Angular Momentum

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 * m2/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= r × mv

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:

L = Iω

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 L with respect to t, or dL/dt, which yields the following result after applying the rules for taking the derivative of a product (known as the product rule):

L = r × mv
dL/dt = dr/dt × mv + r × m dv/dt
dL/dt = v × mv + r × ma
dL/dt = 0 + r × ma
dL/dt = r × ma
dL/dt = r × F = τ

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 τ) equals 0, which means that dL/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.