**Definition: **

*Angular momentum* is the analog of linear momentum when applied to the rotational motion of a system. Like linear momentum, angular momentum is a vector quantity, meaning that it has both a magnitude and a direction. Also like linear momentum, angular momentum is also conserved within a system.

The symbol typically used to measure angular momentum is * L*. It is measured in units of kilograms x meters-squared per second ... or kg * m

^{2}/s.

### Calculating Angular Momentum

Consider for a moment the concept of torque, which is the rotational analog of linear force. This is calculated by taking a force and multiplying it by the radius of the motion. Calculating the angular momentum is very similar, except this time we multiply the momentum vector * p* by the radius of the motion, which yields the equation:

=L×r=p×rmv

Keep in mind, though, that * v* is the linear velocity of the particle in question. It might be more useful to consider the angular velocity,

*ω*, when it is rotating about an axis of symmetry. In this case, it's possible to derive the relationship between angular momentum and angular velocity by determining the moment of inertia,

*I*, and applying the following equation:

=LIω

### Conservation of Angular Momentum

As with linear momentum, angular momentum is a conserved quantity. This can be proven mathematically, by applying calculus. The quantity we're interested in is how angular momentum, * L*, changes over time (represented by the variable

*t*), which is represented in calculus as the derivative of

*with respect to*

**L***t*, or

*d*/

**L***dt*, which yields the following result after applying the rules for taking the derivative of a product (known as the product rule):

=L×rmvd/Ldt=d/rdt×m+v×rm d/vdtd/Ldt=×vm+v×rmad/Ldt= 0 +×rmad/Ldt=×rmad/Ldt=×r=Fτ

Some comments on this derivation: the zero in the fourth line comes from the vector product of * v* with itself. The other transformations come out of the understanding of the relationship of velocity to acceleration, acceleration to force, and force to torque.

So what happens in a case where there is no force (or no torque) applied to an object? Well, this means that * F* (or

*) equals*

**τ***, which means that*

**0***d*/

**L***dt*= 0. So if there is no force applied, then the rate of change of angular momentum is zero. This means that the angular momentum remains constant. Since angular momentum remains constant unless a force is applied, we say that angular momentum is conserved.