Torque and Rotational Dynamics
What this unit covers
The topics below follow the published Physics C: Mech course framework for Unit 5. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.
Lessons in this unit
- Rotational Kinematics13 min · 3 objectivesDefine angular velocity and acceleration as derivatives ω = dθ/dt and α = dω/dt · Apply the constant-angular-acceleration equations · Relate linear and angular quantities with v = rω and a_t = rα
- Torque & Moment of Inertia15 min · 3 objectivesDefine torque as τ = rF sin θ and identify the lever arm · Define the moment of inertia I = ∫r² dm and compute it for a simple body · Explain how the distribution of mass, not just its amount, sets the rotational inertia
- Newton's Second Law for Rotation14 min · 3 objectivesApply the rotational form of Newton's second law, τ_net = Iα · Solve for the angular acceleration of a rigid body under applied torques · Analyze a mass hanging from a pulley that has rotational inertia
- Rolling Without Slipping14 min · 3 objectivesState the rolling constraint v_cm = Rω and a_cm = Rα · Analyze an object rolling down an incline using both force and torque equations · Compare the accelerations of different shapes rolling down the same incline
Formulas in Unit 5
Every term in Unit 5
All 17 terms we publish for Torque and Rotational Dynamics, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.
- Parallel axis theorem
- I = I_cm + Md². Moving the axis a distance d from the center of mass always increases the moment of inertia.
- Rolling without slipping
- v_cm = Rω and a_cm = Rα. Static friction acts at the contact point and does no work, since that point is instantaneously at rest.
- Torque as a cross product
- τ = r × F, with magnitude rF sin θ. Direction is given by the right-hand rule along the rotation axis.
- Moment of inertia by integration
- I = ∫r² dm. The integral runs over the mass distribution, with r measured perpendicular to the axis.
- Standard moments of inertia
- Rod about center ML²/12, rod about end ML²/3, disk ½MR², hoop MR², solid sphere ⅖MR², spherical shell ⅔MR².
- Rotational form of Newton's second law
- Στ = Iα about a fixed axis, or about the center of mass for a body that is also translating.
- Rolling down an incline
- a = g sin θ/(1 + I/MR²). Objects with mass concentrated near the axis accelerate faster, independent of mass and radius.
- Torque about different axes
- Choosing the axis through an unknown force eliminates its torque, which reduces the number of unknowns in equilibrium problems.
- Static equilibrium conditions
- ΣF = 0 and Στ = 0 simultaneously. If ΣF = 0, then Στ has the same value about every axis.
- Deriving a moment of inertia
- Set up I = ∫r² dm, express dm through the mass distribution, and choose limits matching the geometry. For a rod, dm = (M/L)dx.
- Why the axis matters
- A rod is ML²/12 about its center but ML²/3 about its end — four times larger, because more mass sits far from the axis.
- Torque from a distributed force
- Gravity on an extended body acts effectively at the center of mass, which is why a beam's weight is drawn at its midpoint.
- Combining translation and rotation
- Write ΣF = Ma_cm and Στ = I_cm α, then apply the rolling constraint a_cm = Rα to link them.
- Direction of friction in rolling
- For an object rolling down an incline, static friction acts up the incline and supplies the torque that produces angular acceleration.
- Slipping vs rolling
- Rolling requires the needed static friction to stay below μsN. Above that the object slips, kinetic friction acts, and v ≠ Rω.
- Angular kinematics with calculus
- ω = dθ/dt and α = dω/dt. The constant-α equations are the special case, exactly as in linear motion.
- Equilibrium of a leaning ladder
- Three unknown forces require all three equilibrium equations; taking torques about the base eliminates two of them at once.
What examiners penalize here
- Two objects with identical mass can have very different moments of inertia. Always ask where the mass sits relative to the axis: the r² weighting in I = ∫r² dm rewards mass at the rim and penalizes it near the center.
- For rolling problems, always bring in the constraint a_cm = Rα to connect the force equation to the torque equation. Remember that the shape factor I/MR² alone decides the race down an incline — mass and radius drop out.
Practice Physics C: Mech
Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.
Questions about this unit
How much of the AP Physics C: Mechanics exam is Unit 5?
Unit 5, Torque and Rotational Dynamics, is worth 10–15% of the Physics C: Mech multiple-choice section according to the published course framework. Across all 7 units that makes it a substantial share — heavier than an even split would give it.
What topics are covered in Physics C: Mech Unit 5?
Torque and Rotational Dynamics covers Moment of inertia, Torque, Rotational kinematics and Rolling. We publish 17 terms with definitions for this unit, all of them on this page.
How should I study Physics C: Mech Unit 5?
Read the 4 lessons below first — about 55 minutes — then drill the 17 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.
All 7 units of AP Physics C: Mechanics
Unit names, topics and exam weights follow the published College Board course framework for AP Physics C: Mechanics. AP® is a trademark registered by the College Board, which does not endorse this site.