Re: A ball in a dish

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

Assuming there were no trivial omission (in the
derivation), I would like to make some comments.

1) We are no longer talking about the rotating dish.
Neither about hot air, or about a wind whose direction
of flow is governed by the coriollis effect. We are
dealing with trivial horizontal trajectories of an
object sliding without friction over the inner part of
a stationary spherical dish.

2) My preconception was that any circular orbit is
possible. The derived formula, however, states that
it is not so. Orbits of small radius [when cos(TET)
would be larger than one] are impossible. Another
paradox to resolve?

3) The mathematical derivation which was used also
applies to a conical pendulum. It tells us that horizontal
circular trajectories are impossible below some speed.
 Watching a conical pendulum one can see that
trajectories are often "ellipses", not circles.

4) I am using quotation marks because I know that the
intersection on a plane with a sphere is not an ellipse.
Once a circle starts looking like an ellipse the trajectory
must have an up and down component. That component
was ignored in the derivation of the cos(TET) formula.

5) So perhaps we should accept the formula as a correct
description of reality. It tells us that horizontal trajectories
(of suspended bobs or pieces sliding on the inner surface
of a spherical dish) are possible only at v>vmin.

6)We tend to think that (for small v) our trajectories look
like ellipses because the initial conditions (correct speed
and correct orientation) are so difficult to satisfy in practice.
But the cos(TET) formula tells us a different story; it tells
us that circular trajectories are impossible at small angles.

7) Should this tentative conclusion be accepted? If so then
we can go back to what Bob was saying about a particle
placed on the surface of a rotating dish. And about the role
of friction. (The bottom of the "particle" matches the dish
surface; sliding is easier than rolling. Suppose that any
values can be assigned to coefficients of friction.)

8) Referring to a ball in a rotating dish, such as a salad
bawl on a turn-table, Robert Cohen wrote:

> ... the ball will remain stationary (relative to the surface)
> no matter where the ball is placed as long as the ball is
> initially stationary (relative to the surface).

He was actually referring to a parabolic not to a spherical
surface. Perhaps this is more significant than I can see. The
cos(TET) formula was "derived (?)" for any rotationally
symmetric surface. I still suspect that something was wrong
in the derivation. What was it?
Ludwik Kowalski