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### Turning in Dance

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Date: Thu, 30 May 1996 21:31:19 -0700 (PDT)
From: Wendy Fan
Subject: turning in dance

I am interested in finding out how a dancer is able to sustain a turn.
What roles do angular velocity and acceleration play? What could I
do with information gathered about angular velocity, angular
acceleration, centripetal force, and centripetal acceleration?

I plan to begin experimenting with the idea of a dancer turning by
using a top as an example. I was wondering if information on how
a top turns is available.

Thank you.
Wendy Fan
```

```
Date: Fri, 31 May 1996 19:23:45 -0400 (EDT)
From: Dr. Tom
Subject: Re: turning in dance

I'm not positive how to answer this because I don't know your
simple stuff, and then get more complex. You can stop reading
when it stops making sense.

The spinning of a dancer is completely described by physics, but
the physics can get quite complex. There are many differences
between a dancer and the "top" from the point of view of a physics
problem.

For example, the dancer's toe on the ground is not frictionless. The
movement through the air is not frictionless, since there's air drag.
The dancer's body and clothes are not completely rigid during the
turn, and, due to the friction, the dancer's body may not be turning
exactly around its center of mass.

However, for a first approximation, a top is not a bad model.
Depending on the amount of resin on the tip of the toe-shoe, there
may be very little friction, and turning speeds are not too high, so
air friction may be low. (The more billowy the dress, or the more
resin there is, the less like a top the dancer will be. An ice skater in
a skin-tight outfit will be much more like a top than the best
ballerina.)

A top with no friction will spin forever. If it is perfectly
symmetric, and starts straight up, it will stay that way forever. If it
starts tilted, after a while it will end up "precessing" - the axis of
the top will turn around the tip touching the ground. When it's first
released, it will have some "nutation" that will die out - a sort of
bouncing around the "perfect" precession path. I learned some of
the details of this in my first year of college physics, but I really
didn't understand why until my third year when we re-solved all
the problems with some better physical intuition and better
mathematical tools.

An important thing that a dancer can do is change the moment of
inertia by bringing the hands/arms closer to the body during the
spin. As this happens, the moment of inertia decreases, and the
angular velocity has to increase to keep the energy the same (again,
ignoring friction). This works great for skaters who start spinning
with their arms out, and when they bring them in, their spinning
speeds up like crazy.

One cool thing about physics that's not too hard to understand is
that there is a more-or-less perfect analogy between the equations
of linear motion and the equations of rotary motion. If you take
almost any equation of linear motion, and substitute angular
velocity for velocity, angular momentum for momentum, moment
of inertia for mass, angular acceleration for acceleration, torque
for moment of inertia, and so on, the exact same equations hold.

So F = ma (force = mass times acceleration)

becomes:

T = I*alpha (torque = moment of inertia times angular acceleration)

Or E = 1/2*mv^2 (kinetic energy = half the mass times velocity squared)

becomes:

E = 1/2*Iw^2 (k. energy = half mom. of inertia * ang. velocity squared)

This last equation explains the increased speed when the dancer
pulls the arms in. The energy has to stay the same (assuming no
friction), so if I is decreased, w must increase to balance it. And the
equation shows by exactly how much.

Other weird things happen, too. A spinning dancer or skater will
get dizzy without "spotting", so a good dancer will lock his/her
eyes on one thing, turn the neck to keep spotting, and then whip it
around to lock again. The mass of the head is not huge, but it's not
insignificant, either, and a spotting dancer will therefore have an
irregular spinning rate.

You can learn about spinning tops in any good physics book on
mechanics. For a thorough understanding, you'll need a thorough
understanding of physics, however.

I hope this helps.

-Doctor Tom, The Math Forum
```

```
Date: 08/03/2001 at 02:02:35
From: Ard Jonker

The key issue here is 'moment of inertia'. Is this the same as
```

```
Date: 08/03/2001 at 10:01:36
From: Doctor Rick

Hi, Ard.

I had no difficulty finding a definition of "radius of gyration" in
terms of moment of inertia by searching the Web, using Google
(www.google.com) and searching on the phrases

"radius of gyration" "moment of inertia"

One such definition can be found here:

http://www.xrefer.com/entry/644217

I quote:

Symbol R0 or k.

For a rigid body of mass M, rotating about a principal axis, the
radius at which a point mass M would have to be placed to have the
same moment of inertia, I. Thus the radius of gyration, given by
k = [sqroot](I/M), depends on the body's shape, not its density.

The New Penguin Dictionary of Science, copyright M. J. Clugston 1998

I think that says it quite clearly. The two are not the same, but they
are closely related. The radius of gyration is a distance, as the name
suggests, while the moment of inertia has units of mass times distance
squared. You can compute one from the other if you also know the mass
of the object.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
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