This week”s subject is “hanging”. No, I”m not talking capital
punishment here. Football fans will appreciate that the longer a
punter can “hang” the ball in the air, the more time there is for the
tacklers to get down the field to do their job on the punt returner.
Quite a few factors enter into the hang time of the punt; wind
currents, angle of contact between shoe and ball, speed of foot,
whether or not the ball spirals or tumbles end over end, etc. I”m
not sure how much scientific attention has been paid to the
aerodynamics of the punt but it would seem to warrant as much
attention as the knuckleball, discussed in an earlier column.
Another type of hang time in sports is that associated with the
likes of Michael Jordan. Although I”m not a great basketball fan,
I can”t help but be amazed at the moves and grace shown by
Jordan while airbound. But, back to football, the great pass
receivers also exhibit a remarkable example of agility and style
while leaping into the air to snag that pass. Lynn Swann of the
Pittsburgh Steelers was not only a superb receiver but he was also
a student of ballet. Actually, my first taste of ballet was at a
performance of “Swan Lake” at Covent Garden in London in
1968. Aside from the beauty of the dancing and the glorious
Tchaikowsky music, the apparent ability of the dancers to hang in
midair was amazing. Even more amazing to me was the ability of
the dancers to jump into the air and clap their legs together two
or three times. I read somewhere that this maneuver is called the
“batterie”. I found that I could jump in the air and clap my legs
together once and used this ploy to introduce our battery course,
saying we were not discussing that kind of battery. With
advancing age and a plate in my leg, I”ve given up that tradition.
I”m lucky just to jump in the air and land on my feet!
All of which brings me to the dinner I attended last week to meet
the new president of Dickinson College, my undergraduate alma
mater. It was somewhat frightening to find out that he was a
fraternity brother of Harry Trumbore, our Lamb guy. However,
my fears at having such a young guy in charge were quelled by
the obvious enthusiasm and competence displayed by this member
of the Boomer generation. In his little talk, he mentioned the
diversity of the student body and the professorial ranks at
Dickinson. I immediately thought of the article in November”s
issue of Discover magazine about Ken Laws, a professor of
physics at Dickinson.
The article was titled “Spin Doctoring” but had nothing to do
with politics. Rather, the subject was torque and the grand jete* en
tournant (GJET). I realize you sophisticated readers of this
column would be familiar with this term but I was not. In case
you”ve forgotten, the GJET is the ballet move in which the dancer
pushes off one foot more or less facing one way and, with hands
over head and legs together, gracefully executes a midair 180
degree turn and lands facing the opposite way on the other foot.
When executed by someone such as the ballerina Cynthia Harvey
of the American Ballet Theater, the GJET is a thing of grace and
beauty.
So where does Ken Laws come into the picture? Well, at the age
of 40 (he”s now in his 60s), Laws enrolled his young son and
daughter in a ballet class at the Central Pennsylvania Youth
Ballet. Being a most unusual father, he decided to join the class
himself, being about twice as tall as his fellow students! Laws
became hooked on ballet, apparently still takes classes, and has
even performed with the Youth Ballet a few times. His special
interest has been to relate the laws of physics to the various
motions in ballet. In fact, he has written two books on the subject
with the aforementioned Cynthia Harvey. One of them is titled
“Physics, Dance and the Pas De Deux” and was highly
recommended on one Web site I visited. Specifically, Laws has
looked into the roles played by torque, angular momentum and
the moment of inertia. I must admit that my basic physics is a bit
rusty, to put it mildly, and I will now refresh my memory about
these three items before continuing.
I”m back! All refreshed, having consulted my Handbook of
Chemistry and Physics. So, here”s a quick course in Newton”s
laws of motion regarding spinning bodies. First, let”s consider
ordinary momentum, which is the mass of something times its
velocity (M = mv, if your mathematically inclined). What this
means is that a locomotive moving at 20 miles an hour is a bit
harder to stop than a fly moving at the same speed! Newton said
that if no other force acts on a body the momentum stays the
same. In other words, once something is moving it stays moving.
In reality, such things as the force of gravity or friction will slow a
body down and stop it. If you throw a ball into the air it doesn”t
keep going out into space. Gravity pulls it back. If you threw it
fast enough, over about 25,000 miles an hour, aided by a rocket
launch, then we would have a space probe, some of which are
making their way out of our planetary system forever.
With a spinning body there is another kind of momentum called
angular momentum, which is also conserved without any other
forces coming into play. If you start a top spinning, it should spin
forever. Again, frictional forces from the resistance of the air and
the contact with the floor slow down and topple the top. The
angular momentum is a little more complicated than ordinary
momentum and is equal to the angular velocity (how fast the body
is spinning, let”s call it w) times something called the moment of
inertia (let”s call it I). It turns out that I depends on not just the
mass, but how the mass is distributed around the axis of rotation
(the imaginary line around which the body is spinning). The
further out the mass is from this axis of rotation, the more the
inertia. Since the angular momentum of a spinning body equals
the inertia times the rpm (M[angular] = Iw) if we decrease the
inertia the body spins faster (w = M/I). (Purists among you will
worry about radians and a factor of pi, but we”ll worry about that
some other time.)
If this is confusing, consider the classic case of the skater who
starts her spin with arms outstretched. As she pulls her arms in
the mass shifts toward the axis, the inertia is less and the skater
spins faster. All a matter of physics. The same principle applies
to the dancer in our GJET. She starts her spin with arms and legs
more or less outstretched and after pushing off, she puts her
hands over her head and pulls her legs together. Both arms and
legs are now more centered, lowering the inertia and speeding up
her rotation so she lands facing the opposite direction. Lest you
get the impression that she started twisting in midair, that”s
impossible, according to the Discover article. She started the
twist as she pushed off with a force exerted off center; that is, she
used torque to initiate the spinning. Without the torque, no
twisting. Torque is the force times the perpendicular distance
from the axis of rotation where the force is applied. In other
words, the reason your wrench has a long handle is to increase
the distance where you apply the force from the bolt your trying
to tighten or loosen.
But we started out talking about “hanging”. To explain that,
consider our ballerina in a plain old grand jete* (GT). This GT
move doesn”t involve a turn but is the move where the dancer
leaps into the air like a gazelle. Cynthia Harvey is known for her
GT jumps and in particular her great “hang time”. Actually, as
the article points out, the jumping dancer is really like a ballistic
missile and doesn”t hang at all. Her center of gravity moves in a
parabola, just like a missile or a football thrown or punted on a
nice calm day. However, as she gets near the top of the jump she
scissors her legs open and then closes them as she starts to
descend. This causes the legs to take up most of the up and
down vertical motion while, for a very brief time, her head and
upper body actually move horizontally. The audience is focused
on that part of her body, which appears for that instant to be
floating. Hence the “hang” time.
For those wishing to try their own balletic skills, the article goes
on to tell of Ken Laws analysis of the fouette* turn, that amazing
maneuver whereby the dancer turns (up to 32 times in Swan
Lake) on one leg. Laws has explained this as a case of the other
leg storing momentum by the tucking of it in and out during the
spinning, aided by a push from the support leg at the appropriate
point in the move. Spurred by the logic of this argument, I just
got up and tried a fouette* myself. I completed a whole eighth
of a turn! Of course, I attribute my performance, or lack thereof,
to the fact I didn”t have ballet shoes handy and could not achieve
the on point position to cut down on frictional forces. Perhaps
the fact that I”m a klutz on the dance floor also was a factor.
Should I still be around in 2001 for my 55th reunion, I hope to
look up Prof. Laws and have him demonstrate the moves close up
and personal. Or perhaps I should just watch old tapes of
Michael Jordan to see how he used torque and momentum to
accomplish his other worldly moves on the court. And now that
the golfing season is over, I may spend the winter trying to apply
these principles to my golf swing, which is sorely in need of more
angular momentum and less torque!
Allen F. Bortrum
Editor Note: Dr. Bortrum knows that “jete” and “fouette” are
accented. Unfortunately, our computer program isn”t as smart.
