Given a liquid with constant density , introduce coordinates so that the (flat) surface

Chapter 7, Problem 22

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Given a liquid with constant density , introduce coordinates so that the (flat) surface of the liquid is the x y-plane and the z-coordinate measures the depth of the liquid from the surface. (That is, the positive z-axis points down into the liquid.) Then the pressure p inside the fluid due to gravity is given by p(x, y,z) = gz, where g is acceleration due to gravity. Suppose that a solid object is immersed in the liquid. If the object fills out a region D in space, then the total buoyant force on the solid is the total liquid pressure on the boundary surface S = D and is given by B = S pn d S, where n is the outward unit normal to S. (The negative sign arises because the pressure causes a force pointing inward on the object.) Use the previous exercise to demonstrate Archimedes principle: The magnitude of the total buoyant force on an object equals the weight of the liquid displaced.