Thanks JohnD for exposing a typing error (the
"centripetal" instead "centrifugal," as in Millikan
and Gale). I see other typing errors. Let me correct
these errors and post the message again. I will make
some changes based on what JohnD wrote. And I
will use capital letters to comment on this.
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My question was asked in the context of trying to
construct a free-body force diagram for an object
sliding counter-clock-wise along a looping track,
for example in the two o'clock location. The first
arrow, representing the force m*g, and pointing
down has already been drawn. What other arrows
must be drawn and how should they be explain
them to students?
1) The diagram MUST SHOW ALL FORCES ACTING
ON THE SLIDING OBJECT. We assume that forces
due to frictional and Coriolis effects are negligible.
Please be specific, I need this for a Monday class.
2) What is the nature of the force acting on the track?
WHAT CAUSES IT HAS BEEN REMOVED. I know
what Millikan said about this force. In the "A First
Course in Physics," copyright 1906, he and Gale
wrote: "Inertia manifesting itself in this tendency
of the parts of rotating systems to move away from
the center of rotation is called centrifugal force."
Accepting this I would say something like this:
a) the sliding object exerts a force on a track;
it is the force "necessary to overcome inertia,"
it is directed away from the center (not necessarily
along the radius). THANKS FOR THE PHRASE
"force necessary to overcome inertia," JOHN.
b) That centripetal force C ("due" to rotation) does
not act on the sliding object, IT ACTS ON THE
GUIDING TRACK OR ROAD. But the spring-like
reaction to that force must be drawn because it
acts on the sliding object. That reaction force is
labeled as R
THE IDEA OF A ROTATING FRAME HAS NOT
BEEN INTRODUCED IN MY COURSE, OR IN
MILLIKAN'S BOOK. THE FIGURE 25 IN THAT
BOOK SHOWS A LABORATORY APPARATUS
FOR OBSERVING A CENTRIFUGAL EFFECT.
NOWHERE DID I SAY THAT THE CENTRIFUGAL
AND CENTRIPETAL FORCES "BALANCE." THESE
FORCES ACT ON DIFFERENT BODIES.
c) We draw the second force and give it a name,
such as "reaction, " R. The direction of that force
is neither radial nor tangential; I draw it at some
clock-wise angle (say 20 degrees) from the
vertical m*g.
d) The total force, T, is the sum of mg and R. The
radial component of T is associated with the
centripetal acceleration while the tangential
component of T is "responsible for" the change
in the instantaneous speed. THE RADIAL
COMPONENT OF T, CALLED CONSTRAINT
FORCE, IS RESPONSIBLE (ASSOCIATED
WITH) THE CENTRIPETAL ACCELERATION.
WE ASSUME THAT SLIDING IS CONFINED
TO A SINGLE PLANE (2D MOTION)
e) The mass of the object was given. Knowing
the radius of the loop, and the instantaneous
speed, one can calculate the centripetal force at
the two o'clock location. Likewise, knowing the
tangential acceleration one can calculate the
tangential component of R.
Is this an acceptable approach? If not then please
provide a better way for explaining the free-body
diagram. The vertical looping road, and a small
friction-free sliding object, offer the most simple
context for dealing with our issues. If we know how
to handle this simple situation we will know how
to deal with more complex problems, such as
equatorial bulging, tides, centrifuges, etc.
3) Yes, the centripetal force is "only a model," like
many other concepts in physics. According to many
old physics books the "cause" of centrifugal force is
inertia (Newton's first law). Who was the first to
declare that the concept of centrifugal force should
not be part of our vocabulary in physics?
Ludwik Kowalski