Rotational Kinematics

You’ll be able to

Describe rotation with angular position, velocity, and acceleration
Relate linear and angular quantities through v = rω
Apply the rotational kinematic equations that mirror the linear ones

Angular quantities describe spinning

Rotation is described with angular versions of the familiar quantities. Angular position θ (radians) says how far around; angular velocity ω (rad/s) is how fast it turns; angular acceleration α (rad/s²) is how fast ω changes. Every point on a rigid rotating body shares the same ω and α, no matter how far from the axis — the whole object turns together.

Linking angular to linear

A point at radius r from the axis moves along a circle, so its linear speed is v = rω. Points farther from the axis trace bigger circles and therefore move faster, even though every point shares the same angular speed. The same bridge connects accelerations: the tangential acceleration is a_t = rα. This is why the rim of a wheel outruns a point near its hub.

Linear–angular link

v = rω · a_t = rα

r is the distance from the rotation axis. Angular quantities use radians; v and a_t come out in m/s and m/s².

Rotational kinematics (constant α)

ω = ω₀ + αt · θ = ω₀t + ½αt²

Identical in form to the linear equations, with θ→x, ω→v, α→a. Every trick you learned in kinematics carries over.

Worked example

A disk starts from rest and spins up to 12 rad/s in 4 s at constant angular acceleration. A point sits 0.5 m from the axis. Find the angular acceleration and that point’s final linear speed.

1.Angular acceleration: α = (ω − ω₀)/t = (12 − 0)/4 = 3 rad/s².
2.Final angular speed is given: ω = 12 rad/s.
3.Linear speed of the point: v = rω = 0.5 × 12.
4.Compute: v = 6 m/s.

Answer: α = 3 rad/s² and v = 6 m/s

Tip

The rotational kinematic equations are the linear ones in disguise. If you can solve a "starts from rest, accelerates for t seconds" problem for x and v, you can solve the identical one for θ and ω — just swap the symbols.

Checkpoint

A wheel of radius 0.2 m spins at an angular speed of 10 rad/s. What is the linear speed of a point on its rim?

Checkpoint

A turntable starts from rest and reaches 12 rad/s in 4 s at constant angular acceleration. What is its angular acceleration?

On the exam

Watch the difference between angular speed ω (shared by the whole object) and linear speed v (larger the farther you are from the axis). A question about "a point on the rim" almost always wants v = rω, not ω.

Answer the 2 checkpoints as you read.