AP® Physics 1: Algebra-Based · Unit 5 · Topic 5.1
Angular Motion Graphs
Angular motion graphs plot angle, angular velocity, and angular acceleration against time the same way linear motion graphs do, with slope and area carrying the same meaning: slope gives the next quantity's rate, area under a curve gives the next quantity's total change.
Angular position, angular velocity, and angular acceleration graphs relate exactly like position, velocity, and acceleration graphs do. The slope of an angle-vs-time graph is angular velocity. The slope of an angular velocity-vs-time graph is angular acceleration. Flip that around and area does the opposite job: the area under an angular velocity graph is the angle turned through, and the area under an angular acceleration graph is the change in angular velocity. This works because angular velocity is just angle turned per second, so multiplying it by a time interval (the area of a thin rectangle under the curve) gives angle turned during that interval. Add up all those slivers and you get total angle.
Sketching a wheel that spins up then brakes: angular velocity starts at zero, climbs in a straight line (constant angular acceleration) up to some top speed, holds flat while spinning at constant rate, then drops in a straight line back to zero as it brakes. The angular acceleration graph matches: positive and constant while speeding up, zero during the flat coasting section, negative and constant while braking. The angle-vs-time graph curves upward (concave up) while speeding up, becomes a straight line during constant speed, then curves over (concave down) while slowing to a stop.
Sign only tells you direction, never magnitude or motion status. Pick counterclockwise as positive, and a wheel spinning clockwise shows up as negative angular velocity no matter how fast it spins. This is the same convention trick from linear motion graphs in earlier units, now applied to rotation, and it carries forward into torque and angular momentum, where the same positive direction has to stay consistent across every quantity you use.
Key terms
- angular position
- The angle, usually in radians, that marks how far an object has rotated from a reference direction.
- angular velocity
- The rate of change of angular position, equal to the slope of an angle-vs-time graph.
- angular acceleration
- The rate of change of angular velocity, equal to the slope of an angular velocity-vs-time graph.
- area under the curve
- The region between a graph and the time axis, whose value equals the total change in the next lower quantity.
- sign convention
- A chosen positive rotation direction, often counterclockwise, that fixes the meaning of positive and negative values.
Watch it explained
- How it worksHow does the slope of an angular velocity graph produce angular acceleration?
- How it worksWalk me through sketching all three angular motion graphs for a wheel spun up then braked.
- Worked exampleFind the total angle turned from an angular velocity graph rising from 0 to 15 rad/s in 4 s.
- Going furtherWhat does a negative angular velocity look like on a spinning object?
Check yourself
End of topic.