Centre of mass

Key ideas

• The centre of mass of a system is the single point that moves as if all the mass were concentrated there and all external forces acted there.
• For masses on a line, the centre of mass position is the mass-weighted average of their positions:

$x_{COM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$

• Each variable, with sub-bullets:
  • $m_1, m_2$ — the masses (kg)
  • $x_1, x_2$ — their positions measured from a chosen origin (m)
• In two dimensions, do the same calculation separately for the $x$-coordinates and the $y$-coordinates to get the point $(x_{COM}, y_{COM})$.
• With no external force, the centre of mass keeps moving at constant velocity — even if the parts of the system push on each other. This is why the centre of mass is the natural point to track in a collision or explosion.

Why the centre of mass matters

• The momentum of a whole system equals its total mass × the velocity of its centre of mass.
• Internal forces (the parts pushing on each other) never move the centre of mass — only external forces do.

Tips

• Measure every position from the same origin, and keep signs. A mass to the left of your origin has a negative position; forgetting the sign is the most common error.
• If a question involves a collision or explosion, the centre of mass moves at constant velocity throughout — stating that is often worth an explanation mark.