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Re: [Phys-l] football orientation in flight



On 05/21/2008 07:18 PM, Jeffrey Schnick wrote:
Here's a question I got from one of my students that I would appreciate
your input on: Why does a football, in the case of a long pass, tip
over during the course of its flight so that it is always pointing, to
at least a pretty good approximation, in the direction of its velocity,

On 05/21/2008 07:36 PM, Pete Lohstreter wrote:

Smallest cross sectional area is along the long axis of the ball. As it
decends, aerodynamic forces are at a minimum if the orientation is along
the long axis of the ball.

That's why you might /wish/ for it to be aligned with the airflow.
But wishing does not make it so.

It is not required or even likely that an arbitrary object will fall in
the minimum-drag orientation.

==================

Disclaimer: This is probably quite a complicated phenomenon. It is very,
very easy to get the wrong answer in situations like this.

Having said that, here's the first hypothesis that occurs to me, and it seems
to fit the facts.

0) Readers who are not big football fans might be (and should be!) wondering
whether the effect is real. Before trying to explain why something happens,
it would be prudent to check /whether/ it happens. I recommend
http://www.youtube.com/watch?v=QV9oKSmHX3k
which shows the effect pretty clearly in the last few seconds of the clip.
You can skip all but the last few seconds.

1) The ball is given a large amount of spin, so any re-orientation must be analyzed
in terms of addition of angular momenta. Simple non-rotational F=ma arguments
are not going to get the job done.

2) I can never keep track of angular momentum vectors and cross products, and
students generally can't keep track of them either, so I've stopped trying.
I'm going to use the bivector representation. If you're not familiar with
that, trust me, it's worth learning. The payoff is immediate. It makes
angular momenta much less mysterious, much more physical.

Cursory introduction:
http://www.av8n.com/how/htm/motion.html#sec-gyro-rule
More systematic introduction, with equations and references:
http://www.av8n.com/physics/clifford-intro.htm

3) Let's assume that the ball is thrown with a nose-high pitch attitude, and
that later in the flight -- in the absence of aerodynamic forces -- it would
retain that pitch attitude just by simple gyroscopic stability. So in
the middle third and final third of the flight, the ball would naturally
be "too nose high" relative to its velocity vector, and we need to explain
why it would pitch down.

Without loss of generality let's assume the direction of spin is clockwise
as seen from the rear.

4) The dominant force on the ball will be a Magnus force due to the spin.
This is the same force that you observe in tennis or table-tennis, where
a ball with backspin is a "floater" (upward aerodynamic force, opposing
gravity) whereas a ball with forward spin is a "smash" (diving faster
than gravity due to downward aerodynamic force).

In the case of the football in horizontal flight or descending flight
in a nose-high pitch attitude, the Magnus force will be rightward.
There will be a purely rightward overall /force/, but we are not
presently interested in that; we are looking for a /torque/ that
affects the nose of the ball differently from the tail.

5) The next bit of physics is that the airflow in front of the ball
will be relatively "clean" whereas the flow in back of the ball will
be somewhat "spoiled" due to separation and turbulence. The fat
belly of the ball is involved here; if you threw a cylindrical
pipe this effect would be smaller, possibly much smaller.

Therefore the rightward force is larger on the nose than on the
tail. This leads to a rightward torque, i.e. a rightward yawing
moment.

6) The following diagram
http://www.av8n.com/how/htm/motion.html#fig-angular-precession
can (with a little imagination) be used to describe this situation.
Imagine that a line from the lower right of the screen to the upper
left of the screen is a line of sight looking /up/ at the ball in
flight, so that the "new angular momentum" has a relatively more
nose-down pitch attitude relative to the "old angular momentum".

The small gray rectangle represents change in angular momentum,
i.e. torque times time.

You can see it has a rightward component in the front (near the
"y" label) and a leftward component in the rear (near the "w"
label). This is exactly the direction of change in angular
momentum necessary to produce a downward pitch change. Bivectors
are added geometrically edge-to-edge for the same reason that
vectors are added tip-to-tail.

Let's be clear: the law of gyroscopic precession says that
a yaw-wise /torque/ produces a pitch-wise /motion/.

7) The effect is rather small, so you're not going to notice it
much unless the ball has a lot of speed, a lot of spin, and a lot
of hang-time.

===============================

To recap: There is a bunch of good physics here. At a minimum,
we have:
-- gyroscopic stability
-- Magnus force
-- spoilage of the airflow
-- gyroscopic precession, understood in terms of adding
bivectors edge-to-edge.