Inside This Unit: The Full Breakdown
Torque and Rotational Motion extends the concepts of force, mass, and acceleration to rotating objects. This unit covers torque, rotational inertia, angular kinematics, rotational energy, and angular momentum conservation.
Why it matters
Rotational motion is a significant portion of the AP Physics 1 exam. You must draw extended free-body diagrams showing torques, apply τ = Iα, and use conservation of angular momentum. These concepts parallel translational dynamics but require new variables and new intuition.
Key concepts
- Torque (τ = rF sin θ) is the rotational equivalent of force. It depends on the force magnitude, the distance from the pivot, and the angle.
- Rotational inertia (I) is the rotational equivalent of mass. It depends on both mass and the distribution of mass relative to the axis of rotation.
- Newton's second law for rotation: Στ = Iα, where α is angular acceleration.
- Angular momentum (L = Iω) is conserved when no net external torque acts on a system.
Torque and Rotational Equilibrium
Torque is the tendency of a force to cause rotation about a pivot point. Its magnitude is τ = rF sin θ, where r is the distance from the pivot to the point where the force is applied, F is the force magnitude, and θ is the angle between the force and the lever arm. The perpendicular distance from the pivot to the line of action of the force (r sin θ) is called the moment arm. Torque has a sign: counterclockwise is typically positive, clockwise negative. An object is in rotational equilibrium when the net torque about any point is zero (Στ = 0). Combined with translational equilibrium (ΣF = 0), this condition allows you to solve statics problems involving beams, ladders, and other extended objects.
Rotational Kinematics and Dynamics
Rotational motion has direct analogs to translational motion. Angular displacement (θ) corresponds to linear displacement, angular velocity (ω) to linear velocity, and angular acceleration (α) to linear acceleration. The rotational kinematic equations mirror the translational ones: ω = ω₀ + αt, θ = ω₀t + ½αt², and ω² = ω₀² + 2αθ. Rotational inertia (I) is the rotational analog of mass — it measures an object's resistance to angular acceleration. Unlike mass, I depends on how mass is distributed relative to the rotation axis: a hoop (I = MR²) has more rotational inertia than a solid disk (I = ½MR²) of the same mass and radius because its mass is farther from the axis. Newton's second law for rotation is Στ = Iα, directly parallel to ΣF = ma.
Angular Momentum and Rotational Energy
Angular momentum (L = Iω) is the rotational analog of linear momentum. The law of conservation of angular momentum states that if no net external torque acts on a system, its total angular momentum remains constant. This explains why a figure skater spins faster when pulling in their arms: I decreases, so ω must increase to keep L = Iω constant. Rotational kinetic energy is KE_rot = ½Iω², analogous to ½mv². A rolling object has both translational and rotational kinetic energy: KE_total = ½mv² + ½Iω². For rolling without slipping, v = rω, which connects the two. Conservation of energy problems with rolling objects must include both forms of kinetic energy. Angular momentum conservation is a powerful tool for analyzing spinning systems, collisions involving rotation, and planetary orbits.
AP exam tip
On AP Physics 1, rotational problems often require you to draw extended free-body diagrams showing where each force is applied. Always choose your pivot point strategically — picking the point where an unknown force acts eliminates that force from the torque equation, simplifying the algebra.
Connections to other units
- Unit 2 (Dynamics): Rotational dynamics (τ = Iα) directly parallels translational dynamics (F = ma).
- Unit 4 (Energy): Rolling objects have both translational and rotational kinetic energy, requiring both terms in conservation of energy.
- Unit 5 (Momentum): Angular momentum conservation (L = Iω = constant) parallels linear momentum conservation.
