Ancient Egyptians already understood basic concepts related to buoyancy at

Ancient egyptians already understood basic concepts

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Ancient Egyptians already understood basic concepts related to buoyancy, at least at a practical level, and were able to successfully construct barges for transporting various materials down the Nile River. Obviously, such applications are still very important today. In this section we will provide a very brief introduction consisting of the statement of Archimedes’ principle and then an example of applying it. We note that in these lectures we will not consider such important topics as stability of floating objects and buoyancy in flowing fluids. With respect to the latter, however, we comment that the equations of motion we have previously derived are fully capable of handling such phenomena if augmented with equations able to account for density changes in the fluid. We begin with a formal definition of buoyancy in the context of the present treatment. Definition 4.1 The resultant fluid force acting on a submerged, or partially submerged, object is called the buoyant force . We remark that the objects in question need not necessarily be solid; in particular, fluid elements can experience buoyancy forces. As alluded to above, such forces are among the body forces already present in the Navier–Stokes equations. Furthermore, other forces might also be acting on the submerged body simultaneously, the most common being weight of the body due to gravitation. Archimedes’ Principle We begin this subsection with a statement of Archimedes’ principle. Archimedes’ Principle. The buoyant force acting on a submerged, or partially submerged, object has a magnitude equal to the weight of fluid displaced by the object and a direction directly opposite the direction of local acceleration giving rise to body forces. In the most commonly-studied case, the acceleration is the result of a gravitational field, and it is clear that the buoyant force acting on a submerged object having volume V is F b = ρ fluid V g . (4.7) In the case of a partially-submerged (or floating) object, only a fraction of the total volume cor- responding to the percentage below the fluid surface should be used in the above formula for the buoyancy force. We remark that the above statement of Archimedes’ principle is somewhat more complete than that originally given by Archimedes in ancient Greece to permit its application in situations involving arbitrary acceleration fields, e.g. , such as those occurring in orbiting spacecraft when their station-keeping thrustors are activated. Application of Archimedes’ Principle The following example will provide a simple illustration of how to apply the above-stated prin- ciple. It will be evident that little is required beyond making direct use of its contents. EXAMPLE 4.4 We consider a cubical object with sides of length h that is floating in water in such a way that 1 4 h of its vertical side is above the surface of the water, as indicated in Fig. 4.4. It is required to find the density of this cube.
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4.2. BERNOULLI’S EQUATION 109 g F W F 1 b h h 4 Figure 4.4: Application of Archimedes’ principle to the case of a floating object.
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