Rotational Kinematics
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Rotation has its own versions of position, velocity and acceleration: angle θ, angular velocity ω = dθ/dt and angular acceleration α = dω/dt. Every point on a rigid object shares the same ω and α; points farther out move faster: v = rω.
1. Angular and linear quantities
2. Constant angular acceleration
These are the linear equations with θ, ω, α in place of x, v, a. They only work if α is constant; otherwise differentiate or integrate, just like in Unit 1.
Use radians. Convert to revolutions at the end by dividing by 2π.
3. Example, then try it
A wheel’s angle is θ(t) = t³ − 4t (rad). Find ω and α at t = 2 s.
ω = dθ/dt = 3t² − 4, so ω(2) = 8 rad/s.
α = dω/dt = 6t, so α(2) = 12 rad/s².
A disk of radius 0.20 m starts from rest with a constant α = 3.0 rad/s². After 4.0 s, find its angular velocity, how many revolutions it has made, and the speed and acceleration of a point on its rim.
ω: 0 + (3.0)(4.0) = 12 rad/s.
Angle: θ = ½(3.0)(4.0)² = 24 rad = 24 ÷ 2π ≈ 3.8 rev.
Rim speed: v = rω = (0.20)(12) = 2.4 m/s.
Rim acceleration: at = (0.20)(3.0) = 0.60 m/s²; ac = (0.20)(12)² = 28.8 m/s². Total = √(0.60² + 28.8²) ≈ 29 m/s², almost all centripetal.
Answer: 12 rad/s, 3.8 rev, 2.4 m/s, about 29 m/s².
4. Common mistakes
A wheel spins at a constant angular velocity. Which part of a rim point’s acceleration is zero?
α = 0, so at = rα = 0. The centripetal acceleration rω² is still there.
Ready to practice?
5 questions on Rotational Kinematics, each with a hint and an explanation for every wrong answer.