Rotational Kinematics: Differentiate Down, Integrate Up

You'll learnthat angle, angular velocity, and angular acceleration form one derivative chain.

A disk spins up, and its angle, angular velocity, and angular acceleration form one calculus chain — ω = dθ/dt, α = dω/dt going down; integrate to run back up, each step carrying its starting value. The constant-α shortcuts are only the special case where α never changes.

8 STEPS · 6 QUICK CHECKS · v2

SPIN LAB

Before you start

What you're looking at

A disk starting from rest and spinning up, its rate climbing, with dashed preview curves for θ, ω, and α.

The question

How fast does the disk spin after 4 s, and through what total angle does it turn?

Watch for

The three rows filling together: ω is the slope of θ, and α is the slope of ω.

What the symbols mean

θ = angle, ω = angular velocity (rad/s), α = angular acceleration (rad/s²); α = 2 rad/s² is constant here.

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