
T=μv2=103×20×20=0.4 N
Frequency (f)=vλ=200.2=100 Hz
Let A be the amplitude of the wave.
So, average power
Pav=12ω2A2μv=12ω2A2×Tv2×v
50=12×(2πf)2A2×0.420
A2=50×2×200.4×(2π×100)2=0.01266
A0.1125 m=11.25 cm
Hence, all options are correct.

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