AP Physics 1 Unit 6: Energy and Momentum in Rotating Systems

Rotational inertia

The rotational analog of mass; describes how strongly an object resists changes in its rotational motion about a chosen axis.

Moment of inertia (I)

A quantitative measure of rotational inertia that depends on both the object’s mass and how that mass is distributed relative to the axis of rotation.

Point-mass moment of inertia formula

For discrete masses about a specific axis: I = Σ(m r^2), where r is each mass’s perpendicular distance to the axis.

Units of moment of inertia

kg·m^2 (because I involves mass times distance squared).

Radius-squared (r^2) dependence

In I = Σ(m r^2), doubling r makes that mass’s contribution to I four times larger, so mass farther from the axis increases I a lot.

Thin hoop (ring) moment of inertia

About its center axis perpendicular to the plane: I = M R^2.

Solid disk/cylinder moment of inertia

About its center axis perpendicular to the plane: I = (1/2) M R^2.

Solid sphere moment of inertia

About an axis through its center: I = (2/5) M R^2.

Thin rod moment of inertia (about center)

For a rod of length L about its center, axis perpendicular to rod: I = (1/12) M L^2.

Thin rod moment of inertia (about one end)

For a rod of length L about one end, axis perpendicular to rod: I = (1/3) M L^2.

Parallel-axis theorem

Shifts moment of inertia from a center-of-mass axis to a parallel axis a distance d away: I = I_cm + M d^2.

Center-of-mass moment of inertia (I_cm)

The moment of inertia about an axis through the object’s center of mass (often the reference value used in standard formulas).

Rotational kinetic energy

Energy associated with rotational motion (spinning) about an axis.

Rotational kinetic energy formula

For rotation about a fixed axis: K_rot = (1/2) I ω^2, where ω is angular speed (rad/s).

Translational kinetic energy

Energy associated with linear motion: K_trans = (1/2) m v^2.

Mass–inertia analogy (translation vs rotation)

Key analogies: m ↔ I and v ↔ ω, so (1/2)mv^2 mirrors (1/2)Iω^2.

Total kinetic energy for rolling/combined motion

When an object translates and rotates: Ktotal = (1/2) M vcm^2 + (1/2) I_cm ω^2.

Center-of-mass speed (v_cm)

The translational speed of the object’s center of mass; used for the translational part of rolling kinetic energy.

Angular speed (ω)

Rate of rotation, measured in rad/s; appears in K_rot = (1/2)Iω^2 and L = Iω.

Rolling without slipping

A rolling condition where the point of contact is instantaneously at rest relative to the surface (no relative sliding).

No-slip speed constraint

For rolling without slipping: v_cm = ωR.

No-slip acceleration constraint

For rolling without slipping: a_cm = αR, where α is angular acceleration.

Static friction in rolling

Friction that prevents slipping; in ideal rolling on a rigid surface it often does no net work because the contact point does not slide.

Kinetic friction in rolling with slipping

Friction that acts when surfaces slide; it does negative work and dissipates mechanical energy as thermal energy.

Work in rotational motion

Energy transferred by a torque when it causes an angular displacement.

Rotational work formula

For constant torque aligned with rotation: W = τ Δθ (Δθ must be in radians).

Radians requirement (for W = τΔθ)

Angular displacement must be in radians so that W = τΔθ is consistent with standard SI units (joules).

Torque (τ)

The rotational effect of a force about an axis; for a perpendicular force applied at radius r: τ = rF.

Tangential force

The component of force perpendicular to the radius that produces torque and can change rotational motion.

Arc length relation

A point at radius r moving through angle Δθ travels arc length s = rΔθ, linking translational work to rotational work.

Rotational work-energy theorem

Net work done by torques equals the change in rotational kinetic energy: Wnet = ΔKrot.

Net work in rotation (expanded)

Wnet = (1/2)Iωf^2 − (1/2)Iω_i^2 (when I is constant).

Rotational power

Rate of doing rotational work; the power delivered by a torque at angular speed ω.

Power–torque–speed relation

Rotational analog of P = Fv: P = τω.

Angular momentum (L)

Measure of rotational motion about a chosen axis and how difficult it is to change that rotation.

Rigid-body angular momentum

For a rigid object about a fixed axis: L = Iω.

Particle angular momentum magnitude (about a point/axis)

For a particle moving perpendicular to the lever arm: L = m v r, where r is the perpendicular distance from the point to the line of motion.

Choice of axis for angular momentum

Angular momentum is defined “about” a specific axis; the same object can have different L values for different chosen axes.

Net torque–angular momentum relation

Rotational analog of Fnet = Δp/Δt: τnet = ΔL/Δt.

Angular impulse

The change in angular momentum from a torque applied over time; for constant net torque: ΔL = τ_net Δt.

Torque–angular acceleration relation

When I is constant for a rigid object: τnet = Iα (rotational analog of Fnet = ma).

Constant-I assumption

The equation τ_net = Iα assumes the moment of inertia does not change during the motion.

Conservation of angular momentum condition

Angular momentum is conserved when net external torque about the chosen axis is zero (or negligible): τext,net = 0 ⇒ Li = L_f.

Skater effect

Pulling mass inward decreases I; if external torque is negligible, L stays constant so ω increases.

Angular momentum conservation equation (configuration change)

If τext ≈ 0: Ii ωi = If ωf, so ωf = (Ii/If) ω_i.

Rotational kinetic energy not necessarily conserved (configuration change)

Even if L is conserved, K_rot = (1/2)Iω^2 can change because internal work (e.g., muscles) can add energy.

Inelastic rotational collision

A collision where objects stick together; angular momentum about an axis may be conserved, but kinetic energy is typically not conserved.

Putty sticking to a rotating disk (result)

If a disk (Id) with ωi gains putty mass m at radius r and external torque is negligible: ωf = [Id/(Id + mr^2)] ωi.

Compound moment of inertia

Total I for a system is the sum of parts about the same axis (e.g., Itotal = Irod + I_pointmass).

Rolling-down-an-incline final speed (energy method)

For rolling without slipping from height h with no losses: v = sqrt[ 2gh / (1 + I_cm/(MR^2)) ].