Center of mass

Every system has one point that moves as though all of the mass sat there and all of the external forces acted there. The AP Physics 1 CED says it directly: a system can be modeled as a singular object that is located at the system's center of mass.

Along an axis it is a mass weighted average of positions, printed on the equation sheet as $\vec{x}_{\text{cm}} = \left(\sum m_i \vec{x}_i\right) / \sum m_i$. The heavier a part, the closer the center of mass sits to it. For symmetrical mass distributions the CED says it lies on the lines of symmetry. AP Physics C: Mechanics extends the idea to continuous bodies with $\vec{r}_{\text{cm}} = \int \vec{r}\, dm / \int dm$.

Two consequences get used constantly: gravitational force on a system can be treated as acting at its center of mass, and $\vec{v}_{\text{cm}} = \sum \vec{p}_i / \sum m_i$ stays constant when no net external force acts.

Scope, from the Topic 2.1 boundary statement: AP Physics 1 only expects students to calculate the center of mass for systems of five or fewer particles arranged in a two-dimensional configuration or for systems that are highly symmetrical.

Center of gravity is a common alias. The two points coincide whenever the gravitational field is uniform across the system, which covers every AP Physics 1 situation; the CED uses center of mass throughout.