A stone is hung in air from a wire which is stretched over a sonometer . The bridges of the sonometer are 40 cm apart when the wire is in unison with a tuning fork of frequency 256. When the stone is completely immersed in water, the length between the bridges is 22 cm for re-establishing unison. The specific gravity of the material of the stone is :

Options

A. `256 xx (40)/(22)`
B. `((40)^(2))/((40)^(2) + (22)^(2))`
C. `((40)^(2))/((40)^(2) - (22)^(2))`
D. `256 xx (22)/(40)`

Answer

C

Explanation

Here `v = (1)/(2l) sqrt((T)/(m))`
Tension, `" " T = `Weight of stone.
`therefore` In air `v_(1) = (1)/(2l_(1)) sqrt((W_(a))/(m))`
In water `v_(2) = (1)/(2l_(2)) sqrt((W_(w))/(m))`
since ` v_(1) = v_(2) `
`therefore (1)/(2l_(1)) sqrt((W_(a))/(m)) = (1)/(2l_(2)) sqrt((W_(w))/(m))`
`rArr " " (W_(a))/(W_(w)) = (I_(1)^(2))/(I_(2)^(2))`.
Now relative density of stone = ` (w_(a))/(W_(a) - W_(w))`
`therefore` Relative density = `(1)/(1 - (W_(w))/(W_(a)) ) = (1)/( 1 - (I_(2)^(2))/(I_(1)^(2)))`
= `(I_(1)^(2))/(I_(1)^(2)- I_(2)^(2)) = ((40)^(2))/((40)^(2)- (22)^(2) ) `.
Correct choice is c.