Review AP Physics C: Mechanics Unit 6 to build fluency with rotational kinetic energy, angular momentum, conservation laws, rolling motion, and satellite orbits. This unit carries 10-15% of the exam and pulls together energy and momentum tools from earlier units and applies them to spinning and orbiting systems.
Use the topic guides, practice questions, and FRQ practice available on Fiveable to work through each concept before your exam.
Unit 6 asks you to apply two of the most powerful tools in mechanics, energy conservation and momentum conservation, to systems that rotate. The unit builds from the rotational inertia and torque ideas in Unit 5 and adds the energy and momentum accounting needed to solve complex spinning and orbiting problems.
Unit 6 covers rotational kinetic energy, work done by torques, angular momentum and angular impulse, conservation of angular momentum, rolling motion combining translation and rotation, and the orbital mechanics of satellites governed by gravitational conservation laws.
Rotational kinetic energy K_rot = 1/2 I omega^2 is a scalar that adds directly to translational kinetic energy. For a rolling object, K_tot = 1/2 m v_cm^2 + 1/2 I omega^2. Work done by a torque equals the integral of tau d theta, and the rotational work-energy theorem connects net rotational work to the change in K_rot.
Angular momentum is L = I omega for a rigid body about a fixed axis, or L = r x p for any object about a point. Angular impulse equals the integral of tau dt and equals the change in L. When net external torque is zero, total angular momentum is conserved, even if the system changes shape and redistributes mass.
Rolling without slipping links translation and rotation through v_cm = r omega, a_cm = r alpha. Static friction does no work, so mechanical energy is conserved. For satellites, circular orbits have constant L, K, and total mechanical energy E = -GMm/(2r), while elliptical orbits conserve L and total energy but allow K and gravitational potential energy to vary.
The central skill in Unit 6 is recognizing when angular momentum is conserved (zero net external torque) versus when it changes (nonzero net external torque equals angular impulse). The same logic applies to energy: identify whether work is done by external torques or whether the no-slip condition allows mechanical energy conservation. These two questions, is angular momentum conserved? is mechanical energy conserved? drive nearly every problem in the unit.
Defines K_rot = 1/2 I omega^2 and establishes that total kinetic energy is the sum of translational and rotational contributions. Introduces the scalar nature of rotational kinetic energy and common moment-of-inertia values.
Develops W = integral tau d theta as the rotational analog of linear work, including the graphical interpretation as area under a tau-versus-theta curve and the rotational work-energy theorem.
Introduces L = I omega and L = r x p, defines angular impulse as the integral of tau dt, and connects them through the rotational impulse-momentum theorem delta L = integral tau dt.
Establishes that total angular momentum is conserved when net external torque is zero, explains how nonrigid systems change omega by redistributing mass, and emphasizes system boundary selection.
Applies the no-slip constraint v_cm = r omega to combine translational and rotational kinetic energy, distinguishes pure rolling (energy conserved) from slipping (kinetic friction dissipates energy).
Uses conservation of energy and angular momentum to analyze circular and elliptical orbits, deriving orbital speed, total mechanical energy E = -GMm/(2r), and escape velocity v_esc = sqrt(2GM/r).
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Review Motion of Orbiting Satellites with attention to how the concept appears in AP-style source and evidence questions.
Review Angular Momentum and Angular Impulse with attention to how the concept appears in AP-style source and evidence questions.
Review Conservation of Angular Momentum with attention to how the concept appears in AP-style source and evidence questions.
Review Rotational Kinetic Energy with attention to how the concept appears in AP-style source and evidence questions.
A rigid body spinning about any axis has rotational kinetic energy K_rot = 1/2 I omega^2. This is a scalar, so it adds directly to translational kinetic energy. When a rigid body both translates and rotates, K_tot = 1/2 m v_cm^2 + 1/2 I omega^2. A system can have rotational kinetic energy even when its center of mass is stationary, because individual mass elements still have linear speed.
A solid disk (I = 1/2 m r^2) and a hoop (I = m r^2) of equal mass and radius spin at the same omega. Which has more rotational kinetic energy, and by what factor?
| Object | Moment of inertia I | K_rot at same omega |
|---|---|---|
| Solid disk | 1/2 m r^2 | 1/4 m r^2 omega^2 |
| Hoop (thin ring) | m r^2 | 1/2 m r^2 omega^2 |
| Solid sphere | 2/5 m r^2 | 1/5 m r^2 omega^2 |
| Thin rod (center) | 1/12 m L^2 | 1/24 m L^2 omega^2 |
A torque transfers energy into or out of a rotating system when it acts through an angular displacement. The work done is W = integral from theta_1 to theta_2 of tau d theta. For constant torque, W = tau * delta theta. On a torque-versus-angular-position graph, work equals the area under the curve. The rotational work-energy theorem states that the net work done by all torques equals the change in rotational kinetic energy.
A torque that varies as tau = 3 theta (in N*m, theta in radians) acts on a disk from theta = 0 to theta = 2 rad. What is the total work done on the disk?
| Quantity | Linear analog | Rotational version |
|---|---|---|
| Work | W = integral F dx | W = integral tau d theta |
| Constant-force/torque work | W = F * delta x | W = tau * delta theta |
| Work-energy theorem | W_net = delta K_trans | W_net = delta K_rot |
Angular momentum has two useful forms: L = I omega for a rigid body about a fixed axis, and L = r x p (vector cross product) for any object about a chosen point. The choice of reference point matters because it changes the value of L. Even an object moving in a straight line has nonzero angular momentum about any point not on its line of motion. Angular impulse equals the integral of tau dt and equals the change in angular momentum: delta L = integral tau dt. On a torque-versus-time graph, angular impulse is the area under the curve, and the slope of an L-versus-time graph equals the net torque.
A point mass moves in a straight line at constant velocity. Explain why its angular momentum about a point off its path is nonzero and constant.
| Concept | Linear version | Rotational version |
|---|---|---|
| Momentum | p = m v | L = I omega or r x p |
| Impulse | J = integral F dt | Angular impulse = integral tau dt |
| Impulse-momentum theorem | J = delta p | Angular impulse = delta L |
| Newton's second law | F_net = dp/dt | tau_net = dL/dt |
The total angular momentum of a system is the sum of the angular momenta of all its parts. If the net external torque on a system is zero, total angular momentum is conserved. If the net external torque is nonzero, angular momentum is transferred between the system and its environment by an amount equal to the angular impulse. A nonrigid system can change its angular speed without any external torque if it redistributes mass, changing I while L stays constant (the classic spinning-skater scenario). System selection is critical: choose a system boundary where all relevant torques are internal to apply conservation.
A student sits on a frictionless rotating stool holding a spinning bicycle wheel with its axis vertical. The student flips the wheel upside down. Describe what happens to the student's rotation and explain why using angular momentum conservation.
| Condition | Net external torque | Angular momentum |
|---|---|---|
| Isolated system | Zero | Conserved |
| External torque applied | Nonzero | Changes by angular impulse |
| Nonrigid, no external torque | Zero | Conserved; omega changes as I changes |
When an object rolls without slipping, its center-of-mass translation and its rotation are linked by v_cm = r omega and a_cm = r alpha. Total kinetic energy is K_tot = 1/2 m v_cm^2 + 1/2 I omega^2. Because static friction does no work in pure rolling, mechanical energy is conserved and you can use energy methods directly. When an object slips, kinetic friction acts at the contact point, the no-slip constraint breaks down, and kinetic friction dissipates energy from the system.
A solid sphere (I = 2/5 m r^2) and a hollow sphere (I = 2/3 m r^2) of equal mass and radius start from rest at the top of the same incline. Which reaches the bottom first, and why?
| Object | I / (m r^2) | Fraction of K_tot that is rotational |
|---|---|---|
| Hoop | 1 | 1/2 |
| Hollow sphere | 2/3 | 2/5 |
| Solid disk | 1/2 | 1/3 |
| Solid sphere | 2/5 | 2/7 |
For a satellite of negligible mass orbiting a massive central body M, gravity provides centripetal force and two conservation laws govern the motion. In circular orbits, total mechanical energy E = -GMm/(2r), kinetic energy K = GMm/(2r), and gravitational potential energy U = -GMm/r are all constant, as is angular momentum L = m v r. In elliptical orbits, total mechanical energy and angular momentum are still conserved, but K and U each vary as the satellite moves closer to or farther from the central body. Escape velocity from radius r is v_esc = sqrt(2GM/r), which corresponds to total mechanical energy equal to zero.
A satellite in a circular orbit of radius r is given a brief forward thrust that increases its speed. Describe qualitatively what happens to its orbit shape, total energy, and angular momentum immediately after the thrust.
| Orbit type | Total energy E | Angular momentum L | K and U individually |
|---|---|---|---|
| Circular | Constant, -GMm/(2r) | Constant | Both constant |
| Elliptical | Constant (negative) | Constant | Each varies |
| Escape (parabolic) | Zero | Constant | Each varies |
Try AP-style multiple-choice questions and written prompts after you review the notes.
A timed, AP-style test built only from unit 6 questions — multiple choice plus free response, scored with AI feedback. Use the standard format or build your own.
| Term | Definition |
|---|---|
| angular displacement | The angle in radians through which a rigid body rotates about a specified axis, calculated as delta theta = theta minus theta_0. It appears in the rotational work integral W = integral tau d theta. |
| impulse-momentum theorem | In rotational form: the angular impulse delivered to a system equals its change in angular momentum, delta L = integral tau dt. This is the rotational analog of J = delta p. |
| scalar | A physical quantity described by magnitude only, without direction. Rotational kinetic energy and work are scalars, which is why K_rot adds directly to K_trans. |
| two-body gravitational system | A system of a massive central object and an orbiting satellite interacting only via gravity. When the satellite mass is negligible, the central object is treated as stationary and conservation laws govern the satellite's orbit. |
For a rolling object, total kinetic energy is K_tot = 1/2 m v_cm^2 + 1/2 I omega^2. Counting only translational kinetic energy underestimates the total and gives wrong speeds at the bottom of inclines.
Conservation of angular momentum requires zero net external torque about the chosen axis. Friction from a surface or a pivot exerting a torque means angular momentum is not conserved for that system.
L = I omega applies to rigid bodies rotating about a fixed axis. L = r x p applies to any object about any reference point, including a point mass moving in a straight line. Using the wrong form leads to incorrect magnitudes and directions.
For satellites, gravitational potential energy is U = -GMm/r, not mgh. Using mgh is only valid near Earth's surface. Orbital energy calculations require U = -GMm/r and E = -GMm/(2r).
In pure rolling without slipping, the contact point has zero instantaneous velocity, so static friction does no work and mechanical energy is conserved. Only kinetic friction during slipping dissipates energy.
Free-response questions in this unit frequently require you to apply both energy conservation and angular momentum conservation in sequence within the same problem. A common structure is: find the angular speed after a collision or mass redistribution using angular momentum conservation, then use that result to find kinetic energy or height using energy conservation. Keeping track of which quantities are conserved at each stage is the central skill.
Multiple-choice and free-response questions often present torque-versus-time or torque-versus-angle graphs and ask you to extract angular impulse or work as areas under the curve, or to identify net torque as the slope of an angular momentum graph. Practicing these graphical readings directly from the equations W = integral tau d theta and delta L = integral tau dt is essential.
A recurring task is explaining in words why a conservation law does or does not apply to a given system. For angular momentum, you must identify whether net external torque is zero and justify your system boundary choice. For energy in rolling problems, you must state whether the no-slip condition holds and whether friction does work. Clear, equation-linked justifications are expected in free-response scoring.
Use this list to confirm you can handle every major skill before your exam.
Apply K_rot = 1/2 I omega^2 and K_tot = K_trans + K_rot for rigid bodies with known moments of inertia such as disks, hoops, and spheres.
Evaluate W = integral tau d theta analytically for variable torques and graphically as area under a tau-versus-theta curve; apply the rotational work-energy theorem.
Use L = I omega for rigid bodies about a fixed axis and L = r x p (magnitude m v r sin theta) for point masses about any reference point.
Identify whether net external torque is zero, select the correct system boundary, and solve for unknown angular speeds when moment of inertia changes.
Use v_cm = r omega to link translation and rotation, write K_tot = 1/2 m v_cm^2 + 1/2 I omega^2, and apply energy conservation for rolling without slipping on inclines.
Derive circular orbital speed and total energy, use E = -GMm/(2r) and v_esc = sqrt(2GM/r), and explain which quantities are conserved in elliptical versus circular orbits.
Read the Topic 6.1 and 6.2 guides on Fiveable. Practice writing K_rot = 1/2 I omega^2 for disks, hoops, and spheres, then evaluate W = integral tau d theta for both constant and variable torques. Confirm you can read area off a tau-versus-theta graph.
Work through the Topic 6.3 guide. Practice computing L = I omega for rigid bodies and L = r x p for point masses at various reference points. Sketch torque-versus-time graphs and identify angular impulse as the area under the curve.
Use the Topic 6.4 guide to practice identifying zero-external-torque conditions and selecting system boundaries. Solve problems where a skater, diver, or rotating platform changes shape, and verify that L = I omega is conserved while omega changes.
Work through the Topic 6.5 guide. Set up K_tot = 1/2 m v_cm^2 + 1/2 I omega^2 with the no-slip substitution v_cm = r omega for incline problems. Compare final speeds for objects with different I values and explain why the hoop is slowest.
Read the Topic 6.6 guide and derive E = -GMm/(2r) and v_esc = sqrt(2GM/r) from scratch. Then use the available FRQ practice on Fiveable to work multi-part problems that combine rolling, angular momentum conservation, or orbital energy in a single question.
Open the individual guides for Unit 6 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Watch past review streams filtered to Unit 6 when you want a video walkthrough.
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get printablesAP Physics C: Mechanics Unit 6 covers 6 topics: Rotational Kinetic Energy, Torque and Work, Angular Momentum and Angular Impulse, Conservation of Angular Momentum, Rolling, and Motion of Orbiting Satellites. Together they apply energy and momentum concepts to rotating systems, including objects rolling without slipping and orbiting satellites. See the full topic list at /ap-physics-c-mechanics/unit-6.
Unit 6 makes up 10-15% of the AP Physics C: Mechanics exam. That weight comes from topics like Rotational Kinetic Energy, Conservation of Angular Momentum, Rolling, and Motion of Orbiting Satellites. It's a meaningful chunk of the test, so understanding when angular momentum is conserved versus when it changes is especially important.
The AP Physics C: Mechanics Unit 6 progress check includes both MCQ and FRQ parts drawn from all six unit topics: Rotational Kinetic Energy, Torque and Work, Angular Momentum and Angular Impulse, Conservation of Angular Momentum, Rolling, and Motion of Orbiting Satellites. MCQ questions test conceptual understanding and calculation, while the FRQ section asks you to set up and solve multi-part problems involving angular momentum or rolling motion. For matched practice aligned to the progress check, visit /ap-physics-c-mechanics/unit-6.
To practice Unit 6 FRQs, focus on the topics that appear most often in free-response questions: Conservation of Angular Momentum, Rolling without slipping, and Torque and Work. Typical FRQ formats ask you to derive an expression for angular momentum before and after a collision, calculate rotational kinetic energy, or analyze a rolling object on an incline. Work through each part by writing out your setup, applying the correct conservation law, and checking units. Find Unit 6 FRQ practice at /ap-physics-c-mechanics/unit-6.
You can find AP Physics C: Mechanics Unit 6 practice questions, including multiple-choice and practice test problems, at /ap-physics-c-mechanics/unit-6. That page has MCQ sets and FRQs covering all six topics: Rotational Kinetic Energy, Torque and Work, Angular Momentum, Conservation of Angular Momentum, Rolling, and Motion of Orbiting Satellites.
Start Unit 6 by building a solid feel for rotational kinetic energy and how it connects to the linear kinetic energy formulas you already know. Then work through Torque and Work, Angular Momentum and Angular Impulse, and Conservation of Angular Momentum in order, since each concept builds on the last. Rolling without slipping trips up a lot of students, so practice breaking it into translational and rotational parts separately before combining them. Finish with Motion of Orbiting Satellites, which ties angular momentum to gravity. For topic guides and practice sets, go to /ap-physics-c-mechanics/unit-6.