PlanetPhysics/Archimedes Principle

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Archimedes' Principle states that

When a floating body of \htmladdnormallink{mass {http://planetphysics.us/encyclopedia/CosmologicalConstant.html} is in equilibrium with a fluid of constant density, then it displaces a mass of fluid equal to its own mass; .}

Archimedes' principle can be justified via arguments using some elementary classical mechanics. We use a Cartesian coordinate system oriented such that the -axis is normal to the surface of the fluid.

Let be The Gravitational Field (taken to be a constant) and let denote the submerged region of the body. To obtain the net force of buoyancy acting on the object, we integrate the pressure over the boundary of this region Where is the outward pointing normal to the boundary of . The negative sign is there because pressure points in the direction of the inward normal. It is a consequence of Stokes' theorem that for a differentiable scalar field and for any a compact three-manifold with boundary, we have therefore we can write Now, it turns out that where is the volume density of the fluid. Here is why. Imagine a cubical element of fluid whose height is , whose top and bottom surface area is (in the plane), and whose mass is . Let us consider the forces acting on the bottom surface of this fluid element. Let the z-coordinate of its bottom surface be . Then, there is an upward force equal to </math>p(z)\Delta A\mathbf{e}_z-p(z + \Delta z)\Delta A\mathbf{e}_z + \Delta m\mathbf{g}a simple manipulation of this equation along with dividing by gives taking the limit gives Similar arguments for the and directions yield putting this all together we obtain as desired. Substituting this into the integral expression for the buoyant force obtained above using Stokes' theorem, we have where we can pull and outside of the integral since they are assumed to be constant. But notice that is equal to , the mass of the displaced fluid so that But by Newton's second law, the buoyant force must balance the weight of the object which is given by . It follows from the above expression for the buoyant force that which is precisely the statement of Archimedes' Principle.