Let's consider now the interaction of solids and fluids. When a solid object is placed in a liquid, will it sink or will it float?
Consider the solid at a moment when a part of it, the volume \( V_l \), has been submerged. Now imagine the solid was not present and replace the submerged part of the solid by liquid of the same volume, \( V_l \). The force of gravity on that volume of liquid is simply,
$$ F_g^{liquid} = - m_d g = - V_l \rho_l g $$
Where \( \rho_l \) is the density of the liquid. This configuration must be in equilibrium since \( V_l \) is the same liquid and the boundary drawn is entirely fictitious. So the force of gravity must be balanced by an upward force exerted on this volume by the rest of the liquid,
$$ F_B^{liquid} = V_l \rho_l g $$
But, the surrounding liquid could not known whether \( V_l \) is liquid or any other material. So, the upward force exerted on the solid by the surrounding liquid is the same,
$$ F_B^{solid} = V_l \rho_l g $$
This force is known as buoyancy. The force acting on the solid are then the buoyancy, \( F_B = V_l \rho_l g \) pushing it upwards, as well as its weight \( F_g = -m_{solid} g \) pulling it downwards. The volume, \( V_l \) is simply the volume of liquid displaced by the solid. Therefore, a simple way to state the law of buoyancy is to say that "buoyancy is equal to the weight of the liquid displaced by the solid". This is Archimedes' law.