The second rotational kinematic equation relates angular displacement $\Delta \theta$ to initial angular velocity $\omega_i$, constant angular acceleration $\alpha$, and time $t$.

Derivation

We derive this relationship by interpreting the geometric areas within rotational motion graphs.

For a constant angular acceleration $\alpha$, the acceleration-time graph forms a horizontal line. The area under this graph corresponds to the change in angular velocity, $\Delta \omega = \alpha t$.

Because the velocity changes at a constant rate, the graph of angular velocity versus time is a straight line starting at $\omega_i$. The total angular displacement $\Delta \theta$ is equivalent to the area under this velocity-time graph. This graph creates a trapezoidal area that we decompose into two simpler shapes.