Introduction
Angular momentum is conserved when net external torque is zero. This powerful principle explains diverse phenomena from spinning figure skaters to planetary orbits. Like linear momentum conservation, it remains valid in quantum mechanics and is fundamental to understanding rotational motion.
Statement of Conservation
If net external torque about a point is zero (τ_ext = 0), then angular momentum about that point is constant: dL/dt = 0, so L = constant. This means L_initial = L_final. Applies to isolated systems and central force problems. Conservation holds separately for each component if corresponding torque component is zero.
Conditions for Conservation
Angular momentum is conserved when: (1) No external torque acts (isolated system); (2) External forces act through or parallel to reference point (r × F = 0); (3) For central forces (force parallel to r), torque is automatically zero. Internal torques cancel by Newton's Third Law (action-reaction pairs). Changing moment of inertia can change angular velocity while conserving L.
\nIntroduction
Angular momentum is conserved when net external torque is zero. This powerful principle explains diverse phenomena from spinning figure skaters to planetary orbits. Like linear momentum conservation, it remains valid in quantum mechanics and is fundamental to understanding rotational motion.
Statement of Conservation
If net external torque about a point is zero (τ_ext = 0), then angular momentum about that point is constant: dL/dt = 0, so L = constant. This means L_initial = L_final. Applies to isolated systems and central force problems. Conservation holds separately for each component if corresponding torque component is zero.
Conditions for Conservation
Angular momentum is conserved when: (1) No external torque acts (isolated system); (2) External forces act through or parallel to reference point (r × F = 0); (3) For central forces (force parallel to r), torque is automatically zero. Internal torques cancel by Newton's Third Law (action-reaction pairs). Changing moment of inertia can change angular velocity while conserving L.
\nExamples of Conservation
Figure skater: Pulling arms in reduces I, increases ω to conserve L = Iω. Diver: Tucking reduces moment of inertia, increasing spin rate. Collapsing star: Decreasing radius increases rotation rate (pulsars). Planetary motion: L conservation gives Kepler's Second Law (equal areas in equal times). Gyroscope: Resists changes to orientation due to L conservation.
Relation to Central Forces
Central forces (directed toward a fixed point) exert no torque about that point since r × F = 0 (parallel vectors). Therefore angular momentum about the force center is conserved. This explains why planetary orbits remain in fixed planes and why atomic electron orbitals have definite angular momentum. The conservation makes central force problems integrable and analytically tractable.
\nSpin and Orbital Angular Momentum
Total angular momentum can be decomposed: L_total = L_orbital + L_spin. L_orbital = r_cm × p_cm is CM motion about external point. L_spin is rotation about CM (internal angular momentum). Both types are conserved independently in appropriate circumstances. In quantum mechanics, both are quantized. Spin is intrinsic and exists even for point particles.
\nSolved Example: Figure Skater Spin
A figure skater spins at 2 rev/s (4π rad/s) with arms extended (I1 = 5 kg·m2). She pulls arms in, reducing moment of inertia to I2 = 2 kg·m2. Find new angular speed and compare kinetic energies. Solution: Angular momentum conserved: L1 = L2 → I1ω1 = I2ω2 → ω2 = (I1/I2)ω1 = (5/2)×4π = 10π rad/s = 5 rev/s. New speed is 2.5× faster. Initial KE: K1 = 1/2I1ω₲ = 1/2×5×(4π)2 = 1/2×5×16π2 = 40π2 ≈ 395 J. Final KE: K2 = 1/2I2ω22 = 1/2×2×(10π)2 = 1/2×2×100π2 = 100π2 ≈ 987 J. Kinetic energy increased by factor 2.5! Where did extra energy come from? Skater did work pulling arms in against centrifugal force. This demonstrates that conservation laws (L conserved) don't imply all quantities conserved - energy can change via work.
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