The methods used to evaluate multiple integrals make possible the determination of centers of mass and moments of inertia for solids. This leads to the following definition.
Definition 15.7.1 Let a solid occupy a region R such that its density is ρ
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Letting
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where x,y,z are the Cartesian coordinates of the point at x.
The reason the moment of inertia is of interest has to do with the total kinetic energy of a solid occupying the region R which is rotating about the line L. Suppose its angular velocity is ω. Then the kinetic energy of an infinitesimal chunk of volume located at point x is
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Thus in the consideration of a rotating body, the moment of inertia takes the place of mass when angular velocity takes the place of speed.
As to the center of mass, its significance is that it gives the point at which the mass will balance. See Volume 1 to see this explained with point masses. The only difference is that here the sums need to be replaced with integrals.
Example 15.7.2 Let a solid occupy the three dimensional region R and suppose the density is ρ. What is the moment of inertia of this solid about the z axis? What is the center of mass?
Here the little masses would be of the form ρ
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To find the center of mass, sum up rρdV for the points in R and divide by the total mass. In Cartesian coordinates, where r =
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Here is a specific example.
Example 15.7.3 Find the moment of inertia about the z axis and center of mass of the solid which occupies the region R defined by 9 −
This moment of inertia is ∫ R
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To find the center of mass, note the x and y coordinates of the center of mass,
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both equal zero because the above shape is symmetric about the z axis and ρ is also symmetric in its values. Thus xρdV will cancel with −xρdV and a similar conclusion will hold for the y coordinate. It only remains to find the z coordinate of the center of mass, z. In polar coordinates, ρ = r and so,
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Thus the center of mass will be