The fundamental frequencies of wires of same material of same length stretched by weights $0.8 k g$ and $0.2 k g$ are in the ratio $1 : 2$ . Find the ratio of their diameters.

4 : 1

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2 : 1

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1 : 4

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1 : 2

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Solution

The correct option is A $4 : 1$
Densities of wires are same $ρ$
Tension in 1st wire $T_{1} = 0.8 k g w t$
Tension in 2nd wire $T_{2} = 0.2 k g w t$
If $f_{1}$ and $f_{2}$ are frequencies, $\frac{f_{1}}{f_{2}} = \frac{1}{2}$
$f_{1} = \frac{1}{2 L} \sqrt{\frac{T_{1}}{m_{1}}}$ and $f_{2} = \frac{1}{2 L} \sqrt{\frac{T_{2}}{m_{2}}}$
$\frac{f_{1}}{f_{2}} = \sqrt{\frac{T_{1} m_{2}}{T_{2} m_{1}}}$ but $\sqrt{\frac{m_{1}}{m_{2}}} = \frac{f_{2}}{f_{1}} \sqrt{\frac{T_{1}}{T_{2}}}$
$\sqrt{\frac{π r_{1}^{2} ρ}{π r_{2}^{2} ρ}} = \frac{2}{1} \sqrt{\frac{0.8}{0.2}} \Rightarrow \frac{r_{1}}{r_{2}} = 4$
Ratio $4 : 1$

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The fundamental frequencies of wires of same material of same length stretched by weights 0.8kg and 0.2kg are in the ratio 1:2. Find the ratio of their diameters.

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