A) \[\frac{1}{n}=\frac{1}{{{n}_{1}}}+\frac{1}{{{n}_{2}}}+\frac{1}{{{n}_{3}}}\]
B) \[\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{{{n}_{1}}}}+\frac{1}{\sqrt{{{n}_{2}}}}+\frac{1}{\sqrt{{{n}_{3}}}}\]
C) \[\sqrt{n}=\sqrt{{{n}_{1}}}+\sqrt{{{n}_{2}}}+\sqrt{{{n}_{3}}}\]
D) \[n={{n}_{1}}+{{n}_{2}}+{{n}_{3}}\]
Correct Answer: A
Solution :
[a] Total length of string \[\ell ={{\ell }_{1}}+{{\ell }_{2}}+{{\ell }_{3}}\] (As string is divided into three segments) \[But\text{ }frequency\propto \frac{1}{length}\] \[\,\left( \because f=\frac{1}{2\ell }\sqrt{\frac{T}{m}} \right)\] so \[\frac{1}{n}=\frac{1}{{{n}_{1}}}+\frac{1}{{{n}_{2}}}+\frac{1}{{{n}_{3}}}\].
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