Center of Mass Example

Let m > 0. Consider the region bounded by y=xm, y=xm+1, x=0, and x=1.

You can (and should) find the center of mass of this region (or, if you prefer, the centroid of the region) to be the point with coordinates
equation 1

The intersection of the moving vertical and horizontal lines in the animation mark the center of mass as m varies over the range from 0.1 to 4.02. You can see that as long as m is small enough, the center of mass is inside the region.

When

equation 2

the center of mass will be on the upper curve. Solving this equation to find this critical value can be done using numerical methods. It turns out that this critical value is

m=2.369724658859673265580878... .

Exercises

  1. Check that the center of mass of the region is as expressed above by working out the appropriate integrals.
  2. Verify that if m=2.3, then the center of mass is inside the region, while if m=2.4 then the center of mass is outside.

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