To describe rotation we need three quantities that parallel the linear trio from Unit 1 — but measured in angles rather than distances.
The angle through which a rigid system rotates. Measured in radians (rad). Dtheta = theta_f - theta_0.
The rate of change of angular position. omega_avg = Dtheta / Dt. Instantaneous omega is the slope of a theta vs. t graph.
The rate of change of angular velocity. alpha_avg = Domega / Dt. Instantaneous alpha is the slope of an omega vs. t graph.
Defined as arc length divided by radius: theta = s/r. A full revolution = 2*pi rad. One revolution = 360° = 2*pi rad.
A rigid system is one that holds its shape — all its parts maintain fixed distances from each other and from the axis of rotation. This is the model we use for spinning wheels, rotating rods, planets, and gears.
Think of a spinning vinyl record. Every point completes one full revolution in the same time — so every point has the same omega. But a point near the outer edge travels a much larger arc length in that same time, so it has a larger linear speed v = r*omega.
Under constant angular acceleration, three equations relate omega, alpha, Dtheta, and t — exactly paralleling the linear kinematic equations from Unit 1:
These are not new equations. Replace x with theta, v with omega, and a with alpha in the Unit 1 equations and you have these exactly. Every strategy you used for linear kinematics — identify knowns, choose the equation with one unknown, solve — works identically here.
Set initial angular velocity, angular acceleration, and time. All three rotational kinematic equations update live. The graph shows omega vs. t — the shaded area is angular displacement.
Set alpha = 0 — omega is constant, Dtheta grows linearly. Set alpha negative — the object decelerates. All three equations still hold.
A wheel starts from rest and reaches an angular velocity of 24 rad/s in 6 seconds under constant angular acceleration. Find alpha and the total angle turned.
Every angular quantity has a linear (tangential) counterpart at any point on a rigid system. The bridge is the radius r — the distance from the axis to that point.
| Linear quantity | Relation | Angular quantity |
|---|---|---|
| s (arc length, m) | s = r*theta | theta (rad) |
| v (tangential speed, m/s) | v = r*omega | omega (rad/s) |
| a_t (tangential accel, m/s²) | a_t = r*alpha | alpha (rad/s²) |
A disk rotates at a constant angular velocity of 8 rad/s. Point A is 0.3 m from the center. Point B is 0.9 m from the center. Find the linear speed of each point and the arc length each travels in 5 seconds.
The three rotational graphs — theta vs. t, omega vs. t, and alpha vs. t — behave exactly like their linear counterparts x vs. t, v vs. t, and a vs. t. The same slope-and-area rules apply.
| Graph | Slope gives | Area gives |
|---|---|---|
| theta vs. t | omega | — |
| omega vs. t | alpha | Dtheta |
| alpha vs. t | — | Domega |
An omega vs. t graph shows a straight line from omega = 12 rad/s at t = 0 to omega = 0 at t = 4 s. Find alpha and the total angular displacement.