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Question
A 200 Hz wave with amplitude 1 mm travels on a long string of linear mass density 6 g m−1 kept under a tension of 60 N. (a) Find the average power transmitted across a given point on the string. (b) Find the total energy associated with the wave in a 2⋅0 m long portion of the string.
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Solution
Given,
Frequency of the wave, f = 200 Hz
Amplitude, A = 1 mm = 10−3 m
Linear mass density, m = 6 gm−3
Applied tension, T = 60 N
Now,
Let the velocity of the wave be v.
Thus, we have:
\[v = \sqrt{\left( \frac{T}{m} \right)} = \sqrt{\frac{\left( 60 \right)}{\left( 6 \times {10}^{- 3} \right)}}\]
\[ = {10}^2 = 100 m/s\]
(a) Average power is given as
\[P_{average} = 2 \pi^2 m\nu A^2 f^2 \]
\[= 2 \times \left( 3 . 14 \right)^2 \times \left( 6 \times {10}^{- 3} \right) \times 100 \times \left( {10}^{- 3} \right) \times {200}^2 \]
\[ = 473 \times {10}^{- 3} = 0 . 47 W\]
(b) Length of the string = 2 m
Time required to cover this distance:
\[t = \frac{2}{100} = 0 . 02 s\]
\[Energy = Power \times t\]
\[ = 0 . 47 \times 0 . 02\]
\[ = 9 . 4 \times {10}^{- 3} J = 9 . 4 mJ\]
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