On 11/16/2003 09:52 AM, Ludwik Kowalski wrote:
> Why does a constraint force appear
> (in the laboratory frame) when an object slides inside a
> vertical loop? Note that I did not ask how to calculate
> the constraint force; the question was qualitative.
>
> We know how this question used to be answered in old
> physics textbooks. Millikan, for example, would say that
> the constraining force is the normal reaction force to the
> centrifugal force. But I am not allowed to say this because
> centrifugal forces "do not exist" in inertial frames of
> reference.
1) We regret that Millikan blurred the distinction between
a rotating object and a rotating frame of reference. That's
a common mistake.
2) Be careful: even when there is a centrifugal force, it
is not necessarily equal to the force of constraint.
> What should I do? How should I explain the
> origin of the second force on my free-body diagram?
We have at least two problems here, each with subproblems:
1) Physics: How to calculate forces.
1a) How to calculate the centrifugal force if any.
1b) How to calculate the force of constraint.
2) Metaphysics: Various questions about "why" and "origins"
and "causes" and "explanations".
> Some suggested to lean on Newton's first law. An object
> has a "tendency" to move along a straight line. The track
> generates a constraint force to prevent rectilinear motion.
> Why is the tendency to "move with constant v" acceptable
> but the "tendency to stay away from the center of rotation"
> is not?
The difference is that in the rotating frame, the a free
particle will *accelerate* away from the center, under the
influence of the centrifugal field. In the nonrotating
frame, a free particle will not accelerate. So that is
a very, very good reason for saying there is no centrifugal
field in the nonrotating frame.
> If we agree that the constraint force can be attributed to
> the law of inertia
We don't agree on that.
To my way of thinking, the "reason" for the constraint
force comes from the statement of the problem: we are
told the object is constrained to follow the track. We
can infer therefrom that a force of constraint must
exist.
Note that once again there is a distinction: I can speak
of what causes the *inference* about forces, but I do not
speak of what causes the forces.
> I do not know how to implement the algorithm.
> Please share the solution of your one equation
> with one unknown. Is it really c = m*v^2/r?
Assuming c has to do with the force of constraint, that's
certainly not right. In a weightless situation (no mg
contribution) that would be right, but not otherwise.
In situations where mv^2/r crops up, don't assume it
is properly called the centrifugal force.
We recognize v^2/r as the formula for the magnitude of
the acceleration of an object with *uniform* velocity
v following a path with radius of curvature r. That
however doesn't do us a bit of good, because the
assigned problem involves non-uniform acceleration.
So we must start over. Write X = X(theta) where
theta=theta(t). Differentiate twice to find the
acceleration. In the special case where the
components of X are
X = [ r cos(theta) ]
[ r sin(theta) ]
we get a fun result: we can resolve the acceleration
into two components, one antiparallel to X and
proprortional to (theta dot)^2, and another perpendicular
to X and proportional to (theta dot dot).
That is, we find that v^2/r is the _radial_ component of
the acceleration, even in cases of non-uniform motion.
I don't immediately see any way to derive this result by
methods suitable for an elementary algebra-based course.
I imagine a lot of textbooks just blithely assume that
v^2/r is the radial component of acceleration, by
"analogy" to the case of uniform motion, but this is
utterly bogus; it's just a lucky guess. I suspect this
bogosity is one of the things bothering Ludwik.
I suppose you could multiply this component by m and
call it the "centripetal force", but I don't recommend
that. I prefer to just call it the radial component of
the acceleration. My reason for this is as follows:
using a fancy word like "centripetal" makes people
think that there is some special "centripetal physics"
that somehow creates or "explains" the force, perhaps
in analogy to a centrifuge where the fact that you're
in a rotating reference frame really does change the
physics. But there is no "centripetal physics" ...
all we've done is project out one component of the
acceleration.
=================
Putting it all together:
1) Figure out v^2 from the energy principle.
2) From this, figure out the direction and magnitude
of the _radial_ component of the acceleration. Don't
bother with the tangential component for now.
3) Draw the force diagram. There are only two forces.
-- we know the direction of mg
-- we know the magnitude of mg
-- we know the direction of C. Draw it as a line
without an arrowhead, to indicate ignorance of
the magnitude.
4) Redraw the force diagram, resolving mg into radial
and tangential components.
5) Use F=ma "backwards" to calculate the total radial
force from the known radial acceleration. Subtract
off the radial component of mg to find the hitherto
unknown force of constraint.
All the forces are now known. End of game.
=====
This is a perfect example of where we can infer the
force _because_ we know the acceleration, not vice
versa. (Again note that I speak of what causes the
inference, not what causes the force.)
===================
Returning to the question of the deep nature of the
force of constraint: As previously stated, if you
look closely enough it involves slight spring-like
deformations of the track and objects, just enough
to produce the required force.
This is consistent with the notion that the force
of constraint does no work, because we assume the
spring constant is so high that negligible energy
is stored in these slight deformations.