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)/(sqrtn) =(1)/(sqrt n _(1)) + (1)/(sqrt(n _(2))) + (1)/( sqrt (n _(3)))`
C. `sqrtn =sqrt (n _(1)) + sqrt (n _(2)) + sqrt (n _(3))`
D. `n = n _(1) + n_(2) + n _(3)`

Answer

A

Explanation

For given string of length, `L, n = (1)/(2L) sqrt ((T)/(mu))`
(1) For first part of length `L_(1) , n _(1) = (1)/(2L_(1)) sqrt((T)/(mu))`
(2) For second part of length `l _(2), n _(2) = (1)/(2L _(2)) sqrt ((T)/( mu))`
(3) For the part of length `L_(3), n_(3) = (1)/(2L _(3)) sqrt ((T)/(mu))`
But `L = L _(1) + L _(2) + L _(3)`
`therefore (1)/(2n ) sqrt ((T)/( mu)) = (1)/( 2n _(1)) sqrt ((T)/( mu ))+ (1)/( 2n _(2)) sqrt ((T)/( mu )) + (1)/( 2n _(3)) sqrt ((T)/( mu ))`
`therefore (1)/(n) = (1)/( n _(1) )+ (1)/(n _(2)) + (1)/( n _(3))`