Rucete ✏ AP Physics C In a Nutshell
6. Rotation I: Kinematics, Force, Work, and Energy
This chapter covers rotational motion concepts including angular kinematics, torque, rotational inertia, rotational work and energy, and rotational analogs to linear motion. It applies these concepts to solve problems involving disks, rods, pulleys, and wheels.
Rotational Kinematics
• Angular displacement (θ), angular velocity (ω), and angular acceleration (α) describe rotational motion.
• Relationships for constant angular acceleration:
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
• Tangential speed: v = rω
• Tangential acceleration: aₜ = rα
• Radial (centripetal) acceleration: aᵣ = v²/r = rω²
Torque and Rotational Dynamics
• Torque (τ) = rF sin(θ)
• Positive torque: counterclockwise; Negative torque: clockwise.
• Newton’s second law for rotation:
τnet = Iα
• I: moment of inertia (rotational analog of mass).
• Moment of inertia depends on mass distribution relative to axis of rotation.
Work and Energy in Rotation
• Rotational work done: W = τθ
• Rotational kinetic energy: KErot = ½Iω²
• Mechanical energy conservation (if no nonconservative forces):
KErot,i + PEi = KErot,f + PEf
Rolling Without Slipping
• Condition: vCM = rω
• Total kinetic energy: KEtotal = ½MvCM² + ½Iω²
• Important in pulley systems, wheels climbing steps, and real-world rolling motion.
Key Applications
Rotating Rods and Falling Objects
• Use rotational inertia and torque about the pivot point.
• Energy conservation often simplifies the calculation (potential energy lost = rotational kinetic energy gained).
Pulleys in Systems
• If pulley has mass, account for its rotational inertia (I = ½MR² for a solid disk).
• Apply Newton’s second law for rotation: τ = Iα, where α = a/r (relating linear and angular acceleration).
Rolling Motion Problems
• For rolling without slipping: Total energy = translational KE + rotational KE + potential energy.
• Friction provides necessary torque to maintain rolling but does no work (if no slipping occurs).
In a Nutshell
Rotational motion mirrors linear motion but introduces torque, moment of inertia, and angular kinematics. Work and energy principles apply to rotational systems, and rolling without slipping links translational and rotational dynamics. Mastering these ideas is key to solving problems involving disks, pulleys, and rotating bodies in physics.