An object will sink if the downward forces on it are larger than the upward forces. There are two downward forces: the force of the liquid above the object and its weight (the force of gravity). The upward force is the force of the liquid below it. Let’s think of a cubic object of height h and area of the top and bottom A. Its volume, V, is then given by V = hA. Density is the mass divided by the volume, or ρ = m/V.

Let’s start by considering the difference in water pressure between the bottom and top of this cube. Pressure is force divided by area, so, using P_bottom = P_top ρ_water gh, we can write F_bottom/A = F_top/A ρ_water gh so F_bottom = F_top ρ_waterghA. Now we recall that hA is the volume of the cube and ρ_waterhA is therefore the mass of the water, m_water, whose place is taken by the cube of matter.

The net downward force on the object in the water is then the force on the top plus the object’s weight less the force on the bottom. That means

From the results of the paragraph above, F_net = m_objectg – m_waterg. Therefore, if the object’s mass is larger than the mass of the water whose place it takes, it sinks. If the mass is smaller, then it will rise. Weight is the mass times the gravitational field strength, g, so the net force is F_net = W_object – W_water. The water whose place the object takes is normally referred to as the “water displaced” by the object.