# Angular momentum

measure of the extent to which an object will continue to rotate in the absence of an applied torque

The angular momentum or rotational momentum (L) of an object rotating about an axis is the product of its moment of inertia and its angular velocity:

The angular momentum of the figure skater is conserved—as she decreases her radius by retracting her arms and legs, her moment of inertia decreases, but her angular velocity increases to compensate.
${\displaystyle L=I\omega }$

where

${\displaystyle I}$ is the moment of inertia (resistance to angular acceleration or deceleration, equal to the product of the mass and the square of its radius measured perpendicularly from the axis of rotation);
${\displaystyle \omega \ }$ is the angular velocity.

There are three kinds of angular momentum: the vibrational angular momentum, the spin angular momentum and the orbital angular momentum.

## Vibrational angular momentum

The vibrational angular momentum is that of photons. Its minimum portion is the Planck quantum of vibration or action:

According to this picture, the creation of photons is to be viewed like the plucking of a guitar—as a sudden increment in the excitation of one of the modes of vibration.

—Davies, Paul. The Forces of Nature CUP, 1979, p. 116

... to an energy quantum of vibration, such as that of Planck, there must correspond an energy quantum of rotation ...

—Birtwistle, George. The Quantum Theory of the Atom CUP, 2015, pp. 2–3

The Planck quantum of action, h, has precisely the dimensions of an angular momentum ...

—Biedenharn, L. C.; Louck, J. D. Angular Momentum in Quantum Physics Addison-Wesley Pub. Co., Advanced Book Program, 1981

## Spin angular momentum

The spin angular momentum is that of an object turning around an axis which passes through the object's centre (for example, a top spinning around its central axis).

An object that is very spread-out from the axis of rotation has a large moment of inertia—it is very hard to make it start spinning, but once it gets going, it is very hard to make it stop. Similarly, it is easier to make an object start spinning at a low angular velocity than to make it start spinning at a high angular velocity. This is why the spin angular momentum depends both on how spread-out the object is (moment of inertia) and how fast it is spinning (angular velocity).

## Orbital angular momentum

The orbital angular momentum is that of an object revolving around an axis which does not pass through the object's centre. For example, the Sun and the Earth orbit each other by revolving around an axis that passes through the Sun, but not through the Sun's centre. The orbital angular momentum measures how hard it would be to make the object stop revolving around the axis.

## Conservation

Angular momentum is a conserved quantity—an object's angular momentum stays constant unless an external torque acts on it.

Angular momentum has both a direction and a magnitude, and both are conserved. Motorcycles, frisbees and rifled bullets all owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes have spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system but does not uniquely determine it.