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
E-mail: 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 ‘non-constant velocity’, therefore ‘non-free body’. This
reasoning is logically correct. It is the modus tollens, [(p=>q) ^ -q] => -p.
‘Non-constant 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
2
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
3
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.
4
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.
5
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?
6
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
7
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
8
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  Fi = 0 because
each Fi = 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
9
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 force-free 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
10
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.
References
Alembert, J. d': 1758, Traité de Dynamique, 2nd edn, Paris, Johnson Reprint Corporation, New
York, London, 1968 (rep.).
Arons, A. B.: 1990, A Guide to Introductory Physics Teaching, Wiley and Sons, New York.
Becker, R. A.: 1954, Introduction to Theoretical Mechanics, McGraw-Hill, New York,
London, Toronto.
Bergmann, L. & Schaefer, C.: 1998, Lehrbuch der Experimentalphysik, Vol. I, Mechanik,
Akustik, Wärme, 11th edn, de Gruyter, Berlin, New York.
Budó, Á.: 1974, Theoretische Mechanik, 7th edn, VEB Deutscher Verlag der Wissenschaften,
Berlin.
Carnot, L.: 1803, Principes fondamentaux de l'équilibre et du mouvement, Paris.
11
Carson, R. & Rowlands, S.: 2005, ‘Mechanics as the Logical Point of Entry for the
Enculturation into Scientific Thinking’, Science & Education 14, 473-493.
Coelho, R. L.: 2009, ‘On the Concept of Force: How Understanding its History can Improve
Physics Teaching’, Science & Education (Online first).
Coriolis, G.: 1844, Traité de la Mécanique des corps solides et du calcul de l'effet des
machines, 2nd edn, Paris.
Cutnell, J. D. & Johnson, K. W.: 1997, Physics, Vol. 1, 4th edn, J. Wiley & Sons, New
York, Chichester, Weinheim.
Daniel, H.: 1997, Physik, Vol. 1, Mechanik, Wellen, Wärme, de Gruyter, Berlin, New York.
Dransfeld, K.; Kienle, P. & Kalvius, G. M.: 2001, Physik I: Mechanik und Wärme, 9th edn,
Oldenbourg, München.
Einstein, A. & Grossman, M.: 1913a, Entwurf einer verallgemeinerten Relativitätstheorie und
einer Theorie der Gravitation, Teubner, Leipzig. In: Einstein, A.: 1995, The Collected Papers
of Albert Einstein. Vol. 4. Princeton: Princeton University Press, pp. 303-343.
Einstein, A.: 1913b, ‘Zum gegenwärtigen Stande des Gravitationsproblems’, Physikalische
Zeitschrift 14, 1249-1262. In: Einstein, A.: 1995, The Collected Papers of Albert Einstein. Vol.
4. Princeton: Princeton University Press, pp. 486-503.
Eisberg, R. & Lerner, L.: 1981, Physics: Foundations and Applications, McGraw, New York.
Euler, L.: 1736, Mechanica sive motus scientia analityce exposita, Saint-Pétersbourg.
Feynman, R. P.; Leighton, R. B.; Sand, M.: 1974, Feynman Vorlesungen über Physik. The
Feynman Lectures on Physics. Vol. 1,1. Oldenburg, München, Wien.
Fließbach, T.: 2003, Lehrbuch zur theoretischen Mechanik. Vol. 1 Mechanik, 4th edn,
Spektrum Akademischer Verlag, Heidelberg.
French, A. P.: 1971, Newtonian Mechanics, W. W. Norton, New York, London.
Halliday, D., Resnick, R. & Walker, J.: 1997, Fundamentals of Physics, 5th edn, John
Wiley, New York.
Hamel, G.: 1912, Elementare Mechanik, Teubner, Leipzig, Berlin.
Hanson, N.R.: 1965, 'Newton's First Law: A Philosopher's Door into Natural Philosophy'.
In: R.G. Colodny (ed.), Beyond the Edge of Certainty, Prentice Hall, Englewood-Cliffs, NJ,
pp.6-28.
Hecht, E.: 2006, ‘There Is No Really Good Definition of Mass’, The Physics Teacher 44, 4045.
Hertz, H.: 1894, Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt, J. A.
Barth, Leipzig.
Hofmann, W. : 1904, Kritische Beleuchtung der beiden Grundbegriffe der Mechanik:
Bewegung und Trägheit und daraus gezogene Folgerungen betreffs der Achsendrehung der
Erde und des Foucault’schen Pendelversuchs. Partial translation by J. B. Barbour. In:
Barbour, J. B. & Pfister, H. (eds.): 1995, Mach’s Principle: From Newton’s Bucket to
Quantum Gravity, Birkhäuser, Boston, Basel, pp. 128-133.
Jammer, M.: 1997 Concepts of Mass in Classical and Modern Physics, Dover Publications,
Mineola, N.Y.
Kalman, C.: 2009, ‘A Role for Experiment in Using the Law of Inertia to Explain the Nature of
Science: A Comment on Lopes Coelho’, Science & Education 18, 473-493.
Kirchhoff G.: 1897, Vorlesungen über Mathematische Physik, Vol. I, 4th edn, Teubner,
Leipzig.
Kohlrausch, F.: 1996, Praktische Physik: zum Gebrauch für Unterricht, Forschung und
Technik, 24th edn. V. Kose & S. Wagner (eds.), Teubner, Stuttgart.
Knudsen, J. M. & Hjorth, P. G.: 1996, Elements of Newtonian Mechanics, 2nd ed.,
12
Springer, Berlin.
Lagrange, J.-L.: 1888-9, Mécanique Analytique. 4th edn, Paris.
Laplace, P. S.: 1799, Traité de Mécanique Céleste, Vol. I. Paris. Culture et Civilisation,
Brussell, 1967 (rep.).
Lenard, P.: 1936, Deutsche Physik. Vol. 1. Einleitung und Mechanik, Lehmanns Verl.,
München.
Ludwig, G.: 1985, Einführung in die Grundlagen der Theoretischen Physik, Vol. I Raum, Zeit,
Mechanik, 3rd edn, Vieweg, Braunschweig, Wiesbaden.
Mach, E.: 1933, Die Mechanik in ihrer Entwicklung, 9th edn, Brockhaus, Leipzig.
Matthews, M. R.: 2008, ‘Teaching the Philosophical and Worldviews Components of
Science’, Science & Education, Online First: DOI 10.1007/s11191-007-9132-4.
Neumann, F.: 1883, Einleitung in die theoretische Physik. Teubner, Leipzig.
Newton, I.: 1726, Isaac Newton's Philosophiae naturalis Principia Mathematica, 3rd edn, A.
Koyré & I. B. Cohen (eds.) Harvard Univ. Press, 1972.
Nielsen, J.: 1935, Vorlesungen über elementare Mechanik. Trans. by W. Fenchel.
Springer, Berlin.
Nolting, W.: 2005, Grundkurs: Theoretische Physik, Vol. 1, Klassische Mechanik, 7th edn,
Vieweg, Braunschweig, Wiesbaden.
Planck, M.: 1916, Einführung in die Allgemeine Mechanik, S. Hirzel, Leipzig.
Platrier, C.: 1954, Mécanique Rationnelle, Tome I, Dunod, Paris.
Poincaré, H.: 1897, ‘Les Idées de Hertz sur la Mécanique’, Revue Générale des Sciences 8,
734-743.
Poisson, S. D. : 1833, Traité de Mécanique, Bachelier, Paris.
Sears and Zemansky 2004 (see Young).
Serway, R. A.: 1997, Physics: for scientists and engineers with modern physics,
Saunders College, Philadelphia.
Thomson, W. & Tait, P.: 1890, Treatise on Natural Philosophy, Part I. University Press,
Cambridge.
Tipler, P.: 2000, Physik. Trans. by M. Baumgartner. Spektrum Akad. Verl., Heidelberg, Berlin,
Oxford.
Voigt, W.: 1901, Elementare Mechanik, Veit & Comp., Leipzig.
Webster, A. G.: 1904, The Dynamics of Particles and of rigid, elastic, and fluid Bodies,
Teubner, Leipzig.
Wilson, J. & Buffo, A.: 1997, College Physics, 3rd edn, Prentice Hall, NJ.
Wolfson, R.; Pasachoff, J. M.: 1990, Physics, Scott, Glenview, Ill.
Young, H. D., Freedman, R. A. & Sears, F.: 2004, Sears and Zemansky’s University
Physics, 11th edn., P. Addison-Wesley, San Francisco.
13