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

Answer

B

Explanation

When the stone is suspended in air :
` n = (1)/( 2 L) sqrt((W_(a))/(m))`
When the stone is suspended in water :
`n = (1)/( 2L') sqrt((W_(w))/(m'))`
Hence `(sqrt (W_(a)))/(L) = ( sqrt(W_(w)))/(L')`
or `(W_(a))/(W_(w)) = (L^(2))/(L'^(2))`
Now , specific gravity of material of the stone
`= ( W_(a))/( W_(a) - W_(w)) = (1) /( 1 - (W_(w))/( W_(a))) = (1)/( 1 - (L'^(2))/(L^(2)))`
` = (L^(2))/(L^(2) - L'^(2)) = ((40)^(2))/((40)^(2) - (22)^(2))`