AP Physics C: Mechanics Unit 5: Torque and Rotational Dynamics 2027
Dedicated Unit 5 chapter hub for 6 current topics. Use it to move through the lesson sequence, connect formulas and labs, and hand off cleanly to the previous or next unit without mixing topic and unit search intent.
Unit 5 map: translation rebuilt for rigid-body rotation
Unit 5 is not a new set of unrelated formulas. It rebuilds Units 1–2 for a rigid system rotating about an axis: x maps to θ, v to ω, a to α, force to torque, mass to rotational inertia, and ΣF=ma to Στ=Iα. The challenge is that rotational effects depend on axis choice and mass distribution, so geometry becomes part of the dynamics.
All six Unit 5 topics
Rotational Kinematics
Use θ, ω=dθ/dt, and α=dω/dt; read angular graphs and derive constant-α relations from calculus.
Current lesson · 5.1 →Connecting Linear and Rotational Motion
Relate arc length, tangential speed, and tangential acceleration to angular quantities and apply rigid-body/no-slip constraints when appropriate.
Open 5.2 →Torque
Use the cross product, lever arm, line of action, and sign/direction conventions to quantify rotational effects about a chosen axis.
Open 5.3 →Rotational Inertia
Build I from discrete masses or I=∫r²dm, compare axes, and use the parallel-axis relationship when the rotation axis is displaced from the center of mass.
Open 5.4 →Rotational Equilibrium
Apply Στ=0 together with translational equilibrium when a rigid body is static or rotates with constant angular velocity.
Open 5.5 →Newton’s Second Law in Rotational Form
Apply Στ=Iα and couple it to ΣF=ma plus kinematic constraints for pulleys, spools, and other translating/rotating systems.
Open 5.6 →Translation ↔ rotation analogy table
| Translation | Rotation about a fixed axis | Important difference |
|---|---|---|
| x, v=dx/dt, a=dv/dt | θ, ω=dθ/dt, α=dω/dt | Angular variables describe the whole rigid body; linear speed depends on radius. |
| Force F⃗ | Torque τ⃗=r⃗×F⃗ | Torque depends on chosen axis and line of action. |
| Mass m | Rotational inertia I | I depends on both mass and its distribution relative to the axis. |
| ΣF=ma | Στ=Iα | For many systems both equations are needed simultaneously. |
| Equilibrium ΣF=0 | Rotational equilibrium Στ=0 | A rigid body in static equilibrium generally needs both conditions. |
Axis choice is part of the model
Torque changes with pivot choice
The same force can produce different torque about different axes. Choose an axis that simplifies unknown forces when possible, but keep that axis fixed through the torque equation.
Rotational inertia changes with axis
I=∫r²dm weights mass by squared perpendicular distance from the chosen axis. A table value about the center of mass cannot be reused blindly about another axis.
Lever arm beats angle confusion
Torque magnitude can be written rF sinφ or F r⊥. The perpendicular lever-arm picture often makes zero-torque forces and strategic pivots immediately visible.
Signs need one convention
For planar problems, choose clockwise or counterclockwise as positive and use it consistently for τ, ω, and α. Do not change sign rules midway to make an answer positive.
Calculus and coupled-equation map
Angular kinematics
ω=dθ/dt and α=dω/dt. Areas under ω–t and α–t graphs give angular displacement and angular-velocity change, just as in Unit 1.
Rotational inertia integral
For a continuous object, write dm from the density description and integrate I=∫r²dm over the actual geometry. The r² factor makes distant mass disproportionately important.
Rotational dynamics
For a rigid system about a fixed axis, Στ=Iα. If I is constant, angular acceleration follows net torque; if a translating mass is coupled to the rotor, also write ΣF=ma.
Constraint equations
A taut string without slipping on a pulley can impose a=αR. A rigid point at radius r has v_t=rω and a_t=rα. Constraints connect otherwise separate linear and angular equations.
Two Unit 5 synthesis problems
Massive pulley with two hanging masses
Draw separate free-body diagrams for the masses and a torque diagram for the pulley. Write m₁g−T₁=m₁a, T₂−m₂g=m₂a with signs chosen from the assumed motion, and (T₁−T₂)R=Iα for the pulley. Use a=αR if the string does not slip. The unequal tensions are not a mistake; they are what provide the pulley’s angular acceleration.
Nonuniform beam in static equilibrium
Choose a pivot, locate each force’s line of action, include the beam’s weight at its center of mass, and impose both ΣF=0 and Στ=0. Recalculate the torque equation about a different pivot to verify that the solved forces are unchanged. This checks axis choice, lever arms, center of mass, and simultaneous translational/rotational equilibrium.
Laboratory and data-analysis tasks
Torque versus lever arm
Apply the same force at different perpendicular distances from an axis and measure angular acceleration. Test whether α scales with τ when I is fixed.
Rotational inertia from dynamics
Apply a known torque to a rotor, measure α, and estimate I from τ_net/α. Compare with a geometric prediction from I=∫r²dm or a known model.
Mass-distribution comparison
Keep total mass fixed but move masses radially on a rotating platform. Compare measured angular acceleration under the same torque and test the r² dependence of I.
Static-equilibrium force test
Use a meter stick/beam with multiple hanging masses and a force sensor. Predict support forces from ΣF=0 and Στ=0, then evaluate residuals with measurement uncertainty.
Unit 5 traps that reveal weak modeling
Torque is a force
Torque has units N·m but is not energy and is not another force. It measures rotational effect about an axis.
Using r instead of perpendicular distance
Only the perpendicular component matters: τ=rF sinφ=F r⊥.
Assuming all points have the same linear speed
Rigid-body points share ω and α, not v. Tangential speed grows with radius.
Using the wrong I
Rotational inertia is axis specific. Check geometry and axis before inserting any standard value.
One equilibrium equation
Static equilibrium of a rigid body generally requires both net force zero and net torque zero.
Equal pulley tensions automatically
For a pulley with nonzero rotational inertia and angular acceleration, different tensions can be required to produce net torque.
Where Unit 5 goes next
Unit 6 · Rotational energy
Once τ, I, ω, and α are established, Unit 6 adds rotational kinetic energy, work by torque, angular momentum, rolling energy, and orbital applications.
Unit 6 · Angular momentum
Torque becomes dL⃗/dt and torque impulse changes angular momentum, mirroring the force–momentum structure from Unit 4.
Unit 7 · Oscillations
Physical pendulums use torque and rotational inertia to build an angular equation of motion. Small-angle SHM is a direct extension of Unit 5 dynamics.
E&M magnetic torque bridge
Later magnetic-force systems can exert torques on current loops or dipoles. The mechanical response still follows the rotational framework developed here.
Unit 5 mastery checklist
Unit 5 lesson sequence
This dedicated unit hub owns the chapter intent. Every lesson keeps its own canonical topic URL and returns here for unit context.
Topic 5.1 · Rotational Kinematics
Rotational kinematics describes how a rigid system turns about a specified axis using angular displacement θ, angular velocity ω=dθ/dt, and angular acceleration α=dω/dt. The calculus and graph rules are direct rotational analogs of one-dimensional linear kinematics, and constant angular acceleration produces the same mathematical family of kinematic equations with x→θ, v→ω, and a→α.
Open Topic 5.1 →Topic 5.2 · Connecting Linear and Rotational Motion
A rigid system’s angular motion is shared by all of its points, but the corresponding linear motion depends on distance r from the axis. Arc length, tangential speed, and tangential acceleration follow s=rθ, v_t=rω, and a_t=rα. These relations explain why points at larger radius move faster linearly even though every point has the same angular velocity and angular acceleration.
Open Topic 5.2 →Topic 5.3 · Torque
Torque measures the rotational effect of a force about a chosen axis. In vector form τ⃗=r⃗×F⃗, so only the component of force perpendicular to the position vector contributes. The same torque magnitude can be written rF sinφ or F times the perpendicular lever arm. Physics C requires both magnitude/sign reasoning and the cross-product/right-hand-rule interpretation.
Open Topic 5.3 →Topic 5.4 · Rotational Inertia
Rotational inertia I measures a rigid system’s resistance to changes in angular velocity about a specified axis. It depends not only on total mass but on how strongly each mass element is weighted by its squared perpendicular distance from the axis: I=Σm_ir_i² or I=∫r²dm. The parallel-axis theorem shifts a known center-of-mass result to a parallel axis.
Open Topic 5.4 →Topic 5.5 · Rotational Equilibrium and Newton’s First Law in Rotational Form
Rotational equilibrium means the net torque about the chosen axis is zero, so angular velocity is constant. This condition is independent of translational equilibrium: Στ=0 does not automatically imply ΣF⃗=0, and ΣF⃗=0 does not automatically imply Στ=0. Physics C problems therefore require careful force/torque diagrams, consistent signs, and strategic pivot choice.
Open Topic 5.5 →Topic 5.6 · Newton’s Second Law in Rotational Form
Newton’s second law for rotation connects a rigid system’s net external torque to its angular acceleration: Στ=Iα about the chosen axis. Physics C problems often require a second, independent translational equation ΣF=ma and a geometric constraint such as a=αR. The central skill is to model the same physical system consistently in both linear and rotational forms.
Open Topic 5.6 →Unit 5 resource network
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