Abstract
There has been much research on principles and fundamental concepts of mechanics. Problems concerning the law of inertia, the concepts of force, fictitious force, weight, mass and the distinction between inertial and gravitational mass are addressed in the first part of the present paper. It is argued in the second that the law of inertia is the source of these problems. Consequences drawn from the law explain the metaphysical concept of force, the problematic concept of fictitious force, the nominal definition of weight and the difficulty with defining mass operationally. The core of this connection between the law and these consequences lies in the fact that acceleration is a sufficient condition for force. The experimental basis of the law in the course of its history shows, however, that the law presupposes acceleration necessarily whereas acceleration does not presuppose the law. Therefore, there is no inconvenience in taking acceleration independently of the law. This is enough to bypass those problems. Taking into account how force is measured by force meters and how mass is basically determined, by comparison with the standard mass, a minimal meaning for both concepts of force and mass is established. All this converges with several solutions proposed in the course of history and increases the communicability of mechanics, as outlined in the final part of this paper.
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Notes
The concept of the inertial system was introduced by Lange (1885) and was of mathematical character. He wrote: “The ideal construction of the inertial system will, therefore, be carried out in the following way. Three material points P1, P2, P3 are thrust forward simultaneously from the same point in space and left by themselves. We must be sure that they are not placed on the same straight line and that we link each one of them with whatever fourth point Q in space. The linking lines, which shall, respectively, be called G1, G2, G3 constitute together a figure of three sides. The figure is left to maintain its shape completely rigid […] a system of coordinates, where the figure maintains the position, is an inertial system” (quoted by Coelho 2007, p. 962).
The equation F = ma is usually called ‘Newton’s second law’. It was, however, discovered by Euler (Coelho 2010, p. 95). Hence, the expression ‘fundamental equation of dynamics’ or the abbreviation ‘FED’ is used in the present study.
Daniel (1997, p. 179), Dransfeld et al. (2001, p. 183), Nolting (2005 , p. 128), Fließbach (2007 , p. 40), Tipler and Mosca (2009, p. 164). French (1971) wrote: “To describe such a force as “fictitious” is therefore somewhat misleading […] the term “pseudoforce” is often used. Even this, however, does not do justice to such forces as experienced by someone who is actually in the accelerating frame” (p. 499).
For this reason, Stachel (2007) proposes the concept ‘non-inertial forces’. “To compensate for the use of a non-inertial frame of reference, so-called “inertial forces” appear in the equations of motion (such forces might better be called “non-inertial”)” (p. 1048).
Stachel (2005) writes: “inertial frames cannot be defined independently of inertial motions, which are in turn defined as force-free motions” (p. 24). If it is admitted that each two bodies attract each other, an inertial frame is not available.
“[…] nothing stands in the way of our arbitrarily establishing the following definition:
All those bodies are bodies of equal mass, which, mutually acting on each other, produce in each other equal and opposite accelerations.
[…] In the general case we proceed similarly. The bodies A and B receive respectively as the result of their mutual action […] the accelerations \( - \varphi \) und \( + \varphi^{\prime } \) […] We say then, B has \( \varphi /\varphi^{\prime } \) times the mass of A. If we take A as our unit, we assign to that body the mass m which imparts to A m times the acceleration that A in the reaction imparts to it” (1974, p. 266).
"La masse d'un corps est le rapport de deux nombres exprimant combien de fois ce corps et un autre corps choisi arbitrairement et constamment le même, contiennent de parties qui, étant séparées et heurtées deux à deux l'une contre l'autre se communiquent, par le choc, des vitesses opposées égales" (§ 81).
In fact, both authors presupposed something beyond acceleration, insofar as they indicate a reason for it. Saint–Venant explains acceleration by the attraction of material points which make up the bodies (1851, §81). This means that there would be no acceleration if there were no actions between those points. Mach refers to the reciprocal action of two bodies. This means that body A acts on B and B on A and because of this there is acceleration (see Poincaré 1897, p. 735; Kibble and Berkshire 2004, p. 10; Roche 2006, p. 1030).
French (1971) points out this difficulty: “Because the measures of force and inertial mass are linked in the single equation F = ma, there is danger of circularity in our definitions” (p. 170).
"Bei einer genaueren physikalischen Diskussion der Gleichung (1.28) hat man die grundsätzlich verschiedene physikalische Herkunft des Massenfaktors m auf beiden Seiten der Gleichung zu berücksichtigen […] Erst die experimentelle Erfahrung zeigt uns die Gleichheit von träger und schwerer Masse" (p. 15).
The implication ‘greater mass implies greater force’ can be written ‘p ⇒ q’. This is equivalent to (−p ∨ q). The negation of this is (p ∧ −q), therefore ‘greater mass and non-greater force’.
"Diese schwere Masse, welche statisch die Feder auszieht, da sie mit der Gravitationskraft G zum Erdmittelpunkt gezogen wird, hat von der Natur dieser Definition her nichts mit der trägen Masse m T zu tun. Letztere war ja allein aus dem Widerstand gegen eine Beschleunigung abgeleitet worden" (p. 92).
"1898 hatte Baron Eötvös eine sehr originelle Idee […] Die Genauigkeit dieses Befundes wurde von Professor R. H. Dicke an der Princeton-Universität noch weiter gesteigert […]" (pp. 94–95).
"In Anbetracht der beobachteten Gleichheit von m s und m T werden auch wir von hier ab nicht mehr zwischen schwerer und träger Masse unterscheiden" (p. 95).
“Quoiqu’il n’y ait point de corps qui conserve éternellement son movement, puisqu’il y a toujours des causes qui le ralentissent peu-à-peu, comme le frottement & la résistance de l’air, cependant nous voyons qu’un corps en movement y persiste d’autant plus long-temps que les causes qui retardent ce movement sont moindres; d’où nous pouvons conclure que le mouvement ne finiroit point, si les causes retardatrices étoient nulles” (p. 9).
“Nous observons sur la terre que les mouvemens se perpétuent plus long-temps, à mesure que les obstacles qui s’y opposent viennent à diminuer; ce qui nous porte à croire que, sans ces obstacles, ils dureroient toujours”(p. 14).
“L’expérience prouve, que si sur une table horizontale parfaitement unie, on place une boule sans lui imprimer aucun mouvement, cette boule restera en repos jusqu’à ce qu’on vienne l’en tirer. […] ce qui a lieu pour un globe placé sur une table horizontale, doit s’étendre à tous les corps possibles, dans toutes les positions possibles, pourvu qu’ils soient dégagés de toute influence étrangère” (pp. 51–52).
“We consider the law to be the probable outcome of most general experience. More strictly, the law is stated as a hypothesis or assumption, which comprises many experiences, which is not contradicted by any experience, but which asserts more than can be proved by definite experience at the present time” (1956, § 315).
“On cite ordinairement l’exemple d’une bille roulant un temps très long sur une table de marbre; mais pourquoi disons-nous qu’elle n’est soumise à aucune force? […] Elle n’est pas cependant plus loin de la Terre que si on la lançait librement dans l’air; et chacun sait que dans ce cas elle subirait l’influence de la pesanteur due à l’attraction de la Terre” (p. 460).
“Every body is subject to the law of conservation of its relative state of motion or rest with respect to all other bodies in space; its actual behavior is then the resultant of all the individual influences” (p. 129).
The consequence of the law is (−q ⟹ −p).
(−q ⟹ −p) ⟺ (−q ∨ −p) ⟺ (−q | p), i.e., ‘acceleration’ is incompatible with ‘free body’. The law of inertia implies, therefore, that acceleration is incompatible with free body.
“En effect, dans une science qui a pour objet les mouvemens et les forces, ce sont les mouvemens qui sont susceptibles d’observation immédiate: les forces sont cachées” (p. 56).
“Die Bewegungen und die Beschleunigungen sind Thatsachen, welche beobachtet werden können […] Wenn man dagegen von Kräften spricht als den Ursachen dieser Bewegungserscheinungen, so weiß man von deren Wesen nichts weiter, als was man eben aus der Beobachtung des Bewegungsvorganges herauslesen kann […] Man kann daher von der Kraft nichts aussagen, was man nicht bereits von der Beschleunigung weiss” (p. 24).
‘Quantity of matter’ is defined by Newton (1726) (Definitio I) as the product of volume and density. As abbreviations for quantity of matter, Newton used the terms ‘body’ or ‘mass’. In the Principia, the term ‘body’ is more used than ‘mass’ (Steinle 1992, p. 95). Thomson and Tait (1888) still used Newton’s definition, even though they pointed out that this is more a definition of density than of mass (p. 220). However, according to Hertz (1956, p. 7), this definition of mass is not logically permissible. It is logically vicious to define mass as the product of volume and density and density as the quotient of mass and volume. Mach criticized Newton’s definition from a different point of view: the expression ‘quantity of matter’ is not clear enough (1974, pp. 265–270). He defended that mass is to be defined dynamically. Mach’s approach has played a role in the course of the history of the foundations of mechanics (Lindsay 1933; Eisenbud 1958; Weinstock 1961; Hecht 2006; Kalman 2010). In some contemporary textbooks, mass appears associated with quantity of matter (Young 1992, p. 91; Fishbane et al. 1996, p. 113).
“Mais on peut, en général, se dispenser de ces mesurages de vitesse et d’accélération, qui sont délicates et difficiles, et estimer promptement les masses […] par le pesage” (§ 88).
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Coelho, R.L. Conceptual Problems in the Foundations of Mechanics. Sci & Educ 21, 1337–1356 (2012). https://doi.org/10.1007/s11191-010-9336-x
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DOI: https://doi.org/10.1007/s11191-010-9336-x