Considering an elliptic torus. The boundary equation is rotate: a,b,c are consta

Question

Considering an elliptic torus. The boundary equation is rotate:

a,b,c are constants with c>a

1. Find the voulme of the object
2. Find the moment of inertia about the z-axis

(hint: density(x,y,z) = 1 and the center of mass is a (0,0,0)

Considering an elliptic torus. The boundary equation is rotate: (sqrt x^2+y^}-c)^2/a^2 + z^2/b^2 =1 about the z-axis. a,b,c are constants with c > a 1. Find the voulme of the object 2. Find the moment of inertia about the z-axis (hint: density(x,y,z) = 1 and the center of mass is a (0,0,0)

Explanation / Answer

I will give two solutions to question 1. You can adapt them to question 2 as you like.

Solution 1

A solid of revolution corresponds to a plane figure in the (r,z) plane, which we should suppose to be supported in the r>0 half-plane. Each point in the figure then contributes 2?r worth of volume. From this the following follows:

Lemma: Suppose that the figure is symmetric through its center of mass (in the sense of area). Then the volume of the corresponding volume of revolution is 2?

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