Rotation & Angular Dynamics · Physics

Turn angle vs time into a usable motion model.

Angular displacement, angular velocity, and angular acceleration describe how rotation changes. Under constant angular acceleration, you get a clean set of kinematics equations.

This topic

Angular Displacement, Velocity, and Acceleration

Define θ, ω, α clearly, then use constant-α equations and connect to tangential motion.

Definitions

Angular displacement (θ)

Angular displacement tracks the change in angle about a chosen axis. Treat it as a signed quantity with a clear positive direction.

Δθ and what it measures (change, not absolute “position”)
Radians and why they simplify formulas
Average vs instantaneous angular change
Interpreting wrap-around and multiple turns

Rate

Angular velocity (ω)

Angular velocity is the rate of change of angle. Its sign tells you the rotation direction when paired with the chosen axis.

Average: ω̄ = Δθ/Δt
Instantaneous: ω = dθ/dt
Units (rad/s) and interpreting negative values
Relating ω to tangential speed: v = rω

Change of rate

Angular acceleration (α)

Angular acceleration tells you how angular velocity changes. It can speed up rotation, slow it down, or reverse it depending on signs.

Average: ᾱ = Δω/Δt
Instantaneous: α = dω/dt = d²θ/dt²
When α and ω have the same sign
Tangential acceleration link: a_t = rα

Model

Constant angular acceleration equations

With constant α, rotational kinematics mirrors 1D constant-acceleration motion. Use these equations when the assumptions match the situation.

ω = ω₀ + αt
θ = θ₀ + ω₀t + ½αt²
ω² = ω₀² + 2α(θ − θ₀)
Recognizing when α is approximately constant

Practice

Practice & Exercises

Practice computing θ, ω, α and applying constant-α equations, then translate results into tangential motion at a radius.

Convert rpm ↔ rad/s and track signs
Use constant-α kinematics in word problems
Find r-dependent v and a_t from ω and α
Distinguish tangential vs centripetal acceleration
Exam-style mixed rotational kinematics sets