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.

Answer 1

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.