Q: A solid body is found floating in water with (α/β)^{th} of its volume submerged. The same solid is found floating in a liquid with (α/β)^{th} of its volume above the liquid surface. The specific gravity of the liquid is

(a) $ \displaystyle \frac{\beta -\alpha}{\alpha} $

(b) $ \displaystyle \frac{\alpha – \beta }{\beta} $

(c) $ \displaystyle \frac{\alpha}{\beta – \alpha} $

(d) $ \displaystyle \frac{\beta}{\alpha -\beta} $

Ans: (c)

Sol: To Float , Weight = Buoyancy

In first case ,

$ \displaystyle Weight = (\frac{\alpha}{\beta})V \rho_w g $ …(i)

In second case ,

$ \displaystyle Weight = (1-\frac{\alpha}{\beta})V \rho_l g $ …(ii)

From (i) & (ii)

$ \displaystyle (\frac{\alpha}{\beta})V \rho_w g = (1-\frac{\alpha}{\beta})V \rho_l g $

$ \displaystyle \frac{\rho_l}{\rho_w} = \frac{\alpha}{\beta – \alpha} $