Answer to Question #121045 in Field Theory for Miranda Latsabidze

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

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