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Michael Feltner
04-21-1997, 02:05 AM
Donald & Fellow BIOMCH-Lers,

There is either the easy "short" explanation for this problem or an
explanation that may be considered more difficult by some students and
which is a little longer. I'll try both although the inabilty to draw
pictures will make this difficult.

For all explanations let's assume the following:

Subject and the turntable are free only to rotate about a vertical axis.
Clockwise (CW) and counterclockwise (CCW) will be used to describe
rotations about a vertical axis assuming an overhead view of the system.

In a front view of the person/wheel system, an axis going from
left-to-right will be called the mediolateral (ML) axis.

Thus a bicycle wheel that is rotating in a sagittal plane will have its
angular momentum vector (H) pointing in the ML direction. Forward
rotation of the wheel would result in H vectors pointing to the person's
left, and backward rotations of the wheel will result in H vectors
pointing to the person's right.

Finally, our inertial reference frame is defined so that its X axis
points to our right (person's left) as we view the system from the front,
the Z axis is vertical, and the Y axis is Z x X and points away from us
as we view the system.

SHORT VERSION

Initially, the person-plus-wheel system has no angular momentum about a
vertical axis (Hz=0). Since any torques made by the person on the wheel
or vice versa are internal torques, they can not change the total H of the
system. Thus as the wheel is rotated so that its angular momentum vector
acquires a vertical component, the remaining parts of the system must
rotate in an opposite direction with an equal amount of H. This preserves
the conditions that system H remains constant in the absence of external
torques.

My "problem" with this explanation is that it makes the rotation of the
person seem "mystical". That disappears in the next explanation.

LONGER VERSION

Again the wheel is rotating in a sagittal (Y,Z) plane. Lets assume its H
vector is pointing to the person's left (our right as we view the system
from the front) (positive X direction). Wheel-plus-person system H is
zero about a vertical axis (Hz=0). As the person rotates the wheel from a
sagittal to a horizontal plane, they must apply a torque to the wheel.
Since torque = rate of change of H [ Sum T = dH/dt ] and only a single
torque is applied by the person to the wheel, examination of the change in
the H vector will tell us the torque applied to the wheel.

Assume the wheel is rotated clockwise about an anteroposterior axis as we
view the system from the front (positive Y rotation). The H vector of the
wheel will change from an initial horizontal orientation to our right
(positive X direction) to an orientation where it will have both a positive
X and a negative Z component. Even though the magnitude of the wheel's H
will remain constant (assuming no friction) its angular momentum vector is
changing it orientation. The change in H vector between the initial and
final positions of the wheel will have a negative X and negative Z
orientation. Thus the torque applied by the person to the wheel had
negative X and negative Z components. By reaction the wheel will make
positive X and positive Z torques on the person. The positive Z torque is
the torque that allows the person's H to become nonzero and the person
begins to rotate about a vertical axis in a CCW direction. The positive X
torque made on the person by the wheel also attempts to cause the person
to do a forward somersault. Luckily, the turntable does not rotate in
this direction and an opposing torque is created to counter this rotation.

While more complex, I favor this explanation because it (a) more
completely explains the underlying mechanics and shows that the rotation
of the person is not "magical" and (b) it also demonstates that even
though the magnitude of a vector quantity is constant - it's derivative
may be non-zero. This helps reinforce to students the concept that
vectors have both magnitude *and* direction.

Michael Feltner

Dept. of Sports Medicine & Physical Ed. | mfeltner@pepperdine.edu
Pepperdine University | Office: 1-310-456-4312
Malibu, CA 90263 USA | FAX: 1-310-317-7270

On Fri, 18 Apr 1997, Donald Sussman wrote:

> Biomch-L:
>
> I am interested in the explanation for a demonstration lab of the
> conservation of angular momentum. I am sure most of the readers are
> familiar with this demonstration.
>
> Let me review it:
>
> A subject sits on the turntable and stool which are not rotating.
> The subject is handed a bicycle wheel oriented vertically (to
> ground). The bicycle wheel is spun -- now the system is the
> person, wheel and turntable. The subject and turntable are still
> not rotating, but the wheel is. The subject is asked to rotate
> the wheel in one direction or the other. The subject and
> turntable now begin to rotate more slowly in a direction opposite
> to the rotating wheel.
>
> The question which is usually asked: Why does the subject and
> rotating table rotate in a direction opposite to the wheel?
>
> This is the question I am asking this group.
>