[FOM] A compedium... from McCarthy to Drago

José Félix Costa fgc at math.ist.utl.pt
Thu Dec 15 13:03:55 EST 2005

About independence of concepts of «mass», «force», «accelaration», I think
there is something else to be added.

Something that goes into Newton's Principia and to Newton's «shoulders». How
did Newton conceived the 3 laws?

Since FOM has a wide audience, let me put things just in a few sentences.

(I) The first law allows the observer to test if his referential is a good
one --- an inertial one. Stars are far away from each other, their
interaction can be negleted. A solid tetrahedron --- Kepler's one --- can be
defined with our sun in one vertice, three stars in the other vertices. This
imaginary solid is not deformable. In cartesian reference frame fixed in the
tetrahedron, another body is seen to move in straight line and at constant
speed. We can conclude that the body is not acted by a force (let as say a
total non zero force).

[[Parenthesis: I ask often this question to universitary students in the
third undergradute year of mathematics (they take 3 / 4 courses on Physics):
can we deduce first law from the second? Always got the wrong answer]]

(II) If a body deviates from straigh line, constant speed, Newton says that
a force acts on it. Force here is cause, but velocity is not effect.
Searching for a mesure of the effect --- this deviation --- we have to
consider first term, i.e., acceleration. Since causes should be (since the
world of Aristotle, were we can conceive a Physics were forces are
proportional to velocities, construct a model and validateit , e.g., in
Aristotle Physics I) proportional prima facies to the effects, we can say
that             force (cause)      is proportional to     acceleration
(effect), or force = K acceleration. How to mesure the constant K  ??

(III) Third law ...  From the second and the third laws we can deduce the
principle (theorem) of momenta. Conservation of momenta goes back to
Descartes law's of collision, being some Descartes laws right and some
wrong. (Newton, indeed, was a sleepwalker in the sense of chosing the right
things from his «shoulders» and wisely removing the wrong ones.) Collision
laws were studied before Newton's Principia and were very popular. From
collisions laws made simple (with two bodies, two billard balls) we can
derive that the quocient of the two constants K and K', relative to the
bodies, is equal to the quocient of modulus of variations of velocities
before and after the collision.

Now, choose one constant K as standard, defined it to be 1 (Kg), put it in
the gallery of standards in Paris. Define each constant K' --- call it
(inertial) mass --- by the dynamic experience of collision. Thus the roll of
the third law is also to bring to the concept of mass a precise
definition --- through a dynamic experience. One good reason to expect mass
(inercial mass) to be different from gravitational mass.

J. Felix Costa
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
tel:      351 - 21 - 841 71 45
fax:     351 - 21 - 841 75 98
e-mail:   fgc at math.ist.utl.pt
www:    http://fgc.math.ist.utl.pt/jfc.htm

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