AP Physics 1 Unit 5 Notes: Torque, Rotational Motion, and Angular Momentum

Torque

A measure of how effectively a force causes rotation about a chosen axis; causes angular acceleration (not the same as force, which causes linear acceleration).

Axis of rotation

The line (real or imagined) about which an object rotates; the choice of axis affects torque calculations and moment of inertia.

Pivot point

The point about which torques are calculated (often a hinge or axle); forces whose lines of action pass through the pivot produce zero torque about it.

Radius vector (r)

The vector (or distance) from the axis/pivot to the point where a force is applied; appears in torque and moment of inertia relationships.

Torque angle (θ)

The angle between the radius vector (from axis to application point) and the force vector in the torque formula.

Torque magnitude (τ = rF sinθ)

Magnitude of torque from a force applied distance r from the axis at angle θ; only the perpendicular component of the force contributes.

Lever arm (moment arm)

The perpendicular distance from the axis to the force’s line of action; torque magnitude can be computed as τ = F r⊥.

Line of action

The infinite line along which a force acts; used to find the lever arm as the perpendicular distance from the axis to this line.

Perpendicular force component

The component of a force perpendicular to the radius vector; this component is what “tries to spin” the object and determines torque.

Torque sign convention (CCW +, CW −)

A consistent choice for torque direction in 2D problems; commonly counterclockwise torques are positive and clockwise torques are negative.

Net torque (τ_net)

The sum of all torques about an axis: τnet = Στi; determines angular acceleration via Στ = Iα.

Moment of inertia (I)

Rotational analog of mass; measures resistance to angular acceleration and depends on how mass is distributed relative to the axis.

Point-mass moment of inertia (I = Σ m r²)

Moment of inertia for a set of particles: sum of each mass times the square of its perpendicular distance to the axis.

Units of moment of inertia

kg·m² (mass times distance squared), reflecting that mass farther from the axis increases I strongly.

Thin hoop (ring) moment of inertia (I = MR²)

Standard result for a thin hoop about its center axis perpendicular to the plane; all mass is at radius R.

Solid disk/solid cylinder moment of inertia (I = 1/2 MR²)

Standard result for a uniform solid disk or cylinder about its center axis perpendicular to the face.

Thin rod about center (I = 1/12 ML²)

Standard result for a thin uniform rod rotating about its center, with axis perpendicular to the rod.

Thin rod about one end (I = 1/3 ML²)

Standard result for a thin uniform rod rotating about an axis through one end, perpendicular to the rod.

Axis dependence of I

Moment of inertia changes if the rotation axis changes; the same object can have different I values about different axes.

Angular displacement (Δθ)

How much an object rotates; measured in radians in rotational kinematics.

Radian

The natural unit of angle in physics; makes relationships like s = rθ, v = rω, and a_t = rα work directly.

Angular velocity (ω)

Rate of change of angular displacement: ω = Δθ/Δt (rad/s); points on a rigid body share the same ω.

Angular acceleration (α)

Rate of change of angular velocity: α = Δω/Δt (rad/s²); points on a rigid body share the same α.

Arc length relation (s = rθ)

For motion along a circle of radius r, arc length s is proportional to angle θ (in radians).

Tangential speed (v = rω)

Linear speed of a point at radius r on a rotating object; larger r gives larger v for the same ω.

Tangential acceleration (a_t = rα)

Acceleration that changes the speed along the circular path; depends on radius r and angular acceleration α.

Centripetal acceleration (a_c = rω²)

Inward acceleration for circular motion; changes direction of velocity (not its magnitude) even if ω is constant.

Constant angular acceleration kinematics

Rotational versions of linear kinematics: ω = ω0 + αt; Δθ = ω0 t + (1/2)αt²; ω² = ω0² + 2αΔθ.

Rotational Newton’s second law (Στ = Iα)

Core rotational dynamics equation: net external torque about an axis equals moment of inertia times angular acceleration.

Zero-torque condition (line of action through axis)

If a force’s line of action passes through the axis/pivot, its lever arm is zero, so it produces zero torque about that axis.

No-slip constraint (a = rα)

Geometric link between linear acceleration and angular acceleration when a string doesn’t slip on a pulley or an object rolls without slipping.

Rotational kinetic energy (K_rot = 1/2 Iω²)

Energy of a rigid body due to rotation about an axis; depends on I and ω.

Rolling total kinetic energy

For an object that both translates and rotates: K_total = (1/2)mv² + (1/2)Iω² (don’t omit either term for rolling).

Work done by a torque (W = τΔθ)

For constant torque over angular displacement Δθ (radians), rotational work equals torque times angular displacement.

Torque–angle graph work interpretation

If torque varies with angle, work is the area under the τ vs. θ graph.

Rotational power (P = τω)

Rate of doing rotational work; for a motor at angular speed ω, producing larger torque requires larger power.

Mechanical energy conservation (rotation)

If nonconservative work is negligible, total mechanical energy is conserved (Ei = Ef), including rotational kinetic energy terms.

Rolling without slipping energy equation

For a drop in height h from rest: mgh = (1/2)mv² + (1/2)Iω², with v = rω used to solve for speeds.

Effective mass term (I/r²)

In many rolling/pulley formulas, I/r² acts like additional “effective mass,” reducing acceleration or final speed compared with pure translation.

Angular momentum (L)

Rotational analog of linear momentum; conserved when net external torque about an axis is zero/negligible.

Rigid-body angular momentum (L = Iω)

Angular momentum of a rigid body rotating about a fixed axis; depends on moment of inertia and angular speed.

Particle angular momentum (L = mvr)

Magnitude of angular momentum for a particle moving in a circle: mass × tangential speed × radius from the axis.

Torque–angular momentum relation (τ_net = ΔL/Δt)

Net external torque equals the time rate of change of angular momentum (rotational analog of F = Δp/Δt).

Angular impulse (τΔt = ΔL)

For constant net torque over time interval Δt, the product τ_netΔt equals the change in angular momentum.

Conservation of angular momentum condition

You may set Li = Lf only if net external torque about the chosen axis is zero or negligible during the interaction.

Skater arms-in relation (Ii ωi = If ωf)

If a rotating system changes I with negligible external torque, angular momentum conservation requires ω to change inversely with I.

Inelastic rotational collision (putty sticks)

During a brief sticking event with negligible external torque, angular momentum is conserved but mechanical energy is generally not.

Translational equilibrium (ΣF = 0)

Condition for no linear acceleration of the center of mass; used with torque equilibrium in static equilibrium problems.

Rotational equilibrium (Στ = 0)

Condition for zero angular acceleration; in equilibrium you can compute torques about any point to simplify unknown forces.

Static friction inequality (fs ≤ μs N)

Static friction adjusts as needed up to a maximum μs N; fs equals μ_s N only at the threshold of slipping.