A sonometer wire resonates with a given tuning fork forming a standing wave with five antinodes between the two bridges when a mass of `9 kg` is suspended from the wire. When this mass is replaced by a mass 'M' kg, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. Find the value of `M`.

Options

A. `12.5` kg
B. `(1//25)` kg
C. 5 kg
D. 25 kg

Answer

D

Explanation

In the first case, there are 5 antinodes ( five loops).
` :. ` The resonating length of the string
` L = (5 lambda_(1))/2 :. lambda_(1) = (2L)/5`
In the second case there are 3 antinodes .
` :. ` The resonating length , ` L = (3 lambda_(2))/2 " or " lambda_(2) = (2L)/3`
For a sonometer wire , ` n = 1/lambda sqrt(T/m) = 1 / lambda sqrt((Mg)/m) `
` :. n_(1) = 5/(2L) sqrt((M_(1)g)/m) " and " n_(2) = 3/(2L) sqrt((M_(2)g)/m) `
But it is given that `n_(1) = n_(2) , M_(1) = 9 kg " and " M_(2) = M ` .
` :. 5/(2L) sqrt((9 g)/m) = 3/(2L) sqrt((Mg)/m) `
` :. 5 xx 3 = 3 sqrtM :. 5 = sqrtM " or " M = 25 ` kg