AP Physics C: Mechanics Unit 5: Torque and Rotational Dynamics Notes

Introduction

Rotational motion becomes predictive when angular change, force geometry, and mass distribution are all measured about the same axis.

Angular position generates angular velocity and acceleration through derivatives. Radius converts those shared angular quantities into different linear motion for different points of a rigid body.

Torque is the moment of force about a chosen axis, while rotational inertia is the second moment of mass about that axis. For a rigid body with fixed inertia, their signed relation determines angular acceleration.

Angular position history

angular velocity and acceleration

Radius from axis

linear motion of a rigid-body point

Force line of action

torque

Continuous mass distribution

rotational inertia

Force and torque balances

static equilibrium

Net external torque

angular acceleration

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

5.1

Rotational Kinematics

How do derivatives and integrals connect angular position, velocity, and acceleration?

Short answer

For rotation about a fixed axis, $\omega=d\theta/dt$ and $\alpha=d\omega/dt$ ; integrating reverses those relationships.

Know firstDifferential kinematics · Graph slopes and areas · Radians

Angular signs describe sense, not speed change

Choose a positive rotational sense before reading any sign. A rigid body gives every point the same $\theta$ , $\omega$ , and $\alpha$ , although points at different radii travel different linear distances. Radians make angular derivatives compatible with arc length.

The slope of $\theta(t)$ is $\omega(t)$ ; the slope of $\omega(t)$ is $\alpha(t)$ . Signed area reverses each step. When $\alpha$ is constant, the familiar linear-kinematics equations apply with $x$ , $v$ , and $a$ replaced by $\theta$ , $\omega$ , and $\alpha$ . Negative $\omega$ identifies rotational sense. Speed decreases only when $\omega$ and $\alpha$ have opposite signs.

Physics model

Angular derivatives describe rigid-body rotation

Advance time and compare angle, angular velocity, and angular acceleration for one rotating radius.

Time2 s

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

angular positionangular velocityangular acceleration

\omega=\frac{d\theta}{dt},\qquad\alpha=\frac{d\omega}{dt}

Read this figure without the graphic

Advance time and compare angle, angular velocity, and angular acceleration for one rotating radius.

Current quantity	Value
Angle	2 rad
Angular velocity	2 rad/s
Angular acceleration	1 rad/s²
Model relation	$\omega=\frac{d\theta}{dt},\qquad\alpha=\frac{d\omega}{dt}$

Ranger spacecraft mounted vertically for a spin test at the Jet Propulsion Laboratory. — A spin test checks rotational behavior before flight
A rigid spacecraft rotates about a test axis while engineers monitor balance and response. Angular position, angular velocity, and angular acceleration describe different parts of that motion.
Use this visual to see
Connect angular kinematics to a real rigid-body test.
What it cannot show
The photograph does not identify the axis precisely or provide time, angle, angular speed, angular acceleration, or torque data.

Formula

Constant-angular-acceleration relations

\omega=\omega_0+\alpha t,\qquad \Delta\theta=\omega_0t+\tfrac12\alpha t^2,\qquad \omega^2=\omega_0^2+2\alpha\Delta\theta

These equations connect angular displacement, angular velocity, time, and a constant angular acceleration.

Use when: $\alpha$ is constant over the interval, time is measured from the stated initial state, and all angular quantities use one sign convention.
Do not assume: For changing $\alpha$ , return to derivatives, integrals, or graph slopes and signed areas.
Variables: $\Delta\theta$ : signed angular displacement (rad) · $\omega_0$ : initial angular velocity (rad s $^{-1}$ ) · $\alpha$ : constant angular acceleration (rad s $^{-2}$ )
Common misuse: Using a constant- $\alpha$ equation when the angular-velocity graph is curved.

Build motion from an angular-position function

System: A rigid body rotating about a fixed axis.
Frame: Axis fixed in an inertial frame; positive sense declared.
Assumptions: $\theta(t)$ is differentiable on the interval.
Givens: $\theta(t)=t^3-3t^2+2t$ .

Differentiate $\theta$ once to obtain $\omega$ .
Differentiate again to obtain $\alpha$ .
At $t=1$ s, evaluate both functions rather than inferring one sign from the other.

Result

\omega=3t^2-6t+2,\qquad \alpha=6t-6

Units:

\theta

in rad,

\omega

in rad/s,

\alpha

in rad/s².

Boundary cases

At $t=1$ , $\alpha=0$ while $\omega=-1$ rad/s.
Zero angular acceleration does not require zero angular velocity.

Successive time derivatives turn angular position into angular velocity and angular acceleration.

Read an angular history

Record the positive rotational sense.
Differentiate for an instantaneous rate; integrate for accumulated change.
Keep radians in calculus-based relations.
Compare the signs of $\omega$ and $\alpha$ to determine speeding or slowing.

Keep these three ideas

Angular variables are linked by calculus.
All rigid-body points share angular state.
Direction and speeding are separate sign questions.

Torque and Rotational Dynamics