Angular Momentum: Single Particle

Angular momentum L is the cross-product of the position vector and linear momentum (mass times velocity). Its magnitude equals mvrsinθ, where θ is the angle between position and velocity vectors. The SI units are kilogram meter squared per second. Direction is determined using the right-hand rule: curl fingers in the rotation direction, and the thumb points along the angular momentum vector.

The lever arm is the perpendicular vector from the origin to the linear momentum vector. If the lever arm and linear momentum are collinear, angular momentum equals zero. The magnitude of angular momentum depends directly on the lever arm length and the choice of origin, making origin selection critical for accurate calculations.

The rate of change of angular momentum equals the net torque acting on the particle. This is the rotational analog of Newton's second law for linear momentum. Mathematically, dL/dt = τ, where τ is the torque from the net force. Both torque and angular momentum must be measured relative to the same origin fixed in an inertial frame.

Angular momentum is directed perpendicular to the plane of rotation, and its magnitude depends on the perpendicular distance from the origin to the linear momentum vector. Different origins produce different position vectors r, which directly affect the cross-product calculation. Therefore, angular momentum values are only meaningful when referenced to a specific, fixed origin in an inertial frame.

The right-hand rule determines angular momentum direction: curl your fingers in the direction of rotation, and your thumb points along the angular momentum vector. This method applies because angular momentum is a cross-product, which always produces a vector perpendicular to both the position and momentum vectors involved in the calculation.

Net torque acting on rotating bodies causes angular momentum to change over time. This relationship mirrors Newton's second law for rotational motion: the time derivative of angular momentum equals the applied torque. Without net torque, angular momentum remains constant, demonstrating the rotational analog of inertia in linear systems.

Angular momentum is zero when the lever arm and linear momentum vector are collinear, meaning they lie along the same line. In this case, the particle moves directly toward or away from the origin, producing no rotational motion about that point. The cross-product of parallel vectors always equals zero, resulting in zero angular momentum.