Center of Mass
- Define the center of mass as the mass-weighted average position
- Locate the center of mass of a two-object system
- Explain how the center of mass moves when no external force acts
The center of mass is a weighted average
The center of mass (CM) is the average position of all the mass in a system, weighted by mass. It sits closer to the heavier object, and for a symmetric uniform body it lies at the geometric center. The CM is the single point where you could balance the whole system, and the point that behaves as if all the mass were concentrated there.
The center of mass follows the net external force
A remarkable fact: the center of mass of a system moves exactly as though the total mass sat at that point and the net external force acted on it. Internal forces (like two parts pushing apart, or a wrench flexing) cannot move the CM. So a wrench tossed spinning across frictionless ice has a CM that travels in a straight line at constant velocity, even while the wrench whirls around it.
A 2 kg mass sits at x = 0 and a 6 kg mass sits at x = 8 m. Where is the center of mass?
- 1.Multiply each mass by its position: 2 × 0 = 0 and 6 × 8 = 48.
- 2.Add these: total = 0 + 48 = 48 kg·m.
- 3.Divide by the total mass: x_cm = 48 ÷ (2 + 6) = 48 ÷ 8.
- 4.Result: x_cm = 6 m — closer to the heavier 6 kg mass, as expected.
The center of mass is not the midpoint unless the masses are equal. It always leans toward the heavier object — always check whether your answer sits closer to the larger mass.
A 1 kg mass is at x = 0 and a 3 kg mass is at x = 4 m. Where is the center of mass?
A wrench is given a spinning shove across frictionless ice. How does its center of mass move?
Internal forces never move the center of mass. Whenever a problem shows an object breaking apart or flexing with no external force, the CM keeps whatever motion it had — a fast way to reason about explosions and recoil.
Answer the 2 checkpoints as you read.
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