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Re: momentum before force (was: friction)



Dan MacIsaac writes:

I was trying to bring up the fact that
dP/dt = m dv/dt + v dm/dt; with which Newton was intimately aware and
which is often overlooked in our more modern treatments of F= ma in
beginning dynamics. We tend to leap at F = m dv/dt right away, which is
incomplete. Newton's language was very cagey about this; better
than most of our own textbooks. However, a distinct idea of P evolved
BEFORE F by some considerable time. No co-evolution here, one distinctly
precedes the other.

While reading this thread, I thought some might never have read Newton's
own explanation of his three laws (I had not until a couple of years ago
- about 15 years into my teaching career). I'm sure some of you know far
more about this than I, nevertheless, here is my quick and dirty summary
of his argument (the first 28 pages of "Principia"):

The first section is "Definitions" where Newton defines:

1. "quantity of matter" (mass) as arising from "density and bulk
[volume] conjointly". In other words, mass = density * volume. He
mentioned that mass is proportional to weight.

2. "quantity of motion" as mass * velocity. He said, "the motion of the
whole is the sum of the motion of the parts." This may be a natural way
to justify (at least initially) momentum as m*v.

3. "inertia" as the force that resists changes in motion (quantity
and/or direction). Newton called inertia a "force of matter" but was
careful to explain that it was a force only when an outside force
endeavored to change the motion of the body. Here is where he first
mentioned that motion and rest are relative.

4. "impressed force" as an action exerted on a body. Newton carefully
explained that impressed forces are exerted *on* a body and do not reside
*in* the body. This is a departure from the term "impressed force" used
by medievals such as Oresme and Marchia that meant a force of motion that
resides in the body (acting as a motor). Newton named 3 origins of
impressed force: percussion, pressure, and centripetal force.

5. "centripetal force" as "that by which bodies are drawn . . . towards
a point as to a centre." Newton mentioned gravity and magnetism here and
launched into a discussion of stones whirled on strings, projectile
motion, and the orbits of satellites.

He then defined various ways of measuring the quantity of a centripetal
force. One is "accelerative force" which I interpret as "field strength"
and another is "motive force" which Newton said is equal to (accelerative
force)*(mass). In other words, F=mg or F=mB (remember, he was speaking
of centripetal forces - attractions to a point).

He finished this section with a discussion of the absolute vs. the
relative nature of 1- time, 2- space, and 3- motion. Very interesting
reading.

Next come the "Axioms or Laws of Motion":

1. law of inertia - examples include a projectile continuing its motion
(ignoring air resistance and gravity), a spinning top (its parts are
drawn by cohesion toward the center, but the top, as a whole, does not
stop spinning except due to resistance), and the orbits of planets and
comets. To use Dan's word, this is indeed cagey as it sets up a major
argument later in the book.

2. "the change in motion is proportional to the motive force impressed"
- my interpretation of this is that Newton was describing instantaneous
change here. In other words, he said dp/dt is proportional to dF/dt
(vector nature is explicit in Newton's explanation).

3. equal reaction etc. - Newton mentions a stone tied to a rope being
pulled by a horse and justifies conservation of momentum. I quote this
passage to my students to stress that Newton defined force as an
interaction.

Newton goes on to discuss 1- parallelogram addition of force vectors, 2-
conservation of momentum of the center of mass of a multi-body system, 3-
equivalence of intertial frames of reference, 4- pendulums, and 5-
elastic and inelastic collisions.



I highly recommend that physics teachers read at least this far (28
pages) into "Principia" if they are not familiar with Newton's manner of
presenting his laws. I also recommend looking at Part III of his book
which explains his "System of the World" - here you find explanation of
orbits of various satellites, comets, and the famous explanation of how
to put up an artificial satellite (in theory).

If this is old hat to many of you, I apologize for taking up bandwidth.

Dave


David J. Hamilton, Ed.D. "And gladly wolde he lerne,
Franklin HS, Portland, OR and gladly teche."
djhamil@teleport.com Geoffrey Chaucer