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Q.

A streched wire resonates with a given tunning fork forming standing waves with five antinodes between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced by a mass ‘M’ the wire resonates with same frequency. Then the density of the material of mass ‘M’ is

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a

$\frac{169}{27} \frac{g m}{c . c}$

b

$1 \frac{g m}{c . c}$

c

$5 \frac{g m}{c . c}$

d

$\frac{196}{27} \frac{g m}{c . c}$

answer is D.

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Detailed Solution

$\begin{aligned} n = \frac{1}{2 l_{1}} \sqrt{\frac{M g}{μ}} in air \\ n = \frac{1}{2 l_{2}} \sqrt{\frac{M g (1 - \frac{P w}{P B})}{μ}} in water \\ l_{2} < l_{1} \Rightarrow l_{2} / l_{1} = \sqrt{1 - \frac{P w}{P B}} \\ \frac{65}{70} = \sqrt{1 - \frac{P w}{P B}} \Rightarrow \frac{13}{14} = 1 - \frac{P w}{P B} \\ = 1 - {[\frac{13}{14}]}^{2} P = \frac{196}{27} gm / cc \end{aligned}$

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A streched wire resonates with a given tunning fork forming standing waves with five antinodes between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced by a mass ‘M’ the wire resonates with same frequency. Then the density of the material of mass ‘M’ is