1. Introduction

The Big Five Rotational Kinematic Equations describe rotational motion when the angular acceleration is constant.

They are the rotational equivalents of the linear kinematic equations used in translational motion.

These equations relate:

• angular displacement
• angular velocity
• angular acceleration
• time

without involving forces or torques.

2. Rotational Variables

Before using the equations, it is important to understand the variables.

Units:

• $\theta$ → radians (rad)
• $\omega$ → rad/s
• $\alpha$ → rad/s²

3. Linear–Rotational Analogy

Rotational kinematics mirrors linear kinematics.

If you know the linear Big Five equations, the rotational versions are almost identical.

4. Big Five Equation #1

Angular Velocity–Time Equation

ω=ω0+αt\omega = \omega_0 + \alpha tω=ω0​+αt

Purpose

Use when:

• time is known
• angular acceleration is constant
• angular displacement is not needed

Variables Present

• $\omega$
• ω0\omega_0ω0​
• $\alpha$
• $t$

Missing variable:

$$\Delta \theta$$

5. Big Five Equation #2

Angular Displacement–Time Equation

Δθ=ω0t+12αt2\Delta\theta = \omega_0 t + \frac12\alpha t^2Δθ=ω0​t+21​αt2

Purpose

Use when:

• displacement is required
• time is known

Variables Present

• $\Delta \theta$
• ω0\omega_0ω0​
• $\alpha$
• $t$

Missing variable:

$$\omega$$

6. Big Five Equation #3

Velocity–Displacement Equation

ω2=ω02+2αΔθ\omega^2 = \omega_0^2 + 2\alpha\Delta\thetaω2=ω02​+2αΔθ

Purpose

Use when:

• time is unknown
• angular displacement is known

Variables Present

• $\omega$
• ω0\omega_0ω0​
• $\alpha$
• $\Delta \theta$

Missing variable:

$$t$$

7. Big Five Equation #4

Average Angular Velocity Equation

Δθ=ω+ω02t\Delta\theta = \frac{\omega+\omega_0}{2}tΔθ=2ω+ω0​​t

Purpose

Use when:

• acceleration is not needed
• initial and final angular velocities are known

Variables Present

• $\Delta \theta$
• $\omega$
• ω0\omega_0ω0​
• $t$

Missing variable:

$$\alpha$$

8. Big Five Equation #5

Alternative Displacement Equation

Substitute:

ω=ω0+αt\omega=\omega_0+\alpha tω=ω0​+αt

into Equation #4:

Δθ=ωt−12αt2\Delta\theta = \omega t – \frac12\alpha t^2Δθ=ωt−21​αt2

Purpose

Useful when:

• final angular velocity is known
• initial angular velocity is unknown

Variables Present

• $\Delta \theta$
• $\omega$
• $\alpha$
• $t$

Missing variable:

ω0\omega_0ω0​

9. Choosing the Correct Equation

The easiest strategy is:

Step 1

List all known variables.

Step 2

Identify the unknown variable.

Step 3

Choose the equation that contains all known variables and excludes unnecessary ones.

10. Example Problem

A wheel starts from rest and rotates with constant angular acceleration:

α=3 rad/s2\alpha = 3\,\text{rad/s}^2α=3rad/s2

for:

$$t=4\text{s}$$

Find the final angular velocity.

Solution

Use Equation #1:

ω=ω0+αt\omega = \omega_0+\alpha tω=ω0​+αt

Since:

ω0=0\omega_0=0ω0​=0 $$\omega =(3)(4)$$ $$\omega =12\text{rad/s}$$

11. Relationship to Tangential Motion

For a point located distance $r$ from the axis:

Tangential Velocity

$$v=r\omega$$

Tangential Acceleration

at=rαa_t=r\alphaat​=rα

Arc Length

$$s=r\theta$$

These equations connect rotational motion to linear motion.

12. Conditions for Using the Big Five

The rotational Big Five equations are valid only when:

$$\alpha =\text{constant}$$

If angular acceleration changes with time:

• calculus methods are required
• the Big Five equations no longer apply directly

13. Common AP Physics C Mistakes

Mistake 1

Using degrees instead of radians.

Always convert angles to radians.

Mistake 2

Using the equations when angular acceleration is not constant.

The Big Five require constant $\alpha$.

Mistake 3

Confusing angular velocity with tangential velocity.

Remember:

$$v=r\omega$$

Summary

The Big Five Rotational Kinematic Equations are:

Equation 1

ω=ω0+αt\omega = \omega_0+\alpha tω=ω0​+αt

Equation 2

Δθ=ω0t+12αt2\Delta\theta = \omega_0 t + \frac12\alpha t^2Δθ=ω0​t+21​αt2

Equation 3

ω2=ω02+2αΔθ\omega^2 = \omega_0^2 + 2\alpha\Delta\thetaω2=ω02​+2αΔθ

Equation 4

Δθ=ω+ω02t\Delta\theta = \frac{\omega+\omega_0}{2}tΔθ=2ω+ω0​​t

Equation 5

Δθ=ωt−12αt2\Delta\theta = \omega t – \frac12\alpha t^2Δθ=ωt−21​αt2

Key ideas:

• rotational motion mirrors linear motion
• all equations require constant angular acceleration
• radians must be used
• these equations form the foundation of rotational problem solving in AP Physics C Mechanics

Mastering the rotational Big Five makes later topics such as torque, rotational dynamics, and angular momentum much easier to understand.