Built on Facts

A few days ago I wrote a pretty basic post about the trajectories of a football.  Dr. Pion suggested the real physics in football was in the inertia tensor of the ball itself.  True enough.  As such, this post is going to be a little tough in places for people who haven’t seen this kind of thing before but I’ll try to keep it at least conceptually tractable for everyone.

Angular momentum and angular velocity are intimately bound up with each other, and in introductory physics they’re essentially treated as the same thing.  Multiply angular velocity by the moment of inertia and you get angular momentum.  This works great for objects spinning about their axis of symmetry but it simply is not correct in general.  Nothing in the world prevents there from being angular momentum components about axes other than the axis of rotation.  We don’t even need to look at particularly weird scenarios to see examples of this.  A tetherball might have an axis of rotation corresponding to the pole, but its angular momentum about its point of connection to the pole is not parallel to the pole.

This has a lot of implications in the study of dynamics.  The most serious of these is that we can no longer describe the relationship between angular velocity and angular momentum by a single number called the moment of inertia.  We have to make the leap to a more complicated mathematical object called the moment of inertia tensor.

Strictly speaking, a tensor is not a matrix.  However, tensors can often be meaningfully written as a matrix and the moment of inertia tensor is no exception.  The moment of inertia tensor in all its glory is usually written as a 3×3 matrix:

where

This is pretty intimidating if you’ve never seen it before, so here’s two things to keep in mind.  The tensor is symmetric about the diagonal, and for a continuous mass the sum over all particles is replaced by an integral of the mass density over the volume.  Since the matrix is Hermitian, it is diagonalizable - this just means there’s some change of coordinate axes we can do that makes the off-diagonal elements zero.  The axes we find upon doing this procedure are called the principle axes.  These are the axes of symmetry if the object is symmetric, and in general the principle axes are the ones you can spin the object smoothly around without it trying to “wobble”.

How does this apply to a football?  Some apologies first: Dr. Pion is probably going to be unhappy about the lack of rigor here, and Carl Brannen is going to want me to focus more on forces. Both are good criticisms!  But I think this is probably the clearest way to think about it without worrying about the mathematical details of the problem.  Besides, I have no idea what equation describes the shape of a football, and I really think the resulting integrals would be too hideously bad to do without approximation anyway.  But let’s look qualitatively.

Lay the football down on the ground, long axis parallel to the floor.  If you spin it gently, it will rotate about its center.  The principle axes of the football will be along the symmetry axes, so that means the long axis is a principle axis.  It’s basically symmetric about that line, and so we can take the other principle axis to be the one going through the center pointing straight upward.  The third principle axis is perpendicular to the other two but it’s not important here.

The kinetic energy of that spinning football will be

Wait, how did that get so bizarre looking?  Why isn’t it just the usual (1/2)Iω²?  Well, now the moment of inertia and the angular velocity are explicitly matrices and vectors, and thus we have to treat them as such and pay attention to how they’re actually supposed to be multiplied.  But!  We can make it simpler.  For our football spinning about the short axis of symmetry, only one component of ω will not be 0, and only one component of the diagonal matrix will multiply that.  Thus if we give that short axis the label “1″, the kinetic energy will be

Now imagine the football balanced with the long axis vertical - the way it is before a field goal.  If it’s set up like that and spinning around the long axis (which we’ll label “2″), the kinetic energy will be

Here’s the important part.  For the football laid horizontally spinning about the short axis, its moment of inertia about that axis is large because most of the football mass is far from the axis.  Remember the inertia tensor equation - it involves an r squared term.  For the football balanced on its tip and spinning like a top, the mass of the football is closer to the axis and thus the inertia about that axis is much smaller.  The #2 moment of inertia is smaller than the #1 moment.  Thus for a given angular velocity, the rotating football lying horizontally will have a larger rotational kinetic energy.

But remember all that Lagrangian stuff we did a few posts back?  The potential energy of the ball is also important.  Lying flat on the ground the potential energy is small because the mass is mostly close to the ground.  Balanced on the tip the potential energy is larger since more of the ball’s mass is farther above the ground.

Let’s put all this together.  There’s nothing adding or subtracting energy from the spinning football, and so the total energy is conserved.  The Lagrangian is

Where K is the kinetic energy from rotation and U is potential energy due to gravity.  For a slowly rotating football, K is small and thus U dominates the motion.  Since the time integral of the Lagrangian is going to be a stationary point with respect to small variations in the orientation of the ball, U must therefore be also a stationary point.  Sure enough, any small tilting of the football away from its horizontal spinning position will raise the potential energy and so the forces will conspire to keep the system horizontal and close to the ground.

But what if the football is spinning very quickly?  K dominates the motion and takes U along for the ride.  Since K is proportional to I, therefore the football will pick out a motion that minimizes I.  That’s going to happen when the football is rotating about the long axis - when it’s spinning like a top balanced on its point.

What about for those speeds of rotation where K and U are comparable?  The situation becomes much more complicated because the football is no longer spinning about a principle axis.  It will start wobbling crazily as we’ve all seen.  So a football spun at a high speed will tend to rise and orient itself to spin like a top on its point.  As it slows, it will wobble crazily and then finally lie horizontally and rotate about the short axis through the center of mass.

Crazily heuristic and thoroughly unmathematically rigorous?  Yes, but mostly true nonetheless. Mostly.  Physicists familiar with this stuff will recognize some serious simplifications and smoothing over.  The biggest thing I’ve ignored is the possibility of gyroscope-type motion, where the football is spinning about the long axis but not oriented vertically.  This is certainly possible in theory, but in the particular case of a football it’s not something that you’re likely to see for long.  The leather surface of the ball is not a low-friction surface and thus a quickly spinning ball will tend to roll about the point of contact in sort of a precession-type motion. It will be close to vertical, but not quite there unless it’s spinning quite fast.

Nonetheless I hope it’s a helpful explanation on why footballs behave as they do.  There’s serious physics lurking in the old pigskin.