force

This article explains the concept of force in physics. For other meanings, see Kraft (disambiguation) .

Physical size
Surname	force
Formula symbol	${\displaystyle F}$, ${\displaystyle K}$
Size and unit system unit dimension SI N  = kg · m · s -2 M · L · T −2 cgs dyn  = g · cm · s -2 M · L · T −2

Force is a fundamental term in physics . In classical physics , it is understood to be an action that can deform and / or accelerate a body . Forces are required to work to perform, with the energy of a body or a physical system changes.

Forces are directed physical quantities that can be represented by vectors . The following applies to two forces that attack at the same point: They are in equilibrium and cancel each other out when they are opposite and equally strong. In general, they can be combined to form a resulting force that is determined according to the force parallelogram . Forces have various causes or effects and are sometimes named after them, such as frictional force , centripetal force and weight . Some types of forces were also named after people who made a significant contribution to their research, such as the Coriolis force , the Coulomb force or the Lorentz force .

The SI unit of force is the newton . The symbol of force is mostly (from Latin fortitudo or English force ) or less often after the beginning of the German word. ${\displaystyle F}$ ${\displaystyle K}$

The physical concept of force goes back essentially to Isaac Newton , who in the 17th century laid the foundations of classical mechanics with the three Newtonian laws . He defined the force as a change in the momentum over time and identified it as the cause of every change in the state of motion of a body. He also recognized that every body that exerts a force on another experiences an opposing, equally large reaction force.

In certain cases, the term force is also used in a figurative sense, synonymous with interaction and detached from the representation by a mechanical force vector. The four fundamental interactions are also known as the basic forces of physics . They are the cause not only of all known manifestations of forces, but of all processes known in physics. One of the four basic forces, gravity , is described in general relativity by the curvature of space- time. The three other basic forces are explained in the standard model of particle physics by the exchange of gauge bosons , which are often referred to as "force particles".

Word and conceptual history

The word "force"

The word “force” is of ancient Germanic origin and was associated with muscle tension .

In physical terminology force (or force ) at the latest in the 17th century with the Latin vis equated.

The Greek word for force, δύναμις, lives on in the physical branch of dynamics , the theory of movement under the influence of forces. It is also based on the force unit dyn of the CGS system of units .

Outside of physics denotes force

• a physical or mental quality that belongs to a particular wearer. Because he has this "power", he is capable of certain actions and can thus achieve certain effects. Examples are, intellectual power, political power, powerful voice, powerful language, etc. Other formulations such as "exert a force", "collapse under the force" are more oriented towards the course of the action and thus come closer to the physical term.
• Since around the end of the 18th century, "strength" can also mean the persons themselves who are the bearers of strength ("armed forces", "workers", "teachers").
• In legal language , force already means " validity " or " effectiveness " in Old High German , which is only expressed in certain formulas: "remain in / out of force / enter / set" (cf. legally binding). By abbreviation, the preposition "kraft" (always with the genitive ) arose from "in / through force" , such as "by virtue of its office" or in the formulation "... the German people by virtue of their constituent power ..." from the preamble to the Basic Law .

The roots of the general and mechanical concept of force

Aristotle

The origin of the concept of force (according to Max Jammer ) lies in the human experience that once an intention has been made, it can be put into practice. The concept of force is therefore closely related to the causal principle from the very beginning : the exercise of force is the result of the decision, the effect produced is the result of the action of force. Starting from the simplest examples, such as moving parts of one's own body or other objects willingly, all processes that change an assumed “natural” state or course of things are then attributed to the action of forces. This also includes the case of preventing a naturally occurring process, for example when the "natural" falling of an object can be prevented by applying a force. Certain forces were held responsible for inexplicable processes, probably since prehistoric times. They were longed for or feared, and sometimes personified, invoked and worshiped in the form of gods of various ranks.

Physical forces in the narrower mechanical sense have been discussed since ancient times in connection with problems of rest and movement of the body. The statics dealt with the question of which forces are to be used to keep a body at rest (or in balance). Even in ancient times discovered Archimedes to this, the Archimedes' principle as the law of buoyancy and the lever principle . The equilibrium of three forces was only indicated in the 16th century by Simon Stevin through the condition of the closed triangle of forces . This is equivalent to the well-known parallelogram of forces . Stevin was also the first to derive the condition of equilibrium on the inclined plane. The “four simple machines” were thus presented in detail in university textbooks on physics well into the 19th century.

Since Aristotle, the prevailing view of the movement of bodies is that a force is only necessary to dissuade a body from its natural form of movement so that it executes a forced movement . The natural movement of celestial bodies was the circular path, and earthly bodies of free fall. A forced movement, such as a crooked throw or a pendulum swing, ends automatically as soon as the moving force ceases to act. The action of the moving force could not take place at a distance , but only in a mechanical way, i.e. H. by impact, traction or friction when two bodies come into direct contact. The thrown stone was believed to be the surrounding air that propelled it forward. The force also determines the speed of the body in motion in a way that was later interpreted as a proportionality to the speed caused. A uniformly applied force was seen as a rapid sequence of imperceptibly small impacts.

In the Middle Ages, the Aristotelian doctrine gave rise to various doctrines of movement, which eventually became part of the impetus theory . Accordingly, a push or throw at the beginning of the movement gives the body an impetus that drives it forward. This impetus imprinted on the body and located in it slackens over time, which is reinforced by the resistance of the medium, for example air. In the impetus theory, every movement ends automatically when the impetus is used up and the body "has no more strength". In contrast to Aristotle's point of view, there was no longer any need for the external mover to act continuously. However, the proportionality of the imprinted impetus and speed was retained, for example.

Today's concept of physical force was replaced by it when it was used in the Renaissance in the 16th and 17th centuries. In the 19th century the movements of earthly and heavenly bodies were explored through more precise and measuring observations. It turned out (e.g. by Nicolaus Copernicus , Galileo Galilei , Johannes Kepler ) that these movements follow simple rules, which Isaac Newton could explain by a common law of motion if one takes a new concept of force as a basis. Newton's concept of force, which became the basis of classical mechanics, is based entirely on motion. As a measure of the applied force, it determines the deviation from the pure inertial movement of the body, which in turn was assumed to be straight and uniform. With this, the weight also lost the property of a property inherent in the individual body and became an impressed force, the strength of which could be determined via the acceleration of gravity. However, Newton himself, like his successors, used the word force in some passages in a different sense until the 19th century. B. sometimes the impetus.

The modern concept of mechanical force

Galileo Galilei

Galileo, too, had been shaped by the Aristotelian tradition, but with his law of inertia came very close to overcoming it. ^{(P. 209)} He recognized that rest and uniform horizontal movement are physically not essentially different (see Galileo invariance ). With this knowledge Christiaan Huygens then derived the conservation of impulse and thus the laws of shock . These laws showed that uniform movement and rest do not differ in that a separate force is necessary for the mere maintenance of the movement, but not for the maintenance of the rest. Rather, it is only the change in the respective state of motion that requires an external influence. A little later Isaac Newton specified this influence in his laws of motion . First he (like Descartes before him ) defined the uninfluenced state of motion as straight and uniform and defined in his new term vis impressa (Latin for "impressed force") the speed with which this state of motion changes as a measure of force. This concept of force still applies after further clarification by Leonhard Euler and Jean-Baptiste le Rond d'Alembert ; it is one of the foundations of classical mechanics . Newton himself did not use the term vis impressa only in this sense and used the same word vis, among other things, as vis inertiae for the endeavor of the body to maintain its state of motion, i.e. its inertia. ^{(P. 262)}

In addition, the word force was used well into the 19th century in other physical meanings that were also not covered by the Newtonian definition, e.g. B. in the meaning of today's terms of momentum and energy . Until the modern concept of energy emerged and prevailed, kinetic energy , for example, was referred to by the expression "living force" ( vis viva ) coined  by Gottfried Wilhelm Leibniz , which was used by Hermann von Helmholtz in the 19th century and at the beginning of the 20th century was used in technical mechanics. This is continued in words like "power station" and "motor vehicle" for machines that provide energy.

“Power” in natural science and natural philosophy

In addition to the concept specified in Newtonian mechanics, force was also one of the most general basic concepts in natural science . Up until the first half of the 19th century, natural science formed the reservoir for the descriptive natural sciences and thus also the empirical basis of natural philosophy . In it, the forces that a thing possesses were considered to be the cause of every change or effect that it can produce. The forces of a thing were seen as the epitome of its nature , whereby nature, according to the original meaning of the word, denoted everything that was not produced artificially, i.e. through human art . In the theory of nature, a cause in the form of a constantly acting force had to be given for the pure change of location of a body. In the case of uniform, rectilinear movement, this was the body's own “inertial force”, which should not be confused with the current term of an inertial force . An acceleration or a change of direction was then only possible through the simultaneous action of an additional second force. This concept of force, which ultimately emerged from ancient philosophy, fitted well with everyday understanding, but was incompatible with Newtonian mechanics. Nevertheless, it can be found in Germany even in physical textbooks until late in the 19th century, two hundred years after Newton and one hundred years after Newtonian mechanics had established itself through its undeniable successes and thus developed into the first universally recognized exact natural science .

Critique of the concept of mechanical force

Newtonian mechanics and in particular its concept of force were criticized from the beginning with various justifications. On the one hand, one could not explain the phenomena in such a way from first principles as was expected of natural philosophy (and philosophy in general) at the time. Force appeared in Newtonian mechanics as a useful, but only mathematical, auxiliary term for the analysis and description of movements, which, however, tends to leave the real causes hidden. Newton himself had wanted to prevent this criticism with the sentence "I am not making any hypotheses" ( Hypotheses non fingo , in his main work Philosophiae Naturalis Principia Mathematica ). In addition - partly rightly - ambiguities in the relationship between the accelerating force (vis impressa) and the inertia force (vis inertiae) were criticized. In the romantic natural philosophy of German idealism ( Friedrich Schelling , Georg Friedrich Hegel ), this criticism culminated in a sustained, fundamental rejection of modern natural science in the manner founded by Newton at the beginning of the 19th century.

A second counter-argument was that with Newton's concept of force one would have to understand gravity as an action at a distance that could penetrate the absolutely empty space. Newton himself violently protested against this conclusion, because it was in contrast to the traditional view (e.g. Descartes ) that no body can do something where it is not. Even with Leonhard Euler , the possibility of such a long-distance effect was considered an absurd assumption.

In the late 19th century, it was objected ( Ernst Mach , Gustav Kirchhoff , Heinrich Hertz ) that the origin of the Newtonian concept of force lies in the context of the pair of concepts cause-effect, which is completely shaped by the human understanding of what happened. Therefore, the concept of force in mechanics should not be one of the basic concepts, but should even be completely eliminated if possible. The development of classical mechanics by (among others) Lagrange and Hamilton had shown that this is actually possible , according to which knowledge of the formulas for the kinetic and potential energy of a mechanical system is sufficient to fully determine its equations of motion. Modern quantum mechanics and quantum field theory have followed this path . Here, force - if this term appears at all in the Newtonian sense - and even potential energy are not basic terms, but are derived as effective quantities under certain conditions characteristic of classical physics . In technical mechanics, however, the central role of force remained unchallenged.

Basic forces of physics

In today's quantum field theory, the creation and annihilation of one elementary particle by another due to one of the fundamental interactions is fundamental for the derivation of force as an effective quantity . One of the conditions that must be complied with is the maintenance of total momentum and total energy . Overall, this comes very close to the earlier picture that mechanical force is a result of small impacts. In the extended sense of the word “force”, these fundamental interactions, together with gravity, are also referred to as “basic forces of nature”; all known physical processes between bodies can be traced back to them.

Measurement of forces

→ Main article : Force measurement

Force measurement with Hooke's law , here in the form ${\displaystyle F_{\mathrm {p} }=D\cdot x}$

A force can be determined via a distance-time measurement if it causes an acceleration. According to Newton's second law , the relationship applies to bodies with constant mass and constant acceleration . This relationship can also be read from the derived unit Newton ( ). In practice, conclusions are often drawn about the force from a known (advantageously linear ) relationship between the acting force and an easily measured variable. Examples of this are the deformation of an elastic material or the change in the electrical resistance of a strain gauge . ${\displaystyle m}$ ${\displaystyle a}$ ${\displaystyle {\vec {F}}=m\cdot {\vec {a}}}$ ${\displaystyle \mathrm {N=kg\cdot {\tfrac {m}{s^{2}}}} }$

A force can be determined in various ways by the deformation it causes. In school lessons and in some simple applications, forces are measured with so-called spring dynamometers via the change in length of coil springs . In this case, which is Hooke's Law used, according to which the expansion of suitable springs to the force applied is proportional to: It applies wherein the change in length of the spring and the spring constant referred to. ${\displaystyle F=D\cdot \Delta l,}$ ${\displaystyle \Delta l}$ ${\displaystyle D}$

The law of leverage can also be used . In this way, an unknown force can be determined by comparing it with a known force, for example the weight of a mass. In the simplest case, a balance is used, the display of which can be converted into the effective force with the aid of the known gravitational acceleration . ${\displaystyle g}$

With the atomic force microscope , forces on a small leaf spring up to about 1  pN can be detected. This can be used for examining surfaces. Forces in the range of approximately have been measured with the help of individual ultra-cold magnesium ions in Paul traps via synchronization with an external radio signal. ${\displaystyle 5\cdot 10^{-24}\,\mathrm {N} }$

See also : magnitude of forces

Force as a vector quantity

Representation of forces

For the description of a force not only its amount (ie its “strength”), but also the indication of the direction in which the force acts is necessary. Such quantities, determined by specifying the numerical value, unit and direction, are called vector quantities; they are indicated by arrows above the symbol. In a three-dimensional Cartesian coordinate system , a force vector has three components:

${\displaystyle {\vec {F}}={\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}}$

• For example, to describe the weight with which a body of mass is attracted to the earth, a coordinate system with a vertical axis can be selected: ${\displaystyle {\vec {F}}_{\mathrm {G} }}$ ${\displaystyle m}$ ${\displaystyle z}$

${\displaystyle {\vec {F}}_{\mathrm {G} }={\begin{pmatrix}\ 0\ \\\ 0\ \\-mg\end{pmatrix}}}$

The body is accelerated downwards (with the acceleration of gravity ), therefore the z-component is negative. ${\displaystyle g}$

Forces can be graphically illustrated by arrows that start at the point of application of the force and indicate the direction of the force. The length of the arrow from the starting point to the arrowhead is a scale for the amount of force.

Point of attack and line of action

In addition to the amount and direction of the force vector, its point of application also determines the force effect. For rigid bodies, forces can be shifted along their line of action without changing their effect. For example, rope forces always act in the direction of the rope (namely in the direction of pull) and have the same effect (with static systems or sufficiently slow processes) regardless of how long the rope is. With surface and volume forces, the line of action and thus the point of application is often not obvious. For example, engage in a floating balloon simply put the weight of the center of gravity of the lift force, however, in so-called form of gravity , which is why here do not coincide with the lines of action of the two forces in general. A righting moment acts on the balloon until its center of gravity is vertically below its center of gravity. Even then, both forces still have different points of application, but lie on one line of action so that they can compensate each other. The stable flight position of balloons is based on this.

Superposition principle

→ Main article : Principle of superposition

Decomposition of the weight force into the components (downhill force) and (counterforce to normal force ) ${\displaystyle {\vec {G}}}$ ${\displaystyle {\vec {F_{1}}}}$ ${\displaystyle {\vec {F_{2}}}}$ ${\displaystyle {\vec {N}}}$

The superposition principle of mechanics which in Newton's work as " lex Quarta is called" states that work to a point (or a rigid body ) several forces , so add these vectorially to a resultant force That is, the same effect as all forces together. ${\displaystyle {\vec {F_{1}}},{\vec {F_{2}}},\dotsc ,{\vec {F_{n}}}}$ ${\displaystyle {\vec {F_{\mathrm {res} }}}={\vec {F_{1}}}+{\vec {F_{2}}}+\dotsb +{\vec {F_{n}}}.}$ ${\displaystyle {\vec {F_{\mathrm {res} }}}}$ ${\displaystyle {\vec {F_{1}}},{\vec {F_{2}}},\dotsc ,{\vec {F_{n}}}}$

• When two attacking at the same point of attack forces and the same size, but are in opposite directions, so the resulting force is zero. One then speaks of an equilibrium of forces . ${\displaystyle {\vec {F_{1}}}}$ ${\displaystyle {\vec {F_{2}}}}$
• Composition of forces (which act on the same point):
  two forces with the amounts and in the same direction, so the amounts add to the amount of total force . If two forces act with amounts and in opposite directions, the amount of the total force results from the fact that the greater amount of force is reduced by the smaller one. The direction of the total force corresponds to the direction of the individual force that has the greater amount . If two forces act in different directions, the direction and magnitude of the resultant result graphically through a force parallelogram . The forces and are added to a parallelogram, the parallelogram diagonal corresponds to the resulting force. The resulting force of several forces in different directions can be determined graphically with a force polygon or arithmetically as the sum of vectors. ${\displaystyle F_{1}}$ ${\displaystyle F_{2}}$ ${\displaystyle F}$ ${\displaystyle F=F_{1}+F_{2}}$
  ${\displaystyle F_{1}}$ ${\displaystyle F_{2}}$ ${\displaystyle F}$ ${\displaystyle F=|F_{1}-F_{2}|}$
  ${\displaystyle {\vec {F_{1}}}}$ ${\displaystyle {\vec {F_{2}}}}$
• Decomposition of forces:
  While the weight force and the normal force compensate each other on a horizontal plane , this can not happen in the case of the inclined plane . The normal force acts perpendicular to the plane upwards and is therefore not exactly opposite to the weight force. In order to be able to indicate which part of the weight force is not compensated by the normal force and thus accelerates the body down the inclined plane as a downhill force , the weight force can be divided into two forces. One points expediently in the opposite direction of the normal force (and is compensated by this ), the second in the direction of the plane - this represents the downhill force . The acceleration of the body can be calculated via . Such a decomposition is always correct if the vector sum of the partial forces results in the original force, so it must apply here. ${\displaystyle {\vec {G}}}$ ${\displaystyle {\vec {N}}}$ ${\displaystyle {\vec {F_{2}}}=-{\vec {N}}}$ ${\displaystyle {\vec {F_{1}}}}$ ${\displaystyle {\vec {F_{1}}}=m\,{\vec {a}}}$ ${\displaystyle {\vec {a}}}$
  ${\displaystyle {\vec {F_{1}}}+{\vec {F_{2}}}={\vec {G}}}$

Strength units

See also : System of units  and Newton (unit)

Depending on the system of units used, a different unit of measurement is used for the force. Instead of such units as dynes , kilopond , pound-force or poundal , the newton [ ˈnjuːtn̩ ] is used in the international system of units (SI) . The Newton was established in 1946 by the General Conference on Weights and Measures in today's sense as a derived unit of the basic units kilogram (kg), meter (m) and second (s):

${\displaystyle 1\,\mathrm {N} =1\,{\frac {\mathrm {kg} \cdot \mathrm {m} }{\mathrm {s} ^{2}}}}$

and named by her in 1948 after Isaac Newton.

Force units and conversion factors

	Newton	Dyn	Kilopond	Pound-force	Poundal
1 N	≡ 1 kg · m / s²	= 10 5 dyn	≈ 0.102 kp	≈ 0.225 lb f	≈ 7.233 pdl
1 dyn	= 10 −5 N	≡ 1 g · cm / s²	= 1/980665 kp	≈ 1/444822 lb f	≈ 1 / 13825.5 pdl
1 kp	= 9.80665 N	= 980665 dyn	≡ g N 1 kg	≈ 2.205 lb f	≈ 70.932 pdl
1 lb f	= 4.4482216152605 N	≈ 444822 dyn	= 0.45359237 kp	≡ g N · 1 lb	≈ 32.174 pdl
1 pdl	= 0.138254954376 N	≈ 13825.5 dyn	≈ 0.0141 kp	≈ 0.0311 lb f	≡ 1 lb · ft / s²

Power in classical mechanics

Force in Newton's Laws

→ Main article : Newton's laws

Sir Isaac Newton

The Newtonian concept of force is based on the following thought: All effects on a body that lead to a change in its state of motion are forces. The force describes the intensity and direction of the interaction between two bodies, not a property of a body. In the case of a force-free movement or if there is an equilibrium of forces, the state of movement of a body does not change, it therefore continues to move in a straight line at a constant speed or it remains at rest. That is the content of the principle of inertia , as Galileo put it.

The principle of action links the force that is exerted on a free body with the change in its momentum : In every infinitesimally short period of time , the body's momentum changes according to The momentum of a body is the product of its mass and speed ; it is important , since the mass of the body in most cases remains virtually constant (exceptions are for example rockets or body at relativistic speeds), to write the second Newtonian Axiom mostly in the form , wherein the acceleration is of the body. The force on the body under consideration corresponds to the product of its mass and its acceleration. ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {p}}}$ ${\displaystyle \mathrm {d} t}$ ${\displaystyle \mathrm {d} {\vec {p}}}$ ${\displaystyle \mathrm {d} {\vec {p}}={\vec {F}}\mathrm {d} t.}$ ${\displaystyle m}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {p}}=m{\vec {v}}.}$ ${\displaystyle {\vec {F}}=m{\tfrac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}=m{\vec {a}}}$ ${\displaystyle {\vec {a}}}$

As a consequence of the conservation of momentum , the reaction principle follows , according to which there is always a force ("actio") from body A to body B, that is , an equally large but exactly opposite force ("reaction") from body B to body A : The reaction is not just a kind of passive resistance, but a force that actively attacks the interaction partner. It is to be distinguished from the equilibrium of forces, because the points of application of and are different, so the two forces cannot compensate each other. ${\displaystyle {\vec {F}}_{A\to B}}$ ${\displaystyle {\vec {F}}_{A\to B}=-{\vec {F}}_{B\to A}.}$ ${\displaystyle {\vec {F}}_{A\to B}}$ ${\displaystyle {\vec {F}}_{B\to A}}$

In modern notation, the version corresponding to Newton's intention would rather read. The use of the word force in Newton's writings is not always straightforward. Force is mostly to be interpreted as an impulse of force that causes an additional impulse . ${\displaystyle \Delta {\vec {p}}\sim {\vec {F}}\cdot \Delta t}$ ${\displaystyle {\vec {F}}\cdot \Delta t}$ ${\displaystyle \Delta {\vec {p}}}$

Balance of forces as a key concept in statics

When several forces act on a body that cancel each other out, i. i.e. if for the vector sum of the forces ${\displaystyle {\vec {F}}_{i}}$

${\displaystyle \sum _{i}{{\vec {F}}_{i}}={\vec {0}}}$

applies, then one speaks of the balance of forces. The observed body does not change its state of motion. This fact used to be called the "first law of the statics of rigid bodies ". The forces are both the impressed forces that arise through interaction with other bodies in the environment and the internal forces that appear between parts of the body. In particular, “constraining forces” occur when parts of the body are subject to geometrical bonds that prevent the body from accelerating, for example through supporting and holding forces. The consideration of the balance of forces is part of the statics . ${\displaystyle {\vec {F}}_{i}}$

In order to make systems (e.g. supporting structures ) accessible for calculation here or more generally in technical mechanics , these are cut free from their environment by replacing all of their interactions with the environment with forces and moments. Connections between the bodies of the system and between the system and its environment, which allow only minor changes in shape, are idealized as “rigid connections”. Such rigid bonds are usually joints between the bodies or bearings . The physical character of these bonds is thus lost, and the mechanical interaction of the bodies caused by these bonds is represented by constraining forces ( bearing reactions ). Forces exerted by stationary rigid bodies do no work on the system because there is no resulting movement. Force systems can be processed analytically or graphically and in equilibrium they fulfill the above-mentioned force equilibrium as well as the moment equilibrium , which was previously called the "second law of the statics of rigid bodies".

• Examples of applied forces with a specified course: weight force, pressure and tensile force (e.g. load and wind suction ), spring force , torque
• Examples of constraining forces resulting from the applied forces: support force , adhesive force ( frictional force )
• Example of internal forces: joint forces , if the body has joints, cutting reactions

The principle of virtual work says that in statics the sum of the virtual (assumed) work of all forces must result in zero, which is the “third law of the statics of rigid bodies”. The d'Alembert principle extends this principle to systems of classical dynamics that are subject to constraining forces and is used to set up equations of motion.

Strength theory and structural engineering

In structural engineering and strength theory , the force cannot always be treated in a simplified way as a point load ( single load ). One takes z. B. when determining the deflection of a beam supported at its ends, the dead weight as a force distributed over its length ( distance ), i.e. as a line force or line load . If the weight force were to be applied as a point load in the beam's center of gravity, the deflection would be significantly too great. An example for the more general case of area force or area load is the traffic load of a two-lane bridge. If the tracks are used unevenly, the force distribution is also different across the width of the bridge.

Volume forces and surface forces

In technical mechanics in particular, a distinction is made between volume forces and surface forces that act on a body.

Differences in free cutting

→ Main article : Cutting principle

With the cutting principle, cutting forces are released at the cutting surface.

Treatment of a complex problem - such as the mechanical interactions of several bodies to each other or the mechanical effects in each other connected elements (for example rigid or deformable body.) - so it is a relevant subsystem intersects mentally from its surroundings free (cut into compounds or in bodies self). Now the individual pieces of the surface of the cut-out subsystem are assigned the forces with which the rest of the system acts on the subsystem at this point. The cut-free subsystem then also acts on the rest of the system with the opposite force. These forces are called surface forces. You belong to the contact forces . In terms of their strength and direction, they depend on the previously defined cuts.

Volume forces are non-contact forces . Like gravitational force or magnetism , they attack the entire volume, i.e. H. also inside a cut body. For example, if a homogeneous block is in a homogeneous gravity field (approximately a small block near the surface of the earth), the volume force acts on it. Cuts to this block conceptually in its center, so the other hand, acts on each part block, only the volume of force In determining the deflection of a bearing at its ends the bar (so) that is transversely in differentially small pieces (each acting on them differentially small proportion of the total volume force) cut. ${\displaystyle F_{\mathrm {g} }}$ ${\displaystyle F_{\mathrm {g} }/2}$

Floating under water and weightlessness in space

→ Main article : Weightlessness

When diving, as in space, a feeling of weightlessness can be experienced when the diver has adjusted his weight to the density of the water with counterweights. Gravity acts as volume force on the diver, even under water . The feeling of weightlessness, similar to weightlessness, occurs underwater through buoyancy - the hydrostatic pressure , which acts as a surface force on the underside of the body, is higher than the pressure acting on the upper side of the body. Since only the volume force, but not the surface force, acts on the organ of equilibrium , one can also distinguish above from below under water .

Weightlessness in space, where the astronaut is also “without weight”, can be viewed as a state in which no other force apart from gravity acts, in particular not the supporting force otherwise exerted by a surface. In this case, the organ of equilibrium does not function (the result is often what is known as space sickness ), and the spaceman cannot orient himself with his eyes closed.

Forces with a non-mechanical cause

Some forces, which were still regarded as different at the time of Newton, turned out to be expressions of electromagnetic forces inside matter. These forces make themselves felt

• as electromotive force driving electrons through a conductor,
• in the compressibility and viscosity of fluids ,
• in the friction between the surfaces of different bodies,
• the resistance that a body opposes to deformation ( spring force , compressibility , shear modulus ).

Force and determinism

Quadratic dependence of the fall distance on the fall time: You can see a freely falling ball in the light of eleven strobe flashes that illuminated it every 0.05 seconds. During the first 0.05 seconds the ball falls through a unit of length (here about 12 mm); four units of length within 0.10 seconds; then nine units of length within 0.15 seconds, and so on, until after 0.50 seconds it has covered a hundred units of length.

→ Main article : determinism

With the help of Newton's laws , it is possible to predict the development of a physical system over time from a given initial situation and the acting forces. This applies not only to individual experiments in the laboratory, but in principle also to the universe as a whole. This conclusion contributed to the spread of a deterministic worldview in the 18th century . Accordingly, all events would basically be predetermined even if the calculations required for a prediction are usually not practicable. However, the early 20th century turned out that the formulas of classical physics at the level of atoms are not applicable. The deterministic world view inferred from the formulas therefore had to be rejected in its original form.

Relationship between strength and work

→ Main article : work

The work of a force can change the energy of a body. One example is the tension energy in the expander . The energy transferred when moving the point of application of a force by a certain distance is also called work and is often referred to as . ${\displaystyle W}$

If you want to do a certain job with less force, this is possible with a force converter . Examples of force converters are pulley blocks , levers or gear shifts . However, the path along which the force must be exerted increases. If, for example, only a quarter of the force required without it is required by using a force converter, this is associated with at least four times the travel. This consequence of the law of conservation of energy is described in the " golden rule of mechanics ".

If the force is constant and is acting in the direction of a straight line path of length , then the work is done by the relationship ${\displaystyle s}$

${\displaystyle W=F\,s}$

certainly. If the force acts at an angle to the line, the work can be calculated using the following formula: ${\displaystyle \alpha }$

${\displaystyle W={\vec {F}}\cdot {\vec {s}}=|{\vec {F}}|\,|{\vec {s}}|\,\cos \alpha }$

In the above equation, the vector is from the start point to the end point of the route. In particular, no work is done if the force forms a right angle with the path : Carrying a load on the plane can be tedious, but the load does not absorb any energy. ${\displaystyle {\vec {s}}}$

In general, the work done is the curve integral of the force along the distance covered:

${\displaystyle W=\int _{{\vec {s}}_{1}}^{{\vec {s}}_{2}}{\vec {F}}({\vec {s}})\cdot \mathrm {d} {\vec {s}}}$

Where and are the position vectors of the start and end point of the path. ${\displaystyle {\vec {s}}_{1}}$ ${\displaystyle {\vec {s}}_{2}}$

Conservative and dissipative forces

→ Main articles : Conservative force , friction  and dissipation

If the expander, to stay with the above example, is fixed on one side and the other end is moved in space, the direction and amount of the force change systematically from point to point. If the movements are carried out slowly, so that no vibrations of the expander are excited, and if internal friction is neglected , the force is only a function of the location (a static vector field ). Each location corresponds to a certain stress state of the expander. It does not matter how the location and the associated state of tension were reached. In such cases one speaks of a conservative force . Work done against a conservative force is independent of the path, it only depends on the starting and ending point. In particular, work done is returned when one reaches the starting point again - on the same or a different path.

The value of the path integral of a conservative force from a fixed reference point is called potential energy , usually also potential, but for a distinction see potentials and potential fields in the main article. It is often easier to start from the potential energy (in the above example from the tension energy stored in the expander) than its negative gradient ${\displaystyle E_{\mathrm {pot} }\,}$ ${\displaystyle {\vec {F}}}$

${\displaystyle {\vec {F}}({\vec {r}})=-\nabla E_{\mathrm {pot} }({\vec {r}})}$

to determine, because the field of potential energy is only a scalar field .

The fact that work done on a system is completely converted into potential energy is never fulfilled in practical cases. Frictional forces also have to be overcome. The work done against them is converted into heat . Sometimes such dissipation is desirable ( parachute , fitness equipment , engine brake ).

Force in the force field

→ Main article : Force field

Against the expander in the example above, the skinny little guy has to use the same force as the heavyweight. In the stair climbing discipline, both work against their respective weight force and in earth orbit both would float in harmony next to each other. When describing movements in fields of force , such as the earth's gravity field , it is often useful to abstract from that property of the body to which the force is proportional. This property (here the mass of the athlete) is commonly called the charge . The abstraction happens by dividing the vector field of the force by the charge. The result ${\displaystyle F_{\mathrm {G} }}$ ${\displaystyle m}$

${\displaystyle {\frac {{\vec {F}}_{\mathrm {G} }({\vec {r}})}{m}}={\vec {g}}({\vec {r}})}$

is called field strength and describes the force field independent of the charge on the specimen . The field strength g of the gravitational field is also called gravitational acceleration.

The scalar field of the potential energy that exists for conservative force fields divided by the charge gives the potential of the force field.

Relationship between force and torque

Law of leverage

→ Main articles : Torque  and Lever

The torque can be understood as the rotational effect of the force . It is the cross product of strength arm and strength: ${\displaystyle {\vec {M}}}$ ${\displaystyle {\vec {F}}}$

${\displaystyle {\vec {M}}={\vec {r}}\times {\vec {F}}}$

The force arm is the position vector from the pivot point to the point at which the force acts (point of application). This means that the greater the distance between the pivot point and the point of application, the greater the torque. In addition, only that component of the force contributes to the torque that is perpendicular to the distance between the pivot point and the point of application. ${\displaystyle {\vec {r}}}$

Torques occur, among other things, when the speed of rotatable bodies increases or decreases . They play a role comparable to that of forces in linear motion . Analogous to the equilibrium of forces , the equilibrium of torque is an important special case.

Relationship between force and pressure

→ Main article : Pressure

When a force acts on a surface , the pressure generated by it is the amount (the vector length) of the force component per area that is perpendicular to this surface : ${\displaystyle p}$ ${\displaystyle {\vec {F}}_{\perp }}$ ${\displaystyle A}$

${\displaystyle p={\frac {|{\vec {F}}_{\perp }|}{A}}}$

The pressure is an intensive state variable of thermodynamic systems and also a linear field variable . This concept is a simplification of the general stress tensor .

In contrast to pressure, compressive stress is not a scalar state variable.

Inertial forces or apparent forces

→ Main article : Inertial force

The change between the Aristotelian and Newtonian conception of force is also noticeable in the term “apparent force” (used synonymously: inertial force). The name "sham" can be misleading; these forces are quite measurable and produce real effects. The name comes from the fact that they only occur in accelerated coordinate systems and do not exist when viewed from an inertial system. A suitable outside observer explains the effects simply by applying the principle of inertia without additional forces.

Another approach to the concept of inertial force is linked to d'Alembert's principle : it converts - to put it simply - the dynamic problem of the moving body into a static problem by introducing a d'Alembert's inertial force . The technical mechanism in which the principle is applied successfully, speaks of a " dynamic equilibrium ". While some specialist books refer to this d'Alembert inertial force as a counterforce in the sense of the principle of interaction , other specialist book authors see it as a contradiction to the principle of interaction, since there is no counterforce to it. The designation pseudo force is also justified by the fact that the inertial force does not meet the definition of Newton, which is to be understood as an acting force . ${\displaystyle {\vec {F}}_{T}=-m\,{\vec {a}}}$

• If a car is braked by a force (extreme case: frontal impact), this force does not act directly on the driver. So, according to the principle of inertia, the driver will be moving straight ahead at a constant speed while the car is slowing down. Only through the restraint systems ( seat belt and airbag ) are constraining forces exerted on the driver, which also slow him down. From his point of view, a forward inertial force is now acting, which pushes him towards the windshield. ${\displaystyle {\vec {F}}}$
• The seat of a chain carousel would move straight ahead through the chain without any force; the circular movement only comes about through the centripetal force directed to the center of the circular path . A person on the seat feels the centrifugal force (centrifugal force) as inertial force.

• Other examples of inertial forces are
  • the Coriolis force
  • the inertial forces in engine construction
  • the gravitation , considered in the context of the general theory of relativity, see gravitational force as inertial force

Force in relativity

→ Main articles : Special Theory of Relativity  and General Theory of Relativity

The special theory of relativity takes the place of the dynamic laws of classical mechanics when the observed speeds are no longer negligible compared to the speed of light . In the special theory of relativity, the momentum has to be generalized to the relativistic momentum , the force then remains calculable, but the momentum can no longer be calculated using the relationship . Instead of the Newtonian relation "force = mass times acceleration", the equation is used ${\displaystyle {\vec {F}}={\tfrac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}}$ ${\displaystyle {\vec {p}}=m\cdot {\vec {v}}}$ ${\displaystyle {\vec {F}}=m\cdot {\vec {a}}}$

${\displaystyle {\vec {F}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {m\cdot {\vec {v}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.}$

The force is further extended to the Minkowski force ("four force"), which is usually written as and can be calculated from the four momentum using the proper time and the Lorentz factor ${\displaystyle K^{\mu }}$ ${\displaystyle K^{\mu }={\tfrac {\mathrm {d} p^{\mu }}{\mathrm {d} \tau }}=\gamma {\tfrac {\mathrm {d} p^{\mu }}{\mathrm {d} t}}}$ ${\displaystyle \tau }$

${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.}$

This equation, the " equation of motion of the special theory of relativity for the four-momentum", describes accelerated movements in an inertial system . There is a connection between and where the spatial part is the force of four ; the newly added temporal part describes a change in energy, more precisely: (see four-pulse ), so that one also speaks of the force-power four-vector. ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {K}}}$ ${\displaystyle {\vec {F}}={\sqrt {1-{\frac {{\vec {v}}^{2}}{c^{2}}}}}\,{\vec {K}},}$ ${\displaystyle {\vec {K}}=(K^{1},K^{2},K^{3})}$ ${\displaystyle K^{\mu }}$ ${\displaystyle {\tfrac {\mathrm {d} {\tfrac {E}{c}}}{\mathrm {d} \tau }}}$

The general theory of relativity is an extension of Newton's theory of gravity ; it contains these as a limit case for sufficiently small mass densities and speeds. Its foundations were largely developed by Albert Einstein at the beginning of the 20th century; it generally describes the interaction between matter (including fields ) on the one hand and space and time on the other.

The gravitational force is understood in the general theory of relativity as a geometric property of the curved four-dimensional space - time : the energy, momentum and pressure of matter influence the geometry of the space-time in which they are located. This influence can be described by the term »space-time curvature«. The spatial and temporal coordinates are regarded as equal, all changes are only treated as a geometric problem. Matter on which a gravitational force is exerted moves in space-time along a geodesic , that is, as it is meant in the naive sense of "straight ahead". The straight line as a model for the straight-ahead movement of the free body only exists in uncurved (i.e. gravitation-free) spaces.

Physically, the movement along a geodesic corresponds to free fall . Much of the force of gravity is attributed to the fact that the earth is accelerated upwards relative to a freely falling object due to the mutual repulsion of the atoms that make up the earth. This roughly corresponds to the experience when braking an elevator going down. Apart from tidal forces , a person on the ground feels almost the same force as if he were standing in a rocket that is accelerated at a steady rate. These tidal forces, which prevail in every gravitational field, show up as deformation forces in an extended object. In the gravitational field of a spherical body (such as the earth), the tidal forces pull the object out in the direction of fall and push it together perpendicular to the direction of fall. Tidal forces follow directly from the curvature of space-time and are particularly strong with very massive objects like a black hole .

Force in Quantum Mechanics

→ Main article : quantum mechanics  and exchange interaction

When the smallest particles interact, experiments provide results that contradict classical mechanics . In particular, certain quantities are quantized, that is, they only appear in certain portions - the so-called " quanta ". While the force itself is not quantized, forces can bring about a quantization of the possible particle energies.

In quantum mechanics, forces are usually not considered explicitly. The phenomena caused by forces are - analogous to classical mechanics - described by the potential .

There are quantum mechanical effects that make themselves felt like a force, but cannot be traced back to one of the basic forces. Examples are the Pauli principle and the exchange interaction .

Force in the quantum field theories

→ Main articles : quantum field theory  and gauge boson

From 1927, attempts were made to apply “quantization” not only to the original objects of quantum mechanics, the particles, but also to fields (e.g. the electric field), from which the quantum field theories arose; one also speaks of the "second quantization". The quantization of fields is also used in solid state physics and in other many-body theories.

In quantum field theory, all forces are traced back to the exchange of virtual bosons ; these particles interacting with each basic force are, so to speak, individual "force particles" or "force carriers".

Concrete quantum field theories are quantum electrodynamics (this describes electrons , positrons and the electromagnetic field ) and quantum chromodynamics (this describes the strong nuclear force , including the internal structure of protons and neutrons ). In addition, the weak nuclear force was merged with quantum electrodynamics to form the theory of the electroweak interaction . The electroweak model, together with quantum chromodynamics, forms the so-called standard model of elementary particle physics . It contains all known particles and can explain most of the known processes. In the standard model, gauge bosons act as force particles to mediate interactions, but the gravitational force is not included. Here, too, such interaction particles are assumed, called gravitons .

Unification of the basic forces

→ Main article : Basic forces of physics

Coupling constants α as a function of energy E ( s trong, sch w surface, e lektro m agnetic interaction, G ravitation)

In today's physics, a distinction is usually made between four basic forces or interactions. Sorted according to increasing relative strength - the coupling constant is usually used as a measure  - these are:

• Gravitational force
• Weak nuclear power
• Electromagnetic force
• Strong nuclear power

One of the goals of physics is to describe all basic forces or interactions in a " great unified theory " in a unified overall concept, as shown in the table. It is assumed that these basic forces were a single force at the time of the Big Bang , which was split into the individual forces as a result of the cooling.

Fundamental interactions and their descriptions
	Strong interaction	Electromagnetic interaction	Weak interaction	Gravity
classic		Electrostatics & magnetostatics , electrodynamics		Newton's law of gravitation , general relativity
quantum theory	Quantum ( standard model )	Quantum electrodynamics	Fermi theory	Quantum gravity  ?
		Electroweak Interaction ( Standard Model )
	Big Unified Theory  ?
	World formula ("theory of everything")?
Theories at an early stage of development are grayed out. The four basic forces are highlighted in green.

There have already been successes on this path, initially with the combination of the electrical interaction and the magnetic interaction to form the electromagnetic interaction through the electrodynamics of James Clerk Maxwell . The interactions between electric and magnetic fields can also be explained relativistically in other ways .

It has also already been possible to describe the electromagnetic interaction and the weak interaction in the quantum field theory of the electroweak interaction in a standardized way.

Static force and dynamic force

In technical mechanics a distinction is made between static and dynamic (or harmonic) forces. Static forces are those that act unchanged over time, dynamic forces are forces that change over time. In the case of very slow changes over time, one speaks of quasi-static forces.

In biomechanics and sports medicine , the static force is the force that a muscle or muscle group can exert arbitrarily against a fixed resistance. The dynamic force "is the arbitrarily exerted movement of a mass within a programmed process", an example is the rapid force .

literature

• Wolfgang Nolting: Classic mechanics. In: Basic Course Theoretical Physics. Vol. 1, 8th edition. Springer, Berlin 2008, ISBN 978-3-540-34832-0 .
• Richard P. Feynman: Feynman lectures on physics. Mechanics, radiation, heat. 5th, improved edition, definitive edition. Oldenbourg, Munich / Vienna 2007, ISBN 978-3-486-58444-8 (= The Feynman Lectures on Physics. Volume 1).
• Paul A. Tipler: Physics. 3rd corrected reprint of the 1st edition. 1994, Spektrum Akademischer Verlag, Heidelberg / Berlin 2000, ISBN 3-86025-122-8 .
• Ludwig Bergmann, Clemens Schaefer: Mechanics - Acoustics - Warmth. In: Textbook of Experimental Physics. Vol. 1, 12th edition. Walter de Gruyter, Berlin 2008, ISBN 978-3-11-019311-4 .
• Max Jammer : Concepts of Force: A Study in the Foundations of Dynamics. Cambridge (Mass): Harvard UP, 1957 New York: Harper, 1962 New York: Dover, 1999. ISBN 0-486-40689-X .

Web links

Wikiquote: Kraft  - Quotes

Wiktionary: Kraft  - explanations of meanings, word origins, synonyms, translations

• Force measurement using Hooke's law. LEIFI , at student level.
• Flash animation for the addition of forces. dwu teaching materials, at student level.
• Cornelis Harm Glimmerveen: The force of dialectics: on the logical and ontological structures concerning the concepts of force in Leibniz, Kant, and Hegel. Diss. Groningen 1992 (on the concept of force in Leibniz, Kant and Hegel).

References and footnotes

1. ↑ Ludwig Bergmann, Clemens Schaefer and others: Textbook of Experimental Physics, Vol. 1. Mechanics, Relativity, Warmth . 11th edition. de Gruyter, 1998, ISBN 978-3-11-012870-3 . , Section 4.1 Mass and Force.
2. ^ Richard P. Feynman, Robert B. Leighton, Matthew Sands: Mechanics . Walter de Gruyter GmbH & Co KG, Berlin / Boston 2015, ISBN 978-3-11-044460-5 , p. 133 ( limited preview in Google Book search).
3. ^ Günther Drosdowski, Paul Grebe: The dictionary of origin. The etymology of the German language. Vol. 7 . Dudenverlag, Mannheim 1963, ISBN 978-3-411-00907-7 , p. 364 .
4. ↑ ^{a b c} Wolfgang Pfeifer (head): Etymological dictionary of German. Unabridged, revised edition. Deutscher Taschenbuch Verlag, Munich 1995. ISBN 3-05-000626-9 ; 7th edition 2004, ISBN 3-423-32511-9 . A digital version of this dictionary is available in the lexical information system: dwds.de.
5. ^ Philosophiae Naturalis Principia Mathematica . 1687, German edition Mathematical Principles of Nature. Translated and explained by Jacob Philip Wolfers, Oppenheim, Berlin 1872. (Unchanged reprint Minerva, 1992, ISBN 3-8102-0939-2 ).
6. ^ Moritz Rühlmann: Lectures on the history of technical mechanics and theoretical machine theory and the related mathematical sciences . Baumgärtner, Leipzig 1885. Reprint: Documenta technica, series 1, presentations on the history of technology, Verlag Olms, Hildesheim 1979.
7. ↑ Hans Peter Sang: History of Physics . Klett, Stuttgart 1999, ISBN 3-12-770230-2 , p. 7 .
8. ↑ Károly Simonyi: cultural history of physics . Harri Deutsch, Thun, Frankfurt a. M. 1995, ISBN 3-8171-1379-X , pp. 77 .
9. ^ Richard S. Westfall: Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century . American Elsevier, New York 1971.
10. ↑ ^{a b} Károly Simonyi: cultural history of physics . Harri Deutsch, Thun, Frankfurt a. M. 1995, ISBN 3-8171-1379-X .
11. ^ Friedrich Albert Carl Gren: Outline of the theory of nature . 3. Edition. Hemmerde & Schwetschke, Halle 1797. p. 1 ff: “§ 1: Nature is the epitome of the forces of a thing. § 2: Force we call any cause of the change in the state of a thing or things. § 3: (...) Power is what brings about or inhibits movement. "
12. ^ Gunter Lind: Physics in the textbook 1700-1850 . Springer, Berlin 1992, ISBN 3-540-55138-7 .
13. ↑ Erhard Scheibe: The Philosophy of Physicists . 2nd Edition. CH Beck, Munich 2012, p. 22nd ff .
14. ↑ Hegel complains ( Encyclopedia of Philosophical Sciences in the Outline. § 270): "... the inundation of physical mechanics with unspeakable metaphysics, which - contrary to experience and concept - has those mathematical determinations as its source alone." See also: Enzyklopädie der philosophischen Sciences in outline. § 137.
15. ↑ Andreas Kleinert: Enlightenment through physics . In: Walter Schmitz / Carsten cell (ed.): Innovation and Transfer . Eckard Richter, Dresden 2004, ISBN 3-933592-37-2 , p. 11-20 . If remote effects are possible, according to Euler, one must fear that the herbs that grow on Saturn will cause digestive problems even without having eaten them.
16. ↑ In this context, at times not the mass , but the force was used as the basic quantity and the other quantity was referred to as the "derived quantity": At that time, the unit of force "1 kilopond" was used as the basic quantity instead of the previously and subsequently usual mass unit "1 kilogram" by prescribing measurement procedures for calibration by law for the corresponding weight forces.
17. ↑ Christian Meier: Interface Physics. Smallest towing service in the world. On: Wissenschaft-online.de. February 21, 2008, accessed March 2, 2015.
18. ↑ S. Knünz, M. Herrmann, V. Batteiger, G. Saathoff, TW Hänsch, K. Vahala, Th. Udem: Injection locking of a trapped-ion phonon laser. In: Physical Review Letters. 105, 2010, 013004.
19. ↑ International Bureau of Weights and Measures (Ed.): The international system of units . US Dept. of Commerce, National Bureau of Standards, 1977, ISBN 0-7456-4974-2 , pp. 17 (English, limited preview in Google Book Search).
20. ↑ H. Schrecker: The way to the physical concept of force from Aristotle to Newton. In: Natural sciences in the classroom physics / chemistry. 36, No. 34, 1988. Abridged version. ( Memento from January 20, 2012 in the Internet Archive ).
21. ↑ ^{a b c} H. Egerer: Engineering Mechanics . Textbook of technical mechanics in mainly graphic treatment. tape 1 . Springer, Berlin, Heidelberg 1919, ISBN 978-3-662-32061-7 , pp. 124 ( google.de [accessed on January 3, 2017]).
22. ^ Carl Hoefer: Causal Determinism. Article in the Stanford Encyclopedia of Philosophy .
23. ↑ Hans J. Paus: Physics in experiments and examples. P. 33, limited preview in Google Book search.
24. ↑ Dietmar Gross, Werner Hauger, Jarg Schrader, Wolfgang A. Wall: Technical Mechanics: Volume 3: Kinetics . 10th edition. Gabler Wissenschaftsverlage, 2008, p. 191 ( limited preview in Google Book Search - Write "We now F-ma = 0 and grasp the negative product of the mass m and the acceleration a formal than a force to which we [...] D'Alembert inertial force F _T call : F _T = −ma. This force is not a force in the Newtonian sense, since there is no counterforce to it (it violates the axiom actio = reactio!); We therefore call it an apparent force. ").
25. ↑ Rolf Isermann: Mechatronic Systems: Basics . 2nd Edition. Gabler Wissenschaftsverlage, 2004, ISBN 3-540-32336-8 , pp. 124 ( limited preview in Google Book search).
26. ↑ Bruno Assmann, Peter Selke: Technical Mechanics Volume 3: Kinematics and Kinetics . 15th edition. Oldenbourg Verlag, 2010, ISBN 3-486-59751-5 , p. 246 ( limited preview in the Google book search - "Newton was the first to try to build up physics systematically. At the beginning of his (...) main work (...) he puts four definitions: (...) Definition 4: An acting force is against it striving exercised by a body to change its state of motion, either that of rest or that of uniform, linear motion. ").
27. ↑ Bruno Assmann, Peter Selke: Technical Mechanics Volume 3: Kinematics and Kinetics . 15th edition. Oldenbourg Verlag, 2010, ISBN 3-486-59751-5 , p. 246 ( limited preview in Google Book search).
28. ↑ Norbert Dragon: Geometry of the theory of relativity. ( Memento from April 19, 2009 in the Internet Archive ) Lecture notes (PDF; 2.4 MB), accessed on July 15, 2014.
29. ↑ ^{a b} Jürgen Grabe, Klaus-Peter Mahutka: Finite element analysis for vibratory driving of piles . In: Structural Engineering . tape 82 , no. 9 . Wiley Online Library, 2005, pp. 632-640 , doi : 10.1002 / bate.200590192 .
30. ↑ Markus Waltering, Danièle Waldmann, Stefan Maas, Arno Zürbes: Investigation of non-linear vibration properties for non-destructive condition testing using the example of reinforced concrete beams . In: Concrete and reinforced concrete construction . tape 102 , no. 9 . Wiley Online Library, 2007, pp. 615–621 , doi : 10.1002 / best.200700572 .
31. ↑ Wildor Hollmann, Heiko K. Strüder: Sports medicine. Basics of physical activity, training and preventive medicine . 5th edition. Schattauer Verlag, 2009, ISBN 978-3-7945-2546-1 ( google.at ).
32. ↑ ^{a b} Freerk T. Baumann: 6 strength training with cancer patients . In: Exercise Therapy and Sports in Cancer. Practical guide; with 22 tables . Deutscher Ärzteverlag, 2008, ISBN 978-3-7691-0564-3 , p. 57 f . (274 p., Limited preview in Google Book search).

This version was added to the list of articles worth reading on August 11, 2009 .