TEX document #1 page 1

1. Center of mass

Suppose that, for i = 1, 2, …, N, there is a particle of mass m_{i} located at position r⃗_{i}. Then

M = ∑_{i=1}^{N} m_{i}

is the total mass. The center of mass is a vector (position) defined by

\overline{r⃗} = r⃗_{cm} = \frac{∑_{i=1}^{N} m_{i}r⃗_{i}}{M} .

[Insert graphic of points labeled by masses and radius vectors. (For the experts, that should be “radii vectores”.)]

This is a weighted average of the positions of the N particles. (If all the masses are equal, it is just the ordinary average of the positions —

\frac{∑_{i=1}^{N} r⃗_{i}}{N}

— the center of the particle cloud.)

In components, we have

\overline{x} = x_{cm} = \frac{∑_{i=1}^{N} m_{i}x_{i}}{M}

[Insert graphic of y–z plane seen on edge, accompanied by two particles, one on each side, with perpendicular distances labeled x_{1} and −x_{2} .] and similarly

\overline{y} = \frac{∑_{i=1}^{N} m_{i}y_{i}}{M} , \overline{z} = \frac{∑_{i=1}^{N} m_{i}z_{i}}{M} .

2. Moment of inertia

Now for something different, but similar …

The moment of inertia of the particles about (or around) the z axis) is

I_{z} = ∑_{i=1}^{N} m_{i}(x_{i}^{2} + y_{i}^{2}) = ∑_{i=1}^{N} m_{i}r_{i}^{2}.

[Insert graphic of z axis pointing out of the plane (⊙), with particles located at distances r_{1} and r_{2}.]

\frac{I_{z}}{M} is the weighted average of the squares of the distances r_{i} of the particles from the z axis. Similarly,

I_{y} = ∑_{i=1}^{N} m_{i}(z_{i}^{2} + x_{i}^{2}), I_{x} = ∑_{i=1}^{N} m_{i}(y_{i}^{2} + z_{i}^{2}).

It is up to you to read the physics textbook to learn what centers of mass and moments of inertia have to do with total momentum, angular momentum, and kinetic energy.