Center of Mass

The center of mass or center of gravity is the point in a rigid object at which it will always balance. It's very useful in physics and engineering because it turns out that it usually works in calculations to assume that all of an object's mass is concentrated at this point, the center of mass. This is much easier than accounting for each bit of the thing individually. For example, think of a book hanging over the edge of a desk. If you want to calculate whether it will stay put or fall, you have to consider that some of the book is solidly on the desk and contributing to stability while some is hanging over and trying to make it fall. In this case, you can either use calculus to sum up all the effects and see what prevails or you can consider a single point: the center of mass. If the center of mass is over the desk, it's stable, and if not, it falls.

There's an easy way to find the center of mass of a flat disk-like object like a plate. Hang it by some point and draw a line straight down from the hanging point. Now repeat from a different point. Those lines intersect at the center of mass. This would actually work with something that wasn't thin and flat, but you'd have to hang it three times, and you'd have to be able to draw planes instead of lines (which would not be easy)

Another trick works for things that are mostly one-dimensional. Maybe it's a hammer, or rake, or pencil, or something similar. You rest the thing on top of your hands, or preferably on just two fingers. You're not holding onto the thing. If you're successful so far, that means the center of mass must be between the two supports. Now, move your hand together. The hand farthest from the center of mass will have less weight on it, so it will have less friction, so it will move more easily. So the hand farthest away is always the one moving, so when they eventually meet it will be at the center of mass. Try it! 

Mathematically, to find the center of mass, you assign three axis to the thing (say x, y, z) at right angles from one another. Pick some arbitrary point as the origin. Now consider them one at a time. For x, sum up each bit of mass times it's distance from the y-z plane. Then divide that sum by the total mass and that's the distance of the center of mass from your origin. Repeat for the other two axis and you have the coordinates of the center of mass.