AP Physics 1 Unit 5 Essentials: Torque and Rotational Dynamics Key Terms & Big Ideas

Big Idea 1

Rotation has a perfect parallel to linear motion

Every linear quantity has a rotational counterpart: position ↔ angle (θ), velocity ↔ angular velocity (ω), acceleration ↔ angular acceleration (α), force ↔ torque (τ), mass ↔ rotational inertia (I). The kinematic equations and Newton's laws translate directly. Master the analogy table and most of Unit 5 follows automatically.

Angular Kinematics Analogy Connection

Big Idea 2

Torque depends on force, distance, AND angle

τ = rF·sin(θ). To produce a torque, you need a force AND a lever arm (distance from pivot) AND a component of force perpendicular to that lever arm. A force directed straight at the pivot produces zero torque, no matter how big the force is. That's why doors have handles far from the hinges — bigger lever arm means more torque for the same push.

Torque Moment Arm τ = rF·sin(θ)

Big Idea 3

Rotational inertia depends on how mass is distributed

Two objects with the same mass can have very different rotational inertias depending on where the mass is. Mass close to the axis spins easily; mass far from the axis is hard to spin up. That's why a tightrope walker uses a long pole (high rotational inertia resists tipping), and why figure skaters spin faster by pulling their arms in (smaller I means faster ω).

Rotational Inertia I = Σmr² Mass Distribution

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Rotational Motion

Movement of an object around an axis of rotation. Every point on the object traces a circular path around that axis. Examples: a spinning wheel, a planet rotating on its axis, a door swinging on hinges.

Foundation

Radian (rad)

The SI unit of angle. One radian = the angle that gives an arc length equal to the radius. One full revolution = 2π radians ≈ 6.28 rad ≈ 360°. Radians are unit-less in calculations (they're a ratio).

Foundation

Angular Displacement (θ)

The angle through which an object has rotated, measured in radians. The rotational analog of linear displacement (Δx). Positive = counterclockwise, negative = clockwise (by convention).

5.1 Rotational Kinematics

Angular Velocity (ω)

How fast something is rotating: ω = Δθ/Δt. Units: rad/s. The rotational analog of velocity. A vector quantity — direction is along the axis of rotation (right-hand rule), but for AP Physics 1 you can treat it as a signed scalar.

5.1 Rotational Kinematics

Angular Acceleration (α)

The rate at which angular velocity changes: α = Δω/Δt. Units: rad/s². The rotational analog of linear acceleration. If a wheel is spinning faster and faster, it has angular acceleration.

5.1 Rotational Kinematics

Rotational Kinematic Equations

Same as Unit 1 kinematic equations, but with rotational variables: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt², ω² = ω₀² + 2αΔθ. Use whenever angular acceleration is constant.

5.1 Rotational Kinematics

Period of Rotation (T)

The time for one full rotation (one revolution). Connected to angular velocity: T = 2π/ω. A wheel with ω = 2π rad/s completes one revolution per second, so T = 1 s.

5.1 Rotational Kinematics

Arc Length (s = rθ)

The distance traveled along a circular path. Equals radius times angle in RADIANS. A point on a rotating disk farther from the axis travels a longer arc per revolution.

5.2 Linear ↔ Rotational

Tangential Velocity (v = rω)

The linear speed of a point on a rotating object, tangent to its circular path. Larger radius = faster tangential speed for the same angular velocity. The tip of a propeller moves much faster than a point near the hub.

5.2 Linear ↔ Rotational

Tangential Acceleration (a_t = rα)

The linear acceleration along the tangential direction (along the path of motion). Caused by angular acceleration of the rotating object. Doesn't change the direction of motion — just the speed.

5.2 Linear ↔ Rotational

Centripetal Acceleration (a_c = ω²r)

The acceleration pointing toward the center of the circular path, caused by the constantly-changing direction of velocity. Always present in any circular motion, even at constant speed. Also written a_c = v²/r.

5.2 Linear ↔ Rotational

Torque (τ = rF·sin(θ))

The "rotational force" — what makes things spin faster or slower. Equals the lever arm (r) times the perpendicular component of force (F·sin θ). Units: N·m. The rotational analog of force.

5.3 Torque

Moment Arm (Lever Arm)

The perpendicular distance from the axis of rotation to the line of action of the force. A longer moment arm = bigger torque for the same force. That's why wrenches have long handles.

5.3 Torque

Pivot / Axis of Rotation

The point or line about which an object rotates. Torques are measured relative to this point. Different choices of pivot give different individual torque values, but the SAME final answer for equilibrium or net torque problems.

5.3 Torque

Torque Direction (Sign)

By convention: counterclockwise = positive, clockwise = negative. When summing torques, add positive contributions and subtract negative ones. For equilibrium, the magnitudes of clockwise and counterclockwise torques must balance.

5.3 Torque

Force Component Perpendicular to r

Only the component of force perpendicular to the lever arm produces torque. A force pulling along the lever arm (toward or away from the pivot) creates zero torque. This is why you push a door perpendicular to its surface, not along it.

5.3 Torque

Rotational Inertia (Moment of Inertia, I)

An object's resistance to changes in rotation. The rotational analog of mass. Units: kg·m². For a point mass: I = mr². For extended objects, I depends on shape AND axis location.

5.4 Rotational Inertia

Mass Distribution

How an object's mass is spread out relative to the axis. Mass close to the axis contributes less to I; mass far away contributes a LOT (since I = Σmr², the r is squared). A solid sphere has less I than a thin hoop of the same mass and radius.

5.4 Rotational Inertia

Parallel Axis Theorem (qualitative)

An object's rotational inertia about ANY axis is greater than its rotational inertia about an axis through its center of mass. The farther from the COM your axis is, the larger the I. AP Physics 1 only requires qualitative understanding.

5.4 Rotational Inertia

Point Mass (I = mr²)

A simple model treating an object as a single point of mass m located at distance r from the axis. Its rotational inertia is just I = mr². For systems of point masses, add them up: I_total = Σm_i·r_i².

5.4 Rotational Inertia

Rotational Equilibrium (Στ = 0)

The condition where the net torque on an object is zero. An object in rotational equilibrium has constant angular velocity — usually zero (not spinning) but could also be spinning at a steady rate.

5.5 Rotational Equilibrium

Newton's First Law (Rotational)

If the net torque on an object is zero, its angular velocity stays constant. The rotational analog of "objects at rest stay at rest." A spinning wheel keeps spinning forever (in the absence of friction); a stationary lever stays still.

5.5 Rotational Equilibrium

Balanced Torques

When clockwise torques exactly cancel counterclockwise torques. The classic example: a seesaw with a heavy kid close to the pivot balancing a lighter kid farther away. Different forces, different lever arms, same torques.

5.5 Rotational Equilibrium

Static Equilibrium

When an object is both translationally AND rotationally at rest: ΣF = 0 AND Στ = 0. Classic test problem: a ladder against a wall, or a uniform beam supported by a cable. Both force and torque conditions must hold.

5.5 Rotational Equilibrium

Newton's Second Law (Rotational): τ_net = Iα

The rotational version of F = ma. Net torque equals rotational inertia times angular acceleration. To spin something up faster, you need more torque or less rotational inertia.

5.6 Newton's 2nd Law (Rot.)

Connection to F = ma

τ_net = Iα is just F = ma rewritten for rotation. Every term has a rotational counterpart: F → τ, m → I, a → α. Same logic, same problem-solving steps — just with angular quantities.

5.6 Newton's 2nd Law (Rot.)

Rotational vs Translational Comparison

Same Newton's laws, same kinematic equations, same energy ideas — applied to rotation instead of straight-line motion. When you see a problem involving rotation, ask: "What's the linear-motion version, and how does each piece translate?"

5.6 Newton's 2nd Law (Rot.)

Analogous Quantities (the Translation Table)

x ↔ θ; v ↔ ω; a ↔ α; m ↔ I; F ↔ τ; F = ma ↔ τ = Iα. Memorize this table. Every Unit 5 problem is a Unit 1 or Unit 2 problem with rotational analogs swapped in.

5.6 Newton's 2nd Law (Rot.)