Let f be the fundamental frequency of the string . If the string is divided into three segments l_(1) , l_(2) and l_(3) such that the fundamental frequencies of each segments be f_(1) , f_(2) and f_(3) , respectively . Show that(1)/(f) = (1)/(f_(1)) + (1)/(f_(2)) + (1)/(f_(3))

Let f be the fundamental frequency of the string . If the string is di

1.	Let f be the fundamental frequency of the string . If the string is divided into three segments l_(1) , l_(2) and l_(3) such that the fundamental frequencies of each segments be f_(1) , f_(2) and f_(3) , respectively . Show that(1)/(f) = (1)/(f_(1)) + (1)/(f_(2)) + (1)/(f_(3))
Answer» Solution :For a fixed TENSION T and mass density `mu`,frequency is inversely proportional to the STRING length i.e. `f prop (1)/(l) implies f = (V)/(2l) implies l = (v)/(2f)` For the first length segment `f_(1) = (v)/(2 l_(1)) implies l_(1) = (v)/(2f_(1))` For the second length segment `f_(2) = (v)/(2l_(2)) implies l_(2) = (v)/(2f_(2))` For the third length segment `f_(3) = (v)/(2l_(3)) implies l_(3) = (v)/(2f_(3))` Therefore , the total length `l = l_(1) + l_(2) + l_(3)` `(v)/(2f) =(v)/(2f_(1)) + (v)/(2f_(2)) + (v)/(2f_(3)) implies (1)/(f) = (1)/(f_(1)) + (1)/(f_(2)) + (1)/(f_(3))`