## Center of mass for a toroidal segment

Posted by peeterjoot on May 15, 2010

# Motivation.

[Click here for a PDF of this post with nicer formatting]Note that this PDF file is formatted in a wide-for-screen layout that is probably not good for printing.

After seeing Iron Man II with Lance earlier, a movie with bountiful toroids, and now that the kids are tucked in, finishing up the toroidal center of mass calculation started earlier seems like it is in order. This is a problem I’d been meaning to try since reading this blog post for the center of mass of a toroidal wire segment

# Center of mass.

With the prep done, we are ready to move on to the original problem. Given a toroidal segment over angle , then the volume of that segment is

Our center of mass position vector is then located at

Evaluating the integrals we loose the and terms and are left with and . This leaves us with

Since , we have a conjugate commutation with the for just

A final reassembly, provides the desired final result for the center of mass vector

Presuming no algebraic errors have been made, how about a couple of sanity checks to see if the correctness of this seems plausible.

We are pointing in the -axis direction as expected by symmetry. Good. For , our center of mass vector is at the origin. Good, that’s also what we expected. If we let , and , we have as also expected for a tiny segment of “wire” at that position. Also good.

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