If `n_(1), n_(2 ) "and" n_(3)` are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency n of the string is given by

Options

A. `(1)/(n)=(1)/(n_1)+(1)/(n_2)+(1)/(n_(3))`
B. `(1)/(sqrt(n))=(1)/(sqrt(n_(1))+(1)/(sqrt(n_(2))+(1)/(sqrt(n_(3))`
C. `sqrt(n)=sqrt(n_(1))+sqrt(n_(2))+sqrt(n_(3))`
D. `n=n_(1)+n_(2)+n_(3)`

Answer

A

Explanation

Consider different segments of the string, we get

`n_(1)=(1)/(2l_(1))sqrt((T)/(mu))` . . . (i)
`n_(2)=(1)/(2l_(2))sqrt((T)/(mu))` . . . (ii) ltbr`n_(3)=(1)/(2l_(3))sqrt((T)/(mu))` . . . (iii)
and `n=(1)/(2l)sqrt((T)/(mu))` . . . (iv)
From Eq. (iv) we get
`(1)/(n)=(2l)/(sqrt((T)/(mu)))`
As `l=l_(1)+l_(2)+l_(3)`
`therefore(1)/(n)-(2(l_(1)+l_(2)+l_(3)))/(sqrt((T)/(mu)))`
`=(2l_(1))/(sqrt((T)/(mu)))+(2l_(2))/(sqrt((T)/(mu)))+(2l_(3))/(sqrt((T)/(mu)))`
`=(1)/(n_(1))+(1)/(n_(2))+(1)/(n_(3))`
[Using Eqs. (i), (ii) and (iii)]