Archimedes' principle

Statement

Archimedes' principle makes sense in the context that a uniform gravitational field is operational in the region (close to the surface of the earth, this is the earth's gravitational field).

It states that the buoyant force exerted by a fluid on a body partially or wholly immersed in it is in the upward direction and its magnitude is the weight of fluid displaced by the immersed part of the body, i.e., it is given by:

(Volume of the fluid displaced by the body) $\times$ (Density of fluid) $\times$ acceleration due to gravity

For a convex body, the volume of the fluid displaced by the body equals the volume of the part of the body immersed in the fluid. For non-convex bodies, such as hollow hemispherical hulls or boies with internal cavities, this need not be true, and part of the fluid may be displaced to air rather than to the body itself.

Particular cases

Convex bodies

We assume a convex body of uniform density and a fluid of uniform density.

Case	What Archimedes' principle tells us
A convex body of uniform density is floating on the surface of the fluid	The density of the body is less than that of the fluid, and the fraction of the body's volume immersed in the fluid is equal to (density of body)/(density of fluid).
A convex body of uniform density is floating, just completely immersed in the surface of the fluid.	The density of the body is the same as that of the fluid.
A convex body of uniform density is sinking into the fluid.	The density of the body is greater than that of the fluid. Assuming that the fluid is an ideal fluid (so no viscous force applies) the downward acceleration is $g$ times (1 - (density of fluid)/(density of body)). Here, $g$ represents acceleration due to gravity.