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By the end of this section, you will be able to:
  • Describe the vector nature of angular momentum
  • Find the total angular momentum and torque about a designated origin of a system of particles
  • Calculate the angular momentum of a rigid body rotating about a fixed axis
  • Calculate the torque on a rigid body rotating about a fixed axis
  • Use conservation of angular momentum in the analysis of objects that change their rotation rate

Why does Earth keep on spinning? What started it spinning to begin with? Why doesn’t Earth’s gravitational attraction not bring the Moon crashing in toward Earth? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster?

The answer to these questions is that just as the total linear motion (momentum) in the universe is conserved, so is the total rotational motion conserved. We call the total rotational motion angular momentum, the rotational counterpart to linear momentum. In this chapter, we first define and then explore angular momentum from a variety of viewpoints. First, however, we investigate the angular momentum of a single particle. This allows us to develop angular momentum for a system of particles and for a rigid body.

Angular momentum of a single particle

[link] shows a particle at a position r with linear momentum p = m v with respect to the origin. Even if the particle is not rotating about the origin, we can still define an angular momentum in terms of the position vector and the linear momentum.

Angular momentum of a particle

The angular momentum     l of a particle is defined as the cross-product of r and p , and is perpendicular to the plane containing r and p :

l = r × p .
An x y z coordinate system is shown in which x points out of the page, y points to the right and z points up. The vector r points from the origin to a point in the x y plane, in the first quadrant. The vector points from the tip of the r vector, at an angle of theta counterclockwise from the r vector direction, as viewed from above. Both r and p vectors are in the x y plane. The vector l points up, and is perpendicular to the x y plane, consistent with the right hand rule. When the right hand has its fingers curling counterclockwise as viewed from above, the thumb points up, in the direction of l. We are also shown the components of the vector r parallel and perpendicular to the p vector. The vector r sub perpendicular is the projection of the r vector perpendicular to the p vector direction.
In three-dimensional space, the position vector r locates a particle in the xy -plane with linear momentum p . The angular momentum with respect to the origin is l = r × p , which is in the z -direction. The direction of l is given by the right-hand rule, as shown.

The intent of choosing the direction of the angular momentum to be perpendicular to the plane containing r and p is similar to choosing the direction of torque to be perpendicular to the plane of r and F , as discussed in Fixed-Axis Rotation . The magnitude of the angular momentum is found from the definition of the cross-product,

l = r p sin θ ,

where θ is the angle between r and p . The units of angular momentum are kg · m 2 / s .

As with the definition of torque, we can define a lever arm r that is the perpendicular distance from the momentum vector p to the origin, r = r sin θ . With this definition, the magnitude of the angular momentum becomes

l = r p = r m v .

We see that if the direction of p is such that it passes through the origin, then θ = 0 , and the angular momentum is zero because the lever arm is zero. In this respect, the magnitude of the angular momentum depends on the choice of origin.

If we take the time derivative of the angular momentum, we arrive at an expression for the torque on the particle:

d l d t = d r d t × p + r × d p d t = v × m v + r × d p d t = r × d p d t .
Practice Key Terms 1

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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