Rotational Newton's Second Law
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
Newton’s second law for rotation: α = τnet ÷ I (or Στ = Iα). A net torque makes an object’s rotation speed up or slow down. More rotational inertia means less angular acceleration for the same torque.
1. Torque causes angular acceleration
2. Count every torque
Add all the torques with signs: the pull of a string, gravity on an off-center mass, and friction at the axle, which always opposes the spin.
Then divide by the rotational inertia about the same axis.
Zero net torque doesn’t mean the object isn’t spinning. It means its angular velocity isn’t changing, just as zero net force means constant velocity.
3. Example, then try it
A string wrapped around a disk (I = 0.50 kg·m², R = 0.20 m) is pulled with a 10 N force. Ignore friction. What is the disk’s angular acceleration?
Torque: τ = FR = (10)(0.20) = 2.0 N·m.
α = τ ÷ I = 2.0 ÷ 0.50 = 4.0 rad/s².
Same disk and pull, but now friction at the axle exerts a 0.50 N·m torque against the spin. Starting from rest, how fast is it spinning after 4.0 s, and how much string has unwound?
Net torque: 2.0 − 0.50 = 1.5 N·m.
Angular acceleration: α = 1.5 ÷ 0.50 = 3.0 rad/s².
After 4.0 s: ω = αt = (3.0)(4.0) = 12 rad/s.
String unwound: θ = ½αt² = ½(3.0)(4.0)² = 24 rad, so s = Rθ = (0.20)(24) = 4.8 m.
Answer: 12 rad/s, with 4.8 m of string unwound.
4. Common mistakes
Two identical wheels get the same force, applied tangentially. On wheel 1 it’s applied at the rim (radius R); on wheel 2, at R/2. How do their angular accelerations compare?
Same force and same I, but wheel 1’s lever arm is twice as long, so its torque, and its angular acceleration α = τ/I, is twice as large.
Ready to practice?
4 questions on Rotational Newton's Second Law, each with a hint and an explanation for every wrong answer.