Q: If *f*_{1}, *f*_{2} and *f*_{3} are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency *f*_{0} of the whole string is

(a) f_{0} = f_{1} + f_{2} +f_{3}

(b) $ \displaystyle \frac{1}{f_0} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3}$

(c) $ \displaystyle \frac{1}{\sqrt{f_0}} = \frac{1}{\sqrt{f_1}} + \frac{1}{\sqrt{f_2}} + \frac{1}{\sqrt{f_3}}$

(d) None of these

Ans: (b)

Sol:Let length of wires are l_{1} , l_{2} , l_{3}

Total length l = l_{1} + l_{2} + l_{3}

$ \displaystyle f\propto \frac{1}{l} $

f l = constant

f l = f_{1} l_{1} = f_{2} l_{2} = f_{3} l_{3} = K

l = K/f , l_{1} = K/ f_{1}

l_{2} = K/f_{2} , l_{3} = K/f_{3}

Since , l = l_{1} + l_{2} + l_{3}

$ \displaystyle \frac{K}{f_0} = \frac{K}{f_1} + \frac{K}{f_2} +\frac{K}{f_3} $

$ \displaystyle \frac{1}{f_0} = \frac{1}{f_1} + \frac{1}{f_2} +\frac{1}{f_3} $