Angular Motion

Angular motion is the motion of an object around a fixed axis point, such as a swinging pendulum or an orbiting planet. By convention, measurements of angular motion are considered positive when the motion is counter-clockwise and negative when the motion is clockwise.

Angular Displacement

Angular displacement (θ\theta) is the change (Δ\Delta) in angle in radians (radrad) of the arc length (SS) traveled by an object along a circular path. The angular displacement of an arc length equal to the radius (rr) of a circle is, by definition, 1 rad1\ rad. Angular displacement can therefore be calculated as θ=ΔS/r\theta = \Delta S/r, though the Δ\Delta is often omitted for brevity.

Angular displacement is the basis for angular motion because it allows all points on a rotating line to be measured, rather than just a single point by solving for the missing variable in the angular displacement calculation.

Angular displacementθ=S/r\theta = S/r
Arc lengthS=rθS = r\theta
Radiusr=S/θr = S/\theta

Angular Velocity

Angular velocity (ω\omega) is the change in angular displacement over the change in time (tt), or ω=Δθ/Δt\omega = \Delta\theta/\Delta t (again, the Δ\Delta is often omitted). Essentially, "radians per second".

Angular velocity can be converted to speed (note that speed is not the same as velocity) by substituting arc length for angular displacement. Given S=rθS = r\theta and ω=θ/t\omega = \theta/t, therefore rω=rθ/tr\omega = r\theta/ t which equals S/tS/t and S/tS/t is distance over time, which is speed.

Angular Acceleration

Angular acceleration (α\alpha) is the change in angular velocity over time, or α=Δω/Δt\alpha=\Delta\omega/\Delta t. Essentially, "radians per second per second" or "radians per second squared".

Angular acceleration can be converted to tangential acceleration (AtanA_{tan}), which is acceleration which doesn't account for the change in direction in circular motion, by substituting arc length for angular displacement. Given S=rθS = r\theta and α=ω/t\alpha=\omega/t, therefore rα=rω/tr\alpha=r\omega/t, which is the change in angular velocity over time.

Conversion Tables

Angular velocitySpeedS/tS/t or rθ/tr\theta/t
Angular accelerationTangential accelerationrω/tr\omega/t or rθ/t2r\theta/t^2

Resources to Lean about Angular Motion

Deeper Knowledge on Angular Motion


Learn about gears and cogs


Learn about torque: Force that causes an object to rotate around an axis.

Broader Topics Related to Angular Motion


The fundamental nature and properties of matter, energy, and motion


The angle which subtends an arc equal to the length of the radius of a circle

Angular Motion Knowledge Graph