Topic 6: Energy and momentum of rotating systems

Once an object spins or rolls, its energy and its 'amount of turning' obey rules that parallel the linear world you already know. This unit shows how kinetic energy splits between sliding and spinning, why a skater speeds up when she pulls her arms in, and how the same physics governs planets sweeping around the Sun.

Rotational kinetic energy

A spinning object stores kinetic energy in its rotation. The amount is K_rot = (1/2) I w^2, where I is the rotational inertia (moment of inertia) and w is the angular speed in rad/s. This mirrors the linear formula (1/2) m v^2: rotational inertia plays the role of mass and angular speed plays the role of velocity. Because I depends on how mass is distributed relative to the axis, two objects of equal mass can store very different amounts of rotational energy at the same w. A hoop, with its mass far from the axis, has a larger I and therefore more rotational kinetic energy than a solid disk of the same mass and radius spinning at the same rate. Units of K_rot are joules (J), since I is in kg m^2 and w^2 is in (rad/s)^2, and the radian is dimensionless.

Work and power in rotation

A torque that acts through an angular displacement does rotational work: W = (torque)(angular displacement) = tau * (delta theta), valid when the torque is constant and along the axis. This is the rotational analog of W = F d. The work-energy theorem still applies: the net rotational work done on a rigid body equals its change in rotational kinetic energy, W_net = (1/2) I w_f^2 - (1/2) I w_i^2. Rotational power is the rate of doing this work, P = tau * w, the analog of P = F v. So a motor producing a fixed torque delivers more power as the shaft spins faster. These relations let you connect a driving torque to how quickly a flywheel spins up, and how much energy it ends up holding.

Rolling without slipping and the energy split

When a round object rolls without slipping, the contact point is instantaneously at rest, so static friction (not kinetic) acts and does no net work. The rolling condition links translation and rotation: v_cm = w R, and a = (alpha) R. The total kinetic energy is the sum of two pieces: K_total = (1/2) m v_cm^2 + (1/2) I w^2. The first term is translation of the center of mass; the second is spin about the center of mass. Using v_cm = w R, the split depends only on the shape's I. For objects released from the same height on an incline, more of the energy goes into rotation when I is larger, leaving less for translation. That is why a solid sphere (small I relative to m R^2) reaches the bottom before a hollow cylinder: the sphere puts a smaller fraction of its energy into spinning.

Angular momentum

Angular momentum measures an object's tendency to keep rotating. For a rigid body rotating about a fixed axis, L = I w, with units of kg m^2/s. For a particle moving with momentum p at a perpendicular distance r from an axis, L = r p (more generally the perpendicular component is used). Angular momentum is a vector; its direction is given by a right-hand rule along the axis. Just as a net force changes linear momentum, a net torque changes angular momentum: the net external torque equals the rate of change of L. When mass moves closer to the axis, I shrinks, and if no torque acts, w must rise to keep L the same. This single idea drives the conservation results that follow.

Conservation of angular momentum (spinning skater)

If the net external torque on a system is zero, its total angular momentum stays constant: L_i = L_f, or I_i w_i = I_f w_f. Consider a figure skater spinning with arms extended. Suppose she has I_i = 4.0 kg m^2 and w_i = 2.0 rad/s, giving L = 8.0 kg m^2/s. She pulls her arms in, reducing her rotational inertia to I_f = 1.6 kg m^2. With no external torque about her vertical axis, L is conserved, so w_f = L / I_f = 8.0 / 1.6 = 5.0 rad/s. Her spin rate climbs by a factor of 2.5. Notice her kinetic energy also changes: K_i = (1/2)(4.0)(2.0)^2 = 8.0 J, while K_f = (1/2)(1.6)(5.0)^2 = 20 J. The extra 12 J comes from the work her muscles do pulling her arms inward against the outward tendency of the moving mass. Angular momentum is conserved; kinetic energy is not.

Orbital motion and gravitation

Gravity supplies the centripetal force that holds a satellite or planet in orbit. For a circular orbit, the gravitational force equals the required centripetal force: G M m / r^2 = m v^2 / r. Canceling m and solving gives the orbital speed v = sqrt(G M / r), which depends only on the central mass M and the orbital radius r, not on the orbiting object's mass. A larger orbit means a slower speed. Because gravity always points toward the central body, it exerts no torque about that center, so a planet's angular momentum about the Sun is conserved throughout its orbit. This conserved angular momentum is the key to understanding non-circular (elliptical) orbits and Kepler's description of them.

Kepler's laws basics

Kepler summarized planetary motion in three laws. First, planets move in ellipses with the Sun at one focus; a circle is the special case of zero eccentricity. Second, the line joining a planet to the Sun sweeps out equal areas in equal times. This 'equal areas' rule is just conservation of angular momentum: when the planet is closer to the Sun it moves faster, and when farther it moves slower, keeping L constant. Third, the square of a planet's orbital period is proportional to the cube of its average orbital distance, T^2 proportional to r^3. For circular orbits this follows directly from setting gravity equal to the centripetal force: T^2 = (4 pi^2 / (G M)) r^3. Together these laws connect geometry, speed, and timing for any object bound by gravity.