AP Physics 1 Unit 5: Torque and Rotational Dynamics Notes

Introduction

Rotational dynamics describes rigid-body motion about an axis by connecting angular variables, torque, mass distribution, and angular acceleration.

Declare the rotation axis and positive rotational sense before using a sign. Every point of a rigid body shares angular position, angular velocity, and angular acceleration, while its linear motion depends on distance from the axis.

Torque measures the rotational effect of a force about that axis. Rotational inertia describes how mass is distributed relative to it. Their relation, $\sum\tau=I\alpha$ , predicts angular acceleration only when every torque and the inertia refer to the same fixed axis.

Angular position over time

angular velocity and acceleration

Radius with angular motion

tangential and radial motion

Force line of action

torque

Mass distribution about an axis

rotational inertia

Zero net force and torque

static equilibrium

Net torque divided by inertia

angular acceleration

Current exam guidance: Aligned to the current 2026–27 AP Physics 1 framework, with algebra-based modeling, experimental reasoning, and explicit assumptions.

5.1

Rotational Kinematics

How do angular position, velocity, and acceleration describe rotation about one axis?

Short answer

Angular velocity is the rate of change of angular position, angular acceleration is the rate of change of angular velocity, and graph slopes and areas connect the three quantities.

Know firstLinear kinematics · Motion graphs · Radians

One angular state describes the whole rigid body

Choose an axis and define counterclockwise or clockwise as positive. A rigid body keeps fixed distances among its points, so every point sweeps through the same angular displacement $\Delta\theta$ in the same time. The average angular velocity is $\Delta\theta/\Delta t$ and average angular acceleration is $\Delta\omega/\Delta t$ . Points farther from the axis cover more linear distance even though their angular variables match.

The slope of a $\theta$ -versus-time graph is $\omega$ ; the slope of an $\omega$ -versus-time graph is $\alpha$ . Signed area under $\omega(t)$ gives angular displacement, and signed area under $\alpha(t)$ gives velocity change. A negative $\alpha$ does not automatically mean slowing: speed decreases only when $\omega$ and $\alpha$ have opposite signs.

Physics model

Angular motion uses the same graph logic as translation

Advance time and compare angular position, angular velocity, and angular acceleration.

Time2 s

Angle: 2 rad
Angular velocity: 2 rad/s
Angular acceleration: 1 rad/s²

angular positionangular velocityangular acceleration

\omega=\frac{\Delta\theta}{\Delta t},\qquad \alpha=\frac{\Delta\omega}{\Delta t}

Read this figure without the graphic

Advance time and compare angular position, angular velocity, and angular acceleration. The table below states the model assumptions and relationships without relying on color or motion.

Quantity	Meaning
Angle	$\theta$ in radians
Angular speed	$\omega$ in rad/s
Angular acceleration	$\alpha$ in rad/s²

Use constant-angular-acceleration equations conditionally

System: A rigid body rotating about one fixed axis.
Frame: Axis fixed in an inertial frame; counterclockwise positive.
Assumptions: Angular acceleration is constant over the interval.
Givens: $\omega_0$ , $\alpha$ , and elapsed time.

Use $\omega=\omega_0+\alpha t$ to update angular velocity.
Average the linearly changing angular velocity or use the displacement equation.
Check signs against the declared rotational sense.

Result

\Delta\theta=\omega_0t+\frac{1}{2}\alpha t^2

Units:

\theta

in radians,

\omega

in rad/s, and

\alpha

in rad/s squared.

Boundary cases

The equations fail when angular acceleration varies over the interval.
$\alpha<0$ with $\omega>0$ can still leave positive rotation.

For constant angular acceleration, angular displacement combines initial angular motion and a one-half-alpha-time-squared change.

Walkthrough

Separate direction from slowing

A rotor begins at $\omega_0=2\ \mathrm{rad/s}$ with $\alpha=-0.5\ \mathrm{rad/s^2}$ for 3 s.

$\omega_f=2+(-0.5)(3)=0.5\ \mathrm{rad/s}$ .
$\Delta\theta=(2)(3)+\tfrac12(-0.5)(3^2)=3.75\ \mathrm{rad}$ .
Both final angular velocity and displacement remain positive, even though the rotor slowed.

Negative angular acceleration reduced the positive angular speed without reversing the rotation.

Read rotational motion

Mark axis and positive sense.
Identify which angular quantity the graph displays.
Use slope for the next rate and signed area for accumulated change.
Use constant- $\alpha$ equations only when the model states constant acceleration.

Measure rotation with an encoder

Do graph slopes and areas agree for a slowly changing turntable?

Variables: Measured: angle versus time; derived: angular velocity and acceleration.
Apparatus: Guarded low-speed turntable, encoder, timer, and data logger.
Procedure boundary: Use no loose clothing or dangling cords near rotating parts.
Uncertainty: Record encoder resolution, zero offset, sampling interval, and bearing-friction drift.

Measure rotation with an encoder lab dataset
Time (s)	Angular velocity (rad/s)
0	2
1	1.5
2	1
3	0.5

Provided accessible table

Measure rotation with an encoder accessible table
Time s	Omega rad/s
0	2
1	1.5
2	1
3	0.5

Keep these three ideas

Angular graph slopes and areas connect state variables.
All rigid-body points share angular motion.
Signs describe sense; compare signs to decide speeding or slowing.

Torque and Rotational Dynamics