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Question
Consider the situation shown in the figure.The wire which has a mass of 4.00 g oscillates in its second harmonic and sets the air column in the tube into vibrations in its fundamental mode. Assuming that the speed of sound in air is 340 m s−1, find the tension in the wire.
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Solution
Given:
Speed of sound in air v = 340 ms−1
Length of the wire l = 40 cm = 0.4 m
Mass of the wire M = 4 g
Mass per unit length of wire \[\left( m \right)\] is given by :
\[m = \frac{\text { Mass }}{\text { Unit length }} = {10}^{- 2} \text { kg/m }\]
\[n_0\]= frequency of the tuning fork
T = tension of the string
Fundamental frequency : \[n_0 = \frac{1}{2L}\sqrt{\frac{T}{m}}\]
For second harmonic,
\[n_1 = 2 n_0\] :
\[n_1 = \frac{2}{2L}\sqrt{\frac{T}{m}} . . . . . \left( i \right)\]
\[n_1 = 2 n_0 = \frac{340}{4} \times 1 = 85 \text { Hz }\]
On substituting the respective values in equation (i), we get :
\[85 = \frac{2}{2 \times 0 . 4}\sqrt{\frac{T}{{10}^{- 2}}}\]
\[ \Rightarrow T = (85 )^2 \times (0 . 4 )^2 \times {10}^{- 2} \]
\[ = 11 . 6 \text { Newton }\]
Hence, the tension in the wire is 11.6 N.
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