Question

Consider the situation shown in 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:

$m=\frac{\mathrm{Mass}}{\mathrm{Unit}\mathrm{length}}={10}^{-2}\mathrm{kg}/\mathrm{m}$

$n_0$ = frequency of the tuning fork
T = tension of the string

Fundamental frequency:
$n_0=\frac{1}{2L}\sqrt{\frac{T}{m}}$

For second harmonic, $n_1=2n_0$:

$n_1=\frac{2}{2L}\sqrt{\frac{T}{m}}.....\left(\mathrm{i}\right)$

$n_1=2n_0=\frac{340}{4}\times 1=85\mathrm{Hz}$
On substituting the respective values in equation (i), we get:

$$\begin{gathered} 85=\frac{2}{2\times 0.4}\sqrt{\frac{T}{{10}^{-2}}} \\ \Rightarrow T=(85{)}^2\times (0.4{)}^2\times {10}^{-2} \\ =11.6\mathrm{Newton} \end{gathered}$$

Hence, the tension in the wire is 11.6 N.