A 400Hz wave with amplitude 2mm travels on a long string of linear mass density 5g/m kept under a tension of 50N. Find the average power and total energy associated with the wave in a 2m long portion of the string.
A
6W , 1J
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B
1W , 6J
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C
6.32W , 0.13J
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D
None of the above
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Solution
The correct option is C6.32W , 0.13J Given that f=400Hz,A=2mm=2×10−3m μ=5g/m=5×10−3kg/m T=50N and length (l)=2m
Average Power (P)=2π2f2A2μv =2π2f2A2√Tμ[∵v=√Tμ] =2π2×(400)2×(2×10−3)2×√50×5×10−3 =6.32W
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×(2×10−3)2×5×10−3×2 =0.1263J ≈0.13J