Question 323: Waves

Question

A sonometer wire is in unison with a tuning fork of frequency ' $n$ ' when it is stretched by a weight of specific gravity ' $d$ '. When the weight is completely immersed in water, ' $x$ ' beats are produced per second, then

Accepted Answer

To understand the behavior of a sonometer wire when it is in unison with a tuning fork and later immersed in water, we consider the following:
Fundamentals:
Relative Density (Specific Gravity) Definition
:
$    \sigma = \frac{ho_{	ext{load}}}{ho_{	ext{water}}}    $
This describes the density of the load relative to water.
Frequency and Tension Relationship
:
$    n \propto \sqrt{T}    $
Here, $ n $ is the frequency and $ T $ is the tension in the wire.
Tension Calculations
:
In Air
:
$      T_{	ext{air}} = ho_{	ext{load}} \cdot V \cdot g      $
In Water
:
$      \begin{aligned}      T_{	ext{water}} &= (ho_{	ext{load}} - ho_{	ext{water}}) \cdot V \cdot g \      &= ho_{	ext{water}} \cdot (\sigma - 1) \cdot V \cdot g      \end{aligned}      $
Frequency Ratio
:
$    \frac{n_{	ext{load in air}}}{n_{	ext{load immersed in water}}} = \frac{\sqrt{T_{	ext{air}}}}{\sqrt{T_{	ext{water}}}}    $
Substituting the tensions:
$    \frac{n_{	ext{load in air}}}{n_{	ext{load immersed in water}}} = \frac{\sqrt{ho_{	ext{load}} \cdot V \cdot g}}{\sqrt{ho_{	ext{water}} \cdot (\sigma - 1) \cdot V \cdot g}} = \sqrt{\frac{\sigma_{	ext{load}}}{\sigma_{	ext{load}} - 1}} \quad 	ext{(Equation i)}    $
Frequency Change with Immersion
:
In air, the frequency is $ n $.
In water, the frequency becomes $ n - x $, where $ x $ is the beat frequency.
Condition for Specific Gravity
:
Given that the specific gravity of the load $\sigma_{	ext{load}} = d$, we combine the above insights to find:
$    \frac{n}{n-x} = \sqrt{\frac{d}{d-1}}    $
This equation represents the relationship between the frequency of the sonometer wire in air and when the weight is immersed in water, taking into account the specific gravity of the load.

Answer

$\frac{\mathrm{n}}{\mathrm{n}-\mathrm{x}}=\frac{\mathrm{d}}{\mathrm{d}-1}$

Answer

$\frac{n}{n-x}=\sqrt{\frac{d}{d-1}}$

Answer

$\frac{\mathrm{n}-\mathrm{x}}{\mathrm{n}}=\frac{\mathrm{d}-1}{\mathrm{~d}}$

Answer

$\frac{\mathrm{n}-\mathrm{x}}{\mathrm{n}}=\sqrt{\frac{\mathrm{d}}{\mathrm{d}-1}}$

Waves

A sonometer wire is in unison with a tuning fork of frequency ' n ' when it is stretched by a weight of specific gravity ' d '. When the weight is completely immersed in water, ' x ' beats are produced per second, then

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