5.1 Angular Kinematics

Angular position tracks orientation using an angle measured from a reference line. As a rigid body rotates, the change in angle gives angular displacement. Dividing that displacement by elapsed time gives average angular velocity, and the instantaneous value is the limit over very small time intervals.

Radians are the natural unit for rotational equations because they connect angle directly to arc length and radius. Using degrees inside rotational kinematics often introduces hidden conversion mistakes, especially when combining angular and linear quantities.

Angular acceleration describes how fast angular velocity changes. When angular acceleration is constant, the rotational equations mirror linear equations term-for-term. Replace displacement with angle, velocity with angular velocity, and acceleration with angular acceleration, and the algebraic structure stays the same.

For any point at radius r from the axis, linear speed and tangential acceleration depend on the same angular state: v = omega r and a_t = alpha r. This means one rotating body can have a single angular velocity but many linear speeds, because points farther from the axis move through larger arcs in the same time.

That radius dependence explains why wheels and gears are so effective. Small changes in radius change linear edge speed without changing the shared angular motion. In practical problems, always identify which radius corresponds to the point being analyzed.