Unit 5 Torque and Rotational Dynamics — AP Physics C Practice Test
Practice AP Physics C: Mechanics Unit 5 with moment of inertia integrals, torque problems, and rotational Newton's second law. Calculus-based FRQ preparation included.
Rotational Mechanics with Integral Calculus
Unit 5 introduces the rotational analogues of translational dynamics and places heavy demand on integral calculus. Computing moments of inertia for continuous mass distributions — a central AP Physics C: Mechanics skill — requires setting up and evaluating integrals over one- and two-dimensional bodies. This unit is among the most calculus-intensive in the course.
Core Concepts in Unit 5
Torque and Rotational Newton's Second Law
Torque is the rotational analogue of force: τ = r × F. For rotation about a fixed axis, Newton's second law in rotational form is τ_net = Iα, where I is the moment of inertia and α is the angular acceleration. This equation is the starting point for all rotational dynamics problems in AP Physics C: Mechanics.
Moment of Inertia via Integration
For a continuous mass distribution, the moment of inertia is I = ∫r² dm, where r is the perpendicular distance from the rotation axis and the integral is taken over the entire object. Students must express dm in terms of a spatial variable using the appropriate density function, then evaluate the integral with correct limits. Standard results derived this way include:
- Uniform thin rod about its center: I = (1/12)ML²
- Uniform thin rod about one end: I = (1/3)ML²
- Uniform solid disc about its central axis: I = (1/2)MR²
- Thin ring about its central axis: I = MR²
The Parallel-Axis Theorem
The parallel-axis theorem — I = I_cm + Md² — allows the moment of inertia about any axis parallel to a center-of-mass axis to be computed without re-evaluating the integral from scratch. AP FRQs commonly require applying this theorem after deriving I_cm via integration.
Angular Kinematics
Angular kinematics mirrors translational kinematics: angular velocity ω = dθ/dt and angular acceleration α = dω/dt. For variable angular acceleration, the same calculus approach used in Unit 1 applies here — integrate α(t) to find ω(t), and integrate ω(t) to find θ(t).
Key AP FRQ Skills for Unit 5
- Deriving the moment of inertia of a rod, disc, or ring from the integral definition.
- Applying the parallel-axis theorem to shifted rotation axes.
- Setting up and solving τ_net = Iα for systems with multiple torques.
- Connecting angular and translational quantities for rolling or constrained systems.
- When setting up an I integral for a rod, use linear mass density λ = M/L to express dm = λ dx.
- For a disc, use the thin-ring element dm = (2πr dr)(M/πR²) to build up the disc from rings.
- Always state the axis of rotation explicitly before computing I.