Rucete ✏ AP Physics 1 In a Nutshell
7. Rotational Motion
This chapter introduces torque, static equilibrium, moment of inertia, angular kinematics, energy of rolling objects, and conservation of angular momentum. It bridges linear and rotational motion through analogies between force, mass, and acceleration versus torque, moment of inertia, and angular acceleration.
Torque and Equilibrium
• Torque (τ) = r·F·sin(θ), where r is the lever arm and θ is the angle between r and F.
• Units: newton·meter (N·m), not to be confused with work (which is scalar).
• Torque is a vector; direction is positive for counterclockwise and negative for clockwise.
• Static equilibrium conditions:
– ΣF = 0 (no net force)
– Στ = 0 (no net torque)
• Choosing pivot points wisely simplifies torque calculations — forces through the pivot produce no torque.
Examples of Torque
• Seesaws balance when clockwise and counterclockwise torques are equal: Fg₁·d₁ = Fg₂·d₂
• Torque from a hanging mass: τ = r·mg (if force is perpendicular)
• Applied torque at an angle: τ = r·F·sin(θ)
Moment of Inertia
• Moment of inertia (I) quantifies rotational inertia: I = Σmr² (for point masses)
• Depends on mass distribution and axis of rotation.
• Units: kg·m²
• Example: rotating a ruler is harder about one end than its center due to larger I.
• You don’t need to calculate I on the AP exam but should interpret how it affects motion.
Angular Kinematics
• Analogous to linear kinematics, with angular variables:
θ (angular position), ω (angular velocity), α (angular acceleration)
• Equations (for constant α):
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
• Tangential speed: v = rω
• Tangential acceleration: a = rα
• Rotational motion and linear motion are deeply connected through r (radius).
Energy in Rolling Motion
• Total mechanical energy includes translational and rotational kinetic energy:
KEtotal = ½mv² + ½Iω²
• For rolling without slipping: v = rω
• Rolling objects with more mass concentrated farther from the axis (larger I) roll more slowly down inclines.
• Potential energy converts into both forms of kinetic energy during rolling motion.
Angular Momentum
• Angular momentum (L) = Iω
• Conserved when no external torque acts: Li = Lf
• Analogous to linear momentum conservation.
• Example: A figure skater spins faster when pulling in arms — I decreases, so ω increases.
• Torque is the rotational version of force: τ = ΔL / Δt
In a Nutshell
Rotational motion extends familiar mechanics principles into circular systems. Torque causes angular acceleration, resisted by moment of inertia. Rotational analogs of velocity, acceleration, and energy mirror linear motion. When no external torque acts, angular momentum is conserved — a principle that governs systems from spinning wheels to orbiting planets.