Non-uniform motion along a circle

In uniform motion along a circle the linear speed is υ = 2πR/T and the angular speed is ω = Δφ/Δt = 2π/T = 2πν (unit rad/s); they are related by υ = ωR. If the angular speed changes, the motion is non-uniform rotation. The ratio of the change of angular speed to the time of that change is the angular acceleration: ε = Δω/Δt, unit rad/s². Uniformly variable rotation (ε = const) has equations like those of straight-line motion: ω = ω₀ + εt, φ = ω₀t + εt²/2, ω² – ω₀² = 2εφ, where s → φ, υ → ω, a → ε. If the magnitude of the linear speed changes, a tangential acceleration aτ = Δυ/Δt = εR appears along the tangent; the change of direction gives the centripetal acceleration aₙ = υ²/R = ω²R along the radius. The total acceleration is the vector sum of these two perpendicular components: a = √(aτ² + aₙ²). The turning angle φ is linked to the number of revolutions N by φ = 2πN.