Classical Mechanics: Two Body Problem

[teme92]

Homework Statement

Consider two objects with masses $m_1$ and $m_2$ exerting forces on each other with magnitude $F$. If no other net forces act on the objects, they obey the equations of motion

$m_1\ddot r_1=F$, $m_2\ddot r_2=-F$

Show that the corresponding equations of motion for the centre of mass $R$ and the relative position $r$ of the two objects in terms of the total mass $M$ and the reduced mass $\mu$, are given by

$M\ddot R=0$, $\mu\ddot r=F$

and that

$r_1=R+\frac{m_2}{M}r$, $r_2=R-\frac{m_1}{M}r$

Homework Equations

The Attempt at a Solution

Using the relation where $F_{12}+F_{21}=0$ I said $m_1\ddot r_1+m_2\ddot r_2=0$. This equates to $M\ddot R=0$ as asked.

To show $\mu\ddot R=F$, I manipulated $m_1a_1=-m_2a_2$ to get $a_2=-\frac{m_1}{m_2}a_1$. Then using

$a_{rel}=a_1-a_2=a_1+\frac{m_1}{m_2}a_1=(1+\frac{m_1}{m_2})a_1$

$=\frac{m_2+m_1}{m_1m_2}m_1a_1$, Therefore $\frac{F}{\mu}=a_{rel}=\ddot r$

And finally $\mu\ddot r=F$

I can't find a way of finding $r_1$ and $r_2$ as asked however. I tried using $m_1r_1+m_2r_2=0$ but I can't find it as I'm asked. Any help here would be greatly appreciated.

 

[sgd37]

Use the defining equations for the center of mass R and relative position r

 

[teme92]

Worked a charm cheers!