Question 328: Waves

Question

A sonometer wire is stretched by hanging a metal bob, the fundamental frequency of the wire is ' $n_1$ '. When the bob is completely immersed in water, the frequency of vibration of wire becomes ' $n_2$ '. The relative density of the metal of the bob is

Accepted Answer

The fundamental frequency of a wire under tension is given by the equation:
$$ n = \frac{1}{2L} \sqrt{\frac{T}{\mu}} $$
where:
$ n $ is the frequency,
$ L $ is the length of the wire,
$ T $ is the tension in the wire,
$ \mu $ is the linear mass density of the wire.
When the bob is not immersed in water, the tension $ T $ is due to the weight of the bob, $ Mg $. Therefore, the initial frequency $ n_1 $ is:
$$ n_1 = \frac{1}{2L} \sqrt{\frac{Mg}{\mu}} $$
When the bob is immersed in water, the apparent weight of the bob changes due to the buoyant force. The new tension $ T' $ is given by:
$$ T' = Mg - 	ext{buoyant force} = Mg - ho_{	ext{water}}Vg $$
where $ ho_{	ext{water}} $ is the density of water and $ V $ is the volume of the bob.
Therefore, the new frequency $ n_2 $ is:
$$ n_2 = \frac{1}{2L} \sqrt{\frac{T'}{\mu}} $$
Substituting the new tension:
$$ n_2 = \frac{1}{2L} \sqrt{\frac{Mg - ho_{	ext{water}}Vg}{\mu}} $$
The frequency relates to the relative density $ d $ (which is the density of the bob divided by the density of water) as:
$$ n_2 = \frac{1}{2L} \sqrt{\frac{Mg(1 - \frac{1}{d})}{\mu}} $$
Given the relationship between $ n_1 $ and $ n_2 $, the ratio of the squares of the frequencies allows determining:
$$ \frac{n_1^2}{n_2^2} = \frac{Mg}{Mg - ho_{	ext{water}}Vg} $$
From the equation:
$$ \frac{n_1^2}{n_2^2} = \frac{d}{d - 1} $$
Solving for $ d $:
$$ d = \frac{n_1^2}{n_1^2 - n_2^2} $$
Thus, the relative density of the metal of the bob corresponds to
Option C
:
$$ \frac{\mathrm{n}_1^2}{\mathrm{n}_1^2-\mathrm{n}_2^2} $$

Answer

$\frac{\mathrm{n}_1-\mathrm{n}_2}{\mathrm{n}_1}$

Answer

$\frac{\mathrm{n}_2}{\mathrm{n}_1-\mathrm{n}_2}$

Answer

$\frac{\mathrm{n}_1^2}{\mathrm{n}_1^2-\mathrm{n}_2^2}$

Answer

$\frac{\mathrm{n}_2^2}{\mathrm{n}_1^2-\mathrm{n}_2^2}$

Waves

A sonometer wire is stretched by hanging a metal bob, the fundamental frequency of the wire is ' n_1 '. When the bob is completely immersed in water, the frequency of vibration of wire becomes ' n_2 '. The relative density of the metal of the bob is

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