A sonometer wire is in unison with a tuning f | vedclass

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(D) The frequency of vibration $v$ of a sonometer wire is given by the formula:$v = \frac{1}{2L} \sqrt{\frac{T}{m}}$where $T$ is the tension (weight $

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$2:1$

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(D) The frequency of vibration $v$ of a sonometer wire is given by the formula:$v = \frac{1}{2L} \sqrt{\frac{T}{m}}$where $T$ is the tension (weight $w$) and $m$ is the mass per unit length.For the first case,the frequency is:$v = \frac{1}{2L_1} \sqrt{\frac{w}{m}} \quad (i)$For the second case,the weight is reduced to $\frac{w}{4}$ and the length becomes $L_2$:$v = \frac{1}{2L_2} \sqrt{\frac{w/4}{m}} = \frac{1}{2L_2} \cdot \frac{1}{2} \sqrt{\frac{w}{m}} = \frac{1}{4L_2} \sqrt{\frac{w}{m}} \quad (ii)$Since the tuning fork frequency $v$ remains constant,we equate $(i)$ and $(ii)$:$\frac{1}{2L_1} \sqrt{\frac{w}{m}} = \frac{1}{4L_2} \sqrt{\frac{w}{m}}$$\frac{1}{2L_1} = \frac{1}{4L_2}$$4L_2 = 2L_1$$\frac{L_1}{L_2} = \frac{4}{2} = 2$Therefore,the ratio $\frac{L_1}{L_2}$ is $2:1$.

$A$ sonometer wire is in unison with a tuning fork when it is stretched by weight $w$ and the corresponding resonating length is $L_1$. If the weight is reduced to $\frac{w}{4}$,the corresponding resonating length becomes $L_2$. The ratio $\frac{L_1}{L_2}$ is:

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