Recall from How Pressure Varies with Depth the three forces acting on a cubic region of water: the pressure from above and below, and the weight of the water. Now suppose we were to replace this water with another cubic object: the pressures $P_t$ and $P_b$ would be the same, but the weight would be different. If the weight of the new object is larger than the weight of the water it displaced, then the object will sink; if it is smaller, then the object will float.
The pressure force $(P_b A − P_t A)\uparrow$ = $(\Delta PA) \uparrow$ points upwards because the bottom pressure is larger than the top. Substituting in the fact that $\Delta P = \rho_f g\Delta y$, we have $\vec B = \rho_f g(A\Delta y) \uparrow$. But $A\Delta y$ is the volume of the object, and so we have
$$\vec B = \rho_f gV \uparrow$$
Buoyancy
\(\vec B\)
N
This is called the buoyancy force and always points opposite gravity: remember that $\rho_f$ is the density of the fluid the object is in, and $V$ is the volume of fluid that the object is displacing. (This is sometimes known as Archimedes’ Principle.)
