Topic 5.1 Notes – Rotational Kinematics

1. Rotation About an Axis and Rigid Systems

When something rotates about an axis, every point in the object moves in a circle around the same fixed line.

Rigid disk rotating about a fixed axis through O

A rigid system:

• Keeps its shape (distances between points stay constant).
• Has different points moving in different directions at any instant.
• Cannot be treated as a single particle when analyzing rotation.

In the diagram, point P moves in a circular path around O. Its velocity is tangent to the circle, and its acceleration has a component pointing toward the center. Every point on the disk behaves this way, but the farther a point is from the center, the larger its speed.

That last point is big. In linear motion, we often treat an object as a dot. In rotation, points at different radii move differently. The edge of a wheel moves faster than a point near the center.

There are times you can treat a rotating system as one object:

• If you only care about the motion of the center of mass, and the rotation doesn’t affect the situation much.
  • Example: When analyzing Earth orbiting the Sun, we ignore Earth’s daily spin.

Direction convention (AP limit)

On the AP exam, direction is limited to:

• Counterclockwise (CCW)
• Clockwise (CW)

You choose one as positive. The other becomes negative. Stay consistent. If you switch signs halfway through a problem, everything falls apart.

2. Angular Displacement, Velocity, and Acceleration

These are the rotational versions of displacement, velocity, and acceleration in 1D motion.

Angular Displacement $\theta$

Angular displacement is the angle through which something rotates.

• Measured in radians.
• Can be positive or negative.
• Not limited to $2\pi$. If something spins 3 full turns, that’s $6\pi$ radians.

A radian is defined by arc length:

$$\theta = \frac{s}{r}$$

If arc length equals radius, that’s 1 radian.

Average Angular Velocity $\omega$

Angular velocity tells you how fast the angle is changing.

$$\omega_{\text{avg}} = \frac{\Delta \theta}{\Delta t}$$

Units: rad/s

If $\omega$ is constant, the object rotates through equal angles in equal time intervals.

Sign matters:

• Positive → rotating in your defined positive direction.
• Negative → rotating the other way.

Average Angular Acceleration $\alpha$

Angular acceleration tells you how fast angular velocity changes.

$$\alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t}$$

Units: rad/s²

The sign logic trips people up. Think carefully:

• $\alpha$ and $\omega$ same sign → speeding up.
• Opposite signs → slowing down.

If a wheel spins CCW (positive ω) and α is negative, it’s slowing down.

3. Constant Angular Acceleration Equations

If angular acceleration is constant, the equations are structurally identical to linear kinematics.

$$\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$$

$$\omega = \omega_0 + \alpha t$$

$$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$$

Just swap:

• $x \rightarrow \theta$
• $v \rightarrow \omega$
• $a \rightarrow \alpha$

Quick example:
A disk starts from rest ($\omega_0 = 0$) and has $\alpha = 3 \text{ rad/s}^2$ for 4 s.

$$\omega = 0 + (3)(4) = 12 \text{ rad/s}$$

Same math you’ve done all year.

4. Angular vs Linear Motion Connections

Even though we describe rotation with angles, each point also has linear motion.

For a point distance $r$ from the axis:

$$v = r\omega$$

$$a_{\text{tangential}} = r\alpha$$

$$\theta = \frac{s}{r}$$

Important ideas:

• All points share the same $\theta, \omega, \alpha$.
• Points farther from the axis move faster linearly.
• Radians are dimensionless, so units work out cleanly.

If two points are at different radii:

• Same ω.
• Different linear speeds.

That shows up often in conceptual multiple-choice questions.

5. Graphs of Angular Motion

These graphs behave exactly like 1D motion graphs. If you understand position, velocity, and acceleration vs time, you already understand these.

$\theta$ vs $t$

• Slope = $\omega$

$\omega$ vs $t$

• Slope = $\alpha$
• Area under curve = $\Delta \theta$

$\alpha$ vs $t$

• Area under curve = $\Delta \omega$

In the example below, the acceleration is constant and positive. That makes the $\omega$ vs $t$ graph a straight line with positive slope, and the $\theta$ vs $t$ graph curves upward because the slope is increasing over time.

The shaded area under the $\omega$ vs $t$ graph represents the change in angle $\Delta \theta$. The shaded area under the $\alpha$ vs $t$ graph represents the change in angular velocity $\Delta \omega$.

When you see a graph question, think slope gives rate, area gives accumulated change. Same logic as linear motion.