# Inertia

In classical mechanics , inertia means  ...

• ... the force on a body , which is assumed in addition to noticeable external forces in order to interpret its dynamics when its movement is described in the context of an accelerated reference system (e.g. relative to the braking car, the rotating turntable on the playground or the surface of the earth ). The inertial force defined in this way occurs in every accelerated frame of reference, even in the absence of external forces. Their strength and direction in a certain place are not fixed quantities, but depend on the choice of the accelerated reference system. In an inertial system , this inertial force does not occur at all. That is why it is often referred to as pseudo power .
• ... the resistance that every body opposes to an actual acceleration of its movement . The accelerated body develops this inertial resistance “from within”, simply because it has mass . It can be expressed by a force, namely by d'Alembert's inertial force , which is opposite and equal to the sum of all external forces. In terms of magnitude and direction, it is equal to the apparent force (according to the definition in the previous point) if the accelerated reference system was selected for the description of the movement, which moves with the accelerated body.

Both the inertial force in the accelerated frame of reference and d'Alembert's inertial force are proportional to the mass of the body. That is why the inertial forces are also called mass forces .

Inertial forces do not meet the principle of actio and reaction , because there is no second body from which they proceed. Well-known manifestations include the inertial force when starting and braking, the centrifugal force and the Coriolis force . In classical mechanics, gravity is one of the noticeable external forces. However, since gravitation is also a mass force according to the equivalence principle and a constant straight-line acceleration cannot be distinguished from the action of a homogeneous gravitational field, it is possible to understand gravitation as an inertial force dependent on the reference system. This is the starting point of general relativity .

Inertial forces are useful quantities in theoretical and technical mechanics for setting up and solving the equations of motion of mechanical systems.

## overview

One of the bases for explaining inertial forces is the principle of inertia , which applies to movements that are described relative to an inertial system. According to this, the movement of a body takes place in a straight line and uniform if no external force acts on it. This includes that a resting body remains at rest without the action of an external force, because rest is to be viewed as movement with zero speed . But when an external force acts, the body no longer moves in a straight line, such a change in the state of motion is called acceleration. An accelerated movement is not only the deceleration or acceleration of a linear movement, but also any movement on a curved path, e.g. B. also when the object is moving at a constant speed, i. H. constant speed, moved on a circular path.

When a body is accelerated by an external force, it opposes this force with inertial resistance. The negative product of mass and acceleration of the body is called d'Alembert's inertial force (after Jean-Baptiste le Rond d'Alembert ), in technical mechanics simply called inertial force , even without the addition . According to Newton's second law (or the basic equation of mechanics ) it is exactly the negative of the external force, so that the sum of both is zero. Together with the external forces, this d'Alembert inertial force therefore forms a dynamic equilibrium . The d'Alembert inertial force is also known as inertial resistance or - because it is caused by the mass of the body and is locally proportional to the density - as inertial force .

Another approach to inertial force results if the movement of a force-free body is not described in relation to an inertial system, but from the point of view of an accelerated reference system . Precisely because this force-free body rests in an inertial system or moves in a straight and uniform manner, it appears in the accelerated reference system in an accelerated movement. If one inferred from this - without considering the acceleration of the reference system - that a force is acting, the result is the inertial force in the accelerated reference system. With their participation, one can explain the acceleration observed in the accelerated reference system according to Newton's second law, without considering the accelerated movement of the reference system itself. The inertial force in the accelerated reference system does not exist, so to speak, "real" like the external forces, which are independent of the movement of the reference system in terms of strength and direction (except in the theory of relativity , where the "real" forces in different reference systems are different), but only for the purpose of describing the motion using Newton's second law in the context of the accelerated frame of reference. It is therefore also referred to as " pseudo force" , "pseudo force" or "fictitious force". In calculations of movements relative to an accelerated reference system, it is treated like another external force, and its effects in this reference system are just as real as those of the “real” external forces.

One often notices the inertial force when one is accelerated against the solid ground. The solid earth's surface forms an inertial system - not exactly, but at least approximately. Often, however, one intuitively chooses one's own body and possibly its immediate surroundings as the reference system for one's observation of rest, movement and acceleration and thus interprets the movement from an accelerated reference system. Examples are the perceived inertia of one's own body when starting or braking the tram or the elevator, the centrifugal force when cornering z. B. in the car, Ferris wheel or chain carousel. The Coriolis force is less intuitively understandable. B. forms large-scale air currents into high and low pressure eddies due to the rotation of the earth's surface. If, however, the relevant movement of the body is viewed from an inertial system, the effects attributed to the inertial force turn out to be without exception a consequence of the inertia principle in connection with external forces emanating from other bodies.

## D'Alembert inertial force

### definition

The concept of d'Alembert's inertial force is based on an inertial system. In classical mechanics, the absolute acceleration shown in this is linked to the totality of external forces by Newton's second law : ${\ displaystyle {\ vec {a}}}$${\ displaystyle {\ vec {F}}}$

${\ displaystyle {\ vec {F}} = m \, {\ vec {a}}}$

or

${\ displaystyle {\ vec {F}} - m {\ vec {a}} = {\ vec {0}}.}$

If it is formally understood as a force , one receives with ${\ displaystyle -m {\ vec {a}}}$${\ displaystyle F _ {\ text {T}}}$

${\ displaystyle {\ vec {F}} + {\ vec {F}} _ {T} = {\ vec {0}}}$

an equation that describes the balance of forces in statics and is known as dynamic balance . The difference is that it is not due to an interaction with another body, but rather an apparent force. is called d'Alembert's inertial force, in technical mechanics it is mostly just inertial force. ${\ displaystyle F _ {\ text {T}}}$${\ displaystyle F _ {\ text {T}}}$

The d'Alembert inertia force is the mathematical specification of the "vis inertiae ", which was introduced by Newton and exists as long as the speed of a body is changed in direction and / or amount by an external force. With this, Newton overcame the older meaning of the vis inertiae , which had existed since ancient times to ascribe the property of inertia to all matter (in order to distinguish it from spirit). This should be expressed by the fact that a body, through its inertia, opposes any movement in general and any change in an existing movement. In addition, Newton defined in his axioms the moving force (" vis motrix ") as the cause of every change in the state of motion, and this gradually became the exact meaning of "force" in mechanics after Newton's mechanics were formulated by Euler. D'Alembert gave Newton's vis inertiae the quantitative definition in the form of the inertial force named after him.

### Relationship to inertial force in accelerated frames of reference

The d'Alembert inertial force determined in the inertial system is exactly as great as the inertial force in the accelerated reference system, which is determined for the case where the accelerated reference system is based on the rest system of the body in question. In general, the concrete treatment of a mechanical question always leads to consistent results, regardless of whether the calculation is carried out with or without the use of d'Alembert's inertial force.

Taking into account d'Alembert's inertial force, the balance of forces in a body always results in zero, as in the case of static equilibrium or force-free movement. It must therefore be emphasized that d'Alembert's inertial force is not a force in the sense of Newton's axioms, in which the force is generally defined as the cause of acceleration.

## Inertial forces in the accelerated frame of reference

### Concept formation

The inertial force in the accelerated reference system (in physics often only briefly referred to as inertial force ) is required to describe the dynamics of bodies in an accelerated reference system. It can be analytically broken down into four parts. The basis of the definition according to Leonhard Euler is the principle of inertia (or Newton's First Law ). Accordingly, among the various reference systems there are those in which every body left to its own devices continues to move in a straight and uniform manner at its current speed (including the special case of zero speed). Any deviation from this force-free, linear, uniform movement is referred to as acceleration and is considered to be evidence that an external force is acting on the body. These reference systems have been called inertial systems since 1886 . An “accelerated reference system” is a reference system that is in accelerated motion compared to an inertial system.

Relative to such an accelerated reference system, the rectilinear uniform movement of the body in the inertial system does not appear to be rectilinearly uniform, i.e. accelerated. According to Euler, these, in a certain sense "apparent" accelerations, are also viewed as the result of an "apparently" acting force. This force is called "inertial force" because it does not arise from the action of other bodies like external forces , but owes its existence solely to the inertia of the body in connection with the choice of an accelerated reference system. The size and direction of the inertial force thus developed are determined from the product of the mass of the body and its acceleration, provided it is not caused by the external force.

In simple cases, if the accelerated reference system is suitably selected, the inertial force results in one of the four forms described below: inertial force during acceleration or deceleration, centrifugal force, Coriolis force, Euler force. In most cases, however, the total inertial force is a sum of all four types of inertial forces. The dependence of the inertial forces on the choice of the reference system is also shown in the fact that they do not even occur in an inertial system and that one and the same process is explained by different combinations of the forms of inertial forces mentioned, depending on the choice of the reference system. There is no such thing as a "real" one. H. independent of the choice of a frame of reference for the inertial force, and also not for the four individual manifestations mentioned above.

If one chooses a reference system in which the body rests for a certain process, the inertial force in the accelerated reference system and the d'Alembert inertial force coincide in terms of magnitude and direction. Nevertheless, both terms must not be equated, because their use is linked to opposing conditions: the d'Alembert inertial force presupposes an inertial system, the inertial force in the accelerated reference system a non-inertial system.

If, in the usual case, other forces are also to be taken into account (which can be seen from the fact that the movement of the body is not rectilinear and uniform, also seen from the inertial system), these are vectorially added to the inertial force in order to obtain the total force. With this total force, Newton's 2nd law then also applies to the movements as observed relative to this accelerated reference system.

The formula-based determination of the individual inertial forces is obtained by considering the given movement from the inertial system and combining the coordinates from the movement of the accelerated reference system in relation to the inertial system and the movement of the body in relation to the accelerated reference system ("composite movement"). The equation for the absolute acceleration in the inertial system is changed in order to obtain the relative acceleration. The expressions for the inertial forces are obtained by multiplying by the mass.

### Inertial force when accelerating or decelerating

In the inertial force in the accelerated reference system, a distinction is made between four contributions, which are illustrated individually in the following paragraphs using the example of a passenger in a vehicle. The moving reference system is permanently connected to the vehicle, and the passenger, who is also the observer here, remains (practically) at rest relative to this reference system. (From other reference systems, looking at the same movement would result in a different inertial force, whereby the individual types can also mix.) The inertial system is connected to the earth.

A vehicle will parallel to its speed at the acceleration speeds ( ) or decelerated ( ). ${\ displaystyle v_ {B}}$${\ displaystyle a_ {B}}$${\ displaystyle a_ {B}> 0}$${\ displaystyle a_ {B} <0}$

Observation in the moving reference system: on a body of mass , e.g. B. a passenger, acts the inertial force ${\ displaystyle m}$

${\ displaystyle F_ {T} = - m \, a_ {B}.}$

The inertial force is opposite to the acceleration of the reference system. When “accelerating” it pushes the passenger back against the backrest, when braking it pushes the passenger forward against the belts. ${\ displaystyle F_ {T}}$

Observation in the inertial system: so that the passenger is accelerated synchronously, the force must act on him . When accelerating, its backrest exerts this force (“ thrust ”). When braking, it is slowed down by the force exerted by the belt on it (“negative thrust”).${\ displaystyle F = + m \, a_ {B}}$

Further examples: impact when falling on the ground or in a rear-end collision, becoming lighter / heavier when starting / braking the elevator, upright objects overturning when the surface accelerates sideways (also in the event of an earthquake), shaking and shaking.

### Centrifugal force

A vehicle travels at constant speed through a curve with a radius . ${\ displaystyle v_ {B}}$${\ displaystyle R}$

Observation in the rotating reference system: The inertial force acts on a body of mass that is moving along with it${\ displaystyle m}$

${\ displaystyle F_ {T} = {\ frac {m \, v_ {B} ^ {2}} {R}}.}$

This inertial force is directed radially outward from the center of the curve and is called centrifugal force. It presses the passenger against the side rest on the outside of the curve.

Observation in the inertial system: In order for the passenger to remain calm relative to his seat, he must follow the same circular path as the vehicle. To do this, the force in the direction of the center of the curve must act on it (centripetal force). Otherwise he would continue to move in a straight line. This force is exerted on him by the external side rest.${\ displaystyle F = {\ frac {m \, v_ {B} ^ {2}} {R}}}$

Other examples: spin dryer, seats pushed outwards in the chain carousel, breaking out of a curve when driving a car or bicycle, the feeling of decreasing weight in the ferris wheel above.

### Coriolis force

A child is sitting in a carousel and wants to throw a ball into a basket that is the center of the carousel. It aims exactly in the middle, but when the carousel turns, the ball still flies past the basket. (Child and basket are at the same height; gravity is disregarded when looking at it.)

Observation in the moving reference system: The ball is thrown off radially inwards at speed and flies at constant speed, but does not move in a straight line. Instead, it describes a curve that is curved to one side. Because the force of inertia acts in the horizontal direction transversely to its direction of speed ${\ displaystyle v '}$

${\ displaystyle F_ {T} = 2m \, v '\, \ omega.}$

Therein is the angular velocity of the carousel. ${\ displaystyle \ omega = {\ frac {2 \ pi} {\ mathrm {Duration \ one \ revolution}}}}$

Observation in the inertial system: The flying ball is force-free and makes a straight, uniform movement with the speed that was given to it at the beginning. According to amount and direction, this is made up of the speed that the child gives the ball in the direction that points radially inward at the moment of throwing it, and the speed with which the child (or the observer) himself to this Time moved in tangential direction with the carousel. These two speeds are at right angles to each other. The direction of the combined total speed points past the basket.${\ displaystyle v '}$${\ displaystyle v_ {B}}$

The Coriolis force always occurs in a rotating reference system when a body does not rest in it, but rather moves relative to it, namely not parallel to the axis of rotation. You can feel it like any inertial force on your own body when you have to “counteract” it to compensate, e.g. B. if you want to go in a straight line on the turntable of the children's playground, without being distracted sideways. The simplest examples of the Coriolis force concern such radial movements. In the general case, the relative speed has not only a radial but also a tangential and an axis-parallel component. The axis-parallel component always has no consequences. The radial velocity component (as in the examples above) creates a tangential Coriolis force. The tangential velocity component, which arises when the body moves around the axis differently than it would simply correspond to the rotation of the reference system, causes a radially directed Coriolis force. This is therefore parallel or antiparallel to the centrifugal force, which continues to exist unchanged solely due to the rotation of the reference system. These two radial forces together result in a radial force which corresponds to the centrifugal force associated with an increased or decreased rotational speed. Considered in the stationary reference system, the body actually moves with this changed rotational speed due to its tangential relative speed. (For example, if you stand still on a turntable, you only feel the centrifugal force and you have to compensate for it with an equally large centripetal force. But if you run against the rotational movement at a constant distance from the axis , then the centrifugal force seems to decrease, although the Disk rotates unchanged. The reason is the additional Coriolis force acting radially inwards. If you now run at the speed of rotation of the disk against its direction of rotation, then the rotor always remains in the same place for the stationary observer outside the turntable because of the running, that is, it rests in the inertial system and is free of forces there. In the rotating reference system the Coriolis force is then exactly twice as large as the centrifugal force. In total, the inwardly directed apparent force arises, which is called the "centripetal force" for the circular path observed from the rotating reference system on the Disk is necessary.) In the general case, tangential and radial K result As a component of the Coriolis force, the Coriolis force is always perpendicular to the direction of speed in the rotating reference system (and on the axis of rotation) and therefore deflects the path of an otherwise force-free body into a circle. This is e.g. B. on the cloud images to see high and low pressure areas. ${\ displaystyle {\ vec {v}} '}$

Further examples: rotation of the pendulum plane in Foucault's pendulum , subtropical trade winds and stratospheric jet stream , east deflection of freely falling bodies and bodies on earth moving horizontally away from the equator.

### Euler force

If the angular velocity of a rotating frame of reference varies in magnitude and / or direction, the Euler force occurs (although this name has not become established). A simple example with changing the amount with a fixed direction of the axis of rotation is approaching a carousel. When describing the movement of the passenger in the frame of reference that begins to rotate with the carousel, it is its angular acceleration and, at a distance from the axis, the force of inertia . It is opposite to the tangential acceleration observed here in the inertial system and does not differ in any way from the inertial force when accelerating or decelerating. ${\ displaystyle {\ dot {\ omega}} = {\ frac {d \ omega} {dt}}}$${\ displaystyle r '}$${\ displaystyle F_ {T} = - m \, {\ dot {\ omega}} \, r '}$${\ displaystyle a = {\ dot {\ omega}} \, r '}$

If the axis of rotation can also change its direction, the Euler force is given by the general formula

${\ displaystyle {\ vec {F}} _ {T} = - m \, {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} {\; '}.}$

The vector is the angular acceleration, i.e. the rate of change of the vectorial angular velocity in terms of direction and magnitude . ${\ displaystyle {\ dot {\ vec {\ omega}}} = {\ frac {d {\ vec {\ omega}}} {dt}}}$${\ displaystyle {\ vec {\ omega}}}$

To illustrate this, consider this inertial force using the example of a mass point that is part of a horizontal, rapidly rotating, rotationally symmetrical top while it is (slow) precession around a vertical axis (see).

Observation in the inertial system: If the top is not precessing, the path of the mass point is circular in a fixed, vertical plane. This circular movement is caused by a corresponding centripetal force, which need not be considered further here. During precession, the plane of the orbit rotates around a vertical axis. The path of the mass point has an additional curvature that is opposite at the upper and lower point and is particularly large because the mass point then passes the axis of rotation of the plane of the path. This curvature can only be caused by an additional external force that is parallel or antiparallel to the axis of the gyro. This additional external force on the mass point, which is required for precession, thus varies with each revolution of the top. Since the gyroscope is rotationally symmetrical, the sum of all mass points means that the additional forces together correspond to a torque . In a reference system in which the gyro axis is fixed, but which does not follow the rapid rotation of the gyro, this torque is constant over time. In order for the precession movement of the top to proceed as observed, this external torque must constantly act on the top axis. The vector of the torque is perpendicular to the (horizontal) axis of the gyroscope and to the (vertical) axis of precession. When the top is stationary, the axis would then simply tilt up or down.
In demonstration experiments with a force-free top (as in), the external torque required for precession is achieved by an attached weight, with the inclined toy top by gravity acting on the center of gravity.

Observation in the moving reference system: If the moving reference system is based on the rest system of the mass point, then it is at rest relative to this, although the external additional force just described acts on it. The reason is that it is compensated by an inertial force of the same magnitude in opposite directions, which arises precisely from the special type of accelerated movement of this reference system. This force is the Euler force.

(The reference system is chosen here in such a way that its axis of rotation changes and thus produces the Euler force. The reference system executes both the rapid rotation around the horizontal axis of the gyro and the slow precession of the axis of the gyro about the vertical axis through the suspension point To explain, a more easily imaginable reference system is often chosen, in which the gyro axis rests, but the gyro rotates. This reference system only shows the precession with its constant angular velocity, so it does not cause any Euler force. Relative to this reference system, however, the mass point and move therefore experiences a Coriolis force. At every point of its orbit it corresponds to the Euler force previously determined - in the rest system of the mass point. For example, the Coriolis force is greatest when the relative speed of the mass point is perpendicular to the axis of rotation of the reference system, i.e. the axis of precession. This happens at the upper and lower point of the circular path, with opposite signs of the Coriolis force.)

Further examples: pan grinder . There, the rotation of the millstones increases the pressure on the surface, which, as in the case of precession, can be explained by an Euler force or a Coriolis force, depending on the choice of the accelerated reference system.

### Formulas

#### notation

In order to differentiate between the sizes of an object (location, speed, acceleration) in two reference systems, the normal notation is used for observations in the inertial system and the same letter with an apostrophe ( prime ) is used for the accelerated reference system . The latter is then also referred to as the “deleted reference system”, and all quantities related to it are given the addition “relative-” for linguistic differentiation. The subindex stands for the origin of the deleted reference system. ${\ displaystyle _ {B}}$

meaning
${\ displaystyle m}$ Mass of the body under consideration.
${\ displaystyle {\ vec {r}}}$ Position of the object in S (inertial system).
${\ displaystyle {\ vec {r}} {\; '}}$ Relative position of the object in S '(non-inertial system).
${\ displaystyle {\ vec {v}} = {\ dot {\ vec {r}}}}$ Speed ​​of the object in S
${\ displaystyle {\ vec {v}} {\; '}}$ Relative speed of the object in S '
${\ displaystyle {\ vec {a}} = {\ dot {\ vec {v}}}}$ Acceleration of the object in S
${\ displaystyle {\ vec {a}} {\; '}}$ Relative acceleration of the object in S '
${\ displaystyle {\ vec {r}} _ {B}}$ Position of the origin of S 'in S
${\ displaystyle {\ vec {v}} _ {B} = {\ dot {\ vec {r}}} _ {B}}$ Velocity of the origin of S 'in S
${\ displaystyle {\ vec {a}} _ {B} = {\ dot {\ vec {v}}} _ {B}}$ Accelerating the origin of S 'in S
${\ displaystyle {\ vec {\ omega}}}$ Angular velocity of the system S 'in S
${\ displaystyle {\ vec {\ alpha}} = {\ dot {\ vec {\ omega}}}}$ Angular acceleration of the system S 'in S

#### Translationally moving reference system

If S 'moves purely translationally in the inertial system S, i.e. without any rotation, then all points that are at rest in S' move parallel to one another at the same speed as the origin. A relative movement in the reference system is added. Hence: ${\ displaystyle {\ vec {v}} _ {B}}$

kinematic quantities in S
position ${\ displaystyle {\ vec {r}} = {\ vec {r}} _ {B} + {\ vec {r}} {\; '}}$
speed ${\ displaystyle {\ vec {v}} = {\ frac {d {\ vec {r}}} {dt}} = {\ vec {v}} _ {B} + {\ vec {v}} {\ ; '}}$
acceleration ${\ displaystyle {\ vec {a}} = {\ frac {d {\ vec {v}}} {dt}} = {\ vec {a}} _ {B} + {\ vec {a}} {\ ; '}}$

If the external force is assumed to be known, Newton's equation of motion applies in the inertial system S. ${\ displaystyle {\ vec {F}}}$

${\ displaystyle m {\ vec {a}} = {\ vec {F}}.}$

If the acceleration is used in Newton's equation of motion, we get: ${\ displaystyle {\ vec {a}}}$

${\ displaystyle m \ left ({\ vec {a}} _ {B} + {\ vec {a}} {\; '} \ right) = {\ vec {F}}}$

For the acceleration , which is unknown in the accelerated reference system, we get: ${\ displaystyle {\ vec {a}} {\; '}}$

${\ displaystyle m {\ vec {a}} {\; '} = {\ vec {F}} - m {\ vec {a}} _ {B} = {\ vec {F}} + {\ vec { F}} _ {T}}$

If the inertial force is taken into account in this form, the entire Newtonian mechanics can also be used in the accelerated reference system. ${\ displaystyle {\ vec {F}} _ {T}}$

#### Generally accelerated frame of reference

When deriving a vector that is given in a rotating reference system, the angular velocity and the angular acceleration of the reference system must be taken into account. The kinematic relationships are: ${\ displaystyle {\ vec {\ omega}}}$${\ displaystyle {\ dot {\ vec {\ omega}}}}$

kinematic quantities in S
position ${\ displaystyle {\ vec {r}} = {\ vec {r}} _ {B} + {\ vec {r}} {\; '}}$
speed ${\ displaystyle {\ vec {v}} = {\ frac {d {\ vec {r}}} {dt}} = {\ vec {v}} _ {B} + {\ vec {\ omega}} \ times {\ vec {r}} {\; '} + {\ vec {v}} {\;'}}$
acceleration ${\ displaystyle {\ vec {a}} = {\ frac {d {\ vec {v}}} {dt}} = {\ vec {a}} _ {B} + {\ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} {\; '}) + {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} {\; '} +2 \, {\ vec {\ omega}} \ times {\ vec {v}} {\;'} + {\ vec {a}} {\; '}}$

Inserting the absolute acceleration into Newton's equation of motion results in: ${\ displaystyle {\ vec {a}}}$

${\ displaystyle m {\ vec {a}} _ {B} + m {\ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} {\; '} ) + m {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} {\; '} + 2m \, {\ vec {\ omega}} \ times {\ vec {v}} {\; '} + m {\ vec {a}} {\;'} = {\ vec {F}}}$

Solved for the term with the relative acceleration, it follows:

${\ displaystyle m {\ vec {a}} {\; '} = {\ vec {F}} \ quad -m {\ vec {a}} _ {B} \ quad \ underbrace {-m {\ vec { \ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} {\; '})} _ {{\ vec {F}} _ {\ mathrm {centrifugal}}} \ quad \ underbrace {-m {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} {\; '}} _ {{\ vec {F}} _ {\ mathrm {Euler}} } \ quad \ underbrace {-2m \, {\ vec {\ omega}} \ times {\ vec {v}} {\; '}} _ {{\ vec {F}} _ {\ mathrm {Coriolis}} } = {\ vec {F}} + {\ vec {F}} _ {T}}$

The term is the inertial force that must be taken into account in addition to the force in the accelerated reference system. ${\ displaystyle {\ vec {F}} _ {\ mathrm {T}} = - m {\ vec {a}} _ {B} + {\ vec {F}} _ {\ mathrm {centrifugal}} + { \ vec {F}} _ {\ mathrm {Euler}} + {\ vec {F}} _ {\ mathrm {Coriolis}}}$${\ displaystyle {\ vec {F}}}$

The expression comes from the acceleration of the frame of reference and has no special name. Next is the centrifugal force . The centrifugal force is zero on an axis that goes through the origin of the reference system and points in the direction of the angular velocity. The term is referred to here as Euler force (in as "linear acceleration force"). The term is the Coriolis force . ${\ displaystyle -m {\ vec {a}} _ {B}}$${\ displaystyle {\ vec {F}} _ {\ mathrm {centrifugal}} = - m {\ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} { \; '})}$${\ displaystyle {\ vec {F}} _ {\ mathrm {Euler}} = - m {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} {\; '}}$${\ displaystyle {\ vec {F}} _ {\ mathrm {Coriolis}} = - 2m ({\ vec {\ omega}} \ times {\ vec {v}} {\; '})}$

##### Force-free movement

If the external force does not apply, the unknown relative movement in the accelerated reference system is calculated exclusively through the inertial force . In order to calculate the inertial force, knowledge of an inertial system is required here: ${\ displaystyle {\ vec {F}}}$${\ displaystyle {\ vec {F}} _ {T}}$

${\ displaystyle m {\ vec {a}} {\; '} = {\ vec {F}} _ {T}}$

Use case: How do the sparks move when they detach from the grinding wheel.

##### Predefined movement

If the relative movement is known, e.g. B. by observing planetary orbits in a system fixed to the earth, conclusions can be drawn about the total force.

${\ displaystyle {\ vec {F}} + {\ vec {F}} _ {T} = m {\ vec {a}} {\; '}}$

Can an external force be excluded or neglected, e.g. B. with the sparks that come off the grinding wheel, the inertial force can be calculated, which is required for the given movement. The movement of the reference system itself does not have to be taken into account.

${\ displaystyle {\ vec {F}} _ {T} = m {\ vec {a}} {\; '}}$

## Inertia and Mach's principle

Within the framework of Newtonian mechanics, it is theoretically possible to ascribe properties such as location, speed, acceleration, inertia and thus also inertial force to a single body in the otherwise empty universe. The conceptual basis for this are the assumptions of an absolute space and an absolute time, which have been recognized as untenable by the special theory of relativity and the general theory of relativity . In a principle named after him , Ernst Mach had already called for the laws of mechanics to be formulated in such a way that only the relative movements of the masses distributed in space play a role. But then the inertia and inertial force of a body must also be based on an interaction with other bodies.

## Gravitational force as inertial force

### Concept formation

The first of the inertial forces described above

${\ displaystyle {\ vec {F}} _ {T} = - m \, {\ vec {a}} _ {B}}$

has all the properties of a homogeneous gravitational field

${\ displaystyle {\ vec {F}} _ {S} = m \, {\ vec {g}}}$.

It is proportional to the mass of the body and does not otherwise depend on any other of its properties. In terms of its effects, it cannot therefore be guided by an effective force of gravity

${\ displaystyle {\ vec {g}} = {\ vec {g}} - {\ vec {a}} _ {B}}$

distinguish.

An inertial force of the form can only be clearly identified if the acceleration due to gravity is a quantity that is fixed from the outset in the reference system under consideration, such as is determined by Newton's law of gravitation or by the determination of acceleration due to gravity , which is common in everyday life and in technology . Otherwise, one could say of a reference system that is accelerated with an inertial system in which there is an acceleration due to gravity . In this sense, a reference system accelerated in a straight line is also valid as an inertial system. ${\ displaystyle {\ vec {g}}}$${\ displaystyle g}$${\ displaystyle {\ vec {F}} _ {T} = - m \, {\ vec {a}} _ {B}}$${\ displaystyle {\ vec {a}} _ {B}}$${\ displaystyle {\ vec {g}} '}$

An accelerated reference system can always be defined for a given gravitational field , in which the effective gravitational acceleration just compensates for the gravitational forces, regardless of the movement and the type of body. For this purpose, this reference system only needs to be accelerated, i.e. that is, it must free fall with respect to the system at rest. Within the falling frame of reference, neither gravitational nor inertial forces would be observed, since they cancel each other out exactly. However, due to the inhomogeneity of every real gravitational field, this only applies locally, i.e. H. approximated in a sufficiently small spatial area. ${\ displaystyle {\ vec {g}}}$${\ displaystyle {\ vec {g}} '}$${\ displaystyle {\ vec {a}} _ {B} = {\ vec {g}}}$

This observation can be reinterpreted by defining the freely falling frame of reference as the only valid inertial frame here. Then the previous frame of reference, in which gravity prevails, is no longer an inertial system, because viewed from the new inertial system it moves in the opposite direction to free fall, i.e. accelerated. Inertial forces then occur in this system, which exactly match the gravitational forces previously determined there and can therefore "explain" them completely. An inertial system is then only understood as one in which there is no gravitation. Gravitational force as an independent phenomenon does not exist in this description. It becomes an inertial force that only occurs in reference systems that are not such inertial systems. This statement is synonymous with the equivalence principle , the basis of general relativity .

In the context of the general theory of relativity, however, the principle must be dropped that an inertial system with Euclidean geometry that is valid for the entire universe can be defined. However, inertial systems can still be defined for sufficiently small areas of space and time. The entire space-time is described by a four-dimensional, curved manifold . The general theory of relativity goes beyond Newton's law of gravitation and is the theory of gravitation recognized today.

### example

As an example, let us explain why a passenger in a braking train on a horizontal route has the same experience as when traveling steadily on a sloping route. In the braking car, the sum of the downward gravitational force and the forward inertial force results in a total force that is directed obliquely forward. In order to be able to stand still, the total force must be directed along the body axis from the head to the feet, which is why you either lean backwards or hold on to bring a third force into play, with which the total force is again perpendicular to the floor of the car. The same thing happens when the car is stationary or driving at constant speed but the track is sloping. Then none of the inertial forces from Newton's mechanics act, but the gravitational force no longer pulls at a right angle to the ground, but obliquely forward. If the gravitational force is also understood as the force of inertia, the explanation is the same in both cases.

## literature

• JW Warren: Understanding Force . John Murray, 1979, ISBN 0-7195-3564-6 . German translation: Problems understanding the concept of force. (PDF; 395 kB), p. 15 ff.
• Istvan Szabo: Introduction to Engineering Mechanics . 8th edition. Springer, Berlin 1975, ISBN 3-540-03679-2 .
• Richard Feynman, Robert Leighton, Matthew Sands: The Feynman Lectures on Physics (Volume I Part 1, eng) . Oldenbourg, Munich 1974, ISBN 3-486-33691-6 .
• Jürgen Dankert, Helga Dankert: Technical Mechanics . 6th edition. Vieweg-Teubner, 2011, ISBN 978-3-8348-1375-6 .
• Martin Mayr: Technical mechanics: statics, kinematics - kinetics - vibrations, strength theory . 6th revised edition. Hanser, 2008, ISBN 978-3-446-41690-1 ( limited preview in the Google book search): "According to D'Alembert we understand the expression in the law of motion (8.1) as an auxiliary worker and call it inertia."${\ displaystyle m {\ vec {a}}}$
• Dieter Meschede: Gerthsen Physics . Ed .: Christian Gerthsen, Dieter Meschede. 24th edition. Gabler Wissenschaftsverlage, 2010, ISBN 978-3-642-12893-6 , p. 41–42 ( limited preview in the Google book search): “Forces that arise from describing the process in a certain reference system and which would not be present in another reference system: Inertial forces […] This common but somewhat misleading classification However, the force as an apparent force does not change anything in terms of its real, often catastrophic consequences. "
• Istvan Szabo: History of Mechanical Principles . 3. Edition. Birkhäuser, Basel 1987, ISBN 3-7643-1735-3 .
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## Remarks

1. Euler started from the question of whether the force law of gravity, which has been derived from the planetary observations, could be falsified by the fact that the observer itself accelerated with the earth.

## Individual evidence

1. a b Cornelius Lanczos: The Variational Principles of Mechanics . Courier Dover Publications, New York 1986, ISBN 0-486-65067-7 , pp. 88–110 ( limited preview in the Google book search): “Accordingly, the force of inertia I has to be defined as the negative rate of change of momentum: I = −d / dt (mv)… The definition of the force of inertia requires 'an absolute reference system' in which the acceleration is measured. This is an inherent difficulty of Newtonian mechanics, keenly felt by Newton and his contemporaries. The solution of this difficulty came in recent times through Einstein's great achievement, the Theory of General Relativity. "
2. Dietmar Gross, Werner Hauger, Jarg Schrader, Wolfgang A. Wall: Technische Mechanik: Volume 3: Kinetik , 10th edition, Gabler Wissenschaftsverlage, 2008, p. 191. ( limited preview in the Google book search) p. 191: “ We now write F - ma = 0 and take the negative product of the mass m and the acceleration a formally as a force that we call […] D'Alembert's inertial force F T : 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 a pseudo-force. "
3. Max Jammer: The concept of mass in physics . Scientific Book Society, Darmstadt 1964.
4. ^ Giulio Maltese: On the Relativity of Motion in Leonhard Euler's Science . In: Archives for History of Exact Sciences Springer-Verlag . tape 54 , 2000, pp. 319-348 .
5. a b
6. The term "Einsteinkraft" is occasionally used, but is used completely differently in a different context: Use of the term Einsteinkraft (p. 5). (PDF; 130 kB).
7. Eckhard Rebhan: Theoretical Physics I . Spectrum, Heidelberg / Berlin 1999, ISBN 3-8274-0246-8 . , P. 66.