Torque and rotational dynamics describe how forces cause or prevent rotation. Torque depends on force magnitude, distance from the axis, and the angle of application. Rotational inertia resists changes in spinning motion. When net torque is zero, angular velocity stays constant. When net torque is nonzero, angular acceleration results according to α = τ_net / I.
Fiveable's AP Physics 1 Unit 5 review covers 6 topics, with guided practice questions for the unit.
Describes rotation using angular displacement Δθ, angular velocity ω, and angular acceleration α. Under constant α, rotational kinematic equations parallel the linear equations from Unit 1. All points in a rigid system share the same ω and α.
Links angular and linear quantities for any point at distance r from the axis: s = rθ, v = rω, and a_T = rα. Points farther from the axis have greater tangential speed and acceleration even though ω and α are the same for the whole rigid system.
Torque is the rotational effect of a force, calculated as τ = rF sinθ. Only the force component perpendicular to the position vector from the axis contributes. The lever arm is the perpendicular distance from the axis to the line of action of the force.
Rotational inertia I = mr² measures resistance to changes in rotation. For discrete systems, I_total = Σmᵢrᵢ². Mass farther from the axis increases I more. The parallel axis theorem I' = I_cm + Md² applies when the axis does not pass through the center of mass.
Rotational equilibrium requires Στ = 0, keeping angular velocity constant. This is the rotational form of Newton's first law. A system can be in rotational equilibrium without translational equilibrium. Choosing a smart pivot point simplifies balance calculations.
A nonzero net torque produces angular acceleration: α = τ_net / I. Angular acceleration is proportional to net torque and inversely proportional to rotational inertia. Systems with both linear and rotational motion require separate analyses connected by a = rα.
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Rotational kinematics uses angular displacement Δθ (in radians), angular velocity ω (rad/s), and angular acceleration α (rad/s²) to describe how a rigid system rotates. These quantities follow the same relationships as linear kinematics, so under constant angular acceleration the same equation structure applies. Counterclockwise is typically positive by convention. A rigid system holds its shape, meaning all points share the same ω and α even though they travel different linear distances.
A wheel starts from rest and reaches ω = 12 rad/s in 4 s under constant angular acceleration. What is α, and how many radians does it rotate through?
| Linear quantity | Symbol | Rotational equivalent | Symbol |
|---|---|---|---|
| Displacement | Δx | Angular displacement | Δθ |
| Velocity | v | Angular velocity | ω |
| Acceleration | a | Angular acceleration | α |
| v = v₀ + at | ω = ω₀ + αt |
For any point at distance r from a fixed axis, linear and angular quantities are connected by three key equations: arc length s = rθ, tangential speed v = rω, and tangential acceleration a_T = rα. Because all points in a rigid system share the same ω and α, a point farther from the axis moves faster and accelerates more than a point closer to the axis. This relationship is essential for analyzing wheels, gears, and rolling objects.
A disk of radius 0.5 m rotates at ω = 6 rad/s. What is the tangential speed of a point on the rim, and what is the tangential speed of a point 0.25 m from the center?
| Equation | Relates | Key point |
|---|---|---|
| s = rθ | Arc length to angular displacement | θ must be in radians |
| v = rω | Tangential speed to angular velocity | Larger r gives larger v |
| a_T = rα | Tangential acceleration to angular acceleration | Distinct from centripetal acceleration |
Torque measures how effectively a force rotates a rigid system about an axis. Only the component of force perpendicular to the position vector from the axis to the point of application produces torque. The magnitude is τ = rF sinθ = rF⊥, where θ is the angle between the force vector and the position vector. The lever arm is the perpendicular distance from the axis to the line of action of the force. Force diagrams, similar to free-body diagrams, show force magnitudes, directions, and points of application relative to the axis.
A 20 N force is applied at 0.4 m from a pivot at an angle of 30° to the position vector. What is the magnitude of the torque?
| Scenario | Lever arm | Torque effect |
|---|---|---|
| Force perpendicular to r (θ = 90°) | r | Maximum torque: τ = rF |
| Force at angle θ to r | r sinθ | Reduced torque: τ = rF sinθ |
| Force parallel to r (θ = 0°) | 0 | Zero torque |
Rotational inertia I is the rotational analog of mass. It measures how strongly a rigid system resists changes in its rotation and depends on both the total mass and how that mass is distributed relative to the axis. For a point mass, I = mr². For a collection of objects, I_total = Σmᵢrᵢ². Mass farther from the axis contributes more to I, which is why a hoop has greater rotational inertia than a solid disk of the same mass and radius. The parallel axis theorem I' = I_cm + Md² gives the rotational inertia about any axis parallel to one through the center of mass.
Two 2 kg masses are placed 0.3 m and 0.6 m from an axis. What is the total rotational inertia of the system? Which mass contributes more, and why?
| Object | Axis location | Relative rotational inertia |
|---|---|---|
| Hoop (mass M, radius R) | Through center | Larger (all mass at R) |
| Solid disk (mass M, radius R) | Through center | Smaller (mass distributed from 0 to R) |
| Rod (mass M, length L) | Through center | Smaller than through end |
| Rod (mass M, length L) | Through end | Larger (mass farther on average) |
A system is in rotational equilibrium when the net torque on it is zero (Στᵢ = 0), meaning its angular velocity remains constant. This is the rotational form of Newton's first law. A system can be in rotational equilibrium without being in translational equilibrium, and vice versa. To solve balance problems, choose a convenient pivot point, identify all torques with their signs, and set the sum equal to zero. Force diagrams show both the forces and their points of application relative to the chosen axis.
A uniform 4 m beam of mass 10 kg is supported at one end. A 30 N downward force is applied 1 m from the free end. What upward force at the free end keeps the beam in rotational equilibrium about the supported end?
| Condition | Net force | Net torque | Result |
|---|---|---|---|
| Full static equilibrium | Zero | Zero | No linear or angular acceleration |
| Rotational equilibrium only | Nonzero | Zero | Linear acceleration, constant ω |
| Translational equilibrium only | Zero | Nonzero | Angular acceleration, constant v_cm |
| Neither equilibrium | Nonzero | Nonzero | Both linear and angular acceleration |
When the net torque on a rigid system is not zero, the system undergoes angular acceleration. The rotational form of Newton's second law is α_sys = τ_net / I_sys. Angular acceleration is directly proportional to net torque and inversely proportional to rotational inertia. For systems that involve both linear and rotational motion, such as a mass hanging from a pulley, you must perform separate linear and rotational analyses and connect them through the constraint equations from Topic 5.2.
A pulley of rotational inertia 0.2 kg·m² has a net torque of 0.8 N·m applied to it. What is its angular acceleration? If the net torque doubles and I stays the same, what happens to α?
| Linear (Unit 2) | Rotational (Unit 5) |
|---|---|
| F_net = ma | τ_net = Iα |
| a = F_net / m | α = τ_net / I |
| Larger m resists acceleration | Larger I resists angular acceleration |
| ΣF = 0 means constant v | Στ = 0 means constant ω |
| Term | Definition |
|---|---|
| axis of rotation | The specified line about which a rigid system rotates; the reference point for measuring torque, lever arm, and rotational inertia. |
| rigid system | A system that holds its shape during rotation, with all points sharing the same angular velocity and angular acceleration even though they travel different linear distances. |
| constant angular acceleration | A condition in which angular velocity changes at a uniform rate, allowing the use of rotational kinematic equations analogous to constant linear acceleration equations from Unit 1. |
| Δs = r Δθ | The arc length a point travels equals its distance from the axis times the angular displacement in radians; the foundation for v = rω and a_T = rα. |
| lever arm | The perpendicular distance from the axis of rotation to the line of action of an applied force; a longer lever arm produces greater torque for the same force magnitude. |
| moment arm | The perpendicular distance from the axis of rotation to the line of action of a force, used interchangeably with lever arm to calculate torque as τ = rF sinθ. |
| force diagram | A diagram similar to a free-body diagram used to analyze torques on a rigid system, showing force magnitudes, directions, and points of application relative to the axis of rotation. |
| parallel axis theorem | I' = I_cm + Md²; gives the rotational inertia about any axis parallel to one through the center of mass, where d is the perpendicular distance between the two axes. Rotational inertia is minimum when the axis passes through the center of mass. |
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All rotational kinematic equations and the arc length formula s = rθ require θ in radians. Converting to degrees before calculating will give wrong answers. Always check your units before substituting.
Only the component of force perpendicular to the position vector from the axis causes rotation. If a force is applied at an angle, you must use τ = rF sinθ, not τ = rF. Forgetting the sinθ factor is one of the most common torque errors.
Rotational inertia depends on both mass and the square of the distance from the axis. Doubling the distance from the axis quadruples I, not doubles it. Always square the radius when calculating I = mr².
A system can have zero net torque (constant ω) while its center of mass accelerates, or zero net force (constant v_cm) while angular velocity changes. These conditions are independent and must be checked separately.
In problems with a hanging mass and a pulley, students often write separate equations but forget to use a = rα to link the linear acceleration of the mass to the angular acceleration of the pulley. Without this constraint, the system is underdetermined.
The AP Physics 1 exam frequently asks how angular acceleration changes when net torque or rotational inertia changes by a given factor. You should be able to use α = τ_net / I to explain, without numbers, that doubling I halves α for the same torque, or that tripling τ_net triples α for the same I. Written justification of these proportional relationships is a core skill tested in both multiple-choice and free-response questions.
Free-response questions in this unit often involve systems where a hanging mass drives a rotating pulley or a force causes both translation and rotation. You are expected to write separate Newton's second law equations for the linear and rotational components, then connect them using the constraint a = rα. Clearly labeling which equation applies to which part of the system and showing the connection step is essential for full credit.
Questions may present a beam, rod, or lever with multiple forces and ask you to find an unknown force or distance that maintains rotational equilibrium. The skill is setting up Στ = 0 about a strategically chosen pivot, correctly computing each torque with its sign, and solving algebraically. You may also need to explain why a system is or is not in rotational equilibrium based on a described or diagrammed scenario.
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get printablesThe AP Physics 1 Unit 5 progress check in AP Classroom includes both MCQ and FRQ parts drawn from all six unit topics: rotational kinematics, connecting linear and rotational motion, torque, rotational inertia, rotational equilibrium, and Newton's Second Law in rotational form. The MCQ section tests conceptual understanding and quantitative reasoning across these topics, while the FRQ part typically asks you to apply Newton's laws in rotational form or analyze a system in rotational equilibrium. For matched practice that mirrors the progress check format, check out /ap-physics-1-revised/unit-5.
AP Physics 1 Unit 5 FRQs most often focus on torque, rotational inertia, and Newton's Second Law in rotational form, asking you to set up equations, justify reasoning, or analyze a physical scenario involving rotational equilibrium. To practice effectively, work through problems where you draw extended free-body diagrams, identify the pivot point, and write out net torque equations step by step. Good habits: always define your sign convention for rotation, show your algebra clearly, and connect back to the physical situation in your explanation. You can find FRQ-style practice problems at /ap-physics-1-revised/unit-5.
The best place to find AP Physics 1 Unit 5 practice questions, including multiple-choice and practice test problems on torque and rotational dynamics, is /ap-physics-1-revised/unit-5. There you'll find MCQ practice covering all six topics, from rotational kinematics and rotational inertia to rotational equilibrium and Newton's Second Law in rotational form. For the most targeted prep, focus your MCQ practice on problems that ask you to compare rotational inertia for different mass distributions and apply torque to find angular acceleration.
Start AP Physics 1 Unit 5 by locking in rotational kinematics (5.1) and the connections to linear motion (5.2), since those relationships, like angular velocity linking to linear velocity, show up throughout the rest of the unit. Then build toward torque and rotational inertia before tackling Newton's laws in rotational form. Here's a concrete study sequence:
1. Review rotational kinematics equations alongside their linear counterparts so the patterns stick.
2. Practice drawing extended free-body diagrams to identify torques and pivot points.
3. Work problems on rotational inertia for different object shapes, since the mass distribution matters.
4. Apply Newton's Second Law in rotational form to systems with multiple torques.
5. Test yourself with mixed MCQ and FRQ practice at /ap-physics-1-revised/unit-5. Since this unit is 10-15% of the exam, even a few focused study sessions here can meaningfully move your score.