Question

The cube is immersed in one liquid by 15 cm and in the other by 40 cm. To what depth will it sink into a liquid whose density is the arithmetic mean of the densities of the first two liquids?

Answer 1

Newton's second law for all three cases:

$\begin{cases} mg=\rho_1gh_1\\ mg=\rho_2gh_2\\ mg=\frac{\rho_1+\rho_2}{2}gH \end{cases}$

we express H and density from these equations:

$\begin{cases} \rho_1 = \frac{mg}{gh_1}\\ \rho_2 = \frac{mg}{gh_2}\\ H=\frac{2mg}{g(\rho_1+\rho_2)} \end{cases}$

then

$H= \frac{2mg}{g(\frac{mg}{gh_1}+\frac{mg}{gh_2})}=\frac{2h_1h_2}{h_1+h_2} = \frac{2\cdot15\cdot40}{15+40} = 54.55 cm$