Angular Position, Velocity and Acceleration

A spinning object does not need to be a point to have a single angle describing it. Pick any rigid body turning about an axis, and theta(t) tells you how far it has rotated from some reference line, usually in radians. Every point on that body shares the same theta, the same omega, and the same alpha, no matter how far it sits from the axis. That is why the inner and outer horses on a carousel stay side by side: they sweep the same angle in the same time, so they share angular speed, even though the outer horse covers more actual distance per second because its linear speed is omega times a bigger radius.

Angular velocity omega(t) is the derivative of theta(t) with respect to time, and angular acceleration alpha(t) is the derivative of omega(t), or the second derivative of theta(t). The process is identical to linear kinematics: differentiate position to get velocity, differentiate velocity to get acceleration. The payoff is that these three functions, theta, omega, and alpha, describe the entire rotational motion of a rigid body with one set of numbers, no matter its shape.

This differentiation chain is the seed for the rest of the unit. Once you can generate omega and alpha from theta, you can plug them into torque equations and moment of inertia relationships to predict how fast something spins up under a given twist. Rotational kinematics mirrors linear kinematics step for step, so every tool you already trust there, derivatives, graphs, constant-acceleration equations, carries over here with angle in place of position.