The Centre of Mass and the Motion of a System
The centre of mass of any system moves as if all the mass sat there and every external force acted on it, whatever the internal forces do. Internal forces cancel in pairs, which is why an exploding shell keeps its parabola.
What a learner can do afterwards
- Locates the centre of mass of a set of particles and of a simple continuous body by integration
- Shows that internal forces cancel and so cannot move the centre of mass
- Predicts the path of the centre of mass of a body that breaks apart in flight
1 · Read
The centre of mass is the mass weighted average of every position in the system. For separate lumps you add each mass times its position and divide by the total mass. For smooth bodies that sum becomes an integral over the shape. Symmetry hands out free answers: uniform rods balance at the middle and rings at the empty centre. Total momentum is simply total mass times the velocity of this point.
Park 2 kg at x = 0 and 4 kg at x = 3: add 0 + 12 and divide by 6 to get x = 2. Inside pushes always cancel in pairs: when two parts shove each other, equal and opposite forces sum to zero. So internal drama leaves total momentum untouched. A trio with 1 kg at 1 and 3 kg at 5 gives (1 + 15) over 4 = 4.
Only outside forces can move the centre of mass. With none acting, it cruises at constant velocity. A shell bursts mid flight: shards spray everywhere, but the centre of mass keeps the original parabola. The breakup is internal, so the average path never notices. Explosions, collisions, and separating stages all obey this quiet rule.
Take the chaos shortcut: find the special point and track one clean trajectory instead of every fragment. For a bursting firework, draw the smooth arc first, then spray shards around it. Always ask which forces are outside before anything else.
Average the mass positions, cancel the inside forces, and let outside forces steer the point.
More to read
9.6 Center of Mass
Crowds move like single bodies if you watch the right point. The centre of mass is the mass weighted average of every position in the system. For separate lumps you sum mass times position and divide by the total. For smooth bodies the sum becomes an integral over the shape. Symmetry often gives the answer free: uniform rods balance at the middle, rings at the empty centre.
9.3 Conservation of Linear Momentum
Inside pushes cancel in pairs. When two parts of a system shove each other, Newton hands back equal and opposite forces that sum to zero. Internal drama therefore leaves total momentum untouched. Only outside forces can change it, and with none the centre of mass glides on undisturbed. Explosions, collisions, and separating stages all obey this quiet rule.
9.1 Linear Momentum
Momentum is motion with mass behind it. Total momentum of a system is simply total mass times the velocity of its centre of mass. That single relation replaces pages of particle tracking. A bursting shell keeps its centre of mass on the original parabola while shards spray around it. Find the special point and the chaos becomes one clean trajectory.
Explore more
2 · Watch
En anglaisTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.