Center of Mass and Motion of Multi-Particle Systems

The center of mass of a system is the weighted average position of its mass, x_cm = Σm_ix_i ÷ Σm_i, and it moves as if all the mass were concentrated there and all external forces acted on that point: F_ext = Ma_cm. In brief: internal forces, such as in explosions or collisions, cannot change the motion of the center of mass. The total momentum of a system equals the total mass times the velocity of its center of mass, P = Mv_cm. A thrown hammer spins, but its center of mass follows a smooth parabola.

The center of mass lets us treat complicated systems, from two skaters to a rotating spacecraft, as a single point. This guide defines the center of mass, explains how to find it for discrete and symmetric objects, shows how it moves, compares the concepts in a table, works through numerical examples including a person on a boat and corrects common mistakes.

What is the center of mass?

For particles of masses m₁, m₂, … at positions x₁, x₂, …, the center of mass is

x_cm = (m₁x₁ + m₂x₂ + …) ÷ (m₁ + m₂ + …),

and the same formula applies to the y and z coordinates. It lies closer to the heavier mass. For a uniform object with symmetry, the center of mass lies at the geometric center: the middle of a rod, the center of a disc or sphere. For irregular objects, it can be found by hanging the object from different points and drawing plumb lines: they intersect at the center of mass. The center of mass does not have to be inside the body (a ring, a boomerang, or a high jumper arching over the bar). An object balances on a point support when the center of mass is directly above it, and tips over when the line from the center of mass falls outside the base.

How does the center of mass move?

The total momentum of a system of particles is P = Σm_iv_i = Mv_cm, and Newton’s second law for the system is F_ext = dP/dt = Ma_cm, where F_ext is the net external force; internal forces cancel in pairs by the third law. Consequences:

• If the net external force is zero, the center of mass moves with constant velocity (or stays at rest), whatever happens inside the system.
• A firework shell that explodes in the air: the center of mass of the fragments continues along the parabola of the shell, while the fragments move apart.
• A person walking in a boat on still water: the center of mass of the person and boat remains fixed, so the boat moves in the opposite direction.
• A rocket exhaust: the center of mass of the rocket and exhaust gases follows the path determined by external forces such as gravity.

Center of mass and center of gravity compared

Worked example: finding and tracking the center of mass

1. Two masses. A 2.0 kg mass at x = 0 and a 6.0 kg mass at x = 4.0 m. x_cm = (2.0 × 0 + 6.0 × 4.0) ÷ 8.0 = 24 ÷ 8.0 = 3.0 m, closer to the heavier mass.
2. Three masses in a plane. Masses of 1.0 kg at (0, 0), 2.0 kg at (3, 0) and 3.0 kg at (0, 4), in meters. Total mass 6.0 kg. x_cm = (0 + 6.0 + 0) ÷ 6.0 = 1.0 m. y_cm = (0 + 0 + 12) ÷ 6.0 = 2.0 m. The center of mass is at (1.0, 2.0) m.
3. Velocity of the center of mass. Two carts, 2.0 kg at +3.0 m/s and 6.0 kg at −1.0 m/s. v_cm = (6.0 − 6.0) ÷ 8.0 = 0. The total momentum is zero. In a collision between them, the center of mass remains at rest.
4. Person in a boat. A 60 kg person stands at one end of a 120 kg boat, 4.0 m long, on still water (ignore the drag). The person walks to the other end. The center of mass of the system stays fixed. The person moves relative to the boat by 4.0 m. If the boat moves by d in the opposite direction, the person moves 4.0 − d relative to the water: 60 × (4.0 − d) = 120 × d, so 240 = 180d and d = 1.33 m. The boat moves 1.33 m backward, and the person moves 2.67 m forward relative to the water.
5. Exploding projectile. A shell of mass 10 kg explodes at the top of its path into two parts, 4.0 kg and 6.0 kg. Both land at the same time. If the 4.0 kg part lands at 100 m horizontally from the launch point, and the shell would have landed at 250 m, the center of mass lands at 250 m, so 4.0 × 100 + 6.0 × x = 10 × 250, and x = (2,500 − 400) ÷ 6.0 = 350 m.

How was the idea developed?

Archimedes introduced the idea of the center of gravity in the third century BCE in his work on levers and floating bodies. Newton showed in the Principia that the center of mass of an isolated system moves uniformly (Corollary IV to the laws of motion). The concept is the basis of the analysis of the motion of rigid bodies, spacecraft and the two-body problem in astronomy, where the Earth and Moon orbit their common center of mass, located about 4,700 km from the Earth’s center, inside the Earth.

Which mistakes are common?

Mistake 1: Placing the center of mass at the middle of the distance

It is a weighted average, closer to the heavier part.

Mistake 2: Thinking that the center of mass must be on the object

It may lie in empty space, as with a ring or a hollow shape.

Mistake 3: Believing that internal forces move the center of mass

Only external forces change the motion of the center of mass.

Mistake 4: Forgetting a coordinate

In two or three dimensions, calculate each coordinate separately.