The fundamental frequency of a sonometer wire is $200 H z$ when a body whose relative density is $4$ , is suspended at the free end of wire. The fundamental frequency of the same wire when the suspended body is made to immerse in water is:

100 \sqrt{2} H z

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100 \sqrt{3} H z

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50 \sqrt{2} H z

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50 \sqrt{3} H z

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Solution

The correct option is B $100 \sqrt{3} H z$
given : Relative density $= 4 = \frac{ρ_{m}}{ρ_{w}}$
$\Rightarrow ρ_{m} = 4 \times 1000 (∵ ρ_{w} = 1000 k g / m^{3})$
$= 4000 k g / m^{3}$
So, tension when the body is outside :
$= ρ_{m} V g = 4000 V g (∵ ρ_{m} = 4000 k g / m^{3})$
tension when the body is in water
$= (ρ_{m} - ρ_{w}) V g$
$= (4000 - 1000) V g$
$= 3000 V g$
since , $f \propto \sqrt{T}$
$s o, \frac{f_{1}}{f_{2}} = \frac{\sqrt{T_{1}}}{\sqrt{T_{2}}}$
$= \sqrt{\frac{4000 V g}{3000 V g}}$
$f_{2} = f_{1} \times \sqrt{\frac{3}{4}}$
$= 200 \times \frac{\sqrt{3}}{2}$
$= 100 \sqrt{3} H Z .$

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The fundamental frequency of a sonometer wire is 200Hz when a body whose relative density is 4, is suspended at the free end of wire. The fundamental frequency of the same wire when the suspended body is made to immerse in water is:

The fundamental frequency of a sonometer wire is $200 H z$ when a body whose relative density is $4$ , is suspended at the free end of wire. The fundamental frequency of the same wire when the suspended body is made to immerse in water is: