Angular Momentum
Angular momentum is the product of an object's moment of inertia (its rotational mass) and its angular velocity. Angular momentum is a vector quantity represented by the variable, L.
L = Iω
The units for angular momentum are: (kg m ^{2})(radians/sec) = kg m ^{2}/sec. Note that although the angular velocity must be expressed in radians/sec, the term radian is dropped when expressing the units for its angular momentum. Remember that ω = 2πf. This expression was first introduced when we studied the sinusoidal equations for SHM.
The vector nature of L is determined by the right hand rule (RHR). When your fingers curl in the direction of the object's angular velocity, your thumb points in the direction of the object's angular momentum. Examining the rotating masses illustrated in the diagrams below, the sphere, disk and cylinder have angular velocities producing angular momentum vectors pointing along the positive yaxis. The angular velocity of the thin ring results in its angular momentum pointing along the positive xaxis.



solid spheres

solid disks and cylinders

thin rings and hoops

I = 2/5 mr^{2} 
I = 1/2 mr^{2} 
I = mr^{2} 
Point Masses Often we are required to determine the angular momentum of a point mass. A few examples of point masses would be: (1) a speck of dust on a spinning CD's surface; (2) a stopper moving in a circle at the end of a string; (3) a planet or asteroid moving in circular orbit about the sun. As you can see, a point mass comes in many sizes. The term applies to the fact that all of an object's mass is constrained to a small radius in comparison to the radius of its circular motion or from the pivot point of the system. That is, it can easily be represented by a single concentration of mass at the object's center of mass.
The moment of inertia for a point mass traveling in a circle is I = mr^{2} and the instantaneous tangential velocity of a point mass, v, equals rω. This relationship between angular and linear velocities can be understood by imagining a rotating platform.
All points on the platform share the same angular velocity (they all pass through the same angular displacement in a stated interval of time), but each one has a unique linear, or tangential, velocity based on how far it is located from the axis of rotation  that is, how large a circumference it must travel through during each revolution. For the three horse figurines shown above, the figurine closest to the central axis would have the least tangential velocity since the fraction of its circumference that it travels during the 1/6^{th} cycle shown is the shortest.
We will now derive an alternative expression for the moment of inertia of a point mass.
L = Iω where I = mr^{2} and ω = v/r.
NOTE: r in these equations represents the radial distance from the axis of rotation to the center of mass of the point mass. It does NOT represent the radius of the point mass. Remember, as discussed earlier in this lesson, that a point mass, by definition, is an object whose "internal radius" is very, very small in comparision with the radius of the circle through which it is moving.
L = (mr^{2})(v/r) L_{point mass} = mvr
NOTE: this expression is the cross product of the object's radial distance, r, and its linear momentum, mv, L = r × mv. That is, the angular momentum of a point mass equals the product of the magnitude of its moment arm  the perpendicular distance from the line of action of the momentum (instantaneous velocity vector) to the central pivot or axis of rotation  times the magnitude of its linear momentum.
