Unit 6 · Energy and Momentum of Rotating Systems

Rotational kinetic energy ● (core concept) — K_rot = ½·I·ω², a scalar. A rigid system's total kinetic energy is its translational kinetic energy (of the center of mass) plus its rotational kinetic energy about the center of mass.

Work done by a torque ● (core concept) — A torque transfers energy over an angular displacement: W = τ·Δθ, equal to the area under a graph of torque versus angular position.

Angular momentum (rigid body) ● (core concept) — For a rigid system rotating about a fixed axis, L = I·ω.

Angular momentum (point mass) ● (core concept) — For a point mass, L = r·m·v·sinθ about a chosen reference point. It depends on the chosen axis, the object's mass and speed, and the angle between the radial direction and the velocity.

Angular impulse ● (core concept) — The product of the torque and the time it acts: τ_avg·Δt, in the rotational direction of the torque. It equals the area under a torque–time graph.

Angular impulse-momentum theorem ● (core concept) — The angular impulse exerted on a system equals its change in angular momentum: this is the rotational form of Newton's second law for constant rotational inertia.

Torque, angular momentum, and time graphs ● (core concept) — On a graph of torque versus time, the area under the curve is the angular impulse. On a graph of angular momentum versus time, the slope is the net torque.

Conservation of angular momentum ● (core concept) — If the net external torque on a system is zero, the system's total angular momentum is constant: internal changes must balance (e.g., a spinning skater pulls her arms in, rotational inertia drops, angular speed rises), and any change comes from angular impulse exerted by the surroundings.

Rolling without slipping ● (core concept) — Rolling in which the contact point is instantaneously at rest: v_cm = r·ω (and a_cm = r·α). Ideally, static friction acts but does no work, so no mechanical energy is lost.

Rolling with slipping ● (core concept) — Rolling in which v_cm ≠ r·ω; the contact point slides, kinetic friction acts, and mechanical energy is dissipated.

Circular orbits ● (core concept) — Satellites in circular orbits have gravity as the only force, providing the centripetal force; their mechanical energy, gravitational potential energy, kinetic energy, and angular momentum all remain constant.

Elliptical orbits ● (core concept) — Satellites in elliptical orbits conserve mechanical energy and angular momentum, with gravitational potential energy and kinetic energy trading off along the orbit.

Escape velocity ● (core concept) — The speed at which a satellite's total mechanical energy is zero, so it coasts to rest at infinite distance from the planet: v_esc = √(2·G·M/r).