Fictitious force

In classical mechanics the inertia forces are expressed in the form of a force consequences resulting from the inertia of a body by itself. The inertial force at a given time depends on the state of motion of the subject body and the state of motion of the reference frame relative to the movement of the body is considered. In particular, the inertial forces do not like the external forces to an external force field or the action of another body back and not therefore fulfill Newton's third law ( balance of action and reaction ).

Depending on the application to physical problems a distinction is the inertial force in the accelerated reference system and the d' Alembert inertial force. Both play an important role as a means for more complicated equations of motion of mechanical systems in theoretical mechanics and engineering mechanics.

In accelerated reference systems, inertial forces are completely equivalent in their effects to external forces. Among its known forms include the inertial force during acceleration and braking, the centrifugal force and the Coriolis force.

• 3.1 concept formation
• 3.2 inertial force when accelerating or decelerating
• 3.3 centrifugal
• 3.4 Coriolis force
• 3.5 Euler force
• 3.6 formulas 3.6.1 definitions
• 3.6.2 translatory moving reference system S '
• 3.6.3 General accelerated reference system S '

• 5.1 concept formation
• 5.2 Example

Brief explanation of the inertial forces

Basis of the explanation of the forces of inertia is the inertia principle that carried straight - uniform movement of a force-free body, where it is described relative to an inertial system. An external force acting on the body, changing the straight - uniform motion from an accelerated movement. [Note 1] The d' Alembert inertial force then referred to the negative product of mass and acceleration of the body. Thus, it is according to Newton's second law (or the fundamental equation of mechanics) exactly the negative of the external force. The d' Alembert inertial force is referred to as inertial resistance or as a mass force in engineering mechanics and absence simply as inertia force.

But from the perspective of an accelerated reference frame, a body also appears in an accelerated movement when it is without external forces and in an inertial system therefore rests or moves geradlig - uniform. The inertial force in the accelerated reference system refers to the force with which one can explain the observation of these acceleration according to Newton's second law. The inertial force in the accelerated reference system so there is no " real" as the external forces that are independent of the motion of the reference system on the strength and direction [note 2] only for the purpose of description of the movement in the accelerated reference system. It is thus referred to as " apparent force ", " pseudo- force " or " fictitious force ". Its effects, however, are as real as the external forces. One notices the inertial force frequently when one is accelerated relative to the fixed ground, all the while intuitively takes your own body and possibly its immediate surroundings to the reference system of his observation of rest and exercise. The solid surface of the earth is approximately an inertial frame. Examples are the perceived inertia of one's body during starting or braking of the tram or the elevator, the centrifugal force when cornering, for example, in the car, Ferris wheel, carousel chain. Less intuitive to the Coriolis force, which forms, for example, large-scale air flow due to rotation of the earth's surface to the high and low pressure vortices. However, considering the respective motion of the body from one inertial system from, the force of inertia effects attributed to prove without exception as a result of external forces emanating from other bodies.

D' Alembert's inertial force

Definition

When term d'Alembert's inertial force ( in engineering mechanics often only briefly referred to as inertial force ) shall be based on an inertial system. Newton's second law

Links the entire external force of the observed acceleration in the inertial frame. After switching to one summarizes the size and power, and calls them d' Alembert's inertial force

In any inertial frame, the d' Alembert inertial force acting on a given body for a particular operation, the same size. It is

The total force from external force and d' Alembertscher inertial force is therefore always zero. Since this equation is formally not be distinguished from a static equilibrium, it will be also referred to as a dynamic equilibrium.

Dynamic equilibrium / d' Alembert's principle

The fact that external force and d' Alembert's inertial force sum to zero, so cancel each other out, is in engineering mechanics as a dynamic equilibrium or d' Alembert's principle [note 3] respectively. Model is static equilibrium, because it is given by the condition. The concept of mechanical equilibrium between the stable static case can thus be extended to systems with arbitrary accelerations By introducing the d' Alembert inertial force. However, may need to be considered when you have further questions about the meaning or interpretation of this power that she does not go back as all external forces to an interaction with another body.

Application

Great practical benefits unfolds the concept of d' Alembert's inertial force in the treatment of movements that are affected by a known external force, but are also limited to predetermined paths or surfaces. The external force is thus made up of two parts. Firstly, the constraining force which acts perpendicularly to the movement direction and the body in its orbit (or area) holds. The other part is the impressed force which accelerates the body along its orbit (or in the area). The equation of motion is according to Newton's 2nd law, ie:

The strength of the constraining forces and active forces that must act in a place that can be determined from this equation if the acceleration at this location is known, so the exact sequence of movement afterwards. But conversely, would only be the size of all forces known to concretely establish by Newtonian mechanics the equation of motion from which the exact sequence of movements is to be determined. A solution to this dilemma has in many cases the principle of virtual work ( in the physics d' Alembert's principle called ), after which the constraint forces on the moving body do not work because the body does not move in the direction of the constraint forces. Accordingly, the prior knowledge of the constraint forces is unnecessary if you are in a suitably chosen system of independent coordinates (generalized coordinates) performs the bill. These coordinates must describe the motion and take into account their limitations already.

Example

To illustrate this approach on a simple example, whether a particular active force and selected as restriction of movement a prescribed circular path with radius. In Cartesian coordinates with the origin at the center of the circle there is a constraint. When calculated in this coordinate system, the acceleration would then depend on the x and y components of the impressed force, and where the urging force, which prevents the body to leave the circular path. Suitable for this purpose are polar coordinates, wherein the receiving constraint, and the only free variable, the coordinate angle. For an infinitesimal variation is determined the corresponding piece of the path and determines the parallel thereto, both components of the impressed force and the acceleration (here). With the d' Alembert inertial force (here) then applies the equation of dynamic equilibrium, from which the desired solution (here) is determined.

Relationship with inertial force in accelerated frames

The determined in the inertial d' Alembert's inertial force is exactly the same as the inertial force in the accelerated reference system that you identified for the case where you just based on sets as an accelerated reference system the rest system of the body in question. In general, the specific 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.

However, the basic interpretation of the d' Alembert inertial force is as a force in conflict with the interpretation of the concept of force according to Newton. For example, involving the d' Alembert inertial force, the total force on a body is always zero, ie, as in the case of a static equilibrium or force-free motion. Thus, one could not say quite generally the case, force is the cause of acceleration. That is the starting point of Newtonian mechanics. In contrast, the concept of d' Alembert's inertial force the older meaning of inertia is taken correctly. This had existed since ancient times and to Newton's precursor Johannes Kepler is ( contrary to the spirit ) attributable to all matter, the property of inertia, which is to be expressed by the fact that a body by an inertial force ( " force of inertia " ) of each movement at all opposed.

Inertial force in the accelerated reference system

Conceptualization

The inertial force in the accelerated frame of reference ( in physics often only briefly referred to as inertial force ) corresponds to the everyday perception of such forces. Basis of the definition according to Leonhard Euler [note 4], the principle of inertia (or Newton's First Law). Consequently, there is among the different reference systems are those in which each to itself, the body in a straight line - uniformly moved with his current speed ( including the special case zero speed ). Any deviation from these force-free, straight - uniform motion is called acceleration and is regarded as evidence that a force acts on the body. Size and direction of the force is given by Newton's 2nd law by the product of mass and acceleration. These reference systems have been identified since 1886 as inertial systems, in contrast to accelerated reference systems, which are themselves compared to the inertial frame in accelerated motion.

Relative to such an accelerated reference system in the inertial straight - uniform motion of the body does not appear straight - uniformly accelerated. After Euler these are also considered in some sense "apparent " acceleration as a result of "apparently" acting force. This force is called inertia force, because it does not arise as the " external forces " from the influence of other bodies, but owes its existence solely to the inertia of the body in conjunction with the special choice of accelerated reference system. Size and direction of this force of inertia are the same as previously determined by the product of mass of the body and its acceleration observed ( minus the acceleration, which is optionally caused by external forces).

In simple cases, the inertial force results in accelerated reference system in the form of the inertial force when accelerating or braking, the centrifugal force, the Coriolis force or the Euler force. In most cases, all of the inertia force is the sum of these four forces of inertia. The dependence of the inertial forces on the choice of the reference system shown by the fact that they do not occur in an inertial system, and that to obtain various combinations of the above forms of inertial forces for a given process depending on the choice of the reference system.

If you choose for a particular operation, the reference system so that it is with moving the center of gravity of the body, is then obtained for this reference system, the inertial force in the accelerated reference system exactly equal to the d' Alembert inertial force, which would be obtained for the same process when he is viewed from an inertial frame of. Nevertheless, both terms must not be equated, because their use is linked to opposite requirements: the d' Alembert inertial force requires an inertial frame, the inertial force in the accelerated reference system is a non- inertial frame.

Are in the ordinary case to consider other forces (which is evident from the fact that as seen from the inertial frame of the movement of the body not straight - uniformly runs ), they are vectorially added to the inertia force. This total force then applies the second Newton's law for the observations relative to this accelerated reference system.

For the defined herein, the term of the inertia force in the accelerated reference system, which is frequently used in physics, you can also access the following way: Instead of looking at a movement in accelerated reference system from the point of view of an accelerated observer, it can be from one inertial system as a " composite " motion will be described, composed of the movement of the body relative to a moving reference system and the movement of this frame of reference with respect to the inertial system. In the composite motion are then in addition to the relative acceleration, which is to be determined solely from the movement of the body relative to the moving reference system to recognize two additional additive shares of acceleration: a guide acceleration, which shows the body to adapt only mitzubewegen with the moving reference system and the Coriolis acceleration, if the moving frame and the rotating body moves to this reference system. The powers that be determined from these latter two units of acceleration by multiplication with the mass, are of opposite sign equal to the inertial forces, as they are to be set according to the usual physics approach for the relative motion in the accelerated frame of reference.

Inertial force when accelerating or decelerating

In the inertial force in the accelerated reference system there are four posts that are presented in the following paragraphs the example of a fellow passenger in a vehicle clearly individually. The moving frame is respectively fixed to the vehicle, and the passenger, who is also the observer here is relative to this reference system ( virtually ) at rest. ( From other reference systems of each would result from the consideration of the same movement a different inertial force, and the individual species can also be mixed. ) The inertial system is connected to the ground.

A vehicle will accelerate parallel to its velocity with the acceleration () or decelerated ().

Observation in the comoving system: At a body of mass inertia force acts

The inertial force is in the opposite direction of the acceleration of the reference system. At the " accelerate" it pushes the passenger back against the backrest, while braking forward against the straps.

Other examples: the impact when dropped on the floor or the rear-end collision, overturning upright items during acceleration of the support ( even with earthquakes), shaking and shaking.

Centrifugal force

A vehicle is traveling at a constant speed through a curve with a radius.

Observation in the comoving frame of reference: In a co-moving body mass inertia force acts

This inertial force is directed radially outward from the center of curvature and is called centrifugal force. She pushes the passenger against the outer side of the curve in back.

Other examples: spin dryer, rudimentary weightlessness at the highest point in the Ferris wheel, outside crowded seats in the whirligig, breaking out of the curve while driving or cycling.

Coriolis force

A child sits in a carousel and wants to throw a ball in a basket that is at the center of the carousel. It is aimed precisely to the center, but as the carousel rotates, the ball still flies next to the basket past. ( Child and basket are at the same height; gravity was left in the consideration ignored. )

Observation in the comoving reference system: the ball is cast off with speed radially inward and flies at a constant speed, but does not make a straight line movement. Instead, it describes a laterally curved curve. For transverse to its velocity direction acts in the horizontal direction, the inertial force

Is the angular velocity of the carousel.

The Coriolis force is always present in a rotating reference system, where a body is not resting in it, but this moves relative to (except only movements parallel to the axis of rotation). They can feel like any inertial force on your body if you have to compensate by holding ( for example, when going in a straight line on the turntable of the playground inside one wants to be distracted without the side). If, in general case, the relative velocity has a radial, tangential and an axis- parallel component, the axis- parallel component has no consequences. The radial velocity component invokes a tangential Coriolis force produced ( as in the first example above). The tangential component of velocity of the body which is equal to an increase or a reduction of its peripheral speed, causes a radial component of the Coriolis force, that is parallel or antiparallel to the centrifugal force which prevails on the rotating disk. Two forces add up to a radial force of inertia, which corresponds exactly to that of the centrifugal force on the body, which acts on the body due to its current ( increased or decreased ) rotational speed. ( If you stand, for example, on a turntable, you can feel only the centrifugal force. However you walk exactly with the rotational speed of the disk opposite to the rotational motion, then a Coriolis force also acts radially inward and highlights these centrifugal force exactly. For the stationary observer outside the hub remains the rotor indeed because of running in the same place, i.e., its own angular velocity in the inertial system ), tangential and radial component of the Coriolis force is zero. together form that the Coriolis force in the rotating reference system is always perpendicular to the velocity direction ( and is on the axis of rotation ), and thus deflects the trajectory of a free body forces otherwise to a circle. This is for example seen in the clouds images to high - and low-pressure areas.

Other examples are: rotation of the oscillation plane when Foucault pendulum, subtropical trade winds and stratospheric jet stream, Ostablenkung freely falling bodies on earth.

Euler force

If the angular velocity of a rotating frame of reference in terms of magnitude and / or direction varies, the Euler force occurs (this name has not been naturalized ). A simple example of changing the amount at a fixed direction of the rotation axis is the start of a carousel. When describing the movement of the passenger in the reference system, which starts to rotate with the carousel, its angular acceleration and the distance from the axis, the inertial force. She is the tangential acceleration, which can be observed in the opposite direction and differs in nothing from the inertial force during acceleration or deceleration in the inertial system here.

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

Is the vector, the angular acceleration, that is the direction and amount of the rate of change of the vector angular velocity.

To illustrate this, consider the example of inertial force of a mass point, which is part of a horizontal, rapidly rotating, rotationally symmetric top, while it executes a (slow ) precesses around a vertical axis (see).

Observation in the moving frame, if I create a moving reference system the rest frame of the point mass basis, then he is at rest relative to this, although the just described additional external force acts on it. The reason is that it is compensated by an equal and opposite large inertia force of this reference system is currently being built from the special type of accelerated motion. This force is the Euler force.

Other examples: Kollermühle. There, the circulation of the millstones increases the pressure on the pad, which can be explained as in the case of precession, depending on the choice of accelerated reference system by an Euler force or a Coriolis force.

Formulas

Definitions

To distinguish between the sizes of an object ( eg, location, speed ) in two reference systems, the normal notation is used in and for the accelerated reference system, respectively the same letter with an apostrophe (English prime ) for the observations in the inertial frame. The latter is then referred to as " coated reference system ", and all related quantities obtained for linguistic distinction marked " relative ". The subindex stands for the observer, who is at the origin of the primed reference system.

When external forces are known as presupposed in the inertial frame S Newton's second axiom applies

The equation can be solved for the unknown acceleration. In the accelerated frame S ' is the same process before another acceleration. But is scheduled similar to Newton's 2nd law:

Or equivalently

This equation is the definition of the inertial force. Is taken into account in this form with, one can apply the whole Newtonian mechanics even in accelerated reference system. To determine in greater detail the relative acceleration must be expressed by the observations in the inertial system.

Translationally moving reference system S '

Moves S ' in the inertial frame S purely translational, ie without any rotation, then move all the points in S' rest, parallel to each other with the same speed as the origin. Relative motion in the reference frame is added additively. Consequently, the following applies:

Is determined from the last equation, and this used in the definition of the equation yields:

General accelerated reference system S '

Wherein a vector is placed in a rotating reference system derivative, the angular velocity and the angular acceleration of the reference system must be considered. The kinematic relations are:

Substituting the absolute acceleration in the Newtonian equation of motion a, we have:

Solving for the term with the relative acceleration follows:

The term is the inertial force, which occurs in addition to the force in the accelerated frame of reference.

The term stems from the acceleration of the reference system and has no special name. Further, the centrifugal force. The term is here after: 103 called the Euler force ( in a " linear acceleration force "). The term is the Coriolis force.

Inertial force and Mach 's Principle

Within the framework of Newtonian mechanics, it is possible theoretically in an otherwise empty universe ascribed characteristics such as location, speed, acceleration, inertia, and thus inertial force has a single body. The conceptual basis for this are the assumptions of absolute space and an absolute time, but they were recognized by the Special Theory of Relativity and the General Theory of Relativity be untenable. Even before Ernst Mach had demanded in a principle named after him, the laws of mechanics drafted so that only the relative movements of the distributed masses in the universe play a role. But then also inertia and inertia of a body must be based on an interaction with other bodies.

Gravitational force as inertial force

Conceptualization

The gravitational force has characteristics of an inertial force: it is proportional to the mass of a body and otherwise depends on no other properties of the body. In fact, one can not distinguish between gravitational and inertial force principle. A gravitational field always be defined, in which the inertia forces compensate for the force of gravity straight, irrespective of the movement of the body and the nature of an accelerated reference. To do this frame of reference relative to the stationary system to perform only a freefall. Within the falling reference system neither gravity nor inertia forces would be observed, since they cancel each other exactly. However, this is only valid locally because of the inhomogeneity of any gravitational field, that is approached in a sufficiently small region of space.

This observation can be reinterpreted by defining the free-falling reference frame as the inertial valid here. Then the previous reference system in which there is gravity, no inertial frame, because when viewed from the new inertial frame of it moves opposite to the free fall accelerated. In this system, then step on inertial forces, that exactly match the previously established there gravitational forces and therefore they can completely replace. Gravitational force does not exist as a separate phenomenon in this specification. It becomes an inertial force, which occurs only in systems of reference that are not inertial. This statement is equivalent to the equivalence principle, the basis of the general theory of relativity.

In the context of general relativity, however, the principle has to be dropped, that a valid for the whole universe with Euclidean geometry inertial system can be defined. For sufficiently small regions of space and time, however, can continue to define inertial frames. The entire space-time is described by a four-dimensional, curved manifold. General relativity goes beyond the Newtonian law of gravitation and is today accepted theory of gravitation.

Example

As an example, explains why a passenger in a decelerating train on a horizontal line has the same experience as with uniform driving on a slope. In the car braking the sum of the downward force of gravity and the forward inertial force resulting in a total force, which is directed obliquely forwards. To stand quietly, the total force but must be directed along the axis of the body from the head to the feet, which is why you tend to either backwards or must bring by holding a third force in the game, with the total force is perpendicular back to the car floor. The same is seen when the car is stationary or moving at a constant speed, but the track is steep. Then no effect of the inertial forces from Newtonian mechanics, but the force of gravity no longer pulls at right angles to the ground, but at an angle to the front. Summing up the gravitational force as inertial force on the explanation in both cases is the same.