Angular Momentum of a Particle

Introduction

Angular momentum L = r × p is the rotational analog of linear momentum. It describes the amount of rotation a particle possesses about a reference point. Angular momentum is fundamental in orbital mechanics, atomic physics, and rotational dynamics, remaining conserved in the absence of external torque.

Definition

Angular momentum of a particle about origin O: L = r × p = r × (mv), where r is position vector from O to particle. Magnitude: L = rp sinθ = mvr sinθ = mv_tangential × r = p × r_perpendicular. Direction: perpendicular to plane containing r and p, given by right-hand rule. Units: kg·m²/s or J·s.

Relation to Torque

The time derivative of angular momentum equals torque: τ = dL/dt. This is the rotational analog of F = dp/dt. If net torque about a point is zero, angular momentum about that point is conserved. For central forces (directed toward/away from origin), torque is zero and angular momentum is conserved.

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Introduction

Angular momentum L = r × p is the rotational analog of linear momentum. It describes the amount of rotation a particle possesses about a reference point. Angular momentum is fundamental in orbital mechanics, atomic physics, and rotational dynamics, remaining conserved in the absence of external torque.

Definition

Angular momentum of a particle about origin O: L = r × p = r × (mv), where r is position vector from O to particle. Magnitude: L = rp sinθ = mvr sinθ = mv_tangential × r = p × r_perpendicular. Direction: perpendicular to plane containing r and p, given by right-hand rule. Units: kg·m²/s or J·s.

Relation to Torque

The time derivative of angular momentum equals torque: τ = dL/dt. This is the rotational analog of F = dp/dt. If net torque about a point is zero, angular momentum about that point is conserved. For central forces (directed toward/away from origin), torque is zero and angular momentum is conserved.

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Angular Momentum in Circular Motion

For circular motion about origin at center: r is perpendicular to v, so L = mvr = mr²ω = mrω × r. Direction is along rotation axis. For uniform circular motion, L is constant in magnitude and direction. For elliptical orbits, L is still conserved (central force) but r and v vary, maintaining constant L = mrv_tangential.

Components and General Motion

In component form: L_x = yp_z - zp_y, L_y = zp_x - xp_z, L_z = xp_y - yp_x. For planar motion in xy-plane, only L_z is non-zero: L_z = m(xv_y - yv_x). Angular momentum can be defined about any point, not just origin. Change of reference point changes angular momentum by R × P where R is shift vector and P is total momentum.

Applications

Applications: Planetary motion (conserved L means orbits lie in fixed plane, equal areas swept in equal times); Scattering experiments (angular momentum determines trajectory shape); Atomic physics (electron angular momentum is quantized); Gyroscopes (conservation maintains orientation); Helicopter tail rotors (counteract main rotor torque).

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Solved Example: Satellite Angular Momentum

A 1000 kg satellite orbits Earth in circular orbit of radius 8×10^6 m (about 1600 km altitude) with period 2 hours. Find its angular momentum about Earth's center. Solution: First find velocity: v = 2πr/T = 2π × 8×10^6 / (2×3600) = 6981 m/s. Angular momentum magnitude: L = mvr = 1000 × 6981 × 8×10^6 = 5.58×10^13 kg·m²/s. Direction is perpendicular to orbital plane. For circular orbit, L = mr²ω where ω = 2π/T = 8.73×10^-4 rad/s. L = 1000 × (8×10^6)² × 8.73×10^-4 = 5.58×10^13 kg·m²/s ✓. This angular momentum is conserved throughout orbit. If orbit were elliptical, L would still be same constant value, but r and v would vary reciprocally to maintain constant L.

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