Review AP Physics C: Mechanics Unit 5 to build fluency with torque, rotational inertia, and Newton's second law in rotational form. This unit translates the force-and-motion framework from Units 1 and 2 into its rotational counterparts, connecting angular kinematics to dynamics through calculus-based derivations.
Use the topic guides, practice questions, FRQ practice, and AP score calculator available on this page to work through every concept before exam day.
Unit 5 is the rotational counterpart to the translational dynamics you studied in Units 1 and 2. Every linear quantity gets a rotational analog: displacement becomes angular displacement, force becomes torque, mass becomes rotational inertia, and F = ma becomes sum-tau = I-alpha.
Topics 5.1 and 5.2 establish angular displacement (delta-theta), angular velocity (omega = d-theta/dt), and angular acceleration (alpha = d-omega/dt), then link them to linear quantities at a point a distance r from the axis: s = r-theta, v = r-omega, and a_T = r-alpha. All points on a rigid body share the same omega and alpha, but their linear speeds differ by radius.
Topic 5.3 defines torque as the cross product tau = r x F, with magnitude rF sin-theta and direction from the right-hand rule. Topic 5.4 defines rotational inertia I as the resistance to rotational change: I = mr^2 for a point mass, I = integral r^2 dm for a continuous solid, and I' = I_cm + Md^2 via the parallel axis theorem.
Topic 5.5 states that when net torque is zero, angular velocity is constant (rotational Newton's first law). Topic 5.6 states that when net torque is nonzero, alpha = sum-tau / I_sys. Combined linear and rotational analyses are often needed for systems like a mass on a string wrapped around a pulley.
Every concept in Unit 5 has a direct translational analog. Recognizing those pairings (theta with x, omega with v, alpha with a, tau with F, I with m, sum-tau = I-alpha with F = ma) lets you transfer problem-solving strategies you already know from Units 1 and 2 directly into rotational contexts, and it prepares you for the rotational energy and angular momentum work in Unit 6.
Defines angular displacement (delta-theta in radians), angular velocity (omega = d-theta/dt), and angular acceleration (alpha = d-omega/dt). For constant alpha, the four kinematic equations apply directly. Graph interpretation follows the same slope-and-area rules as linear kinematics.
Links angular quantities to the linear motion of a point at radius r: s = r*theta, v = r*omega, a_T = r*alpha. All points on a rigid body share omega and alpha, but linear speeds differ by radius. The no-slip condition v_cm = R*omega applies to rolling objects.
Torque is the cross product tau = r x F with magnitude rF sin-theta. Only the force component perpendicular to r produces torque. The lever arm is the perpendicular distance from the axis to the line of action of the force. Direction is determined by the right-hand rule.
Rotational inertia I = mr^2 for a point mass, I = integral r^2 dm for continuous solids, and I' = I_cm + Md^2 via the parallel axis theorem. Mass farther from the axis contributes more. AP Physics C requires calculus-based derivations for rods, disks, cylindrical shells, and annular rings.
When sum-tau = 0, angular velocity is constant: this is Newton's first law for rotation. Rotational and translational equilibrium are independent conditions. Choosing a strategic pivot eliminates unknown forces from the torque equation.
When net torque is nonzero, alpha = sum-tau / I_sys. For combined translational and rotational systems, write separate sum-F = ma and sum-tau = I*alpha equations, then connect them with a constraint equation such as a = alpha*R.
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Review Rotational Inertia with attention to how the concept appears in AP-style source and evidence questions.
Review Rotational Kinematics with attention to how the concept appears in AP-style source and evidence questions.
Review Torque with attention to how the concept appears in AP-style source and evidence questions.
Review Rotational Equilibrium and Newton's First Law in Rotational Form with attention to how the concept appears in AP-style source and evidence questions.
A rigid body rotating about a fixed axis is described by angular displacement delta-theta (in radians), angular velocity omega = d-theta/dt, and angular acceleration alpha = d-omega/dt. For constant alpha, the same four kinematic equations from linear motion apply with theta, omega, and alpha replacing x, v, and a. Graphs of theta(t), omega(t), and alpha(t) follow the same slope-and-area logic as their linear counterparts.
| Linear quantity | Symbol | Rotational analog | Symbol |
|---|---|---|---|
| Displacement | x | Angular displacement | theta |
| Velocity | v | Angular velocity | omega |
| Acceleration | a | Angular acceleration | alpha |
| Mass | m | Rotational inertia | I |
| Force | F | Torque | tau |
For any point at distance r from a fixed axis, the arc length is s = r*theta, the tangential speed is v = r*omega, and the tangential acceleration is a_T = r*alpha. All points on a rigid body share the same omega and alpha, but their linear speeds scale with r. The centripetal (radial) acceleration at that point is a_r = r*omega^2, directed toward the axis.
Torque is the rotational effect of a force. Only the component of force perpendicular to the position vector from the axis to the point of application produces torque. The magnitude is tau = r*F*sin-theta, where theta is the angle between r and F. The lever arm is the perpendicular distance from the axis to the line of action of the force, giving the equivalent form tau = F * (lever arm). Direction follows the right-hand rule from the cross product tau = r x F.
Rotational inertia I measures a rigid system's resistance to changes in rotation. It depends on both total mass and how that mass is distributed relative to the axis. For a point mass, I = mr^2. For a collection of point masses, I_total = sum(m_i * r_i^2). For a continuous solid, I = integral r^2 dm, where r is the perpendicular distance from each mass element to the axis. The parallel axis theorem I' = I_cm + Md^2 shifts the axis from the center of mass to any parallel axis a distance d away.
| Object | Axis | I |
|---|---|---|
| Point mass | Distance r from mass | mr^2 |
| Solid disk | Through center, perpendicular | (1/2)MR^2 |
| Thin cylindrical shell | Through center, perpendicular | MR^2 |
| Thin rod | Through center, perpendicular | (1/12)ML^2 |
| Thin rod | Through one end, perpendicular | (1/3)ML^2 |
A system is in rotational equilibrium when the net torque about any chosen axis is zero (sum-tau = 0), which means its angular velocity is constant. This is the rotational analog of Newton's first law. A system can be in rotational equilibrium without being in translational equilibrium, and vice versa. When solving equilibrium problems, choose the pivot point strategically to eliminate unknown torques from the equation.
When net torque on a rigid system is not zero, the system has angular acceleration given by alpha = sum-tau / I_sys, the rotational form of Newton's second law. This is the direct analog of F = ma. For systems that both translate and rotate (such as a mass hanging from a string wrapped around a pulley), you must write a separate sum-F = ma equation for the translational motion and a separate sum-tau = I*alpha equation for the rotational motion, then connect them with a constraint equation such as a = alpha*R.
Try AP-style multiple-choice questions and written prompts after you review the notes.
1. A uniform disk of mass M and radius R is mounted on a frictionless horizontal axle through its center, as shown in Figure 1. A string is wrapped around the disk, and a block of mass m is attached to the free end of the string. The block is released from rest at time t = 0, and as it falls, the string unwinds without slipping, causing the disk to rotate. The rotational inertia of the disk about its center is . As the system moves, the tension in the string varies with time.
Figure 1. Disk on a frictionless axle with a string wrapped around the rim and a hanging block.
Figure 2. Free-body diagrams for the disk and the block (forces only).
On Figure 2, draw and label arrows to represent all forces acting on the disk and on the block. The tails of the arrows should be at the points where the forces are exerted. Draw arrows of lengths proportional to the magnitudes of the forces.
Derive an expression for the magnitude of the angular acceleration of the disk as the system moves. Express your answer in terms of M, m, R, g, 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 angular velocity of the disk at time . Express your answer in terms of M, R, , , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. In a different scenario, the disk is initially at rest and a time-varying torque is applied to the disk about its axis of rotation, where is a positive constant and is a positive constant angular frequency. The torque is applied from time t = 0 to time . There is no string or hanging mass in this scenario.
4. A uniform solid disk of mass kg and radius m is mounted on a frictionless horizontal axle through its center, as shown in Figure 1. A light string is wrapped around the disk. One end of the string is attached to a block of mass kg that hangs vertically. The other end of the string passes over the disk and is attached to a second block of mass kg that also hangs vertically on the opposite side of the disk. The rotational inertia of a uniform solid disk about its central axis is . The system is released from rest at time .
Figure 1. Two-block string wrapped on a uniform solid disk (MD = 2.0 kg, R = 0.30 m) mounted on a frictionless axle.
While the system is accelerating, the magnitudes of the tension forces in the string on each side of the disk are (on the side with block 1) and (on the side with block 2).
Indicate whether is greater than, less than, or equal to by writing one of the following.
Justify your answer using qualitative reasoning beyond referencing equations.
Derive an expression for the magnitude of the linear acceleration of block 1. 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.
Figure 2. Same disk-and-string setup as Figure 1, but the right-hand block is replaced with m3 = 1.0 kg.
Indicate whether the magnitude of the angular acceleration of the disk in this scenario is greater than, less than, or equal to the magnitude of the angular acceleration of the disk in the original scenario. In a different scenario, block 2 is replaced with a block of mass kg, as shown in Figure 2. The system is again released from rest. The disk has the same mass kg and radius m as in the original scenario.
Briefly justify your answer.
2. A uniform solid disk of mass M = 2.0 kg and radius R = 0.40 m is mounted on a horizontal frictionless axle through its center, as shown in Figure 1. The disk is initially at rest. A string is wrapped around the outer edge of the disk and attached to a block of mass m = 0.50 kg that hangs vertically. The block is released from rest at time t = 0 s and falls a distance h = 1.5 m before hitting the ground at time t₁. After the block hits the ground, the string goes slack and the disk continues to rotate freely.
Figure 1. Disk–pulley (axle) system with hanging block and measured drop distance.
Figure 2. Force-vector templates for the disk and for the block (vectors drawn from the central dot).
Draw and label free-body diagrams showing all the forces exerted on the disk and on the block while the block is falling. Use Figure 2 to draw each force vector starting on the dot and pointing in the direction of the force. The length of each vector should be proportional to the magnitude of the force. Clearly label each force using standard notation (e.g., for tension, for gravitational force). While the block is falling and before it hits the ground, the string exerts forces on both the disk and the block.
Derive an expression for the magnitude of the angular acceleration of the disk while the block is falling. Express your answer in terms of M, m, R, g, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The block falls with acceleration in the downward direction. The rotational inertia of the disk about its central axis is .
Figure 3. Axes for sketching angular velocity ω versus time t, with t₁ marked.
Sketch a graph of the angular velocity of the disk as a function of time t from t = 0 to a time significantly after using Figure 3. Your graph should show the behavior both before and after the block hits the ground. Clearly indicate any discontinuities or changes in the relationship between and t. The angular velocity of the disk increases as the block falls. At time , the block hits the ground and the string goes slack. After , the disk continues to rotate freely about the axle with no friction.
Figure 4. Rotational inertia options for three disks.
Indicate which disk (Disk 1, Disk 2, or Disk 3) will have the greatest angular velocity immediately after the block hits the ground. If two or more disks will have the same angular velocity, indicate this explicitly. The experiment is repeated three times using different disks, each with radius R = 0.40 m. The same block of mass m = 0.50 kg falls the same distance h = 1.5 m in each trial. The properties of the three disks are shown in Figure 4.
Disk 1 Disk 2 Disk 3 Two or more disks tie
Briefly justify your answer using physics principles, referencing relevant equations or relationships.
| Term | Definition |
|---|---|
| angular displacement | The angle in radians through which a rigid body rotates about a specified axis, calculated as delta-theta = theta - theta_0. |
| radian | The standard unit of angular measurement in rotational kinematics; arc length equals radius times angle in radians. |
| tangential acceleration | The component of linear acceleration directed tangent to the circular path of a point on a rotating body; a_T = r*alpha. |
| lever arm | The perpendicular distance from the axis of rotation to the line of action of an applied force; multiplying force by lever arm gives torque magnitude. |
| Moment arm | The perpendicular distance from the axis of rotation to the line of action of a force; equivalent to lever arm and used interchangeably in torque calculations. |
| cross product | The vector operation tau = r x F that defines torque; its magnitude is rF sin-theta and its direction is perpendicular to both r and F by the right-hand rule. |
| position vector | The vector r from the axis of rotation to the point where a force is applied; its magnitude and the angle it makes with the force determine the torque. |
| mass distribution | The spatial arrangement of mass relative to the axis of rotation; mass farther from the axis contributes more to rotational inertia because I scales as r^2. |
| non-uniform mass distribution | A mass distribution where density varies with position; requires I = integral r^2 dm with a position-dependent expression for dm. |
| translational equilibrium | The condition sum-F = 0; independent of rotational equilibrium and must be checked separately in static problems. |
| rolling motion | Combined translational and rotational motion where the contact point has zero velocity; governed by the no-slip constraint v_cm = R*omega. |
| no-slip condition | The constraint that the contact point between a rolling object and a surface has zero relative velocity, giving v_cm = R*omega and a_cm = R*alpha. |
| constraint equation | A kinematic relationship linking linear and angular motion, such as a = alpha*R for a string on a pulley, used to connect the translational and rotational equations in a combined problem. |
Omega and alpha describe the whole rigid body; v and a_T describe a specific point at radius r. Plugging omega directly into a linear equation without multiplying by r is one of the most frequent errors in Unit 5 problems.
Only the force component perpendicular to the position vector r produces torque. If a force is applied at an angle, you must use F*sin-theta (or equivalently the lever arm), not the full magnitude of F.
I' = I_cm + Md^2 requires I_cm, the rotational inertia about the center of mass, not about some other convenient axis. Using the wrong base value of I leads to an incorrect result.
Sum-F = 0 and sum-tau = 0 are independent equations. A beam can have zero net torque but a nonzero net force (or vice versa). Both conditions must be checked separately in static equilibrium problems.
In a pulley-mass system, if you define downward as positive for the hanging mass, you must define the corresponding rotation direction as positive for the pulley. Mixing sign conventions across the two equations produces incorrect constraint equations.
AP Physics C: Mechanics frequently presents systems where a mass accelerates linearly while a pulley or spool rotates. You are expected to write separate sum-F = ma and sum-tau = I*alpha equations, identify the correct rotational inertia, and apply a constraint equation to solve for angular acceleration, linear acceleration, or tension. Showing each equation separately before substituting is important for earning full credit.
Free-response questions may ask you to derive I for a rod of uniform or nonuniform density, a disk, or an annular ring using I = integral r^2 dm. You are expected to set up the integral explicitly, express dm in terms of a spatial variable and the object's geometry, evaluate the integral, and apply the parallel axis theorem if the axis is not through the center of mass.
Problems involving beams, hinges, or suspended rods require both a force diagram and a torque equation. You are expected to choose a pivot strategically, correctly identify each torque's sign and lever arm, and recognize that satisfying sum-tau = 0 does not automatically satisfy sum-F = 0. Justifying your pivot choice and showing the torque sum explicitly are standard expectations.
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.
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open calculatorAP Physics C: Mechanics Unit 5 covers 6 topics: Rotational Kinematics, Connecting Linear and Rotational Motion, Torque, Rotational Inertia, Rotational Equilibrium and Newton's First Law in Rotational Form, and Newton's Second Law in Rotational Form. Together they build the rotational counterparts to the linear dynamics you studied earlier in the course. See the full topic breakdown at AP Physics C: Mechanics Unit 5.
Unit 5 makes up 10-15% of the AP Physics C: Mechanics exam. That weight covers Torque and Rotational Dynamics, including Rotational Kinematics, Torque, Rotational Inertia, Rotational Equilibrium, and Newton's Second Law in Rotational Form. It's a meaningful chunk of the exam, so strong fluency with these concepts pays off on both the MCQ and FRQ sections.
The AP Physics C: Mechanics 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 part tests conceptual understanding and calculation, while the FRQ part typically asks you to derive expressions, apply Newton's Second Law in rotational form, or analyze rotational equilibrium scenarios. For matched practice before the progress check, visit AP Physics C: Mechanics Unit 5.
The best way to practice Unit 5 FRQs is to work through problems that mirror the three most common question types: deriving rotational inertia expressions, applying Newton's Second Law in rotational form to systems with torque, and analyzing rotational equilibrium. These topics, especially 5.4 Rotational Inertia and 5.6 Newton's Second Law in Rotational Form, show up most often in free-response questions on the AP Physics C: Mechanics exam. Start by writing out full solutions with clear diagrams and labeled torque directions, then check your reasoning step by step. You'll find FRQ-aligned practice at AP Physics C: Mechanics Unit 5.
For AP Physics C: Mechanics Unit 5 practice questions, including multiple-choice and practice test problems on Torque and Rotational Dynamics, head to AP Physics C: Mechanics Unit 5. You'll find MCQ-style questions covering Rotational Kinematics, Torque, Rotational Inertia, and Newton's Second Law in Rotational Form, plus FRQ practice that mirrors what College Board tests on the actual exam.
Start with Rotational Kinematics (5.1) and Connecting Linear and Rotational Motion (5.2) to build the foundation, since every later topic depends on those analogies between linear and rotational quantities. Then work through Torque (5.3) and Rotational Inertia (5.4) together, practicing the integral definitions of inertia for common shapes. Once those feel solid, tackle Rotational Equilibrium (5.5) and Newton's Second Law in Rotational Form (5.6) with full free-body and torque diagrams on every problem. A few concrete steps that help: draw the rotational analog table (x vs. theta, v vs. omega, F vs. tau) and keep it visible while you practice. Do at least one FRQ per topic from past exams, writing out every step rather than just checking the answer. Visit AP Physics C: Mechanics Unit 5 for organized practice by topic.