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} $