Continuous Mathematical Models
1. Naval architecture. Consider a boat with a hull whose cross section
that is close to being a circle:
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
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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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...