A) \[n\]
B) \[\frac{n}{\sqrt{2}}\]
C) \[\frac{n}{2}\]
D) \[\frac{n}{2\sqrt{2}}\]
Correct Answer: D
Solution :
Frequency of sonometer wire is given by \[n=\frac{1}{2l}\sqrt{\frac{T}{m}}\] where m is mass of string per unit length, and is tension in the string. Also, \[m=\pi {{r}^{2}}d\] r being radius of string and d is the density of material of string. So, \[n=\frac{1}{2l}\sqrt{\frac{T}{\pi {{r}^{2}}d}}\] or \[n\propto \frac{\sqrt{T}}{r}\] or \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\times \left( \frac{{{r}_{2}}}{{{r}_{1}}} \right)\] Given, \[{{r}_{2}}=2{{r}_{1}},\,{{T}_{2}}=\frac{{{T}_{1}}}{2},\]\[{{n}_{1}}=n\] Hence, \[\frac{n}{{{n}_{2}}}=\sqrt{2}\times 2\] or \[{{n}_{2}}=\frac{n}{2\sqrt{2}}\]
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