Unit 5: Rotation

📚Study Guide: Rotation

Unit 5: Rotation

Rotation in AP Physics C: Mechanics is treated with the same rigor and calculus-based approach as linear motion. Angular position θ, angular velocity ω, and angular acceleration α are related by derivatives and integrals: ω = dθ/dt and α = dω/dt. For constant angular acceleration, the kinematic equations are directly analogous to their linear counterparts. The dynamical quantity that causes rotation is torque, τ = r × F, a vector cross product. Its magnitude is rF sinθ, and its direction is given by the right-hand rule. Rotational inertia (moment of inertia) I = ∫ r² dm is computed by integrating over the mass distribution of a continuous object. You must know the moments of inertia for common shapes (rod, disk, sphere, cylinder) and be able to apply the parallel axis theorem, I = I_cm + Md², to find I about an axis parallel to the center-of-mass axis. Newton's Second Law for rotation is τ_net = Iα, the rotational analog of F = ma. When a net torque acts on a rigid body, it produces an angular acceleration proportional to the torque and inversely proportional to the rotational inertia. Rolling without slipping is a critical constraint linking linear and angular motion: v_cm = rω and a_cm = rα. In rolling problems, the total kinetic energy is the sum of translational and rotational terms: K = ½Mv_cm² + ½Iω². Angular momentum L = r × p = Iω is conserved when the net external torque is zero. This conservation law explains gyroscopic precession, spinning ice skaters, and the stability of bicycle wheels. On the AP Exam, rotation problems often require you to compute I via integration, set up simultaneous equations for translation and rotation, and apply energy conservation to rolling objects. Vector cross products and the right-hand rule are essential for determining directions.

Key Concepts

• Angular Variables and Calculus: ω = dθ/dt, α = dω/dt. Integrate α to find ω; integrate ω to find θ. Constants of integration are set by initial conditions.
• Moment of Inertia Integral: I = ∫ r² dm. For a continuous body, choose a mass element dm, express r in terms of the integration variable, and integrate over the body.
• Parallel Axis Theorem: I = I_cm + Md². Quickly finds I about any axis parallel to an axis through the center of mass.
• Torque Vector: τ = r × F. Direction is perpendicular to the plane formed by r and F, given by the right-hand rule.
• Rotational Dynamics: τ_net = Iα. The net torque about a fixed axis equals the moment of inertia about that axis times the angular acceleration.
• Rolling Without Slipping: The no-slip condition v_cm = rω links translation and rotation. The contact point is instantaneously at rest relative to the surface.
• Angular Momentum: L = Iω for rotation about a fixed axis. Conserved if τ_net_ext = 0.

Vocabulary

• Angular Displacement (θ): The angle through which an object rotates. Unit: radians.
• Angular Velocity (ω): The rate of change of angular displacement. Unit: rad/s.
• Angular Acceleration (α): The rate of change of angular velocity. Unit: rad/s².
• Moment of Inertia (I): The rotational analog of mass; depends on mass distribution relative to the axis.
• Parallel Axis Theorem: A theorem relating the moment of inertia about any axis to that about a parallel axis through the center of mass.
• Torque (τ): The rotational analog of force, causing angular acceleration.
• Cross Product: A vector operation producing a vector perpendicular to both input vectors, with magnitude equal to the product of magnitudes times the sine of the angle between them.
• Precession: The slow rotation of the axis of a spinning object around another axis due to an external torque.

Essential Formulas

• ω = dθ/dt
• α = dω/dt
• θ = θ0 + ω0*t + ½*α*t²
• I = ∫ r² dm
• I = I_cm + M*d² (parallel axis)
• τ = r × F; |τ| = r*F*sinθ
• τ_net = I*α
• L = I*ω
• v = r*ω; a = r*α

Common Mistakes

• Using Linear Kinematic Equations for Rotation: Do not substitute x for θ without converting. The rotational equations have the same form but different variables and units.
• Wrong Axis in Moment of Inertia Integral: r is the perpendicular distance from the axis of rotation to the mass element dm, not the distance from the origin.
• Forgetting Parallel Axis Theorem: If the axis does not pass through the CM, you must add Md². Do not use I_cm alone.
• Confusing Torque Magnitude with Component: The full torque magnitude is rF sinθ. The perpendicular component of force is F sinθ, and the lever arm is r sinθ. Both give the same result.

AP Exam Strategies

• Identify the Axis First: All rotational quantities (I, τ, α, L) are defined relative to an axis. State your axis explicitly at the beginning of the problem.
• Use the Analog Table Linear↔Angular: x→θ, v→ω, a→α, m→I, F→τ. This helps you remember rotational formulas.
• Compute I via Integration or Table + Parallel Axis: For standard shapes, use memorized I_cm and the parallel axis theorem. For non-standard shapes, set up the integral carefully.
• For Rolling, Use the Constraint v = rω: This eliminates one variable and links the translational and rotational equations.

Real-World Applications

• Gyroscopes: Conservation of angular momentum keeps a spinning gyroscope stable, making it invaluable for navigation in aircraft and spacecraft.
• Flywheels: These store rotational kinetic energy (½Iω²) and are used in energy storage systems and automotive regenerative braking.
• Engine Torque Curves: Automotive engineers optimize engine torque as a function of RPM to maximize power delivery and acceleration.