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 uses the same ideas as straight-line motion. Angle θ plays the role of position, angular velocity ω plays the role of velocity, and angular acceleration α plays the role of acceleration. With constant α, the kinematics equations are the same, as long as you use radians.
1. Radians
Rotation equations only work with angles in radians. Convert first: one revolution = 2π rad.
ω is in rad/s: how fast the angle changes. α is in rad/s²: how fast ω changes.
2. Same equations, new letters
Average angular velocity: ωavg = Δθ ÷ Δt. Angular acceleration: α = Δω ÷ Δt.
Don’t mix up α with centripetal acceleration. α changes how fast something spins. Centripetal acceleration changes the direction a point moves, and it exists even when the spin rate is constant.
3. Example, then try it
A fan spins at 10 revolutions per second. What is its angular velocity in rad/s?
ω = (10 rev/s)(2π rad/rev) = 20π ≈ 63 rad/s.
A wheel starts from rest and speeds up with α = 4.0 rad/s² for 5.0 s. Find its final angular velocity and how many revolutions it turns.
Final ω: ω = ω0 + αt = 0 + (4.0)(5.0) = 20 rad/s.
Angle: Δθ = ω0t + ½αt² = 0 + ½(4.0)(5.0)² = 50 rad.
Revolutions: 50 rad ÷ 2π rad/rev ≈ 8.0 rev.
Answer: 20 rad/s, after about 8 revolutions.
4. Common mistakes
A wheel’s angular velocity drops from 10 rad/s to 4.0 rad/s in 3.0 s. What is its angular acceleration?
α = Δω ÷ Δt = (4.0 − 10) ÷ 3.0 = −2.0 rad/s². The minus sign means the wheel is slowing down.
Ready to practice?
5 questions on Rotational Kinematics, each with a hint and an explanation for every wrong answer.