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Continuous Mathematical Models 1. Naval architecture. Consider a boat with a hull whose cross section that is close to being a circle: superstructure water center of center of bouyancy hull mass In this example, we show that the center of mass above the center of buoyancy. Show that as the boat rotates about the center of the circle, the center of buoyancy remains in the same place, however the center of mass rotates with the ship. Looking at the torques about the center of buoyancy, show that rotations result in a torque the causes the boat to rotate further, which is unstable. What configuration of center of mass and center of buoyancy is stable? You will need to look at the total torque acting on the boat about the center of mass of the boat: laot x X (-pn) ds. You should apply the divergence theorem to c laox X (-pn) ds for a constant vector c, and replacing p with Po P8x3. 2. Elasticity. The Navier-Cauchy equations for linear elasticity are If we look for wave solutions of the form u(t,x) = uncos(wf- Q) show that k is either parallel to uo with one characteristic wave speed (a pressure- or P-wave), or k is perpendicular to un with a dif- ferent speed (a shear- or S-wave). With some research, explain how the speed difference has been used by geologists to identify struc- tures deep inside the Earth.

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Question 1

The center of buoyancy is by definition the CM (center of mass) of the fluid that is displaced by the submerged part of the ship. Since in this case the submerged part of the ship is cylindrical, it results that for any rotation of the ship the section submerged still remains circular (or the same part of the circle) which means that the CM of the fluid displaced (= center of buoyancy) remains unchanged.
Because the entire ship is not symmetrical for all axes through CM (that is, it also contains a superstructure which adds to the cylindrical part that is submerged) it...
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