Game Development Reference
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F t = (ρ g h t ) (s 2 )
Similarly, the force acting upward on the bottom of the cube is equal to the pressure at
the bottom times the surface area of the bottom.
F b = P b A b
F b = (ρ g h b ) (s 2 )
The net vertical force (buoyancy) equals the difference between the top and bottom
F B = F b - F t
F B = (ρ g h b ) (s 2 ) - (ρ g h t ) (s 2 )
F B = (ρ g) (s 2 ) (h b - h t )
This formula gives the magnitude of the buoyancy force. Its direction is straight up,
counteracting the weight of the object.
There is an important observation we need to make here. Notice that ( h b - h t ) is simply
the height of the cube, which is s in this case. Substituting s in place of ( h b - h t ) reveals
that the buoyancy force is a function of the volume of the cube.
F B = (ρ g) (s 3 )
This is great since it means that all you need to do in order to calculate buoyancy is to
first calculate the volume of the object and then multiply that volume by the specific
weight 3 g ) of the fluid. In truth, that's a little easier said than done for all but the
simplest geometries. If you're dealing with spheres, cubes, cylinders, and the like, then
calculating volume is easy. However, if you're dealing with any arbitrary geometry, then
the volume calculation becomes more difficult. There are two ways to handle this dif‐
ficulty. The first way is to simply divide the object into a number of smaller objects of
simpler geometry, calculate their volumes, and then add them all up. The second way
is to use numerical integration techniques to calculate volume by integrating over the
surface of the object.
You should also note that buoyancy is a function of fluid density, and you don't have to
be in a fluid as dense as water to experience the force of buoyancy. In fact, there are
buoyant forces acting on you right now, although they are very small, due to the fact
that you are immersed in air. Water is many times more dense than air, which is why
you notice the force of buoyancy when in water and not in air. Keep in mind, though,
3. Specific weight is density times the acceleration due to gravity. Typical units are lbs/ft 3 and N/m 3 .
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