AP Physics C: Mechanics Unit 2 Cheatsheet: Force and Translational Dynamics

System Definition: An object or collection of objects chosen for analysis, treated as distinct from its environment; a system may be a single object (often modeled as a point particle with no internal structure), and forces are classified as internal (between parts of the system) or external (from the environment).

• Treat as single object at center of mass when internal structure is not important
• Internal structure affects analysis; external changes may alter substructure

Center of Mass Location:

• Discrete: $\vec{x}_{\mathrm{cm}}=\frac{\sum m_{i} \vec{x}_{i}}{\sum m_{i}}$
• Continuous: $\vec{r}_{\mathrm{cm}}=\frac{\int \vec{r} \, dm}{\int dm}$
• On lines of symmetry for uniform distributions

Linear Mass Density: $\lambda=\frac{d}{d\ell}m(\ell)$; Total mass: $M=\int \rho(r) \, dV$

Force Fundamentals:

• Vector describing interaction between two objects—never self-exerted
• Contact forces are macroscopic effects of interatomic electric forces

Free-Body Diagram Rules:

• Show all forces exerted on object by environment
• Draw as vectors originating from center of mass (dot)
• Individual forces as separate arrows, side-by-side if parallel
• Choose axes parallel to acceleration direction (e.g., along incline)

Newton's Third Law: $\vec{F}_{A \text{ on } B}=-\vec{F}_{B \text{ on } A}$

• Internal forces do not affect center-of-mass motion

Ideal Spring Force:

• Force proportional to displacement from equilibrium
• Always directed toward equilibrium
• Ideal: negligible mass, linear response

Springs in Series:

Springs in Parallel:

Series vs. Parallel: Series $k_{\mathrm{eq}}$ is smaller than any individual $k$; parallel $k_{\mathrm{eq}}$ is larger.