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# Buoyancy

Consider the air/water system described in the previous section. Let be some volume, bounded by a closed surface , that straddles the plane , and is thus partially occupied by water, and partially by air. The -component of the net force acting on the fluid (i.e., either water or air) contained within is written (see Section 1.3) (2.5)

where (2.6)

is the stress tensor for a static fluid (see Section 1.5), and (2.7)

the gravitational force density. (Recall that the indices , , and refer to the -, -, and -axes, respectively. Thus, , et cetera.) The first term on the right-hand side of Equation (2.5) represents the net surface force acting across , whereas the second term represents the net volume force distributed throughout . Making use of the tensor divergence theorem (see Section B.4), Equations (2.5)-(2.7) yield the following expression for the net force: (2.8)

where (2.9)

and  (2.10)  (2.11)

Here, is the net surface force, and the net volume force.

It follows from Equations (2.4) and (2.9) that (2.12)

where . Here, is the volume of that part of which lies below the waterline, and the total mass of water contained within . Moreover, from Equations (2.2), (2.10), and (2.11), (2.13)

It can be seen that the net surface force, , is directed vertically upward, and exactly balances the net volume force, , which is directed vertically downward. Of course, is the weight of the water contained within . On the other hand, , which is generally known as the buoyancy force, is the resultant pressure of the water immediately surrounding . We conclude that, in equilibrium, the net buoyancy force acting across exactly balances the weight of the water inside , so that the total force acting on the contents of is zero, as must be the case for a system in mechanical equilibrium. We can also deduce that the line of action of (which is vertical) passes through the center of gravity of the water inside . Otherwise, a net torque would act on the contents of , which would contradict our assumption that the system is in mechanical equilibrium.   Next: Equilibrium of Floating Bodies Up: Hydrostatics Previous: Hydrostatic Pressure
Richard Fitzpatrick 2016-03-31