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.

#### 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.