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Re: Newton's Second Law of Motion



On Mon, 8 Apr 2002, Brian Whatcott wrote:

Perhaps it would be helpful to consider how the proposition was framed by
Newton [at least, in English translation]
Law II
The change of motion is proportional to the motive force impressed; and is
made in the direction of the right line in which that force is impressed.

- If any force generates a motion, a double force will generate double the
motion, a triple force triple the motion, whether that force be
impressed altogether and at once, or gradually and successively. And this
motion (being always directed the same way with the generating force), if
the body moved before, is added to or subtracted from the former motion,
according as they directly conspire with or are directly contrary to each
other; or obliquely joined, when they are oblique, so as to produce a new
motion compounded from the determination of both. [end of Law II]

Newton, Principia Axioms, transl: A Motte: Revised: F. Cajori

In my admittedly amateur reading of the Principia this passage is
typical in showing a certain amount of confusion--at least in
terminology but perhaps even in Newton's own mind--between what we
now call "force" and "impulse." I think it is abundantly clear
that Newton's use of the word "force" above is intended to refer
to "impulse." I have a pet theory that Newton was far less
comfortable with the ideas of differential calculus (which he was,
of course, helping to develop) than he was with integral calculus.
All of his proofs in dynamics are couched in terms of the addition
of a large number of small impulses that cause the
"motion"--which, for Newton, meant "momentum"--to CHANGE. He
always seems awkward about making the transition to the idea of a
CONSTANTLY changING "motion" and its connection to what we call
"force."

We often say that Newton expressed his second law as F = dp/dt,
but this is simply not the case. According to Cajori, he wrote
"The change of motion is proportional to the motive force
impressed; and is made in the direction of the right line in which
that force is impressed"--i.e., "delta p = integral of F dt."

My pet theory goes on to suggest that Newton's own enhanced
comfort level with the integrated form of the second law which
deals only with changES in velocity rather than changING velocity
probably extends to our students as well who would be far better
served by an introduction to mechanics which completely avoids the
use of the word acceleration until the connection of impulse to
change in momentum is more fully grasped.

John
----------------------------------------------------------
A. John Mallinckrodt http://www.csupomona.edu/~ajm
Professor of Physics mailto:ajm@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223