AP® Physics C: Mechanics · Unit 2 · Topic 2.1
Center of Mass of Point Masses
The center of mass is the single point that moves as if all the system's mass were concentrated there — found by averaging each mass's position, weighted by how much mass it carries.
Throw a wrench spinning across a table and most of it traces a messy path. But one point sails through a smooth straight line, unbothered by the tumbling. That point is the center of mass: the average position of all the mass in the system, weighted by how much mass sits where. Heavy chunks pull the average toward themselves; light chunks barely tug it at all.
The formula just formalizes that weighting: x_cm = (m1·x1 + m2·x2 + ... ) / (m1 + m2 + ...). Each mass votes for its own location, and its vote counts in proportion to its size. In two or three dimensions, you do this separately for each coordinate — the x-coordinates give you x_cm, the y-coordinates give you y_cm, completely independent calculations that happen to describe the same point.
This idea is the quiet foundation under everything else in this unit. When you write F_net = ma for a system of many particles, the a that shows up is the acceleration of the center of mass — that is the only reason treating a spinning, tumbling, deforming object as a single point ever works. Later, when mass is spread continuously through an object rather than sitting at discrete points, this same weighted average becomes an integral, but the idea underneath stays identical: find where the mass balances.
Key terms
- Center of mass
- The mass-weighted average position of every particle in a system, which moves as a single point under the net external force.
- Point mass
- An idealized object whose entire mass is treated as concentrated at one location.
- Mass-weighted average
- An average where each position is multiplied by its mass before summing, then divided by the total mass.
Watch it explained
- How it worksHow does a mass-weighted average of positions locate the center of mass?
- Worked exampleFind the center of mass of a 2 kg block at x = 0 and a 6 kg block at x = 4 m.
- Worked exampleFind the center of mass of 1 kg, 3 kg and 2 kg placed at (0,0), (4,0) and (0,3) meters.
- Common mistakeWhy isn't the center of mass always somewhere inside the object itself?
Check yourself
End of topic.