Unit 5: Torque and Rotational Dynamics

What Unit 5 Covers

Unit 5 is where AP Physics 1 turns from straight-line motion to spinning objects: how fast they rotate, what changes their rotation, and what keeps them balanced. It carries 10 to 15% of the multiple-choice section, the same weighting band as kinematics and momentum, so expect several questions on exam day.

The six topics build in a deliberate order. Topics 5.1 and 5.2 give you the language of rotation (angular position, angular velocity, angular acceleration) and connect it back to the linear motion you already know. Topic 5.3 introduces torque, the rotational version of force. Topic 5.4 covers rotational inertia, the rotational version of mass. Topics 5.5 and 5.6 then rebuild Newton's first and second laws in rotational form. If you were comfortable with Unit 2, this unit follows the same script with new symbols.

Topics 5.1 and 5.2: Rotational Kinematics

Rotational kinematics is linear kinematics with new symbols. Angular position $\theta$ replaces $x$, angular velocity $\omega$ replaces $v$, and angular acceleration $\alpha$ replaces $a$. Under constant angular acceleration, the same three kinematic equations apply with those swaps, for example $\omega = \omega_0 + \alpha t$.

Topic 5.2 connects the two descriptions. A point at radius $r$ on a rotating object moves with speed $v = r\omega$ and tangential acceleration $a_t = r\alpha$. Every point on the object shares the same $\omega$, but points farther from the axis move faster, a favorite conceptual question. Because each point travels a circle, it also has centripetal acceleration $a_c = v^2/r$ pointing toward the axis, which links back to circular motion from Unit 2. Work through the rotational kinematics guide for the full equation set with practice problems.

Topic 5.3: Torque

Torque measures how effectively a force makes an object rotate about a chosen axis. The equation sheet gives it as $\tau = r_{\perp}F = rF\sin\theta$, where $r$ is the distance from the axis to the point where the force acts and $\theta$ is the angle between $r$ and the force. Push a door at the handle and it swings easily; push near the hinge and almost nothing happens, because torque scales with the lever arm $r_{\perp}$.

Three habits pay off early. State your axis before computing anything, since torque depends on that choice. Use only the force component perpendicular to $r$; a force directed straight through the axis produces zero torque. Pick a positive rotation direction (usually counterclockwise) and keep signs consistent. The torque guide walks through each setup, and the torque calculator lets you check answers while you practice.

Topic 5.4: Rotational Inertia

Rotational inertia $I$ plays the role that mass plays in linear motion: it measures how hard it is to change an object's rotation. Unlike mass, it depends on where the mass sits relative to the axis. A point mass $m$ at radius $r$ contributes $I = mr^2$, so doubling the distance quadruples the rotational inertia.

That $r^2$ dependence drives most of the conceptual questions. A hoop resists spinning more than a uniform disk of the same mass and radius because the hoop's mass all sits at the rim. A skater pulling in her arms lowers her $I$. Expect ranking tasks: same mass, different shapes or different axes, which arrangement has the largest rotational inertia? Reason from mass distribution instead of memorizing a formula for every shape. Watch the axis too: the same object has different rotational inertia about different axes, and moving the axis away from the center of mass always increases it.

Topics 5.5 and 5.6: Equilibrium and the Rotational Second Law

Topic 5.5 is Newton's first law in rotational form: if the net torque on a rigid object is zero, its angular velocity stays constant. In practice this means balance problems, such as a plank on a pivot or a beam held by a cable and a hinge. You write $\Sigma\tau = 0$ about a chosen axis and solve. The key trick: put the axis at the location of an unknown force, and that force drops out of the equation because its lever arm is zero.

Topic 5.6 is the rotational second law, $\Sigma\tau = I\alpha$. It handles anything whose spin speeds up or slows down: a wheel with a rope wrapped around it, a rod released from horizontal, a pulley with real mass connected to hanging blocks. The strategy mirrors net force problems: identify each force and its lever arm, sum torques with signs, then solve for $\alpha$. For a massive pulley you combine $\Sigma\tau = I\alpha$ on the pulley with $F = ma$ on the hanging masses, tied together by $a = r\alpha$.

The Linear-to-Rotational Dictionary

Almost every equation in this unit is a linear equation you already know with symbols swapped. Learn the dictionary once and half the unit comes free.

The dictionary keeps going in Unit 6: momentum $p = mv$ becomes angular momentum $L = I\omega$, and kinetic energy picks up a rotational term. Spend ten minutes translating equations in both directions and rotational problems stop feeling like a new subject.

How Unit 5 Is Tested and Where to Practice

On the multiple-choice section (42 questions in 85 minutes), Unit 5 shows up as quick torque calculations, rotational inertia rankings, and graph questions where you read $\omega$ or $\alpha$ from an angular position or angular velocity graph. The 10 to 15% weight applies to this section.

On the free-response section (4 questions in 95 minutes), rotation fits all four question types: Mathematical Routines (a pulley or beam calculation), Translation Between Representations (torque diagrams and motion graphs), Experimental Design and Analysis (for example, a setup to determine an object's rotational inertia), and Qualitative/Quantitative Translation. A calculator is allowed on both sections.

To practice: start with the torque guide and the rotational kinematics guide, check your numbers with the torque calculator, and keep the AP Physics 1 formula sheet open so you know exactly which rotational equations are printed for you. Once you are solid here, Unit 6 extends rotation to energy and angular momentum.