A sonometer wire of length `L_(1) ` and vibrating in its fundamental mode is in unison with a tuning fork of frequency n. When the vibrating length of the same wire is reduced to `L_(2)`, it produces x beats/second with the fork.
What is the frequency of teh fork ?

Options

A. `L_(1)/((L_(1) - L_(2)) ) x`
B. `((L_(2))/(L_(1) - L_(2)))x`
C. `((L_(1) - L_(2))/(L_(1)x))`
D. `((L_(1)- L_(2))/(L_(2)x))`

Answer

B

Explanation

For the length `L_(1) , n_(1) = 1/(2L_(1)) sqrt(T/m) = n ` ….(1)
where m = mass per unit length
For the length `L_(2) , n_(2) = 1/(2L_(2)) sqrt(T/m) ` ….(2)
` :. N_(2)/n_(1) = L_(1)/L_(2)` ….(3)
but ` L_(2) lt L_(1) :. N_(2) gt n_(1) `
` :. n_(2) - n_(1) = x " or " n_(2) = n_(1) + x ` ....(4)
` :. " form (3) " , (n_(1) + x)/n_(1) = L_(1)/L_(2) `
`:. 1 + x/n_(1) = L_(1)/L_(2) :. x/n_(1) = L_(1)/L_(2) - 1 = (L_(1) - L_(2))/L_(2) `
` :. n_(1) = (xL_(2))/(L_(1) - L_(2)) `
` :. ` The frequency of the fork `n = n_(1) = (L_(2)/(L_(1) - L_(2))) x `