8.4 Center of Mass and Motion of Systems

Center of Mass Concepts

The center of mass (COM) is the average position of all the mass in a system. It lets you treat a complex object or collection of particles as a single point particle when applying Newton's laws. This simplification is what makes problems involving collisions, explosions, and rocket propulsion manageable.

A few things worth knowing about the COM:

• It's unique for any given system at any instant.
• It can be located outside the physical boundaries of an object. A boomerang's COM, for example, sits in empty space between its arms.
• For symmetric objects with uniform density, the COM sits right at the geometric center.

Center of Mass Calculations

For a system of discrete particles, you calculate the COM as a weighted average of positions:

$\vec{r}_{CM} = \frac{\sum_{i=1}^n m_i \vec{r}_i}{\sum_{i=1}^n m_i}$

Here, $m_i$ is the mass of each particle and $\vec{r}_i$ is its position vector. The denominator is just the total mass $M$ of the system.

In practice, you'll usually break this into components:

$x_{CM} = \frac{\sum m_i x_i}{M}, \quad y_{CM} = \frac{\sum m_i y_i}{M}$

For continuous objects, the sums become integrals:

$\vec{r}_{CM} = \frac{\int \vec{r} \, dm}{\int dm}$

If the object has uniform density, $dm$ can be rewritten using the density ($dm = \rho \, dV$ for a volume, or $\lambda \, dx$ for a rod, etc.), and the COM often coincides with the geometric center. A uniform sphere, cube, or cylinder all have their COM at their center of symmetry.

Symmetry shortcut: If an object has an axis of symmetry and uniform density, the COM must lie on that axis. This can save you a lot of calculation.

Motion of Systems

How Newton's Laws Apply to the COM

The real power of the COM is this: the center of mass of a system moves as though all the mass were concentrated there and all external forces acted on that point. Mathematically:

$\vec{F}_{net,\, ext} = M\vec{a}_{CM}$

This means internal forces between parts of the system don't affect the COM's motion at all. They always come in Newton's Third Law pairs that cancel out. Two ice skaters pushing apart, fragments of an explosion, billiard balls colliding: in every case, the COM keeps moving exactly as it would have without the internal interaction.

Momentum of a System

The total linear momentum of a system equals the total mass times the velocity of the COM:

$\vec{P} = M\vec{v}_{CM}$

If no net external force acts on the system, $\vec{P}$ is conserved, which means $\vec{v}_{CM}$ stays constant. This is why, during a collision between two billiard balls, the COM of the two-ball system glides along at the same velocity before, during, and after impact.

Kinetic Energy of a System

The total kinetic energy can be split into two parts:

• Translational KE of the COM: $KE_{trans} = \frac{1}{2}Mv_{CM}^2$
• Rotational KE about the COM: $KE_{rot} = \frac{1}{2}I\omega^2$

The translational piece tracks the bulk motion of the system, while the rotational piece accounts for spinning. For this unit, you'll mostly focus on the translational part.

Key Applications

• Collisions and explosions: Internal forces don't change $\vec{v}_{CM}$. You can track the COM to check your answers or simplify analysis.
• Rocket propulsion: A rocket expels mass, so both its mass and velocity change over time. But if the only external force is gravity, the COM of the entire system (rocket + exhaust) follows the same parabolic trajectory it would have without the engine firing.
• Variable mass systems: When mass enters or leaves a system (sand falling off a conveyor belt, a chain piling on a scale), apply conservation of momentum carefully, accounting for the changing mass.