PC1672 Advanced dynamics

3.5 Reduced mass

Newton's third law tells us that the forces that two objects exert on each other are equal in magnitude and opposite in direction. The equations of motion for two objects of mass and , with no external forces, are thus

$\displaystyle m_1\ddot{\bf x}_1$	$\textstyle =$	$\displaystyle -{\bf F}({\bf r})$
$\displaystyle m_2\ddot{\bf x}_2$	$\textstyle =$	$\displaystyle +{\bf F}({\bf r})$

where ${\bf r}={\bf x}_2-{\bf x}_1$ is their relative position.

Adding these gives the equation of motion for the centre of mass of the system

$\begin{displaymath}M\ddot{\bf X}=0\end{displaymath}$

where

is the total mass and

$\begin{displaymath}{\bf X}={m_1{\bf x}_1+m_2{\bf x}_2\over m_1+m_2}\end{displaymath}$

is the position of the centre of mass. This shows that in the absence of external forces the centre of mass moves with constant velocity or, equivalently, the total momentum is conserved.

We can also combine the two equations to get one describing the relative motion

$\begin{displaymath}\mu\ddot{\bf r}={\bf F}({\bf r})\end{displaymath}$

where the reduced mass is given by

$\begin{displaymath}{1\over\mu}={1\over m_1}+{1\over m_2}\end{displaymath}$

This equation looks just like that for a single object of mass $\mu$ moving in a force field ${\bf F}({\bf r})$ .

In the limit where one object is much lighter than the other, $m_2\ll m_1$ , the reduced mass becomes approximately equal to the lighter mass, $\mu\simeq m_2$ . The heavy object hardly recoils in this limit, and so the lighter object is effectively moving in a fixed force field.

Textbook references

K&K 9.2
F&C 7.3
B&O 6.1
M 8.2
M&T 8.2

Mike Birse
17th May 2000