Question
When a string is divided into three segments of lengths $${l_1},{l_2}$$ and $${l_3},$$ the fundamental frequencies of these three segments are $${\nu _1},{\nu _2}$$ and $${\nu _3}$$ respectively. The original fundamental frequency $$\left( \nu \right)$$ of the string is
A.
$$\sqrt \nu = \sqrt {{\nu _1}} + \sqrt {{\nu _2}} + \sqrt {{\nu _3}} $$
B.
$$\nu = {\nu _1} + {\nu _2} + {\nu _3}$$
C.
$$\frac{1}{\nu } = \frac{1}{{{\nu _1}}} + \frac{1}{{{\nu _2}}} + \frac{1}{{{\nu _3}}}$$
D.
$$\frac{1}{{\sqrt \nu }} = \frac{1}{{\sqrt {{\nu _1}} }} + \frac{1}{{\sqrt {{\nu _2}} }} + \frac{1}{{\sqrt {{\nu _3}} }}$$
Answer :
$$\frac{1}{\nu } = \frac{1}{{{\nu _1}}} + \frac{1}{{{\nu _2}}} + \frac{1}{{{\nu _3}}}$$
Solution :
The fundamental frequency of string
$$\eqalign{
& \nu = \frac{1}{{2l}}\sqrt {\frac{T}{m}} \cr
& \therefore {\nu _1}{l_1} = {\nu _2}{l_2} = {\nu _3}{l_3} = k\,......\left( {\text{i}} \right) \cr
& {\text{From}}\,{\text{Eq}}{\text{.}}\,\left( {\text{i}} \right) \cr
& {l_1} = \frac{k}{{{\nu _1}}},{l_2} = \frac{k}{{{\nu _2}}},{l_3} = \frac{k}{{{\nu _3}}} \cr
& {\text{Original length,}} \cr
& l = \frac{k}{\nu } \cr
& {\text{Here,}}\,l = {l_1} + {l_2} + {l_3} \cr
& \frac{k}{\nu } = \frac{k}{{{\nu _1}}} + \frac{k}{{{\nu _2}}} + \frac{k}{{{\nu _3}}} \cr
& \frac{1}{\nu } = \frac{1}{{{\nu _1}}} + \frac{1}{{{\nu _2}}} + \frac{1}{{{\nu _3}}} \cr} $$