Unit 6 — Energy and Momentum of Rotating Systems

Rotational Kinetic Energy

A spinning object has kinetic energy due to the motion of all its parts around the axis.

K_rot = ½ · I · ω²
K_total = ½·m·v_cm² + ½·I·ω²      (translation + rotation)

• Rotational kinetic energy is a scalar in joules, and it is never negative.
• It depends on ω²: doubling ω quadruples K_rot.
• For the same ω, a larger I stores more energy. This is why flywheels concentrate mass at the rim.
• Energy conservation includes this term: Ktrans + Krot + U is conserved in an isolated system with no dissipative forces.

Torque and Work

A torque acting through an angular displacement transfers energy to or from a rotating system.

W = τ · Δθ                 (constant torque, θ in radians)
W_net = ΔK_rot = ½Iω_f² − ½Iω_i²
P = τ · ω                  (rotational power)

• Work is positive when the torque and the angular displacement have the same sign (the object speeds up), negative when they are opposite.
• On a graph of torque vs. angle, the area under the curve is the work done.
• Compare with linear motion: W = Fd becomes W = τΔθ, and P = Fv becomes P = τω.
• Only the torque component producing rotation does work; the force at a fixed axis does no work because that point does not move.

Angular Momentum and Angular Impulse

Angular momentum, L, is the rotational analogue of linear momentum. Its units are kg·m²/s.

L = I · ω                       (rigid object spinning about its axis)
L = m · v · r⊥ = r · p · sin θ   (point mass moving relative to a chosen point)
Angular impulse:  τ · Δt = ΔL
Στ = ΔL / Δt

• Angular momentum is a vector; in these problems use + for counterclockwise and − for clockwise.
• For a moving particle, only the velocity component perpendicular to the line from the reference point matters. A particle moving straight toward the point has L = 0.
• A particle moving in a straight line still has angular momentum about a point not on its path.
• Angular impulse (torque × time) equals the change in angular momentum, just as impulse equals change in linear momentum.
• Total L of a system is the sum of the angular momenta of its parts, about the same axis or point.

Conservation of Angular Momentum

If the net external torque on a system is zero, its total angular momentum stays constant.

L_i = L_f        →        I_i·ω_i = I_f·ω_f

• If I decreases, ω increases (and vice versa). A skater or diver pulls in their limbs to spin faster.
• When I changes, kinetic energy also changes. A skater pulling arms in does work, so K = ½Iω² = L²/(2I) increases.
• Rotational collisions (e.g. a non-spinning disk dropped onto a spinning one) conserve L but usually not K: they behave like perfectly inelastic collisions.
• Internal torques cannot change the total angular momentum; only an external torque (or angular impulse) can.

Rolling

An object that rolls without slipping combines translation and rotation, linked by the condition:

v_cm = ω·R        a_cm = α·R
K_total = ½·m·v² + ½·I·ω²

• The point of contact with the ground is instantaneously at rest, so static friction acts there and does no work. Mechanical energy is conserved when rolling without slipping.
• The top of the wheel moves at 2vcm and the center at vcm.
• Rolling down an incline from height h: mgh = ½mv² + ½Iω², so v = √(2gh / (1 + I/mR²)).
• An object with a smaller I/mR² puts less energy into rotation and reaches the bottom faster. The result is independent of mass and radius.

Motion of Orbiting Satellites

For a satellite of mass m in a circular orbit of radius r around a body of mass M, gravity is the only force and provides the centripetal force.

G·M·m / r² = m·v² / r   →   v = √(G·M / r)
T = 2π·r / v              →   T² = (4π² / G·M) · r³
E = K + U = −G·M·m / (2r)   (K = GMm/2r, U = −GMm/r)

• Orbital speed depends on the central mass and the radius, and not on the satellite's mass. Larger r means slower speed and longer period.
• Total energy of a bound orbit is negative; a larger orbit has a higher (less negative) energy.
• Astronauts feel "weightless" because they and the spacecraft are in free fall together, not because gravity is absent.

Elliptical orbits (Kepler's laws):

• 1st: orbits are ellipses with the central body at one focus.
• 2nd: a line from the central body to the satellite sweeps out equal areas in equal times. This follows from conservation of angular momentum, since gravity is a central force that exerts no torque about the central body.
• 3rd: T² ∝ a³ (a = semi-major axis).
• The satellite moves fastest at closest approach (perigee) and slowest at farthest point (apogee): m·vp·rp = m·va·ra. Kinetic energy and potential energy trade off while total energy stays constant.