I am bothered by saying the static friction drops... at least I don't like
saying that the algebraic equation changes.
On the ramp, the static friction equation is <= mu*mgcosT both when the hoop
is laying on its side and when it is in rolling position. This fundamental
relation does not change for either orientation (unless the hoop begins to
slide).
I don't see the ramp question as any different than a box sitting on level
surface, so let's look at that. We can take a specific example of mu = 0.6
and m = 100 kg giving friction <= 0.6mg = 589 newtons.
(1) The box is just sitting there. No one is pushing or pulling on it. How
much is the static friction? Answer = zero.
(2) The box is being pushed with a horizontal force of 75 N to the right.
How much is static friction? Answer = 75 N to the left.
(3) The box is being pushed with a horizontal force of 400 N to the right.
How much is static friction? Answer = 400 N to the left.
(4) The box is being pushed with a horizontal force of 650 N to the right.
How much is static friction? Answer = it isn't; the box has broken free and
we need to use the kinetic coefficient of friction to find the force of
friction.
With the hoop on its side the static friction rises (if it can) to match the
component of the hoop's weight that is parallel to the plane. With the hoop
in position to roll, the static friction rises to match the tangential force
needed to give the angular acceleration needed for the hoop to roll. In
this problem there are two different equations for what the static friction
is reacting to, but there is only one equation for calculating the limit of
how much this reaction can be.
The point is that the actual amount of static friction depends on what it is
reacting to. I guess the answer to the question of "how does it know" would
be that it doesn't... it just matches the opposing force, unless it can't.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817