Short answers, pulled from the story.
The Earth loses angular momentum to the Moon at a rate of 3.82 centimeters per year, which lengthens our days by roughly 1.7 milliseconds per century. This transfer occurs through tidal torques that move angular momentum from Earth's rotation to the Moon's orbit. The process causes the Moon to recede and the Earth's day to lengthen over geological time.
Angular momentum is the rotational analog of linear momentum and is defined as the cross product of a particle's position vector and its linear momentum vector. This calculation yields a pseudovector perpendicular to the plane of motion that depends on the observer's chosen origin point. For a rigid body, the total angular momentum is the sum of the angular momenta of all its constituent particles.
Johannes Kepler determined the laws of planetary motion without knowing the concept of conservation of momentum, yet his second law is a direct consequence of angular momentum conservation. Isaac Newton provided a geometric proof of this law of areas in his Principia, demonstrating that a planet sweeps out equal areas in equal intervals of time. Newton did not fully investigate the subject, considering it too tedious to demonstrate every particular.
The term angular momentum was introduced by R.B. Hayward in 1856. William J. M. Rankine defined angular momentum in the modern sense in 1858 as a line whose length is proportional to the magnitude and whose direction is perpendicular to the plane of motion. Before this, angular momentum was typically referred to as the momentum of rotation.
In quantum mechanics, angular momentum undergoes a radical transformation and becomes quantized, existing only in discrete steps rather than a continuous range. Niels Bohr first postulated this quantization in his model of the atom, and Erwin Schrödinger later predicted it through his wave equation. Quantum particles possess an intrinsic property called spin, which is an inherent form of angular momentum unrelated to any spatial motion.
Léon Foucault used a gyroscope to experimentally display the Earth's rotation in 1852. This experiment proved that a spinning object resists changes to its axis of rotation due to the conservation of angular momentum. This resistance allows gyroscopes to be used in inertial navigation systems to navigate under polar ice caps and maintain course without external reference points.