1. Introduction to Rotational Kinematics

Rotational Kinematics describes the motion of objects that rotate about a fixed axis without considering the forces or torques that cause the motion.

It is the rotational equivalent of linear kinematics.

Just as linear kinematics uses:

• displacement
• velocity
• acceleration

rotational kinematics uses:

• angular displacement
• angular velocity
• angular acceleration

2. Angular Position

Definition

The angular position of an object specifies its orientation relative to a reference line.

It is represented by:

$$\theta$$

and is measured in radians (rad).

Radian Measure

One radian is defined as the angle subtended by an arc whose length equals the radius.

Relationship between arc length and angle:

$$s=r\theta$$

where:

• $s$ = arc length
• $r$ = radius
• $\theta$ = angle in radians

3. Angular Displacement

Definition

Angular displacement is the change in angular position.

Δθ=θf−θi\Delta\theta = \theta_f-\theta_iΔθ=θf​−θi​

where:

• θi\theta_iθi​ = initial angle
• θf\theta_fθf​ = final angle

Angular displacement is a vector quantity whose direction is determined using the right-hand rule.

4. Angular Velocity

Definition

Angular velocity measures how rapidly angular position changes.

Average angular velocity:

ωavg=ΔθΔt\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}ωavg​=ΔtΔθ​

Instantaneous angular velocity:

ω=dθdt\omega = \frac{d\theta}{dt}ω=dtdθ​

where:

• $\omega$ = angular velocity

Units

$$\text{rad/s}$$

Although radians are dimensionless, angular velocity is conventionally written in radians per second.

5. Angular Acceleration

Definition

Angular acceleration measures how rapidly angular velocity changes.

Average angular acceleration:

αavg=ΔωΔt\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t}αavg​=ΔtΔω​

Instantaneous angular acceleration:

α=dωdt\alpha = \frac{d\omega}{dt}α=dtdω​

where:

• $\alpha$ = angular acceleration

Units

rad/s2\text{rad/s}^2rad/s2

6. Constant Angular Acceleration

Many AP Physics C rotational problems assume constant angular acceleration.

Under this assumption, rotational motion follows equations analogous to the Big Five kinematic equations used in linear motion.

7. Rotational Kinematic Equations

Equation 1

Angular velocity as a function of time:

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

where:

• ω0\omega_0ω0​ = initial angular velocity

Equation 2

Angular displacement as a function of time:

θ=θ0+ω0t+12αt2\theta = \theta_0 + \omega_0 t + \frac12\alpha t^2θ=θ0​+ω0​t+21​αt2

Equation 3

Angular velocity and displacement:

ω2=ω02+2α(θ−θ0)\omega^2 = \omega_0^2 + 2\alpha(\theta-\theta_0)ω2=ω02​+2α(θ−θ0​)

Equation 4

Average angular velocity form:

θ−θ0=ω+ω02t\theta-\theta_0 = \frac{\omega+\omega_0}{2}tθ−θ0​=2ω+ω0​​t

8. Correspondence with Linear Motion

Rotational and linear quantities are closely related.

Understanding these analogies simplifies the study of rotational dynamics.

9. Tangential Quantities

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

Tangential Speed

$$v=r\omega$$

This gives the linear speed of the point.

Tangential Acceleration

at=rαa_t=r\alphaat​=rα

This acceleration changes the magnitude of the tangential velocity.

10. Centripetal Acceleration

Even when angular velocity is constant, an object moving in a circle experiences centripetal acceleration.

ac=v2ra_c = \frac{v^2}{r}ac​=rv2​

Using $v=r\omega$:

ac=rω2a_c = r\omega^2ac​=rω2

Centripetal acceleration always points toward the center of rotation.

11. Rotational Motion Example

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

α=4 rad/s2\alpha = 4\,\text{rad/s}^2α=4rad/s2

for:

$$t=3\text{s}$$

Using:

ω=ω0+αt\omega = \omega_0+\alpha tω=ω0​+αt $$\omega =0+(4)(3)$$ $$\omega =12\text{rad/s}$$

The wheel’s angular velocity after 3 seconds is:

$$12\text{rad/s}$$

12. Right-Hand Rule

The direction of angular quantities is determined using the right-hand rule.

Procedure

1. Curl the fingers of your right hand in the direction of rotation.
2. Extend your thumb.

Your thumb points in the positive angular direction.

13. Common AP Physics C Mistakes

Mistake 1

Using degrees instead of radians.

All rotational equations require angles in radians.

Mistake 2

Confusing angular velocity with tangential velocity.

ω≠v\omega \neq vω=v

They are related by:

$$v=r\omega$$

Mistake 3

Ignoring centripetal acceleration when angular velocity is constant.

Constant angular velocity does not imply zero acceleration.

Summary

Angular displacement:

Δθ=θf−θi\Delta\theta=\theta_f-\theta_iΔθ=θf​−θi​

Angular velocity:

ω=dθdt\omega=\frac{d\theta}{dt}ω=dtdθ​

Angular acceleration:

α=dωdt\alpha=\frac{d\omega}{dt}α=dtdω​

Rotational kinematic equations:

ω=ω0+αt\omega=\omega_0+\alpha tω=ω0​+αt θ=θ0+ω0t+12αt2\theta=\theta_0+\omega_0t+\frac12\alpha t^2θ=θ0​+ω0​t+21​αt2 ω2=ω02+2α(θ−θ0)\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)ω2=ω02​+2α(θ−θ0​)

Linear–rotational relationships:

$$v=r\omega$$ at=rαa_t=r\alphaat​=rα ac=rω2a_c=r\omega^2ac​=rω2

Key ideas:

• rotational kinematics describes rotational motion without considering forces
• rotational quantities mirror linear kinematic quantities
• radians are required in all rotational equations
• angular and tangential quantities are directly related through the radius

Rotational Kinematics forms the foundation for studying torque, rotational dynamics, and angular momentum in AP Physics C Mechanics.