A) \[\sqrt{v}=\sqrt{{{v}_{1}}}+\sqrt{{{v}_{2}}}+\sqrt{{{v}_{3}}}\]
B) \[v={{v}_{1}}+{{v}_{2}}+{{v}_{3}}\]
C) \[\frac{1}{v}=\frac{1}{{{v}_{1}}}+\frac{1}{{{v}_{2}}}+\frac{1}{{{v}_{3}}}\]
D) \[\frac{1}{\sqrt{v}}=\frac{1}{\sqrt{{{v}_{1}}}}+\frac{1}{\sqrt{{{v}_{2}}}}+\frac{1}{\sqrt{{{v}_{3}}}}\]
Correct Answer: C
Solution :
| The fundamental frequency of string |
| \[v=\frac{1}{2l}\sqrt{\frac{T}{m}}\] |
| \[\therefore \] \[{{v}_{1}}{{l}_{1}}={{v}_{2}}{{l}_{2}}={{v}_{2}}{{l}_{3}}=k\] (i) |
| From Eq. (i) |
| \[{{l}_{1}}=\frac{k}{{{v}_{1}}},{{l}_{2}}=\frac{k}{{{v}_{2}}},{{l}_{3}}=\frac{k}{{{v}_{3}}}\] |
| Original length \[l=\frac{k}{v}\] |
| Here, \[l={{l}_{1}}+{{l}_{2}}+{{l}_{3}}\] |
| \[\frac{k}{v}=\frac{k}{{{v}_{1}}}+\frac{k}{{{v}_{2}}}+\frac{k}{{{v}_{3}}}\] |
| \[\frac{1}{v}=\frac{1}{{{v}_{1}}}+\frac{1}{{{v}_{2}}}+\frac{1}{{{v}_{3}}}\] |
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