The fundamental physicallaw controlling the static behavior of a body wholly or partially immersed in a fluid is known as Archimedes' Principle which, as normally expressed, states that a body immersed in a fluid is buoyed up by a force that equals the weight of the displaced fluid." Thus, the weight is considered to be a downward force that is proportional to the body's mass; the equal buoyant force is proportional to the mass of the displaced fluid.

Consider a body of fluid such as water, with a free surface, at rest. The fluid is of constant mass density, (i.e., mass per unit volume). At any point P, a distance t below the free surface, the mass of fluid above the point is At, where A is the cross sectional area parallel to the free surface of the column of fluid. In general, a fluid cannot support shear forces. Therefore, if the fluid be in a state of static equilibrium, it is necessary that equal forces be experienced in all directions at any such point. Since the gravitational force resulting from the mass of the fluid above is equal to its mass , the pressure force experienced by the fluid at that point is A t or the weight of the column of fluid above P. If a rigid body is afloat in the water in static equilibrium, Figure below,

a consequence of the above reasoning is that the same pressure forces are directed normal to the surface of the body. The integration of the vertical component of all such pressures experienced by the surface S of the body is the buoyant force,

where is the inclination of any part of S from the horizontal. But represents the volume of the body beneath a plane coincident with the free surface. The weight of fluid would occupy this volume in the absence of the body is identically equal to multiplied by the volume.

For the body to be in equilibrium, the integration of upward components of hydrostatic pressures over the surface of the body, or buoyancy, must be exactly balanced by the gravitational force of the body's mass, directed downward, i.e., its weight. Therefore, the weight of a ship and its contents is equal to the weight of displaced water, or displacement. Likewise, the mass of a ship and its contents is equal to the mass of displaced water. Hence, displacement can be expressed in either weight or mass units.

It is evident that a fully immersed rigid body, such as a submarine, also experiences an upward buoyant force equal and opposite to the weight of the water it displaces. A totally submerged body may weigh either more or less than the displaced water. For the body to be in equilibrium in its submerged position, it would have to receive, in the first case, an additional upward force, and in the second case, an additional downward force. When submerged and not resting on the bottom, a body may remain stationary, without rising or falling, only in the unusual case when its mass exactly equals the mass of the water it displaces.

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