AP Physics 1: Rotational Motion in 3 Key Equations (no calculus required)
Rotational motion is roughly 20% of the AP Physics 1 exam and the section students fear most. The reason is misleading: rotational motion is just mechanics with different variable names. If you can do linear kinematics, you can do rotational kinematics. Here are the three core equations and the linear-rotational mapping that makes everything click.
The linear-rotational mapping (memorize this)
| Linear | Rotational | Relationship |
|---|---|---|
| position x | angle θ | x = rθ |
| velocity v | angular velocity ω | v = rω |
| acceleration a | angular acceleration α | a = rα |
| mass m | moment of inertia I | I depends on shape |
| force F | torque τ | τ = rF sinθ |
| F = ma | τ = Iα | Newton's 2nd for rotation |
| KE = ½ mv² | KE = ½ Iω² | rotational kinetic energy |
Every rotational problem becomes a linear problem if you translate the variables.
Equation 1 — Rotational kinematics
The kinematic equations are identical to linear ones with rotational variables:
- ωf = ωi + αt
- θf = θi + ωit + ½αt²
- ωf² = ωi² + 2αΔθ
Use these when the question asks how long, how fast, or how far something rotates given an angular acceleration.
Equation 2 — Newton's 2nd law for rotation
Στ = Iα. Net torque on an object equals its moment of inertia times its angular acceleration. The torque is τ = rF sinθ, where r is the lever arm, F is the force, and θ is the angle between them.
This is what you use for “a force is applied to a rotating object, find the angular acceleration” problems.
Equation 3 — Conservation of angular momentum
L = Iω. When no external torque acts on a system, angular momentum is conserved: Iiωi = Ifωf.
Classic AP problem: a figure skater pulls her arms in. Her moment of inertia I decreases, so her angular velocity ω increases. This is a one-equation problem if you spot conservation.
How to recognize which equation to use
- Kinematics if the problem gives you angular velocity, angular acceleration, or time and asks for one of those.
- Newton's 2nd for rotation if the problem mentions torque, forces causing rotation, or angular acceleration with a force applied.
- Conservation of angular momentum if the system changes shape (skater pulling arms in, person walking to the edge of a rotating disk) with no external torque.
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