Question

When a string is divided into three segments of
lengths `l_(1),l_(2) and l_(3)` the fundamental frequencies of
these three segments are `v_(1), v_(2) and v_(3)` respectively.
The original fundamental frequency (v) of the string is
A. `sqrt(v)= sqrt(v_(1))+sqrt(v_(2))+sqrt(v_(3))`
B. `v= v_(1) + v_(2)+v_(3)`
C. `1/v=1/v_(1) + 1/v_(2)+1/v_(3)`
D. `1/sqrt(v)= 1/sqrt(v_(1))+1/sqrt(v_(2))+1/sqrt(v_(3))`

Answer 1

Correct Answer - c
The fundamental frequency of string
`v=1/(2l)sqrt(T/m)`
`therefore v_(1)l_(1)=v_(2)l_(2)=v_(3)l_(3)=k ... (i) `
Form Eq. (i), `l_(1)=k/v_(1), l_(2)=k/v_(2), l_(3)=k/v_(3)`
Original length, `l=k/v`
Here, `l=l_(1)+L_(2)+l_(3)`
`k/v=k/v_(1)+k/v_(2)+k/v_(3)`
`rArr 1/v=1/v_(1)+1/v_(2)+1/v_(3)`