How Are Distances from the Center of Mass Determined in Classical Mechanics?

[velo city]

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Homework Statement

I am reading a classical mechanics textbook and I don't understand how they found that.

r₁' = -\frac{m_{2}}{m_{1}+m_{2}}r

and

r₂' = \frac{m_{1}}{m_{1}+m_{2}}r

r₁' is the vector from the center of mass R to m₁ and r₂' is the vector from the center of mass R to m₂.

Homework Equations

Center of mass = \frac{m_{1}r_{1}+m_{2}r_{2}}{m_{1}+m_{2}}

The Attempt at a Solution

 

[DrClaude]

If the origin is at the center of mass (which it is for $r_1'$ and $r_2'$) then by definition of the center of mass
$$
\sum_i m_i r_i = 0
$$
The distance between masses 1 and 2 being $r \equiv r_2' - r_1'$, we have
$$
\begin{align}
m_1 r_1' &= -m_2 r_2' \\
m_1 r_1' &= -m_2 (r + r_1') \\
r_1' (m_1 + m_2) &= - m_2 r \\
r_1' &= -\frac{m_2}{m_1 + m_2} r
\end{align}
$$