Rotational kinematics
Also known as: angular kinematics
Rotational kinematics describes the motion of rotating objects using angular position, angular velocity, and angular acceleration. Its equations mirror the linear kinematic equations, with each linear quantity replaced by its angular counterpart.
Rotational kinematics answers the same questions as linear kinematics — where something is, how fast it is moving, and how quickly that is changing — but for objects turning about an axis. Angular position θ is measured in radians, angular velocity ω is the rate of change of θ in radians per second, and angular acceleration α is the rate of change of ω. As with linear motion, kinematics describes the motion without asking what causes it; torque and rotational dynamics take up that question.
For constant angular acceleration, the equations are direct analogs of the linear ones: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt², and ω² = ω₀² + 2αΔθ. Anything you can solve for a car accelerating in a straight line you can solve for a wheel spinning up, using the same algebra with different symbols.
The bridge between angular and linear quantities is the radius. A point a distance r from the axis travels an arc length s = rθ, moves with tangential speed v = rω, and has tangential acceleration a = rα — all of which require θ to be in radians rather than degrees. A key consequence for rigid bodies is that every point shares the same angular velocity and angular acceleration, but points farther from the axis move faster linearly. That is why the outer edge of a merry-go-round covers more distance per revolution than a point near the center. A point moving in a circle also has centripetal acceleration a = v²/r = ω²r directed toward the axis, which exists even when the rotation rate is constant.
AP Physics 1 tests rotational kinematics in the torque and rotational motion unit. Expect problems that require converting between linear and angular quantities, applying the constant-angular-acceleration equations, reading angular velocity from a graph, and recognizing that all points on a rigid rotating body share one angular velocity.
Key takeaways
- Rotational kinematics uses angular position θ, angular velocity ω, and angular acceleration α to describe rotating motion.
- For constant α the equations mirror linear kinematics: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt², and ω² = ω₀² + 2αΔθ.
- Linear and angular quantities connect through the radius: s = rθ, v = rω, and a = rα, with angles in radians.
- Every point on a rigid rotating body has the same ω and α, but linear speed increases with distance from the axis.
- Circular motion also involves centripetal acceleration a = v²/r, present even at constant angular velocity.