Re: The old loop-de-loop

["John S. Denker" <jsd@AV8N.COM>]

I agree with at least 98% of what Hugh wrote, but
let me pick on one part that may benefit from
clarification:


On 11/15/2003 05:19 PM, Hugh Haskell wrote:
> I see exactly two forces
> acting on the object, gravity, mg, and what Ludwik has sensibly
> called a constraint force, C. That's it. No other forces.

In some sense that's true:  those are the only two
forces crossing the boundary of the sliding object,
the only mechanisms by which momentum flows across
the boundary.

However, we have not been given the magnitude of C
on a silver platter.  (The direction of C is relatively
easy to obtain.)  So drawing a diagram with C on it
doesn't solve Ludwik's problem.

This exercise suffers from a bit of a chicken-and-egg
problem.  Let me say it in the most confusing possible
terms, to illustrate the nature of the confusion: we
wish to find the acceleration, but we cannot find the
acceleration until we know C, but since C is a force
of constraint we don't know C until we know what the
acceleration "would have been" ... or something like
that.....

Now, to cut to the chase:  This is why algebra was
invented.  I don't see any way to attack this
problem in "calculator mode" where you punch in
a value for mg and then punch in a value for C
and subtract.  I think you need to write down
one equation in one unknown and solve it.

So here's what we know:
 -- local tangent direction of track
 -- local curvature of track.
 -- direction mg
 -- magnitude of mg
 -- direction of C
 -- constraint:  object stays on track
 -- assumption:  two-dimensional problem,
    confined to a single vertical plane.

Here's what we don't know:
 -- magnitude of C.

In the absence of a boldface font, let C be the
vector and define c := |C| to be its magnitude.

Algorithm:

1) Find the net force on the object in terms of
mg, known track parameters, and the unknown c.

2) Find the corresponding (note I didn't say
resulting) acceleration of the object.  Again
this will involve the unknown c as well as the
various knowns.

3) Find the instantaneous radius of curvature of
the said accelerated motion.

4) Set that equal to the known local curvature of
the track.

5) Solve for c.

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

I changed the Subject: line of this thread, because
we were requested not to use a rotating frame, so
there cannot be any centrifugal field.