Review AP Physics 1 Unit 5 to build your understanding of torque, rotational inertia, and Newton's laws applied to spinning systems. This unit extends the force and motion framework from Units 1 and 2 into rotational motion, covering everything from angular kinematics to the rotational form of Newton's second law.
Use the topic guides, key terms, and practice questions available for this unit to work through each concept before your exam.
Unit 5 is the rotational counterpart to the linear dynamics you studied in Units 1 and 2. Every major idea has a direct parallel: angular displacement mirrors linear displacement, torque mirrors force, and rotational inertia mirrors mass. The unit builds from describing how things spin, to explaining why they spin the way they do.
Angular displacement (Δθ), angular velocity (ω), and angular acceleration (α) follow the same mathematical structure as their linear counterparts. Under constant angular acceleration, you can use the rotational kinematic equations ω = ω₀ + αt and θ = θ₀ + ω₀t + ½αt² just as you used v = v₀ + at in Unit 1.
Only the component of force perpendicular to the position vector from the axis causes rotation. The magnitude is τ = rF sinθ, where r is the distance from the axis to the point of application and θ is the angle between the force and that position vector. A longer lever arm produces more torque for the same force.
Rotational inertia I = mr² for a point mass tells you that mass farther from the axis contributes more resistance to rotation. For a system of objects, I_total = Σmᵢrᵢ². The parallel axis theorem I' = I_cm + Md² lets you find rotational inertia about any axis parallel to one through the center of mass.
Every linear concept in Units 1 and 2 has a rotational equivalent in Unit 5. Force becomes torque, mass becomes rotational inertia, and linear acceleration becomes angular acceleration. Newton's second law in rotational form, α = τ_net / I, is the engine of this unit. When net torque is zero, the system is in rotational equilibrium and angular velocity does not change, exactly as zero net force keeps linear velocity constant. Recognizing these parallels lets you transfer problem-solving strategies you already know directly into rotational scenarios.
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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Review Torque with attention to how the concept appears in AP-style source and evidence questions.
Review Rotational Inertia with attention to how the concept appears in AP-style source and evidence questions.
Review Connecting Linear and Rotational Motion with attention to how the concept appears in AP-style source and evidence questions.
Review Newton's Second Law in Rotational Form with attention to how the concept appears in AP-style source and evidence questions.
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.
| 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.
| 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.
| 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.
| 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.
| 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.
| 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 |
Try stimulus-based AP practice questions and written prompts after you review the notes.
A satellite, modeled as a uniform rod of length , is initially at rest. Two thrusters apply constant forces of magnitude and perpendicular to the rod at the positions shown in the figure.
Wheel A rolls without slipping at a constant speed along a horizontal surface. Wheel B rolls without slipping down an inclined plane, increasing in speed as it moves.
3. Students are investigating how torque affects the angular acceleration of a rotating system.
Figure 1. Disk-pulley apparatus for angular acceleration
Figure 2. Grid for linearized relationship plot
Table 1. Hanging mass and angular acceleration data
Describe an experimental procedure the students could use to determine the angular acceleration α of the disk when a hanging mass m is released from rest. Include the measurements to be made, how the measurements are used to determine α using angular kinematics, and one method to reduce experimental uncertainty.
The students model the system as follows. The hanging mass m accelerates downward with linear acceleration a while the disk-and-pulley rotate with angular acceleration α. The string does not slip on the pulley, so a = αR. The torque on the rotating system due to the string is τ = TR, where T is the string tension. The net torque about the axle is related to angular acceleration by τ = Iα. Using Newton’s second law for the hanging mass and the relationships above, the students determine that
α = (m g R) / (I + mR^2)
Describe how the students could use measurements of α for different values of m to create a linear graph and how the slope of that graph could be used to determine the rotational inertia I. Clearly identify what quantities are plotted on each axis and how the slope is related to I.
m (kg) | α (rad/s^2) |
|---|---|
0.050 | 1.45 |
0.070 | 1.96 |
0.090 | 2.44 |
0.110 | 2.88 |
0.130 | 3.33 |
The students perform five trials using different hanging masses m. For each trial, the students release the system from rest and measure the time Δt for the disk to rotate through an angular displacement of Δθ = 6.00 rad. The students use rotational kinematics (Δθ = (1/2)α(Δt)^2) to determine α for each trial. The values of m and the calculated α are shown in Table 1.
The students decide to graph 1/m on the horizontal axis.
Indicate what measured or calculated quantity could be plotted on the vertical axis to yield a linear graph whose slope can be used to calculate an experimental value for the rotational inertia I.
Vertical axis: Horizontal axis: 1/m
On the blank grid provided, create a graph of the quantities indicated in part C(i) that can be used to determine I.
Use Table 2 to record the data points or calculated quantities that you will plot.
Clearly label the vertical axis, including units as appropriate.
Plot the points you recorded in Table 2.
Draw a straight best-fit line for the data graphed in part C(ii).
Using the best-fit line that you drew in part C(iii), calculate an experimental value for the rotational inertia I of the disk-and-pulley system. Use g = 9.80 m/s^2 and R = 0.0250 m.
4. In Scenario 1, a uniform solid disk of mass and radius can rotate without friction about a fixed horizontal axle through its center. A light string is wrapped around the rim of the disk and is pulled tangentially with a constant force of magnitude , as shown in Figure 1. The disk starts from rest and has an initial angular acceleration of magnitude . All frictional forces are negligible, and the string does not slip on the disk.
In Scenario 2, the same disk rotates about the same axle. A light string is wrapped around a smaller radius pulley rigidly attached to the disk so that the string is tangent to a circle of radius . The string is pulled tangentially with the same constant force magnitude , as shown in Figure 2. The disk again starts from rest and has an initial angular acceleration of magnitude . All frictional forces are negligible, and the string does not slip.
Figure 1. Scenario 1: A constant tangential pull F₁ = 6.0 N applied at the outer rim (radius R = 0.40 m) of a uniform solid disk (mass M = 2.0 kg).
Figure 2. Scenario 2: The same disk (M = 2.0 kg, R = 0.40 m) with a rigidly attached inner pulley (radius r = 0.20 m) pulled by the same constant tangential force F₂ = 6.0 N.
Refer to Figure 2. Refer to Figure 1. Indicate whether is greater than, less than, or equal to by writing one of the following in your answer booklet.
•
•
•
Justify your answer in terms of ALL torques exerted on the disk about the axle in each scenario. Use qualitative reasoning beyond referencing equations.
Starting with Newton's second law for rotation, derive an expression for the initial angular acceleration magnitude of the disk. Express your answer in terms of , , , , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider the general case of the same uniform solid disk (mass , radius ) pulled tangentially by a string with constant force magnitude applied at a radius from the axle (where ). The disk rotates about a fixed axle through its center, and friction is negligible.
Indicate whether the expression for you derived in part B is or is not consistent with the claim made in part A. Briefly justify your answer by referencing your derivation in part B.
1. A uniform solid disk of radius 0.20 m and mass 2.0 kg is mounted on a low-friction axle through its center, as shown in Figure 1. A light string is wrapped around the rim of the disk and is pulled by a student with a constant force of 8.0 N, causing the disk to rotate.
Figure 1. Uniform solid disk with tangential force
Figure 2. Axes for angular velocity vs. time
On the axes shown in Figure 2, sketch a graph of the disk’s angular velocity ω as a function of time t from t = 0 to t = 3.0 s.
Derive an expression for the magnitude of the angular acceleration α of the disk during the time interval from t = 0 to t = 1.5 s in terms of the given quantities and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Derive an expression for the magnitude of the linear speed v of a point on the rim of the disk at t = 1.5 s in terms of the given quantities and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Indicate whether the angular speed of the disk-plus-point-mass system at t = 1.5 s is greater than, less than, or equal to the angular speed of the disk alone at t = 1.5 s. The same student repeats the experiment, but now a point mass of 0.50 kg is attached to the disk at the rim (0.20 m from the axle). The student again pulls with the same constant force of 8.0 N, tangent to the rim, for the same time interval from t = 0 to t = 1.5 s. The disk again starts from rest, and frictional torques remain negligible.
Greater than
Less than
Equal to
Justify your response.
| 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. |
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.
Open the individual guides for Unit 5 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Physics 1 Unit 5 covers 6 topics in torque and rotational dynamics: Rotational Kinematics (5.1), Connecting Linear and Rotational Motion (5.2), Torque (5.3), Rotational Inertia (5.4), Rotational Equilibrium and Newton's First Law in Rotational Form (5.5), and Newton's Second Law in Rotational Form (5.6). These topics build directly on linear motion and force concepts, translating them into their rotational equivalents. By the end of the unit, you can analyze systems that combine both linear and rotational motion. See all six topics at /ap-physics-1-revised/unit-5.
AP Physics 1 Unit 5 makes up 10-15% of the AP exam, making torque and rotational dynamics one of the more heavily tested concept areas. That means you can expect a meaningful number of multiple-choice questions and potentially an FRQ touching on topics like rotational inertia, Newton's Second Law in rotational form, and rotational equilibrium. Given that weight, it's worth spending solid time here, especially on connecting rotational kinematics to the linear motion concepts you already know.
The 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.