Center of Mass

To locate a center of mass, you weight each piece of mass by its distance from some reference point, add those up, and divide by the total mass: x_cm = (m1 x1 + m2 x2 + ...) / (m1 + m2 + ...). Heavier chunks pull the center of mass toward them. Two equal masses balance exactly in the middle; a heavy block near a light one drags the average position close to itself. This works in one dimension, along a rod, or in two or three dimensions by doing the same weighted average separately for x, y, and z.

The center of mass matters because it is the one point in a system that moves as if all the external force were applied there and all the mass were concentrated there, no matter how the object spins or deforms internally. A high jumper can arch her body over the bar so that her arms, legs, and head all pass above it, while her center of mass actually traces a path under the bar, because the curved shape of her body shifts the average position outside her torso. The same idea explains the spinning wrench: every point on it follows a complicated looping path, but its center of mass sails through the air in a smooth parabola, exactly like a thrown point mass, because gravity acts on the whole wrench as if it were concentrated there.

This idea sets up everything else in the unit. Free-body diagrams and Newton's second law, F = ma, apply cleanly to the center of mass of a system even when the system is made of multiple parts or is rotating, which is why physicists often simplify a complicated object into a single point before analyzing forces.