# inertia

In physics, inertia , also the ability to persist , is the effort of physical bodies to remain in their state of motion as long as no external forces or torques act on them. Such a movement is called an inertial movement . The simplest example of inertial motion is straight ahead motion at constant speed. Another example is the rotation of the earth, which is almost unchanged over very long periods of time. In practice, it is typical for inertial movements that they are slowed down sooner or later by frictional forces ( tidal friction when the earth rotates ).

The measure of the inertia of a body in relation to accelerations of its center of mass is its mass . In relation to the angular acceleration of rotations around the center of mass, it is its rotational mass ( moment of inertia ).

## Significance for important principles of mechanics

Inertia plays an important role in many of the central principles of mechanics. It is the basis of the principle of inertia, which is an axiom of Newtonian mechanics and thus a basis of classical mechanics. The principle of relativity, which is closely related to inertia, is important both in classical mechanics and the basis of the special theory of relativity . The equivalence principle states that inertial mass and heavy mass are equivalent and is one of the central foundations of general relativity . The considerations Ernst Mach to the cause of inertia were of general relativity as a major inspiration in the development.

### Principle of inertia

The principle of inertia states that a uniformly moving body maintains its rectilinear, uniform motion as long as no force is exerted on it. Accordingly, a resting body also remains at rest. A reference system in which the principle of inertia applies is called an inertial system .

Newton formulated the principle in his 1st axiom and specified that the uniform motion is meant with respect to an absolute space .

Originally, this principle was only related to the movement of free bodies and for impact processes, since the concept of a force that could act at a distance did not exist. Galileo Galilei was the first to recognize the principle of inertia at the beginning of the 17th century and already formulated that the force-free movement could continue in a straight line as far as desired. He used this for the first correct treatment of the movements of bodies on earth in free fall, inclined throw and on the inclined plane. René Descartes gave the first clear formulation as a general principle of force-free movements in 1644, but it was only Newton that applied the principle of inertia to the movements of extraterrestrial bodies.

In the theory of relativity , the principle of inertia is expanded in that not only the mass, but every form of energy behaves inertly. In this way, an analogue of the principle of inertia can also be formulated for the energy of an electromagnetic field.

### Equivalence principle

The inertial mass is the measure of the inertia of a body, i.e. a measure of how great a force must be in order to cause a certain acceleration of a body. A large inertial mass ensures, for example, that a car rolling on a flat stretch can only be stopped with great effort. In contrast to this, the heavy mass is responsible for how strong the gravitational force is that a body exerts on other bodies and experiences through them. In the meantime it has been measured with a very high degree of accuracy that these two masses are equivalent, that is, can be regarded as one and the same quantity. The assumption that inertial and heavy mass are exactly the same is called the equivalence principle.

In classical mechanics there is no explanation for the experimentally very well confirmed equivalence principle. It was an authoritative basis for Einstein's formulation of general relativity. This theory is formulated in such a way that the equivalence of heavy and inert mass follows directly from the formulation, so the principle of equivalence is a fundamental part of the theory. According to the general theory of relativity, the mass of a body causes a space-time curvature, and according to the extended principle of inertia, other bodies now move along the geodesics of the curved space-time, so that the passive gravity of these bodies is identical to their inertia.

### Mach's principle

Ernst Mach assumed that inertia and all related properties of a body were caused by the other bodies in the universe.

In 1918, Josef Lense and Hans Thirring derived from Einstein's general theory of relativity, published a few years earlier, that the gravitation of a rotating body pulls other bodies along with it. Einstein saw this lense thirring effect as confirmation of Mach's views and coined the term “Mach's principle” for it. The current interpretation, however, regards the effect as a modification of the gravitational field by the rotation and not as a modification of the inertia.

## Inertial forces

Inertial forces occur in non-inertial systems. One example is the passengers on a chain carousel . They are forced into a circular path in their seats by the chains instead of moving in a straight line, as would be the case if the chain broke. A person on the seat feels like they are being pulled outward by centrifugal force . An observer standing next to the carousel sees that the chains deflect the seat with the person on it from a straight trajectory, that is, the seat exerts a centripetal force on the person in the direction of the axis of rotation of the carousel. Both are correct but different perspectives on the same issue.

In rotating reference systems, the centrifugal force and the Coriolis force appear as inertial forces. The inertial forces in other accelerated frames of reference usually have no special names.

Inertia was understood by Newton as the internal force of the body with which he resisted acceleration. The amount of this force is the product of the acceleration in the inertial system and the mass of the body; its direction is opposite to the acceleration. Since it is defined as the result of an acceleration and not its cause, it is called the force of inertia or, more precisely, d'Alembert's force of inertia .

## history

### Ancient theories on movement

Before the Renaissance in the 15th century, the theory of the movement of Aristotle , which this in the 3rd century BC was in Europe . u. Z. was generally accepted. According to this theory, an object in motion without force will slow down and eventually come to rest, so that sustained force is necessary to keep an object moving. Aristotle explained the movement of a thrown object by means of a force that the surrounding medium exerts on it. Therefore, Aristotle came to the conclusion that such a violent movement in a vacuum was impossible, since there was no medium available to keep the body moving against the resistance of its gravity. A body that is in non-violent motion in a vacuum, on the other hand, must move forever unaffected.

Despite its success and widespread acceptance, Aristotle's teaching of the movement has been repeatedly challenged by philosophers. Lucretius claimed, for example, that the basic state of a body is movement, not rest. In the 6th century AD, Johannes Philoponos took the view that Aristotle's explanation of non-violent movement in a vacuum resulted in a medium slowing down such a body, which contradicts the thesis that the medium is the movement of a body maintain. He therefore suggested that movement was not sustained by the medium, but by some property of the body that was created when it was set in motion. Averroes and many scholastic philosophers opposed this view and supported Aristotle's view. In the Islamic world , however, Philoponus' view found numerous supporters who further developed his theses.

### Impetus theory

In the 14th century Johannes Buridan postulated a property that causes movement, which he called impetus , and assumed that the impetus would not decrease by itself. Instead, he suspected that the air resistance and weight of a body counteracted his impetus. Buridan further postulated that impetus increases with speed; his conception of impetus was thus similar to the modern concept of impulse. He saw his theory, however, only as a modification of Aristotle's philosophy and stuck to other teachings of the Peripatos . He continued to assume that there was a fundamental difference between a body at rest and a body in motion. In addition to an ordinary straight impetus, he also postulated a circular impetus, which causes celestial bodies to move on circular paths.

Buridan's pupil Albert von Rickmersdorf (1316-1390) and a philosophical school in Oxford pursued the impetus theory and carried out some experiments, the results of which were in contradiction to Aristotle's teaching. Nikolaus von Oresme worked on the impetus theory and for the first time presented the laws of motion and relationships of other quantities in graphical form.

Shortly before Galileo put forward his theory of inertia, Giovanni Battista Benedetti modified the impetus theory so that it only included linear motion. He mentions the movement of a stone on a string as an example of a linear movement that is converted into a circular movement by external force. In addition, Benedetti contradicted Aristotle's teaching for the first time that bodies fall the faster the heavier they are, with a thought experiment: If two falling balls are connected with a (massless) rod, nothing changes in the speed of fall, although the mass of the entire body increases .

### European renaissance

The law of the inertia of the mass replaced the idea going back to Aristotle , according to which a constant force is necessary to maintain a movement. The concept of inertia was developed by many physicists during the 17th century.

Thus, in the writings of Galileo Galileo, a first formulation of the related principle of relativity can be found. However, Galileo's drop and pendulum experiments were unsuitable for determining the inertial mass of bodies, as their influence on the measurement result is precisely compensated by that of the heavy mass, because both are proportional to each other. This equivalence, which was only established later, is called the equivalence principle. A contemporary and letter partner of Galileo, Giovanni Battista Baliani , already speculated that the mass independence of the fall times stems from the fact that the mass acts both as an "agens" and a "passum", which corresponds to the later introduced concepts of inert and heavy mass .

One of the first formulations of the principle of inertia can be found in the Principia philosophiae by René Descartes . Christiaan Huygens raised the principle of inertia as well as a clear formulation of the principle of relativity to axioms, which in particular underlie his treatment of collision processes.

The first formulation of the law of inertia in its present form goes back to Isaac Newton , who postulated in his first axiom in 1687 :

"A body remains in its state of rest or uniform, straight-line movement as long as the sum of all forces acting on it is zero."

Newton saw the source of inertia in a triple alliance of absolute mass, time and space and designed the water bucket thought experiment: In absolutely empty space, a bucket of water rotates around its axis of symmetry. The inertial forces now ensure that the water is pushed to the edge of the bucket and a parabolically curved surface is created.

According to Newton's interpretation, absolute space plays a central role here. Without it, one would not be able to determine any circular acceleration (the whole system is at rest when viewed from the bucket ), against which the particles resist with their inertia. The room is assumed to be a real, absolute reference system, regardless of the rotating bucket.

Ernst Mach took a different approach . He suspected that indolence is exercised by all masses on one another. A single particle in an empty universe would therefore have no inertia. This view was described by Albert Einstein among others as the Mach principle .

Einstein's special theory of relativity meant the end of the absoluteness of mass, time and space postulated by Newton. In the space-time continuum of the special theory of relativity, only the space-time distances are absolute. Spatial and temporal distances as well as the inertia depend on the state of motion. In particular, when approaching the speed of light , the inertia increases so quickly that it cannot be exceeded. ${\ displaystyle ds ^ {2} = dx ^ {2} + dy ^ {2} + dz ^ {2} -c ^ {2} dt ^ {2}}$

In Newtonian theory, the equality of inert and heavy mass is a "coincidence" that cannot be further explained, while in general relativity it is postulated as the principle of equivalence . From the equivalence of mass and energy it follows that every form of energy has inertia.