On the Web of the Law of Inertia
Ricardo Lopes Coelho
Faculty of Sciences, University of Lisbon
Centre for Studies in History and Philosophy of Science
Email: rlc@fc.ul.pt
Abstract Since the beginning of the twentieth century, it has been pointed out that it is impossible to do an experiment in compliance with the law of inertia. This difficulty has led to criticism on the law and some adjustments have been proposed. Anyway, the law of inertia has been admitted and consequently taught. A recent study has shown that the law has, as a consequence, the concept of force as cause of acceleration. This is the most common definition of force in contemporary textbooks. However, it has been criticised for more than two centuries. Many studies have shown that the concept of force is a problem for teaching. The concept of force as the real cause of acceleration led to the concept of “fictitious force”. In textbooks, it is in general pointed out that fictitious force has real effects. This remark is necessary, since one does not expect that a fictitious cause has real effects. The action of forces on bodies led physicists to consider the response of the bodies to forces and thus to the distinction between inertial and gravitational mass. This distinction causes some difficulty since very precise experiments do not corroborate it. In this paper, it will be shown that the law of inertia, as it has been understood since the beginning of the twentieth century, leads to logical or theoretical consequences, which represent difficulties concerning the connection with phenomena and, consequently, for teaching.
Key words: Law of inertia, force, mass
1. Introduction
In this paper, I will try to show that the law of inertia leads us to consequences, which represent difficulties concerning the connection with phenomena and, consequently, for teaching
Newton’s first law states:
“Every body perseveres in its state of resting or of moving uniformly in a straight line, as far as it is not compelled to change that state by impressed forces” (1726, 13).
The first sentence lays down the mechanical state at which a body remains: it maintains constant velocity. The second one, which begins with “as far as”, points out that a body does not move as indicated, only if it is disturbed by an impressed force. Nowadays, we do not speak of ‘impressed force’ but of ‘force’. It still holds, however, that there is a natural state of a body, whose change can only arise from the exterior of the body. In logical terms, ‘free body’ is a sufficient condition for constant velocity.
If ‘free body’ is a sufficient condition for constant velocity, it follows that if the velocity is not constant, the moving body is not free. Rewriting this: ‘free body’ => ‘constant velocity’ and ‘nonconstant velocity’, therefore ‘nonfree body’. This reasoning is logically correct. It is the modus tollens, [(p=>q) ^ q] => p.
‘Nonconstant velocity’ means acceleration. Acceleration is, therefore, necessarily connected with an external something acting on the body. According to this connection, whenever there is acceleration, that acting thing must be there. In Mechanics, ‘force’ plays the role of that acting something or, in other words, ‘force’ has the properties required by the law of inertia for that acting something. Force is exterior to the body, acts on it and changes its natural state. It follows that whenever there is acceleration, force must be there.
In applying the theory to phenomena, we try to look for the cause of acceleration in these phenomena to be conform with it. This has led to problems.
2. Problems
The concept, force is the cause of acceleration, has been accepted in Mechanics for decades: Euler 1736, Lagrange 1787 (1888), Poisson 1833, Coriolis 1844, F. Neumann 1883, Thomson and Tait 1890, Webster 1904, Planck 1916, Lenard 1936, Feynman 1974, Eisberg and Lerner 1981, Wolfson and Pasachoff 1990, Knudsen and Hjorth 1996, Sears and Zemansky 2004, among many others. In characterizing force in this way, physicists are logically consistent with the starting principle, the law of inertia. Difficulties arise, if one tries to “see” force in a motion.
Understanding force as the cause of change of the natural motion of a body, force must be a real entity: only something real can change the motion of a body. This characterization of force has led to questions such as where the real thing which causes acceleration in phenomena is or what “to cause” means. D’Alembert 1743 (1758), L. Carnot 1803, Kirchhoff 1876 (1897) and Hertz 1894 restructured mechanics in different ways in order to avoid the concept of force as the cause of acceleration. Poincaré asserts categorically “to say that force is the cause of acceleration is to do metaphysics” (1897, 734). Hamel 1912, Platrier 1954, Ludwig 1985 respectively defend in their textbooks that force is a thing of thought, is only a human concept or force does not belong to “real text”. Even though most physicists present force as the cause of the acceleration and conceive force as something real, there are others who tried to avoid that concept or do not consider force as a real entity. Let us continue in considering the most common definition of force in textbooks.
If force is the cause of acceleration and we are observing an accelerated motion, then it can be concluded that a force is there. If not, we would have an effect without a cause or, in other words, a motion of a body that a body could never have by itself and that was not caused by something else. It follows, therefore, from the concept of force as the cause of acceleration that an accelerated motion is enough to infer the existence of force. In some textbooks, it is even added that acceleration is the only sign we have for force (Bergmann and Schaefer 1998, Nolting 2005). From an experimental point of view, acceleration is, therefore, the proof for the existence of force. A problem arises if acceleration is observed and no agent is acting upon the accelerated body.
Let us consider a table, which is accelerated with a ball on it. The ball rolls in the direction contrary to the table’s motion and is accelerated as well. As the ball’s motion is accelerated, a force must be there. However, no force is acting upon it. There is nothing similar to a spring, which makes an oscillation on a body. It also cannot be said that an unobservable force, like gravity, is acting by pulling the ball away from its natural state, for we know that this is not the case. We are, therefore, faced with the following situation: there is an accelerated motion and no action is exerted upon the accelerated body. As a body by itself cannot have an accelerated motion, a force must be admitted; as there is nothing exerting an action upon the body, no acting force, the force is called “fictitious”.
Let us suppose that instead of ‘fictitious force’ we admit “no force is acting upon the moving body”. In this case, a body could have an accelerated motion by itself. If a body could have both a uniform motion and an accelerated one by itself, the law of inertia would not hold.
Textbooks on Mechanics usually turn students’ attention to the point: fictitious forces have real effects (Daniel 1997, Tipler 2000, Dransfeld 2001, Fließbach 2003, Nolting 2005). This is understandable as the characterisation of force suggests the contrary: if something is fictitious, it is not real and being not real, it can do nothing. Hence, it cannot have any effect. As there is an exception to this, the remark is justified. On the other hand, one could ask the question of why call it ‘fictitious’ since effects are real. There is, in fact, a reason for this term, which is founded in the law of inertia.
By falling, the acting force is not observable. It is, however, admitted that a force is acting on the falling body. This topic was highlighted by d’Alembert in 1743: it was said that the weight was the cause of falling but what could be seen was the motion and not the cause. He had, however, to admit a force, since he had accepted the law of inertia. Einstein 1913a pointed out another aspect concerning the determination of the force.
Halliday, Resnick and Walker’s illustration, 2003.
His thought experiment of an observer enclosed in a box shows us that such an observer can in no way decide whether his moving body is acted on by gravity or not. Consequently, he can neither decide whether he is dealing with a gravitational mass or not. Let us consider if the thought experiment can be interpreted as follows.
Let us imagine that we have to study a motion about which we have no further information. To study this motion, we have to observe it. The best means of achieving this goal are certainly stroboscopic images or filming. The result of this is some tens of images. Thanks to them, we can measure the piece of the path in each interval of time, determine the respective velocity and calculate the acceleration. This is all, however, we can draw from the data. We can say nothing about the force or mass of the moving body without further information.
Bergmann and Schaefer’s stroboscopic picture of falling, 1998.
The observer enclosed in that box is only able to observe the motion. Thanks to this observation, he can do no more than determine the places and times of the motion. Based on these data, he can verify whether the motion is accelerated or not, but can say nothing about the force or the mass of the body. Thus, the situation highlighted by the thought experiment is not a particular but a general one. In fact, the thought experiment points out, in an expressive way, that the only inference drawn from one motion alone concerns acceleration and neither force nor mass.
In some textbooks, it is added that if one takes two balls instead of one, it would be possible to distinguish between the two kinds of force considered in the thought experiment. The second ball enables us to obtain more information. This stresses the above thesis’ statement. From the observation of the one motion, it cannot be decided if the body is in a gravitational field or in an accelerated box.
Let us move on to the case in which it is generally considered that force is observed and take the linear oscillation of a body connected with a spring as an example. We say that the spring exerts force f on the body. This complies with the law of inertia. There is a cause of the observed acceleration. It is the force exerted by the spring. However, it is difficult, perhaps impossible, to distinguish between cause and effect there. Through observation,
it can be verified that the body and the spring are involved in this motion. The motion of the body is accelerated as well as each element of the spring. Since acceleration characterizes the effect and, if cause and effect are to be distinguished, acceleration cannot be characteristic of the cause. Hence, the cause in the phenomenon is not assignable. The interpretation of the phenomenon by the theory is not completely satisfactory, therefore. How can it be explained then that force f, which is said to be exerted by the spring, is successful in the anticipation of the motion or of the mass of the body?
In order to assert that the force of spring A is f, a set of experiments has to be carried out (Arons 1990, Kohlrausch 1996). In using a calibrated spring in a new situation, we assume that it moves as it moved in those experiments. The element, which retains the information drawn from those experiments, is force f. Introducing f into the equation F=ma, where m and a refer now to the moving body, some results can be predicted. This understanding of force f, which is based on our dealing with the phenomena, avoids the difficulties referred to above. This matters for teaching.
Carson and Rowland 2005 write: “The problem is that we do not observe or experience ‘force’ as such” (p. 474). They add: “it is difficult to see how force can be abstracted from experience” (p. 479)
. In fact, if it is taught that force is there, where the motion is accelerated, a student will try to find in that motion, through the observation of it, what does not come from there. If it is taught that force was gained from other experiments, the student will understand it without difficulty. Thus, instead of “spring A exerts force f”, it could be written in exercises: ‘Due to a set of experiments, force f was ascribed to spring A’. Let us move on to the concept of mass.
In spite of the concerted efforts of physicists, philosophers, mathematicians and logicians, no final clarification of the concept of mass has been reached, according to Jammer 1997 in his book on the history of the concept. Hecht 2006 expresses the situation in the title of his article “There Is No Really Good Definition of Mass”. The difficulty in defining mass emerged, when physicists tried to find a solution for the problem of force. To define mass in conformity with its measurement process, acceleration is required. According to mechanics, acceleration is caused by force. If mass implies force, it cannot be used to define it. The attempt to define force, thanks to mass and by making recourse to the fundamental equation of dynamics, ran into difficulties and provoked criticism. Let us consider Mach’s definition, which has been used in many textbooks on mechanics according to Hecht.
Mach 1883 (1933) claims in his Mechanics that mass is to be defined dynamically. The starting proposition of his proposal states that bodies in interaction cause reciprocal acceleration. Taking one body as a unit, the mass of any other is measured through the proportion of the accelerations due to the interaction of both bodies. This complies with the processes of measurement of mass within classical mechanics in so far as an accelerated body or an accelerating environment is required to determine the mass of a body. Mechanics teaches us, however, that acceleration is caused by force. This means that the definition of mass presupposes the definition of force. This disturbs Mach’s sequence: acceleration, mass, force. Let us come back to the point which causes the difficulty and ask the question of why we say that force causes acceleration.
We know that because, without force, the velocity of a body is constant. This means, we claim that a force is acting upon a body, accelerating it, because we start from the law of inertia. If we do not claim that, we are not logically coherent with the law.
In conclusion, a definition of mass based on its measurement process is not useful to define force within the theoretical framework founded in the law of inertia. A difficulty with that definition lies therefore in this law.
Consequences of the law of inertia enable us, therefore, to understand problems of mechanics which have existed for decades and cause difficulties in teaching. Let us consider now the problems of the premise itself.
Since the beginning of the twentieth century, it has been said that it is impossible to verify the law of inertia experimentally. Voigt 1901, Planck 1916, Nielsen 1935, Becker 1954, French 1971, Budò 1974, Bergmann and Schaefer 1998, among many others defended this thesis. The difficulty concerning the law consists of the following. The law tells us how a free body moves. As, however, a free body does not exist, it is impossible to carry out an experiment conform to the law. The impossibility of carrying out an experiment results from the following. Such an experiment requires a spatial structure to localise the body, a chronometer, a device for registration of the motion and obviously the moving body. According to gravitation, any two bodies attract each other. Therefore, the observed body is not free (Hanson 1965 highlighted the logical component of this problem). Thus, the condition stated in the law of inertia is not satisfied.
In some textbooks, inertia is presented as a natural tendency of bodies to remain in their states of resting or moving rectilinearly and uniformly (Cutnell and Johnson 1997, Wilson and Buffo 1997). Since, however, it is impossible to do an experiment conform to the law, we can never verify the consequence of the property. We have, therefore, no experimental reason to attribute the property of inertia to bodies.
In some textbooks, the law of inertia is presented as a special case of Newton’s second law: if the resultant of the forces acting on bodies is equal to zero, the bodies remain at rest or in uniform and rectilinear motion (Serway 1997, Halliday, Resnick and Walker 2003, Sears and Zemansky 2004). There is, however, a difference between the law of inertia and the special case of Newton’s second law. The law tells us what happens when there are no forces at all. The special case tells us what happens when the addition of the acting forces is zero. Thus, the statement, which is a consequence of Newton’s second law, can express the law of inertia, only and if only F_{i} = 0 because each F_{i} = 0. This requirement  there is no force at all  leads us again to the same experimentally impossible situation.
Kalman 2009 suggests deducing the law from the principle of least action, as Landau and Lifshitz did. This way of justifying the law involves the meaning of Hamilton’s principle and deserves special attention. It will be considered at another opportunity. Kalman suggests as well reconsidering Galileo´s experiments. The reintroducing of these experiments requires, however, a change in the contemporary law of inertia, as we will see.
According to Newton, d’Alembert, Laplace, Carnot, Poisson and many others, a ball on a flat table justifies the law of inertia. In fact, it can be observed that it stays at rest or moves rectilinearly and uniformly if it is not disturbed by an impressed force. The difficulty at that time concerned the uniformity, which could not be observed. For this reason, the staying at rest and moving rectilinearly were laws of nature in d’Alembert’s theory and the uniformity of motion was a corollary, as it could not be observed but only inferred. Nowadays, a ball on a flat table is not free.
Thus, whereas in the past, “free body“ meant a body which is free only in some directions and not in all thinkable ones, nowadays “free body” means a body without any constraint. Due to this change in meaning, it is now impossible to outline an experiment in compliance with the law, whereas the law was justified by experiments in the past.
Even though there was a change in the meaning of the law, the structure of the statement has been maintained. There is a disjunction between the free body´s motion and the accelerated motion, which is not a motion of a body by itself.
In 1904, the Viennese mathematician, Wenzel Hofmann, gave to the law of inertia a form which contradicts this disjunction:
“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).
In this case, either the constant velocity of the body or the accelerated motion depends upon other bodies. Thus, that kind of motion which depends upon a body by itself and not upon other bodies does not exist anymore. Therefore, the disjunction inherent in the law of inertia  between the motion of a body by itself and by other bodies – is contradicted.
Hofmann’s point of view is referred to by Einstein 1913b, who integrated it into “the hypothesis of the relativity of inertia”. Einstein neither admits a motion of a body by itself. Hofmann’s statement has, however, not been adopted by mechanics for which it is in fact not adequate. This topic will be dealt with at another opportunity.
Following this summary of positions concerning the contemporary meaning of the law of inertia, we have to face Matthews’ radical question:
“we never see forcefree behaviour in nature, nor can it be experimentally induced, so what is the source and justification of our knowledge of bodies without impressed forces?” (2008, p. 10)
A comparative study of the Newtonian and Hertzian mechanics has shown that the concepts of force in these theories are different. The starting laws are also different but the concepts of force are conceived in the same way (Coelho 2009). This means the following. The classical theory starts from the uniform and rectilinear motion and Hertz from the uniform motion along the path with the minimal curvature. In the Newtonian theory, there is force if the motion is not uniform or the path not a straight line and in the Hertzian one, there is force if the curvature of the path is not a minimum or the motion is not uniform. Thus, in both theories, force is a deviation from a certain special motion. This motion is indicated by the starting law of each of the theories. Euler’s theory on the motion constrained by a surface and Lagrange’s equations of motion could be interpreted in the same way: force is a deviation from a certain motion, which can be called the motion of reference of the theory. The law of inertia is then the statement which presents the motion of reference of the Newtonian mechanics. Interpreting the law in such a way, we are only expressing in words what physicists, who developed classical mechanics, did. In other words, the law of inertia is understood by the function it plays in the theory. This understanding of the law enables us to overcome difficulties.
1. The difficulty concerning the law of inertia itself disappears, since we do not need to prove what we cannot do.
2. If force is conceived as a deviation from a motion of reference, the criticism of the concept of force can be overcome, since it is not necessary anymore to consider force as the cause of acceleration and to try to observe it.
3. Since force is not the necessary cause of acceleration, the difficulty in defining mass through the measurement process can be overcome.
Thus, the fundamental equation of dynamics, F=ma, becomes clear and simple in its meaning.
Thank you very much for your attention.
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