2.1 Systems and Center of Mass

A system is whatever you draw an imaginary boundary around. A single block can be a system, and so can two blocks tied together, or a ball plus Earth. Everything outside the boundary is the environment.

The choice matters because it decides which interactions are internal (between parts of the system) and which are external (between the system and its environment). Only external forces can change how the system as a whole moves. Energy and mass can also cross the boundary, for example when a person does work on a box or a rocket throws out exhaust.

A system's properties come from the interactions inside it. A spring is springy because of forces between its atoms. Some changes outside a system can change its internal structure too, as when a hard push crushes a soda can.

If the parts of a system move together and their internal details don't affect the question, treat the system as a single object. A car rolling down a road is one object for most problems.

A smart choice of system can make a problem easier. For two blocks joined by a string, choosing both blocks as the system makes the string's tension internal, so it drops out of the equation for the system's acceleration. Choosing one block alone brings the tension back as an external force, which is how you then find it.

Sometimes the internal structure does matter. Parts of a system can move differently from each other and from the system as a whole. When a firework bursts, the pieces fly in every direction, but the system's center of mass keeps following the same smooth path it was on, as long as gravity is the only outside force.

The center of mass is the average position of a system's mass, with heavier parts counting more. Along the x-axis:

x_cm = (m₁x₁ + m₂x₂ + m₃x₃ + …) ÷ (m₁ + m₂ + m₃ + …)

Do the same with y-coordinates to get y_cm in two dimensions. The center of mass always lies closer to the heavier mass. For two equal masses it's exactly halfway between them.

For a symmetric object with evenly spread mass, the center of mass lies on every line of symmetry, so it's at the geometric center of a uniform rod, sphere, disk or rectangle. It doesn't have to be inside the material: a ring's center of mass is at the empty middle.

To combine objects, replace each symmetric piece with a single point mass at its own center, then use the formula.

AP Physics 1 only expects center-of-mass calculations for up to five particles arranged in two dimensions, or for highly symmetric objects. For anything else, you only need to reason about roughly where it is.