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When a string is divided into three segments of length $l_1$, $l_2$ and $l_3$, the fundamental frequencies of these three segments are $\nu_1$, $\nu_2$ and $\nu_3$ respectively. The original fundamental frequency ($\nu$) of the string is
A
$\frac{1}{\sqrt\nu} = \frac{1}{\sqrt{\nu_1}} + \frac{1}{\sqrt{\nu_2}} + \frac{1}{\sqrt{\nu_3}}$
B
$\sqrt\nu = \sqrt{\nu_1} + \sqrt{\nu_2} + \sqrt{\nu_3}$
C
$\nu = \nu_1 + \nu_2 + \nu_3$
D
$\frac{1}{\nu} = \frac{1}{\nu_1} + \frac{1}{\nu_2} + \frac{1}{\nu_3}$
Detailed Solution
Fundamental frequency of a string: $\nu = \frac{v}{2l} \Rightarrow l = \frac{v}{2\nu}$
Similarly $l_1 = \frac{v}{2\nu_1}$, $l_2 = \frac{v}{2\nu_2}$, $l_3 = \frac{v}{2\nu_3}$
$l = l_1 + l_2 + l_3 \Rightarrow \frac{v}{2\nu} = \frac{v}{2\nu_1} + \frac{v}{2\nu_2} + \frac{v}{2\nu_3}$
$\frac{1}{\nu} = \frac{1}{\nu_1} + \frac{1}{\nu_2} + \frac{1}{\nu_3}$
Similarly $l_1 = \frac{v}{2\nu_1}$, $l_2 = \frac{v}{2\nu_2}$, $l_3 = \frac{v}{2\nu_3}$
$l = l_1 + l_2 + l_3 \Rightarrow \frac{v}{2\nu} = \frac{v}{2\nu_1} + \frac{v}{2\nu_2} + \frac{v}{2\nu_3}$
$\frac{1}{\nu} = \frac{1}{\nu_1} + \frac{1}{\nu_2} + \frac{1}{\nu_3}$