Internal Forces Never Move the Centre of Mass
The centre of mass of a system moves as if all the mass were concentrated at it and all the external force acted on it. Internal forces, whether attraction, spring or string tension, cannot change its motion.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Zero external force means a fixed centre of mass
Two particles initially at rest that attract each other have speeds v and 2v at some instant, but their centre of mass speed is zero, since their momenta are equal and opposite. Two spheres of masses M and 5M with radii R and 2R, released in free space with centres 12R apart, meet when the centres are 3R apart, so together they have covered 9R. Because the centre of mass does not move, the distances covered are inversely proportional to the masses, so the lighter sphere covers 5/6 of 9R, which is 7.5R.
Acceleration of the centre of mass
For masses 15 kg and 5 kg over a pulley, each has acceleration of magnitude g/2 = 5 m/s², one down and one up, so a_cm = (15(-5) + 5(5))/20 = -2.5 m/s²: the centre of mass accelerates downward at 2.5 m/s² even though the system is internally tied together.
OlympIQ exam lens
- ✓a_cm = F_external / M; internal forces cancel in pairs
- ✓Zero external force and initial rest: the centre of mass stays at rest
- ✓Mutually attracting bodies: distances moved are inversely proportional to the masses
- ✓a_cm = Σ m a / Σ m with signs for direction
Do not set the acceleration of the centre of mass equal to the acceleration of each body. When one body accelerates down and the other up, the centre of mass has a smaller acceleration given by the mass-weighted average with signs.
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