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2006 Preliminary Exam Continuum Mechanics 1) Consider a sea of incompressible fluid of density f , at rest, subject to a uniform gravitational field gê3 . Use the balance of linear momentum and a Cauchy stress of the form T = -p I to show that the pressure distribution in the fluid is p ( x1 , x 2 , x3 ) p 0 f gx3 where the origin of coordinates has been chosen at the surface of the sea, the x3 axis is perpendicular to the surface of the sea and p 0 is the atmospheric pressure at x3 = 0. 2) Consider a rigid solid object completely submerged in the sea. a) Calculate the resultant force on the solid to prove Archimedes law: the resultant force on the solid equals the weight of the displaced fluid (Hint: use the divergence theorem). b) Calculate the resultant moment on the solid with respect to an arbitrary point x0, due to the applied pressure. Use this to find the point x0 with the property that the resultant moment on that point is zero. Note that the force distribution of the fluid on the solid is equivalent to a point force applied at x0. 3) Generalize your results in 2 to a partly submerged body. a) Prove again Archimedes law. (Hint: cut the solid at the sea surface into submerged and exposed pieces.) Explain what is meant by the "displaced fluid". b) Calculate the point x0 at which the buoyancy force acts. Show that this buoyancy force acts through the center of buoyancy, which is the centroid of the displaced volume. 4) Assume that the partially submerged rigid body solid has a mass density solid ( x1 , x2 , x3 ) . Discuss qualitatively (in words and/or equations) the equilibrium and stability of the partly submerged rigid solid in the uniform gravitational field. To get started consider a solid with uniform density. When is equilibrium possible? Next think about a boat hull, with center of gravity and center of buoyancy as shown below. What happens as the boat rocks? Expand your discussion to more general perturbations as well as to non-uniform density of the solid. 5) Now suppose the solid is in equilibrium, fully submerged. Suppose it is a compressible elastic solid, so that its volume can decrease under increasing pressure. Discuss qualitatively the equilibrium and stability of the solid.