Re: TIDES, pedagogy.

["John S. Denker" <jsd@MONMOUTH.COM>]

At 11:18 PM 9/4/01 -0400, Ludwik Kowalski wrote:

> ... how to explain the phenomenon in introductory
> courses. Arguments based on vector calculus can not be used
> in such courses;
....
> Can this question be answered without using advanced calculus?
> I do not think so.

Let's not give up just yet.

> Those who disagree are challenged to produce a pedagogically
> sound explanation for an introductory physics course

Try this as a possible step in the right direction:

I'll bet not everyone in the class can visualize a saddle-shaped potential
very easily.  So here's something that may be of some use:

1) Take a piece of paper.  Ordinary 8x11.5 copier paper will do.

2) Fold it in half.  Cut it for about 1/2 inch perpendicular to the middle
of the fold, as shown by the Xs.

        aaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaa
        +++++++++++++++++++++X+++++++++++++++++++++  fold
        |       .            X            .       |
        |                   .X.                   |
        |                                         |
        |                                         |
        |                                         |
        |                                         |
        |                                         |
        |_________________________________________|  edges

3) Make darts as follows: Fold the paper on each side of the cut, so that a
crease runs from the end of the cut to the outside corner of the fold,
along the path suggested by the dots in the diagram.  Tape down the
triangular flaps so they lie flat permanently.

4) Unfold the thing.  Fold it in half along the perpendicular axis and make
another set of darts.

5) Unfold the thing.  It will probably fall into a saddle shape of its own
accord, but you can help it a little if necessary.

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

Show it to the class.
 -- Hold it edge-on, and point out the downward curvature of the edge
nearest them, and of the bottom of the saddle, and of the far-side edge.
 -- Hold it edge-on the other way, and point out the upward curvature of
the near edge, the saddle, and the far edge.

Explain how it creates a squashing effect in one direction, and a
stretching effect in the perpendicular direction.

Write out the following in words (no calculus)
        potential = constant piece
                + planar piece
                + saddle-shaped piece
                + higher-order terms

Call it a polynomial if they know what a polynomial is.  Or just skip this
step.

The constant piece is uninteresting.  Produces no forces.

The planar piece produces a uniform gravitational pull.  Holds earth in its
orbit around the moon.  Hold up a tilted flat piece of cardboard to
illustrate.  Let a marble roll down the tilted plane, to illustrate the
relationship between tilted potential and force.

The saddle-shaped piece produces tides.

Pass around the saddle.  (Better yet, pass out paper and tape and let the
kids make their own.)

Explain that by putting together about 6 saddles you could create a model
of the funnel-shaped 1/|R| potential.  (Better yet, collect some saddles
and tape them together.)

You get the idea.  You can fill in the rest of the story.  I gotta run.


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

> Oh yes, they also know
> that water will position itself to minimize potential energy. So
> why is the energy minimized when two bulges have equal size?

It's not.  Energy minimization is great for statics.  It's great when
there's lots of damping.  Orbits aren't static.  They don't have much
damping.  Energy would be minimized if the earth sat on top of the moon,
but that's not going to happen any time soon.