AP® Physics 1: Algebra-Based · Unit 2 · Topic 2.1
Motion of the Center of Mass
The center of mass of a system moves as if all its mass were one point, pushed only by the net external force. Internal forces, like an explosion or a spring pushing two carts apart, never change that motion.
Every object or group of objects has a center of mass: the mass-weighted average position of everything in the system. That point obeys Newton's second law all by itself. Add up every external force acting on the system, divide by the total mass, and you get the acceleration of the center of mass. Forces between parts of the system - a spring pushing two carts apart, an explosion flinging fragments outward, a person walking inside a boat - cancel out in pairs because of Newton's third law. They can rearrange where the mass is distributed, but they cannot shift the center of mass off its path.
Take two carts connected by a compressed spring, starting at rest. Release the spring and the carts fly apart in opposite directions. Each cart feels a force, but those two forces are equal and opposite internal forces, so the net external force on the two-cart system is still zero. The center of mass does not move at all; it sits frozen in place while the carts slide away from it symmetrically in momentum terms, even if their masses differ.
This idea is the geometric twin of momentum conservation. If the net external force is zero, the center of mass moves at constant velocity, which is exactly the statement that total momentum is conserved. In a collision, the two objects exert internal forces on each other, so the center of mass velocity before the collision equals the center of mass velocity after. Center-of-mass motion is what makes momentum conservation visual: picture that one averaged point gliding smoothly through the whole messy event.
Key terms
- Center of mass
- The mass-weighted average position of all the mass in a system, which moves as if a single point particle.
- Internal force
- A force between two parts of the same system, which always comes in equal and opposite pairs by Newton's third law.
- External force
- A force exerted on the system by something outside it, the only kind that can change the center of mass motion.
- Net external force
- The vector sum of all outside forces acting on a system, which determines the center of mass acceleration.
Watch it explained
- How it worksHow does the net external force control the center of mass and ignore everything internal?
- How it worksWalk me through the center of mass of two carts pushed apart by a compressed spring.
- Worked exampleFind the velocity of the center of mass of a 3 kg cart at 4 m/s and a 5 kg cart at rest.
- Ties togetherHow does center-of-mass motion connect to conservation of momentum in a collision?
Check yourself
End of topic.