Rotational kinematics equations

Intermediate

Analogues of linear equations with θ, ω, α.

Key relationω = ω₀ + αt

With constant α the equations mirror the linear ones: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ. Replace x→θ, v→ω, a→α.

Worked example

A disc at 10 rad/s decelerates at 2 rad/s²: stops after 5 s turning 25 rad.

Common mistake

Use radians consistently.

In more detail

Rotational kinematics equations match the linear ones: ω = ω0 + αt, θ = ω0 t + ½αt², and ω² = ω0² + 2αθ, valid for constant angular acceleration.

Calculating how many turns a flywheel makes as it slows to a stop.
Estimating the time for a spinning drill to reach operating speed.
A flywheel in a motor slows down with constant angular deceleration, and the rotational equations predict how many turns it makes before stopping.

How do I convert between linear and angular quantities?

Use s = rθ, v = rω and a = rα.

Can I use these for variable angular acceleration?

Only over intervals where the angular acceleration is roughly constant.

How do I know which rotational equation to use?

Match the known quantities: angle, angular velocity, angular acceleration and time, just as with linear motion.