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Question

A 400 Hz wave with amplitude 2 mm travels on a long string of linear mass density 5 g/m kept under a tension of 50 N. Find the average power and total energy associated with the wave in a 2 m long portion of the string.

A
6 W , 1 J
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B
1 W , 6 J
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C
6.32 W , 0.13 J
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D
None of the above
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Solution

The correct option is C 6.32 W , 0.13 J
Given that
f=400 Hz,A=2 mm=2×103 m
μ=5 g/m=5×103 kg/m
T=50 N and length (l)=2 m

Average Power (P)=2π2f2A2μv
=2π2 f2A2Tμ[v=Tμ]
=2π2×(400)2×(2×103)2×50×5×103
=6.32 W

And, Energy density U = 12ρω2A2
Total energy = U×volume
=12×(2πf)2×A2×μ ×l
[μ×l= total mass = volume ×ρ]
=12×(2π×400)2×(2×103)2×5×103×2
=0.1263 J
0.13 J

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