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.
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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.
| 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.
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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.