The old centrifugal force

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

YES, I KNOW, THIS TOPIC ALSO HAS BEEN
DEBATED IN THE PAST; SEVERAL TIMES.

On Thursday, Nov 13, 2003, I wrote:

> The textbook I am using has a provocative
> conceptual question: "Explain why Earth is not
> spherical in shape, but bulges at the equator."
> What answer is expected?

And here is another puzzle. In the same chapter
a roller coaster problem is presented. Instead
of analyzing the roller coster let me refer to a small
object sliding (negligible friction) along the inner
surface of a vertical loop. Suppose the sliding is
counter-clock-wise, as in the textbook illustration.

The object is in the 5 o'clock position. I draw the
m*g as a vertical vector pointing down. I decompose
it into two components: radial (away from the center)
and tangential (clock-wise direction). Then I draw
the normal force vector which has the same length
as the radial component of m*g but the opposite
direction. So far all is the same as when r is infinite
(an object sliding along an inclined plane).

But if I do just that then the net radial force is
zero. This is not possible. An additional radial,
force directed toward the center (m*v^2/r) must
be added to "account for" (or to "explain," or to
"cause," if you prefer) circular motion. But what is
the nature of this force? Keep in mind that the
the normal force, due to the radial component of
m*g, has already been introduced. Another normal
force must be present. What kind of constraint
force is it? It can not be another reaction to the
radial component of mg.

In the rotating frame of reference I would say
that the centripetal normal force is the reaction
force due to the centrifugal force. The centrifugal
force acts on the circular road and the constraining
(centripetal) force is nothing else but the reaction
force from the road. But how can the above
question be answered without introducing the
concept of the centrifugal force?

Please share your answer. But before doing
this draw the free-body diagram for the same
object at the two o'clock location. Here m*g
does not naturally decompose into radial and
tangential components. Show that your way
of explaining things work for both positions?
Ludwik Kowalski