A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and corresponding resonating length is L 1 . If the weight reduced to w 4 , the corresponding resonating length becom

Question

A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and corresponding resonating length is L 1 . If the weight reduced to w 4 , the corresponding resonating length becomes L 2 . The ratio L 1 L 2

TestHub · Accepted Answer

When sonometer is stretched by weight w (tension), then frequency of vibration in the string
$v = \frac{1}{2 l} \sqrt{\frac{T}{m}}$
$v = \frac{1}{2 L_{1}} \sqrt{\frac{w}{m}}$ … (i)
where, $L_{1}$ is the resonating length and m is mass per unit length of the string.
Similarly, when weight reduces to $\frac{w}{4}$ and
Resonating lengths is $L_{2},$ then
$v = \frac{1}{2 L_{2}} \sqrt{\frac{\frac{w}{4}}{m}} \Rightarrow v = \frac{1}{4 L_{2}} \sqrt{\frac{w}{m}}$ …(ii)
From Eqs. (i) and (ii), we have
$\frac{1}{2 L_{1}} \sqrt{\frac{w}{m}} \times \frac{1}{4 L_{2}} \sqrt{\frac{w}{m}} \Rightarrow L_{1} = 2 L_{2}$
$∴ \frac{L_{1}}{L_{2}} = 2$
$∴ L_{1} : L_{2} = 2 : 1$

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