Question

A U-tube having unequal arm-lengths has water in it. A tuning fork of frequency 440 Hz can set up the air in the shorter arm in its fundamental mode of vibration and the same tuning fork can set up the air in the longer arm in its first overtone vibration. Find the length of the air columns. Neglect any end effect and assume that the speed of sound in air = 330 m s⁻¹.

Answer 1

Given:
Speed of sound in air v = 330 ms⁻¹
Frequency of the tuning fork f = 440 Hz

For the shorter arm:
Let the length of the shorter arm of the tube be L_{1 .}
Frequency of fundamental mode is given by:
$f=\frac{v}{4L_1}$

On substituting the respective values, we get:
$$\begin{gathered} 440=\frac{330}{4L_1} \\ \Rightarrow L_1=\frac{330}{440\times 4}=0.1875\mathrm{m}=18.8\mathrm{cm} \end{gathered}$$

For the longer arm:
Let the length of the longer arm of the tube be L_{2 .}
Frequency of the first overtone f = 440 Hz
Frequency of the first overtone is given by:
$f=\frac{3v}{4L_2}$

On substituting the respective values, we get:

$$\begin{gathered} 440=\frac{3\times 330}{4L_2} \\ \Rightarrow L_2=\frac{3\times 330}{440\times 4}=0.563\mathrm{m}=56.3\mathrm{cm} \end{gathered}$$