# 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?

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: