*is*rotating (or

*may*be rotating) is a line drawn from the object to the chosen axis. Thus a train rolling on a straight track possesses angular momentum with respect to a water tower off to one side of the track. (2) An object may possess angular momentum relative to one axis but zero angular momentum relative to another (if it is headed straight toward or away from the latter).

Physicists distinguish two kinds of angular momentum: orbital and spin. Earth serves to illustrate both. Earth possesses orbital angular momentum because of its path around the Sun, and possesses spin angular momentum (often called just spin) because of its daily rotation about its own axis. The distinction extends into the submicroscopic world, where electrons, protons, and many other particles possess intrinsic spin and may also possess orbital angular momentum. In that domain, the quantization (“lumpiness”) of angular momentum becomes important. The size of an orbital-angular momentum “lump” is Planck’s constant *h* divided by 2**π**, a quantity abbreviated *ħ* (“h bar”). The smallest unit of spin is half as great, *½ħ.*

In order to be dealt with quantitatively and manipulated in equations, every measurable concept in physics must be associated with some mathematical “object,” either a number or a vector or something more abstract. What mathematical object successfully describes angular momentum? Does angular momentum have a direction as well as a magnitude? It does, but *not *the direction of motion of the material object. The direction that is assigned to angular momentum is an *axial* direction, a direction along the chosen reference axis. For circular motion, a convenient axis is evidently a line perpendicular to the plane of the motion passing through the center of the circle. For the straight-line motion of a train the axis can be chosen to be a vertical line through the center of an adjacent water tower.

Since an axis has two directions, it is necessary to specify a positive direction. The arbitrary convention, by general agreement (yes, international agreement is possible), follows a right-hand rule. If the curved fingers of the right hand follow the rotational motion or the rotational tendency, the right thumb designates the positive direction along the axis. Thus the angular momentum of the wheel of an automobile moving forward is directed horizontally to the left. In reverse, the wheel’s angular momentum is directed horizontally to the right. The spin angular momentum of the Earth about its own axis is directed toward the North Star, and the orbital angular momenta of the Moon with respect to the Earth and the Earth with respect to the Sun have approximately the same direction. In some applications, a slightly modified version of the right-hand rule can be helpful. Let the straight fingers of the right hand point from the chosen axis to the moving object in such a way that the palm “pushes” the line joining them in its direction of motion. Then the right thumb indicates the direction of the angular-momentum vector.

To the question, Why does angular momentum have this directional property?, there is only the solid pragmatic answer that it is fruitful to *define *angular momentum in this way. Then it behaves experimentally like a mathematical vector. If the angular momentum of a system is conserved (as it always is for an isolated system), the angular momenta of its parts combine—or add—as vectors, not as numbers, and the total angular momentum is conserved as a vector, both in magnitude and direction. When angular momentum is not conserved, the law of its change is expressed by a vector equation that involves the concept of torque.

These arguments about the vector nature of angular momentum may seem rather arbitrary. They can be supplemented by this plausibility argument. For a particle circling around a point, the only possible *fixed* direction associated with the motion is along a line through the central point perpendicular to the plane of the orbit. Within the plane itself, the velocity and momentum vectors take on all possible directions. Throughout the motion, however, the plane of the orbit remains fixed and therefore the perpendicular direction to the plane remains fixed. It is natural to take this axial direction as the direction characterizing the rotational motion. The direction along the axis chosen as the positive direction is entirely arbitrary. No argument can show that one direction is preferable to the other. The conventional choice defined by the right-hand rule is merely a historical accident standardized to avoid confusion.

Since angular momentum is a vector quantity, two or more angular momenta add as vectors. If two gears rotating about perpendicular axes are meshed, for example, their total angular momentum will have a direction distinct from the direction of either gear axis. Two equal and opposite angular momenta, just like two equal and opposite forces or momenta, can add to give zero.

To wrap up, here are some main points about angular momentum: (1) The angular momentum of a particle with respect to a point, a measure of the strength of its rotational tendency about that point, is defined in terms of the mass and velocity of the particle and its distance from the reference point (through which an imaginary axis is drawn). (2) A particle moving at fixed speed in a circle has constant angular momentum with respect to the center of the circle. (3) A particle moving at constant velocity in a straight line has constant angular momentum with respect to *any* chosen point. (4) Angular momentum is a vector quantity. Its direction is the axial direction as defined by the right-hand rule. (5) The angular momentum of a system is the vector sum of the angular momenta of the constituent parts of the system. (6) The angular momentum of a material object or a collection of particles can be separated into a spin part and an orbital part. Its spin is its angular momentum with respect to its center of mass. Its orbital angular momentum is the additional angular momentum arising from motion of the center of mass and is calculated as if all of the mass of the entity were concentrated at the center of mass.