4.5 Center of Mass

Defining center of mass

The center of mass (COM) is the point representing the average position of a system's mass, weighted by each piece's mass and location: x_cm = (Σmᵢxᵢ)/M for discrete masses, or x_cm = (1/M)∫x dm for continuous objects. It behaves as if all the mass and external force were concentrated there — useful for analyzing complex systems as a single particle.

Motion of the center of mass

The COM moves according to Newton's second law applied to the whole system: F_net,ext = M·a_cm. Internal forces (like collisions or explosions) don't affect the COM's motion — only external forces do. If no external force acts, the COM moves at constant velocity, even if individual parts speed up, slow down, or fly apart.

Symmetry and integration

For symmetric, uniform-density objects, the COM lies at the geometric center. For non-uniform or irregular shapes, use integration: x_cm = (1/M)∫x dm, often converting dm using linear density λ, area density σ, or volume density ρ. Setting up limits along the correct axis is key to solving these problems.

Key terms

Center of mass (discrete system)

x_cm = (Σmᵢxᵢ)/M — the mass-weighted average position of all particles in the system.

Center of mass (continuous object)

x_cm = (1/M)∫x dm, integrating over the object's mass distribution.

Newton's second law for a system

F_net,ext = M·a_cm — the net external force determines the acceleration of the center of mass, regardless of internal forces.

Effect of internal forces on COM

Internal forces (collisions, explosions) can change individual particle motion but never change the motion of the system's center of mass.

COM of a symmetric uniform object

Located at the geometric center (centroid) when density is uniform and the shape is symmetric.

Try an AP-style question

A 2 kg mass is at x = 0 and a 3 kg mass is at x = 5 m. Where is the center of mass?

1. x = 5 m
2. x = 3 m
3. x = 2 m
4. x = 2.5 m