Computing the center of mass

The center of mass is an important feature of single objects, and perhaps even more important for collections of objects.

Exactly where does gravity pull an object? At its center of mass
If I need to compute the gravitational force between two objects, what is the R I should use in the formula? The distance between the centers of mass.
If we want to design a plane so that the engine will push the plane forward, without any rotation, where should be put the engine? Behind the center of mass.
If if a system consists of several pieces which break up or come together under mutual forces only, then the center of mass of the system will continue to move with a constant velocity; that can be useful when dealing with objects which fly through the air while exploding

Image courtesy of Gizmodo

Now, when an object is a compact sphere, the center of mass is easy to find: it's just the geometric center of the body. But what happens if we need to find the center of mass of a system consisting of several objects?

Computing the center of mass of discrete little objects

What about extended objects? Some of the nice rules you've learned so far in the class, such as



    " ... in projectile motion, the x-velocity remains 
      constant while the y-velocity changes linearly with
      time.  The trajectory of an object thus traces
      a parabolic curve ...."

don't seem to work if you pick any arbitrary part of an extended body. For example, if I toss a baseball bat up into the air, a movie might show this:

Speaking of real movies, look at this real motion of a sort of dumbbell-shaped object:

But how can we find the center of mass of an extended object?

Computing the center of mass of an extended object