Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 587

To compute the center of mass mathematically, we imagine the object
being divided up into a very large number of small “mass elements.” If
there are n such elements, and we denote the mass and position of the ith
element as m i and r i , respectively, then the center of mass r c is simplify
the weighted average of the positions of all the mass elements.

Calculating the Center of Mass

n

1
M

r c = m i r i . (12.22)

i

In Equation (12.22), M is the total mass of the object

n

M = m i .

i

For our purposes, the most important property of the center of mass is that
if the object rotates, it will rotate about its center of mass. This assumes,
of course, that the object is freely rotating and there isn't a constraint
compelling it to rotate about some other point.
As an example, consider a sledge hammer. Clearly, the center of mass
of the sledge hammer is close to the heavy end of the sledge, not in the
middle of the handle. Assume we throw the hammer across the room. As
it tumbles through space, any arbitrary point on the hammer will trace out
a complicated spiraling shape. The center of mass, however, moves in a
parabola, in perfect agreement with the kinematics equations we learned in
Chapter 11.
The authors couldn't resist the opportunity to chuck big objects around,
so we verified this hypothesis experimentally, and you can, too. 13 We
started with the odd-shaped piece of particle board, whose center of mass
had been experimentally located and clearly marked. Next, the fun part:

13 If you do decide to do this, take this advice: (1) Please be safe. Seriously, we assume
no liability for people being dumb. (Speaking of dumb, one author had to replace his
father's lawn mower wheel, which was found to have a perfectly parabolic trajectory,
but alas unable to withstand the landing impact.) (2) We took still pictures at about
4 Hz, but using a nicer camera capable of taking more frames per second, or perhaps
extracting frames from video might work better. (3) Please send us your pictures at
gamemath.com!