### Center of Mass

The *center of mass* (or *centroid*) of a rigid body is
found by averaging the spatial points of the body
weighted by the mass of those points:^{B.12}

*mass-weighted average location*of the object. For a continuous mass distribution totaling up to , we can write

*mass density*at the point . The total mass is

A nice property of the center of mass is that gravity acts on a
far-away object as if all its mass were concentrated at its center of
mass. For this reason, the center of mass is often called the
*center of gravity*.

#### Linear Momentum of the Center of Mass

Consider a system of point-masses , each traveling with vector velocity , and not necessarily rigidly attached to each other. Then the total momentum of the system is

Thus, the *momentum*
of any collection of masses
(including rigid bodies) equals the total mass times the
*velocity of the center-of-mass*.

####

Whoops, No Angular Momentum!

The previous result might be surprising since we said at the outset
that we were going to *decompose* the total momentum into a sum
of linear plus angular momentum. Instead, we found that the total
momentum is simply that of the center of mass, which means any angular
momentum that might have been present just went away. (The center of
mass is just a point that cannot rotate in a measurable way.) Angular
momentum does not contribute to linear momentum, so it provides three
new ``degrees of freedom'' (three new energy storage dimensions, in 3D
space) that are ``missed'' when considering only linear momentum.

To obtain the desired decomposition of momentum into linear plus
angular momentum, we will choose a fixed reference point in space
(usually the center of mass) and then, with respect to that reference
point, decompose an arbitrary mass-particle travel direction into the
sum of two mutually orthogonal vector components: one will be the
vector component pointing *radially* with respect to the fixed
point (for the ``linear momentum'' component), and the other will be
the vector component pointing *tangentially* with respect to the
fixed point (for the ``angular momentum''), as shown in
Fig.B.3. When the reference point is the center of mass, the
resultant radial force component gives us the force on the center of
mass, which creates *linear momentum*, while the net tangential
component (times distance from the center-of-mass) give us a resultant
*torque* about the reference point, which creates *angular
momentum*. As we saw above, because the tangential force component
does not contribute to linear momentum, we can simply sum the external
force vectors and get the same result as summing their radial
components. These topics will be discussed further below, after some
elementary preliminaries.

**Next Section:**

Translational Kinetic Energy

**Previous Section:**

Conservation of Momentum