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center of mass for a continuous bodies

What is the center of mass of a uniform sphere? That isn't too bad. By symmetry it's at the sphere's center. But what if the sphere is not of uniform density? The top hemisphere is made out of balsa wood and the bottom hemisphere is made out of roquefort cheese? That's a bit more tricky, and to solve this sort of problem, we'll have to formulate it in terms of integration. Let's start off with the one dimensional case. Suppose that you have a rod that has a mass per unit length tex2html_wrap_inline1201 that can vary with position x. We choose coordinates so that the left hand side of the rod is at 0 and the right hand side is at L. To obtain the center of mass, we break up the rod into millions of tiny section each of length tex2html_wrap_inline1209 . Then the center of mass is the sum of all the individual masses tex2html_wrap_inline1211 times their corresponding position tex2html_wrap_inline1213 . So

equation163

Now one of these little masses tex2html_wrap_inline1215 so

equation166

You might remember from calculus expressions of this kind. If we take the limit as tex2html_wrap_inline1217 then these become integrals

  equation170

Since tex2html_wrap_inline1219 , an infinitessimal of mass, this integral can be more elegantly, but less usefully written as

equation174





Joshua Deutsch
Fri Jan 17 12:19:41 PST 1997