Topic 2.1 Notes – Systems and Center of Mass

1. What a System Is in Physics

A system is the object or group of objects you choose to analyze. Everything else is the environment. You draw a boundary (real or imaginary) around the system, and that choice controls your equations.

System properties come from interactions

The behavior of a system depends on how its parts interact.

• Gas molecules collide randomly → the gas has pressure and temperature.
• Atoms bonded in different ways → materials have different strengths.
• Springs inside a cart compress → energy is stored internally.

If the internal details do not matter for the question, you can treat the whole system as a single object.

• A car moving down the road → model as one mass.
• Two carts stuck together → treat as one combined mass.
• A projectile that spins → analyze the motion of its center of mass.

Internal vs external forces

This is one of the most tested ideas in Unit 2.

• External forces act across the boundary.
  → They can change the system’s total momentum and motion.
• Internal forces are between parts inside the boundary.
  → They cannot change the total momentum of the system.

If you redraw the boundary, forces can switch categories. For example, friction between two objects:

• If you analyze one object alone → friction is external.
• If both objects are in the system → friction is internal.

That switch is a classic FRQ move. Always ask yourself, what’s inside my boundary?

Energy and mass transfer

A system can:

• Gain or lose energy (heat, work, radiation).
• Gain or lose mass (matter crossing the boundary).

If energy crosses the boundary, the system’s total energy changes.
If mass crosses, the total mass changes.

System vs individual objects

Objects inside a system can behave differently from the system as a whole.

• In an explosion, fragments fly apart differently.
• The center of mass may move smoothly while parts rotate wildly.

Internal structure matters when:

• Parts move relative to each other.
• The object deforms.
• The arrangement changes (like melting or compressing).

2. The Center of Mass

The center of mass (CM) is the mass‑weighted average position of all the mass in a system.

It is the point where you can pretend all the mass is concentrated when analyzing motion.

Big idea:

$$\sum F_{\text{ext}} = M a_{cm}$$

Only external forces determine the acceleration of the center of mass.

So even if parts are moving in complicated ways, the center of mass follows Newton’s Second Law like a single particle.

A spinning football in projectile motion still has a parabolic CM path.

This is exactly how the AP likes to test this idea.

3. How to Find the Center of Mass

A. Symmetry

If mass is distributed symmetrically, the CM lies on the line of symmetry.

• Uniform rod → midpoint
• Uniform disk or sphere → geometric center
• Uniform rectangle → intersection of diagonals

Always check symmetry first. It saves time.

B. Discrete particles in 1D

For particles along an axis:

$$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$$

This is a weighted average.

For two masses:

$$x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$$

Heavier masses pull the CM closer.

Quick example:
A 2 kg mass at 0 m and a 6 kg mass at 4 m.

$$x_{cm} = \frac{(2)(0) + (6)(4)}{8} = \frac{24}{8} = 3 \text{ m}$$

The CM is closer to the 6 kg mass, which makes sense physically.

C. Discrete particles in 2D

Do the same process separately in x and y:

$$x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \qquad y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$$

Final answer is a coordinate pair.

AP scope limit:

• At most five particles in 2D.
• Or highly symmetrical objects.

You will not be asked to integrate continuous mass distributions in Physics 1.

4. Modeling a System at Its Center of Mass

When you treat the system as a single object at its CM:

1. Add all masses → total mass $M$.
2. Identify only external forces.
3. Apply $\sum F_{\text{ext}} = M a_{cm}$.

Internal forces cancel because of Newton’s Third Law pairs.

This is especially powerful in:

• Collision problems
• Explosion scenarios
• Multi‑object motion questions

Often the fastest path to the answer is choosing a smart system boundary.