centerofmass - Math 21a Supplement on Center of Mass...

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Math 21a Supplement on Center of Mass Integrals over volumes occur naturally when studying the motions of extended objects. The fact is that the ‘point particle’ approximation in Physics is often far from accurate, and in these cases, the spatial extent of an object must be taken into account. In particular, when studying the dynamics of an extended object, three common volume integrals arise; these being The integral of the mass density to obtain the total mass. The integral of the position vector to obtain the center of mass. The integral of the 3 × 3 matrix M whose entry in the i’th column and j’th row is the product of the i’th and j’th components of the position vector. One might call this the moment of inertia matrix. (1) After setting the stage, I shall consider these points in turn. To set the stage, consider an extended object in a volume, V, in R 3 . (Here, I will allow for the case V = R 3 .) The mass distribution in this object is specified by giving a function, σ (x, y, z) on V whose value at any given point specifies the mass density at that point. Thus, for example, σ might be given in the units of kilograms per cubic meter. As an aside, of current intense interest to astronomers is the
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This note was uploaded on 02/18/2012 for the course MATH 310 taught by Professor Gregoryj.galloway during the Fall '11 term at University of Miami.

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centerofmass - Math 21a Supplement on Center of Mass...

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