I was similarly the distance from the axis of rotation of the particular mass m or mass chunk dm. We considered this to be one dimensional rotation because the axis of rotation did not change, all rotation was about that one fixed axis.

This is, alas, not terribly general. We started to see that at the end when we talked about the parallel and perpendicular axis theorem and the possibility of several “moments of inertia” for a single rigid object around di erent rotation axes. However, it is really even worse than that. Torque (as we shall see) is a vector quantity, and it acts to change another vector quantity, the angular momentum of not just a rigid object, but an arbitrary collection of particles, much as force did when we considered the center of mass. This will have a profound e ect on our understanding of certain kinds of phenomena. Let’s get started.

We have already identified the axis of rotation as being a suitable “direction” for a one-dimensional torque, and have adopted the right-hand-rule as a means of selecting which of the two directions along the axis will be considered “positive” by convention. We therefore begin by simply generalizing this rule to three dimensions and writing:

~

where (recall) ~r × F is the cross product of the two vectors and where ~r is the vector from the origin of coordinates or pivot point, not the axis of rotation to the point where the force

~

F is being applied.

as the angular momentum vector of a particle of mass m and momentum p~ located at a vector position ~r with respect to the origin of coordinates.

In that case Newton’s Second Law for a point mass being rotated by a vector torque is:

~

~τ = dL (570) dt

which precisely resembles Newtson’s Second Law for a point mass being translated by a vector torque.

This is good for a single particle, but what if there are many particles? In that case we have to recapitulate our work at the beginning of the center of mass chapter/week.

6.2: Total Torque

simply cannot see their z-components. Also, this picture is just enough to help us visualize, but be thinking 3, 4...N as we proceed.

If and only if the total vector torque acting on any system of particles is zero, then the total angular momentum of the system is a constant vector.

Note that (like the Law of Conservation of Momentum) this is a conditional law – angular momentum is conserved if and only if the net torque acting on a system is zero (so if angular amomentum is conserved, you may conclude that the total torque is zero as that is the only way it could come about).

Just as was the case for Conservation of Momentum, our primary use at this point for Conservation of Angular Momentum will be to help analyze collisions. Clearly the internal forces in two-body collisions in the impulse approximation (which allows us to ignore the torques exerted by external forces during the tiny time t of the impact) can exert no net torque, therefore we expect both linear momentum and angular momentum to be conserved during a collision.

Before we proceed to analyze collisions, however, we need to understand angular momentum (the conserved quantity) in more detail, because it, like momentum, is a very important quantity in nature. In part this is because many elementary particles (such as quarks, electrons, heavy vector bosons) and many microscopic composite particles (such as protons and neutrons, atomic nuclei, atoms, and even molecules) can have a net intrinsic angular momentum, called spin114 .

This spin angular momentum is not classical and does not arise from the physical motion of mass in some kind of path around an axis – and hence is largely beyond the scope of this class, but we certainly need to know how to evaluate and alter (via a torque) the angular momentum of macroscopic objects and collections of particles as they rotate about fixed axes.

6.3: The Angular Momentum of a Symmetric Rotating Rigid

Object

One very important aspect of both vector torque and vector angular momentum is that ~r in the definition of both is measured from a pivot that is a single point, not measured from a pivot axis as we imagined it to be last week when considering only one dimensional rotations. We would very much like to see how the two general descriptions of rotation are related, though, especially as at this point we should intuitively feel (given the strong correspondance between onedimensional linear motion equations and one-dimensional angular motion equations) that something like Lz = Iωz ought to hold to relate angular momentum to the moment of inertia. Our intuition is mostly correct, as it turns out, but things are a little more complicated than that.

~

From the derivation and definitions above, we expect angular momentum L to have three components just like a spatial vector. We also expect ω~ to be a vector (that points in the direction of the right-handed axis of rotation that passes through the pivot point). We expect there to be a linear relationship between angular velocity and angular momentum. Finally, based on our observation of an extremely consistent analogy between quantities in one dimensional linear motion and one dimensional rotation, we expect the moment of inertia to be a quantity that transforms the angular velocity into the angular momentum by some sort of multiplication.

To work out all of these relationships, we need to start by indexing the particular axes in the coordinate system we are considering with e.g. a = x, y, z and label things like the components of

114Wikipedia: http://www.wikipedia.org/wiki/Spin (physics). Physics majors should probably take a peek at this link, as well as chem majors who plan to or are taking physical chemistry. I foresee the learning of Quantum Theory in Your Futures, and believe me, you want to preload your neocortex with lots of quantum cartoons and glances at the algebra of angular momentum in quantum theory ahead of time...