Angular momentum

Intuitively, angular momentum is the measure of how much the linear momentum is directed around an Origin. Since angular momentum depends upon the origin of choice, one must be carefule when discussing angular momentum to specify what origin and not to combine angular momenta about different origins.

The mathematical definition of the angular momentum of a particle about some origin is:

L = r×p

where L is angular momentum of the particle, r is the position of the particle expressed as a displacement vector, and p is the linear momentum of the particle.

For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the vector nature of angular momentum, and treat it like a scalar where is is positive when it corresponds to a counter clock-wise rotations, and negative clock-wise. To do this, just take the definition of the cross product and discard the unit vector, so that angular momentum becomes:

L = |r||p|sinθ

where θ is the angle between r and p measured from r to p; an important distinction because without it, the sign of the cross product would be meaningless. From the above, it is possible to reformulate the definition to either of the following:

L = ±|p||r_{perpendicular}|

where r_{perpendicular} is called the lever arm distance</b> to p. The easiest way to conceptualize this is to consider the lever arm distance to be the distance from the origin to the line that p rests on. With this defintion, it is necessary to consider the direction of p (pointed clock-wise or counter clock-wise) to figure out the sign of L). Equivalently:

L = ±|r||p_{perpendicular}|

where p_{perpendicular} is the component of p that is perpendicular to r. As above, the sign is decided base on the sense of rotation.

Remarkably, angular momentum is a [conserved quantity]? as long as there is no net Torque applied to the particle. What's more, this conservation can be generalized to a system of particles under most conditions so that:

L_system = constant
iff ∑τ_external = 0

where τ_external is any torque applied to the system of particles.

The conservation of angular momentum is used extensively in analyzing what is called <i>central force motion. In central force motion, two bodies form an isolated system not influenced by outside forces, and the origin is placed somewhere on the line between the two bodies. Since any force the bodies exert on each other must be directed along this line, there can be no net torque, with respect to the afore-mentioned origin, on either body. Thus, angular momentum is conserved. Constant angular momentum is extremely useful when dealing with the orbits of planets and satelites, and also when analyzing the Bhor model of the atom.