All of us have heard about Archimedes’ Principle, that the force exerted on a body submerged in a liquid is equal to the weight of the displaced liquid. However, many of us may not know the proper derivation of the principle. This lesson is about using a mathematical method to derive the principle.
ARCHIMEDES' PRINCIPLE.2
May want to check out:
ARCHIMEDES' PRINCIPLE.1
EXAMPLE - HYDROSTATIC FORCE ON A GATE
PRESSURE VARIATION IN FLUID WITH RIGID-BODY MOTION
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VIDEO LECTURE part 2
MAIN CONCEPTS
The buoyant force is given by  and it passes through the centroid of the displaced volume.Get the Printer Friendly Version COMMENTS
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LESSON
When a body is completely or partially submerged in a fluid, there is a resultant force acting on it called a buoyant force. The net upward vertical force results because pressure increases with depth and the pressure forces acting from below are larger than the pressure forces acting from above.
Consider a body having an arbitrary shape, with a volume We enclosed the body in a parallelepiped and draw a free-body diagram of the parallelepiped with the body removed as shown in (b). Forces The idea here is to find
Assuming specific weight of the fluid to be constant, then
where A is the horizontal area of the upper and lower surface of the parallelepiped. Combining both equations gives us
Observing that terms
where To determine the location of the line of action of the buoyant force, we sum up the moments of the forces shown in the free-body diagram with respect to some convenient axis. Summing moments about an axis perpendicular to the paper through point D gives us
and on substituting for the various forces
where we define the new variable
These same results apply to floating bodies which are only partially submerged, if the specific weight of the fluid above the liquid surface is very small compared with the liquid in which the body floats in (d). Usually, the fluid above the surface is air and so for practical purposes this condition is satisfied. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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, that is immersed in a fluid as shown below as (a).
and 
are simply the forces exerted on the plane surfaces of the parallelepiped, ignoring forces in the x direction for simplicity. W is the weight of the shaded fluid volume, parallelepiped minus body, and 
is the force the body is exerting on the fluid.
and by Newton’s third law equate this force to that exerted on the body. The forces on the vertical surfaces, such as 
and 
are all equal and cancel so the equilibrium equation of interest is in the z direction.
cancel out, we arrive at the expression for the buoyant force
is the specific weight of the fluid and 
is the volume of the body. The direction of the buoyant force, which is the force of the fluid on the body, is opposite to that shown on the free-body diagram. Therefore, the buoyant force has a magnitude equal to 
but is directed vertically upwards. This result is commonly referred to as Archimedes’ principle in honor of Archimedes, 287-212 B.C., way before Euler’s time if you guys might be wondering.
as the total volume 
. The right-hand side is the first moment of displaced volume 
with respect to the x-z plane so that 
is equal to the y coordinate of the centroid of the volume 
. Similarly, the x coordinate of the buoyant force coincides with the x coordiate of the centroid. Thus, we conclude that the buoyant force passes through the centroid of the displaced volume as shown below (c). This point through which the buoyant force acts is called the center of buoyancy.
