# Center of Mass

For any system of particles one can define a center of mass. What is a system of particles you ask? Any solid object can be regarded as a system of particles---so can a sample of gas or liquid---so can a set of free particles like the bodies of a solar system. What is the center of mass? It is the mass-weighted mean position of the particles:

``` R_CM=(sum m_i r_i)/(sum m_i) ,
```

where i indexes the particle, ``sum'' means sum on index i, CM means center of mass, and the positions r and R are vectors. If the particles are on a line, one can forget the vector nature of position and just use linear measure.

• Example (a)---What is the center of mass of the Earth-Moon system. Well taking the Earth center as the origin

``` x_Ea=0  ,  x_Mo= 60 R_Ea,   m_Ea=1,  and m_Mo=0.0123,
```

where Earth mass as a unit of mass and the Earth's radius R_Ea as a unit of distance (Ab-14-4). The Moon's position is only a mean position. Behold

```R_CM=(60*0.0123)/(1+0.0123)=0.73  R_Ea  .
```

Well the center of mass of the system is inside the Earth. This is because the Earth is so much more massive than the Moon that it dominates the center of mass location.

Regarding the Earth as a point mass, one can approximately say that the Earth is located at the center of mass. Thus one can say approximately the Moon orbits the Earth although more precisely both bodies orbit the center of mass in nearly elliptical orbits. Similarly, the Sun is dominates the mass of the solar system that it is approximately the center of mass of the solar system and one can say to good approximation that the planets orbit the Sun. The planetary system mass is only

```1.342*10-3 solar masses
```
(Ab-14-2).