A) 300 cm
B) 200 cm
C) 100 cm
D) 50 cm
Correct Answer: C
Solution :
The frequency of the sonometer \[n=\frac{1}{2l}\sqrt{\frac{T}{m}}\] or \[n=\frac{1}{2l}\sqrt{\frac{Mg}{\pi {{r}^{2}}d}}\] \[(\because T=Mg,\,m=\pi {{r}^{2}}d)\] \[\Rightarrow \]\[{{n}_{1}}=\frac{1}{2{{l}_{1}}}\sqrt{\frac{{{M}_{1}}g}{\pi {{r}^{2}}d}}\]and \[{{n}_{2}}=\frac{2}{2{{l}_{2}}}\sqrt{\frac{{{M}_{2g}}}{\pi {{r}^{2}}d}}\] From the question, \[{{n}_{1}}={{n}_{2}}\] \[\frac{{{l}_{2}}}{{{l}_{1}}}=2\sqrt{\frac{{{M}_{2}}}{{{M}_{1}}}}\] \[\Rightarrow \] \[\frac{100}{{{l}_{1}}}=2\sqrt{\frac{3}{12}}=2\sqrt{\frac{1}{4}}\] \[\Rightarrow \] \[=2\times \frac{1}{2}\] \[\Rightarrow \] \[{{l}_{1}}=100\,cm\]
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