(Symbol is **L.** Also called “rotational momentum” or “moment of momentum.”) Angular momentum is the momentum or oomph which an object has as it: 1) spins about its axis, or 2) travels in a curved path around a central point like a planet around the sun. One way to measure angular momentum would be to measure the amount of force needed to stop something from rotating (spinning about its axis of traveling in a curved path). This is the definition of “angular momentum” within classical physics as opposed to its definition within quantum physics.

Within quantum physics, angular momentum takes two forms: spin and orbital angular momentum.

**Example of Angular Momentum**

If you’ve played tetherball, you’ve seen angular momentum in action. A rope is tied to the top of a pole and a ball is attached to the bottom end of the rope. Hit the ball to make it fly off horizontally and it will orbit around the pole. A physicist could measure the force with which the ball is moving to determine its oomph as it winds around the pole, that is, its angular momentum.

As a note, in addition to circles, angular momentum applies to objects that travel in an ellipse or in another curved path.

**Calculating Angular Momentum**

Angular momentum, the oomph of a rotational motion, depends on three factors: 1) mass of the rotating object, 2) velocity of the object, and 3) the radius of the “orbit.”

Here are examples for each of these factors:

- Mass—A lead cannon ball swinging around a pole would have more angular momentum than a tetherball filled with air. Imagine the impact or oomph that the cannon ball would have! The distribution of the mass also makes a difference. For example, a figure skater gains angular momentum when she pulls her arms and legs in tight towards her body.
- Velocity—Imagine the impact or oomph you would feel if hit by a tetherball swinging around the pole at 200 miles per hour rather than 10 miles per hour.
- Radius—If the tetherball were swinging around the pole on a 100-mile rope rather than a 6-foot rope, and both swung at the same speed, the tetherball on the 100-mile rope would have more angular momentum.

These factors, mass and velocity and radius of the circular path, are multiplied times each other to find the angular momentum. The equation is:

**m ****· v ****· r = L **where **m** = mass; **v** = velocity; **r** = radius of the circular path; and L = angular momentum.

Note: When angular momentum is calculated, the distribution of the mass must be taken into account as well as the total amount of mass. For example, an ice skater can gain angular momentum by pulling in her arms and legs, pulling her mass closer to the axis on which she is spinning.

**Measurement Units for Angular Momentum**

The amount of angular momentum can be expressed in three ways:

- Kilogram-meters squared per second, symbolized
**kg-m**. This can be interpreted as the amount of torque per second.^{2}/s - Newton-meter-seconds, symbolized
**N-m-s**. This can be interpreted as work-seconds. This type of measurement is in the genre of a man-hour (the amount of work that a person can do in an hour). - Joule-seconds, symbolized
**J-s**. This can be interpreted as energy-seconds.

**Linear Momentum vs. Angular Momentum**

Linear momentum is the oomph of an object which is not rotating. Linear momentum of an object is calculated as its mass times its velocity. When people say “momentum” with no modifier, they usually mean linear momentum.

**Conservation of Angular Momentum**

Angular momentum is conserved. This means that the angular momentum of all the bodies within an isolated system remains constant. If a rotating tetherball happened to collide with a stray baseball in flight, the tetherball would lose some of its angular momentum and slow down. On the other hand, the baseball would be knocked off course and would gain angular momentum. There would be an exchange of angular momentum. If the amount lost by the tetherball were added to the amount gained by the baseball, the resulting angular momentum would total the amount that the two balls had prior to their collision. Angular momentum would, in other words, be conserved.

**Quantum Angular Momentum**

The term “angular momentum” is used in quantum physics on analogy with angular momentum in classical physics. In quantum physics, there are two kinds of angular momentum: spin and orbital angular momentum. In quantum physics, in an isolated system, the sum of spin plus orbital angular momentum, when considered together, is a conserved quantity.

**Spin—Intrinsic Angular Momentum**

Spin is an inherent property of subatomic particles. They are “born” with it. This is in contrast to an object like a tether ball, which must be hit to get it rotating. Spin is sometimes called “inherent angular momentum” or “intrinsic angular momentum.” When physicists first developed the idea of spin, they thought that particles might have volume and might spin about their own axes. However, this visualization has since been abandoned—and not replaced with another visualization. Spin is describable with mathematical equations but scientists have not been able to come up with a visualization or description expressed in words.

**Orbital Angular Momentum**

Orbital angular momentum for an electron is visualized as relating to the shape of its orbital. The orbital is the region where the electron will be found upon detection. Orbital angular momentum is best described mathematically rather than visualized as a physical motion. However, to the extent it can be visualized, angular orbital momentum relates in some way to the shape of the electron’s orbital.

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