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 m²)(radians/sec) = kg m²/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 y-axis. The angular velocity of the thin ring results in its angular momentum pointing along the positive x-axis.

 

solid spheres	solid disks and cylinders	thin rings and hoops
I = 2/5 mr2	I = 1/2 mr2	I = mr2

 

 

 

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 = mr² 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 = mr² 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 = (mr²)(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.

 

 

 

 

Angular momentum is conserved within a system whenever there are no external forces exerting torques on the objects in the system.

 

 

An example of this occurs in skating. A skater spinning with arms out has a greater moment of inertia, but a smaller angular velocity compared to when she is spinning with her arms folded in (I is small, ω is large). Unless she drags her blades against the ice, her angular momentum is a constant.

 

Conservation of angular momentum also justifies the relationship shown in Kepler's Second Law: a line from the planet to the sun sweeps out equal areas of space in equal intervals of time.

 

physlet animation

 

Remember that gravity is an internal force within this gravitational system which provides the center-seeking force behind the planet's centripetal acceleration. It is not an external force exerting a torque on the satellite. The subscript P represents behavior at the perihelion (the closest position on the left) and the subscript A represents behavior at the aphelion (the most distant position on the right).

 

 

Thus, the satellite’s speed is inversely proportional to its average distance from the sun.