Last time we solved the motion of two bodies numerically from an inertial reference frame.
Now we introduce the notion of the center of mass, or barycenter, of the system, which is just a sort of weighted average of the positions.
\[ {\bf R}_\mathrm{COG} = \frac{m_0 {\bf r}_0 + m_1 {\bf r}_1}{m_0 + m_1} \]
It follows that the velocity and acceleration are:
\[ \begin{align} {\bf \dot{R}}_\mathrm{COG} &= \frac{m_0 {\bf \dot{r}}_0 + m_1 {\bf \dot{r}}_1}{m_0 + m_1} \\ {\bf \ddot{R}}_\mathrm{COG} &= \frac{m_0 {\bf \ddot{r}}_0 + m_1 {\bf \ddot{r}}_1}{m_0 + m_1} \\ \end{align} \]
Since the only force we consider here is the gravitational attraction between the masses, we found previously that \({\bf F}_0 = -{\bf F}_1\), so substituting \({\bf F}_i = m {\bf \ddot{r}}_i\):
\[ m_0 {\bf \ddot{r}}_0 + m_1 {\bf \ddot{r}}_1 = 0 \]
So we see that \({\bf \ddot{R}}_\mathrm{COG} = 0\): the barycenter is not accelerating, and is therefore an inertial reference frame.
Example
Our previous example written more compactly:
df <- data.frame(m = c(1.0e26, 5.0e25), x = c(100, 150), y = c(120, 50))The barycenter lies on the line between the masses and a little closer to the larger of the two, \(m_0\).
<- Reference frames | Relative motion ->