5.3 Rotational Kinematics

Angular variables mirror linear motion

Rotational kinematics uses angular position (θ), angular velocity (ω), and angular acceleration (α), which parallel x, v, and a. When α is constant, the same kinematic equations from linear motion apply: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt², and ω² = ω₀² + 2αΔθ.

Linking angular and linear quantities

For a point at radius r from the rotation axis: arc length s = rθ, tangential velocity v = rω, and tangential acceleration a_t = rα. There's also centripetal acceleration a_c = ω²r = v²/r, directed toward the center, which exists even at constant ω.

Calculus connections

Just like linear kinematics, angular velocity is the derivative of angular position (ω = dθ/dt) and angular acceleration is the derivative of angular velocity (α = dω/dt). Integrating α over time gives ω, and integrating ω gives θ — essential for non-constant angular acceleration problems.

Key terms

Angular velocity (ω)

Rate of change of angular position; ω = dθ/dt, analogous to linear velocity.

Angular acceleration (α)

Rate of change of angular velocity; α = dω/dt, analogous to linear acceleration.

v = rω

Relates tangential (linear) velocity of a point to angular velocity and its radius from the axis.

Centripetal acceleration

a_c = ω²r = v²/r; directed toward the center, present even with constant angular velocity.

Constant-α kinematic equations

ω = ω₀ + αt; θ = θ₀ + ω₀t + ½αt²; ω² = ω₀² + 2αΔθ — rotational analogs of linear kinematics.

Try an AP-style question

θ(t) = 2t³ (rad). What is ω at t = 1 s?

1. 6 rad/s
2. 2 rad/s
3. 3 rad/s
4. 12 rad/s