If f1, f2 and f3 are the fundamental frequencies of three segments into which a string is divided….

Q: If f1, f2 and f3 are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency f0 of the whole string is

(a) f0 = f1 + f2 +f3

(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 l1 , l2 , l3

Total length l = l1 + l2 + l3

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

f l = constant

f l = f1 l1 = f2 l2 = f3 l3 = K

l = K/f , l1 = K/ f1

l2 = K/f2 , l3 = K/f3

Since , l = l1 + l2 + l3

$ \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} $