Angular Motion Graphs

Rotational motion graphs work exactly like linear motion graphs, just with angles instead of distances. Plot angular position theta versus time, and the slope at any instant is the angular velocity omega. Plot omega versus time, and the slope is the angular acceleration alpha. The three graphs form a ladder: each one's slope produces the graph below it, and each one's area produces the graph above it. That's why a single stack of three graphs can tell you a spinning object's entire story, from where it started to how fast it was speeding up at any moment.

The area relationship runs the other direction. The area under an alpha-vs-time graph gives the change in omega, and the area under an omega-vs-time graph gives the total angle turned, delta theta. If omega increases steadily from 0, the region under the graph is a triangle; if it then holds constant, the next region is a rectangle. Add the pieces and you have the total rotation. This mirrors exactly how area under a velocity-time graph gives displacement in linear kinematics, because theta, omega, and alpha are the rotational twins of x, v, and a.

A direction reversal shows up as omega crossing zero and going negative, while alpha might stay constant sign the whole time if the object decelerates, stops, then speeds up the other way. This graph reading sets up the rotational kinematic equations, which are just the constant-acceleration formulas from linear motion rewritten with theta, omega, and alpha, used whenever alpha is constant rather than read off a graph.