What are the forces of inertia. What will we do with the received material?

In order for Newton's second law to be fulfilled in non-inertial frames of reference, in addition to the forces that act on bodies, inertial forces are introduced.

Definition and formula of the force of inertia

DEFINITION

By the force of inertia is called a force that is introduced only because the coordinate system in which the motion of bodies is considered is non-inertial.

The emergence of inertial forces is not associated with the action of any bodies. Recall that non-inertial reference frames are any frame moving with acceleration relative to inertial frames.

Newton's third law for inertial forces is not fulfilled.

Let the acceleration of the body relative to the inertial reference frame be . Usually such acceleration is called absolute, while the acceleration of the body relative to a non-inertial frame of reference is called relative (). Newton's second law for the inertial frame of reference can be written as:

where is the resultant force applied to a body of mass m. In a non-inertial frame of reference:

because the:

Let's add inertial forces to the right side of expression (2) so that Newton's second law is fulfilled in a non-inertial frame of reference:

In this case, we get that the force of inertia is equal to:

Formula (5) for the force of inertia gives a correct description of motion in a non-inertial frame of reference. In this case, finding the difference between the relative and absolute accelerations is a kinematic problem. It can be solved if the nature of the motion of the non-inertial reference frame relative to the inertial one is known.

Frames of reference moving in a straight line with constant acceleration

A reference frame that moves in a straight line with constant acceleration is simplest case non-inertial system. Let us consider a non-inertial frame of reference, which moves in a straight line with constant acceleration (transfer acceleration) relative to the inertial frame of reference. Then:

According to formula (5), the inertia force is equal to:

Rotating reference frame

Consider a frame of reference rotating about a fixed axis with constant speed. For a body at rest in such a frame of reference, the formula for the force of inertia can be written as:

where is the radius vector, equal in magnitude to the distance from the axis of rotation to the body under consideration, directed from the center to the body. The force of inertia (8) is called centrifugal force inertia.

All bodies on the surface of the Earth experience the action of the centrifugal force of inertia.

Note that any problem can be solved in an inertial frame of reference. The use of non-inertial systems is dictated by considerations of the convenience of using non-inertial systems.

Examples of solving problems on the topic "Force of inertia"

EXAMPLE 1

Exercise	What is the strength normal pressure body (weight) to the surface of the Earth, if the body is motionless, has a mass m. Located at latitude. Consider the radius of the Earth equal to R.
Solution	Let's make a drawing. Let's connect the reference system with the Earth. Forces act on the load in this reference system: gravity (); support reaction force (); static friction force (). In addition to these forces, since in our case we will not consider the frame of reference associated with the Earth as inertial, the centrifugal force of inertia () acts. We take the formula for calculating the force of inertia: where is the radius of the trajectory (circle) along which the load moves. We choose the coordinate system so that its origin coincides with the center of the body, the Y axis will be perpendicular to the Earth's surface, the X axis is tangent to the Earth's surface (see Fig. 1). Since the body does not move relative to the Earth, we write Newton's second law as: In projections on the X and Y axes, expressions (1.2), taking into account (1.1), we have: Since the weight of the body (P) is equal to (N) in magnitude, we express it from the first equation of the system (1.3), we get:
Answer

Inertial and non-inertial frames of reference

Newton's laws are valid only in inertial frames of reference. With respect to all inertial frames, the given body moves with the same acceleration $w$. Any non-inertial frame of reference moves relative to inertial frames with some acceleration, so the acceleration of the body in the non-inertial frame $w"$ will be different from $w$. Let's denote the difference in the accelerations of the body and the inertial and non-inertial frames by the symbol $a$:

For a progressively moving non-inertial frame $a$ is the same for all points of space $a=const$ and represents the acceleration of the non-inertial reference frame.

For a rotating non-inertial frame, $a$ will be different at different points in space ($a=a(r")$, where $r"$ is the radius vector that determines the position of the point relative to the non-inertial frame of reference).

Let the resultant of all forces due to the action on the given body by other bodies be equal to $F$. Then, according to Newton's second law, the acceleration of the body relative to any inertial frame of reference is:

The acceleration of the body relative to some non-inertial system can be represented as:

Hence it follows that even at $F=0$ the body will move with respect to the non-inertial frame of reference with acceleration $-a$, i.e., as if a force equal to $-ma$ acts on it.

This means that when describing motion in non-inertial frames of reference, Newton's equations can be used if, along with the forces due to the action of bodies on each other, the so-called inertial forces $F_(in) $ are taken into account, which should be assumed to be equal to the product of the body's mass and taken with opposite sign, the difference of its accelerations with respect to the inertial and non-inertial frames of reference:

Accordingly, the equation of Newton's second law in a non-inertial frame of reference will have the form:

Let us explain our assertion with the following example. Consider a trolley with a bracket mounted on it, to which a ball is suspended on a thread.

Picture 1.

As long as the cart is at rest or moving without acceleration, the thread is vertical and the force of gravity $P$ is balanced by the reaction of the thread $F_(r)$. Now let's bring the cart into translational motion and acceleration $a$. The thread will deviate from the vertical by such an angle that the resultant forces $P$ and $F_(r)$ give the ball an acceleration equal to $a$. Relative to the frame of reference associated with the cart, the ball is at rest, despite the fact that the resultant forces $P$ and $F_(r)$ are nonzero. The lack of acceleration of the ball with respect to this frame of reference can be formally explained by the fact that, in addition to the forces $P$ and $F_(r) $, equal in total to $ma$, the ball is also affected by the inertial force $F_(in) = -ma$.

Forces of inertia and their properties

The introduction of inertial forces makes it possible to describe the motion of bodies in any (both inertial and non-inertial) frames of reference using the same equations of motion.

Remark 1

It should be clearly understood that the forces of inertia cannot be put on a par with forces such as elastic, gravitational forces and friction forces, i.e., forces due to the impact on the body from other bodies. The forces of inertia are due to the properties of the reference frame in which mechanical phenomena are considered. In this sense, they can be called fictitious forces.

Introduction to the consideration of inertial forces is not fundamentally necessary. In principle, any movement can always be considered in relation to an inertial frame of reference. However, in practice, it is often the motion of bodies with respect to non-inertial frames of reference, for example, with respect to the earth's surface, that is of interest.

The use of inertial forces makes it possible to solve the corresponding problem directly with respect to such a frame of reference, which often turns out to be much simpler than considering motion in an inertial frame.

A characteristic property of the forces of inertia is their proportionality to the mass of the body. Due to this property, the forces of inertia are similar to the forces of gravity. Imagine that we are in a closed cabin remote from all external bodies, which moves with an acceleration g in the direction that we will call "top".

Figure 2.

Then all the bodies inside the cabin will behave as if they were acted upon by the inertial force $F_(in) =-ma$. In particular, a spring, to the end of which a body of mass $m$ is suspended, will stretch so that elastic force balanced the force of inertia $-mg$. However, the same phenomena would also be observed if the cabin were stationary and located near the surface of the Earth. Not being able to "look" outside the cabin, no experiments carried out inside the cabin, we could not establish what caused the $-mg$ force - the accelerated movement of the cabin or the action of the Earth's gravitational field. On this basis, one speaks of the equivalence of the forces of inertia and gravity. This equivalence underlies Einstein's general theory of relativity.

Example 1

A body freely falls from a height of $200$ m to the Earth. Determine the deviation of the body to the east under the influence of the Coriolis force of inertia caused by the rotation of the Earth. The latitude of the impact site is $60^\circ$.

Given: $h=200$m, $\varphi =60$?.

Find: $l-$?

Solution: In the earth's frame of reference, a freely falling body is affected by the Coriolis force of inertia:

\, \]

where $\omega =\frac(2\pi )(T) =7.29\cdot 10^(-6) $rad/s is the angular velocity of the Earth's rotation, and $v_(r) $ is the velocity of the body relative to Earth.

The Coriolis force of inertia is many times less than the gravitational force of the body to the Earth. Therefore, in the first approximation, when determining $F_(k) $, we can assume that the velocity $v_(r) $ is directed along the radius of the Earth and is numerically equal to:

where $t$$ $ is the duration of the fall.

Figure 3

The figure shows the direction of the force, then:

Since $a_(k) =\frac(dv)(dt) =\frac(d^(2) l)(dt^(2) ) $,

where $v$ - numerical value component of the body's velocity, tangent to the Earth's surface, $l$ is the displacement of a freely falling body to the east, then:

$v=\omega gt^(2) \cos \varphi +C_(1) $ and $l=\frac(1)(3) \omega gt^(3) \cos \varphi +C_(1) t+ C_(2)$.

At the beginning of the fall of the body $t=0,v=0,l=0$, so the integration constants are equal to zero and then we have:

Duration of free fall of a body from height $h$:

so the desired deviation of the body to the east:

$l=\frac(2)(3) \omega h\sqrt(\frac(2h)(g) ) \cos \varphi =0.3\cdot 10^(-2) $m.

Answer: $l=0.3\cdot 10^(-2) $m.

Newton's laws are valid only in inertial frames of reference. With respect to all inertial frames, the given body moves with the same acceleration w. Any non-inertial frame of reference moves relative to inertial frames with some acceleration, so the acceleration of a body in a non-inertial frame of reference will be different from

For a progressively moving non-inertial frame, a is the same for all points in space and represents the acceleration of a non-inertial frame of reference. For a rotating non-inertial frame, a will be different at different points in space, where is the radius vector that determines the position of the point relative to the non-inertial frame of reference).

Let the resultant of all forces due to the action on a given body from other bodies be equal to F. Then, according to Newton's second law, the acceleration of the body relative to any inertial frame of reference is equal to

The acceleration of the body relative to some non-inertial system can be represented in accordance with (32.1) in the form.

It follows from this that even when the body will move with respect to the non-inertial frame of reference with acceleration - a, i.e., as if a force equal to acted on it.

The foregoing means that when describing motion in non-inertial frames of reference, Newton's equations can be used, if, along with the forces due to the action of bodies on each other, the so-called forces and inertia are taken into account, which should be assumed to be equal to the product of the body's mass and the difference of its accelerations taken with the opposite sign along relation to inertial and non-inertial frames of reference:

Accordingly, the equation of Newton's second law in a non-inertial frame of reference will have the form

Let us explain our assertion with the following example. Consider a trolley with a bracket attached to it, to which a ball is suspended on a thread (Fig. 32.1). While the cart is at rest or moving without acceleration, the thread is vertical and the force of gravity P is balanced by the reaction of the thread. Now let's bring the cart into translational motion and acceleration a. The thread will deviate from the vertical by such an angle that the resultant force imparts an acceleration to the ball equal to . Relative to the frame of reference associated with the trolley, the ball is at rest, despite the fact that the resultant force is different from the rudder. The lack of acceleration of the ball with respect to this frame of reference can be formally explained by the fact that, in addition to the forces P and F, equal in total to m, the ball is also affected by the force of inertia

The introduction of inertial forces makes it possible to describe the motion of bodies in any (both inertial and non-inertial) frames of reference using some of the equations of motion.

It should be clearly understood that the forces of inertia cannot be put on a par with forces such as elastic, gravitational forces and friction forces, i.e., forces due to the impact on the body from other bodies. Sids of inertia are due to the properties of the frame of reference in which mechanical phenomena are considered. In this sense, they can be called fictitious forces.

Introduction to the consideration of inertial forces is not fundamentally necessary. In principle, any movement can always be considered in relation to an inertial frame of reference. However, in practice it is often the motion of bodies with respect to non-inertial frames of reference, for example, with respect to the earth's surface, that is of interest.

The use of inertial forces makes it possible to solve the corresponding problem directly with respect to such a frame of reference, which often turns out to be much simpler than considering motion in an inertial frame.

A characteristic property of the forces of inertia is their proportionality to the mass of the body. Due to this property, the forces of inertia are similar to the forces of gravity. Imagine that we are in a closed cabin remote from all external bodies, which moves with an acceleration g in a direction that we will call "top" (Fig. 32.2). Then all the bodies inside the cabin will behave as if they were acted upon by the inertia force -mg. In particular, a spring, to the end of which a body of mass is suspended, will stretch so that the elastic force balances the inertia force -mg. However, the same phenomena would also be observed if the cabin were stationary and located close to the Earth's surface. Not being able to "look out" outside the cabin, no experiments carried out inside the cabin, We could not establish what caused the force -mg by the accelerated movement of the cabin or the action of the Earth's gravitational field. On this basis, they will talk about the equivalence of the forces of inertia and gravity. This equivalence lies at the core of Einstein's general theory of relativity.

force d "inertia. In other languages, the name of the force more clearly indicates its fictitiousness: in German it. Scheinkrafte("imaginary", "apparent", "visible", "false", "fictitious" force), in English English. pseudo force("pseudo-strength") or eng. fictitious force("fictitious force"). Less commonly used in English are the names " d'Alembert's power" (eng. d'Alembert force) and "inertial force" (eng. inertial force ).

The variety of names is explained by the fact that in Russian the term "force of inertia" is used for descriptions of three various forces:

As a result of the ambiguity of the term, “a confusion has arisen that continues to this day, and there are ongoing disputes about whether the forces of inertia are real or unreal (fictitious) and whether they have a counteraction” .

In addition to the name, all meanings of the term are also united by a vector quantity. It is equal to the product of the mass of the body and its acceleration and is directed opposite to the acceleration. Brief definitions inertial forces sometimes reflect this common property all meanings of the term:

A vector quantity equal to the product of the mass of a material point and its acceleration and directed opposite to the acceleration is called the force of inertia.

Real and fictitious forces

The literature also uses the terms "fictitious" and "real" forces (the latter term is rarely used in Russian literature). Different authors put different meanings into these words:

Depending on the chosen definition, the forces of inertia turn out to be real or fictitious, therefore, some authors consider the use of such terminology unsuccessful and recommend simply avoiding it in the educational process.

Forces

Force is a vector physical quantity, which is a measure of the intensity of the impact of other bodies or fields on a given body. The force applied to a massive body is the cause of a change in its speed or the occurrence of deformations in it. Force, as a vector quantity, is characterized by its modulus, direction, and "point" of force application.

Newton's first law

Newton's first law introduces the concept of inertial frames of reference, and gives reason to talk about non-inertial ones:

There are such systems of reference, relative to which a material point in the absence of external influences (or with their mutual compensation) maintains a state of rest or uniform rectilinear motion.

Newton's second law

It consists in the statement that there is a direct proportionality between the force and the acceleration caused by it, which is written as:

Here, the scalar included in the coefficient of proportionality is the inertial mass .

It has been experimentally proven that for any body the mass included in the expression of Newton's Second Law and in his law of universal gravitation are completely equivalent:

The equality of inertial and inertial masses is, as seen in Special Relativity, a fundamental property of spacetime. Its consideration goes beyond the scope of classical mechanics.

Therefore, below the body weight will be denoted without indices as .

The considered body with a mass (more precisely, an inertial mass) acquires an acceleration that differs from zero at the same moment when a force begins to act on it (Newton's second law:). However, it is also true that it takes some time to reach a non-zero speed in accordance with the definition of the momentum of force : . Or, in other words, the speed of the body does not change by itself, without a reason, but it begins to change immediately how the force is acting on it. Thus, there are no grounds for introducing ideas about any resistance to influence or about some kind of "inertness property".

It is generally accepted that the Second Law is valid only in inertial FRs and is not satisfied in non-inertial systems. Taking into account the fact that inertial systems are fundamentally unrealizable, it would be logical to consider the Second Law also never fulfilled. However, the idea underlying the proportionality of the acceleration received by the body everyone forces acting on it, regardless of their origin, allows, by taking into account the "fictitious" forces of inertia, to extend the action of Newtonian axiomatics to mechanics real movements real bodies.

Like other statements subject to experimental verification, the Second Law can be valid only if the quantities included in it can be measured independently, each separately. Modern experimental technique provides enough high precision measurements of both force and mass and acceleration. These measurements invariably confirm experimentally (within the framework of classical mechanics) the validity of the said Second Law extrapolation.

Newton's third law

He claims that the forces acting from some bodies on others always have the nature of interaction, that is, if the first body changes the speed of the second, then the second changes the speed of the first. At the same time, in any kind of force interaction and regardless of whether the distance between the bodies changes and whether they move at all, the condition is always satisfied:

That is, the accelerations communicated by the bodies to each other, during the interaction of two bodies, are directed towards each other, and are inversely proportional to the masses of the bodies.

Introducing the definition for the inertial mass of bodies from the Second Law into expression (4), we arrive at the generally accepted notation of Newton's third law in its own formulation:

An action always has an equal and opposite reaction, otherwise: the interactions of two bodies against each other are equal and directed in opposite directions

Newton's mechanics is invariant with respect to the arrow of time - it allows the movement of bodies in both direct and reverse sequences with respect to time. This finds its expression in the Third Law, which implies the simultaneous emergence of an action force and a reaction force, regardless of the prehistory of the described physical process.

However, in nature there is a causal order between occurring events, due to which they are located in a certain sequence in time (in cosmic scale there may not be a causal relationship due to the finite propagation velocity of any force interaction, which is the starting point of the special theory of relativity). And therefore, in the interaction of two bodies, it seems logical that one of them, which experienced an acceleration generated by the action of the other, should be considered passive, that is accelerating, and the other is active, that is accelerating. .

From the point of view of analyzing the dynamics of motion, it is important to know in which system of the two systems considered below the observer (recording device) is located and, most importantly, to know (if the observer is in the second, moving system) whether this system is inertial, or not.

Newtonian forces of inertia

Some authors use the term "inertia force" to refer to the reaction force from Newton's third law. The concept was introduced by Newton in his "Mathematical Principles of Natural Philosophy": "The innate force of matter is its inherent ability of resistance, according to which any individual body, since it is left to itself, maintains its state of rest or uniform rectilinear motion", and the term itself " force of inertia "was, according to Euler, first used in this sense by Kepler (, with reference to E. L. Nicolai).

To denote this reaction force, some authors propose to use the term "Newtonian force of inertia" in order to avoid confusion with fictitious forces used in calculations in non-inertial frames of reference and when using the d'Alembert principle.

An echo of Newton's choice of the word "resistance" to describe inertia is also the idea of ​​a certain force that allegedly realizes this property in the form resistance changes in movement parameters. In this connection, Maxwell observed that one could just as well say that coffee resists becoming sweet, since it does not become sweet on its own, but only after sugar is put into it.

Existence of inertial frames of reference

Newton proceeded from the assumption that inertial reference systems exist and among these systems there is the most preferable one (Newton himself associated it with the ether that fills all space). Further development physics showed that there is no such system, but this led to the need to go beyond classical physics. Moreover, the presence of an omnipresent gravitational field, from which there is no protection, excludes in principle the possibility of implementing the reference systems indicated in the First Law, which remain only an abstraction, the adoption of which is associated with a conscious assumption of errors in the result.

Movement in inertial CO

After performing a trivial mathematical operation in the expression of Newton's third law (5) and transferring the term from the right side to the left, we obtain a mathematically flawless notation:

From a physical point of view, the addition of force vectors results in a resultant force.

In this case, the expression (6) read from the point of view of Newton's second law means, on the one hand, that the resultant of forces is equal to zero and, therefore, the system of these two bodies does not move with acceleration. On the other hand, there are no prohibitions on the accelerated movement of the bodies themselves.

The fact is that the concept of a resultant arises only in the case of estimating the joint action of several forces on same body. In this case, although the forces are equal in magnitude and opposite in direction, but applied to different bodies and therefore, regarding each of their considered bodies separately, they do not balance each other, since each of the interacting bodies is affected only by one of them. Equality (6) does not indicate the mutual neutralization of their action for each of the bodies, it speaks of the system as a whole.

A material point in two Cartesian coordinate systems: fixed O, considered to be inertial, and moving O"

The equation that expresses Newton's second law in an inertial frame of reference is widely used:

If there is a resultant of all real forces acting on a body, then this expression, which is the canonical record of the Second Law, is simply the statement that the acceleration received by the body is proportional to this force and the mass of the body. Both expressions in each part of this equality refer to the same body.

But expression (7) can be, like (6), rewritten as:

For an outside observer who is in an inertial frame and analyzes the acceleration of a body, on the basis of what has been said above, such a notation has a physical meaning only if the terms on the left side of the equation refer to forces that arise simultaneously, but belong to different bodies. And in (8) the second term on the left is the force of the same magnitude, but directed in opposite side and applied to another body, namely the force, i.e.

In the case when it turns out to be appropriate to divide the interacting bodies into accelerated and accelerating and, in order to distinguish the forces then acting on the basis of the Third Law, those that act from the accelerated body on the accelerating are called the forces of inertia or "Newtonian forces of inertia", which corresponds to the notation expressions (5) for the Third Law in the new notation:

It is essential that the force of action of the accelerating body on the accelerated and the force of inertia have the same origin, and if the masses of the interacting bodies are so close to each other that the accelerations they receive are comparable in magnitude, then the introduction of a special name "inertia force" is only a consequence of the achieved agreements. It is just as arbitrary as the very division of forces into action and reaction.

The situation is different when the masses of the interacting bodies are incomparable among themselves (a person and a solid floor, starting from which he goes). In this case, the division of bodies into accelerating and accelerating becomes quite distinct, and the accelerating body can be considered as a mechanical connection that accelerates the body, but is not accelerated by itself.

In an inertial frame of reference inertia force attached not to the accelerated body, but to the connection.

Euler forces of inertia

Movement in non-inertial CO

Twice differentiating both sides of the equality with respect to time, we obtain:

is the acceleration of the body in the inertial FR, further called the absolute acceleration. is the acceleration of a non-inertial CO in an inertial CO, hereinafter referred to as the transfer acceleration. is the acceleration of the body in non-inertial FR, further called the relative acceleration.

It is essential that this acceleration depends not only on the force acting on the body, but also on the acceleration of the reference frame in which this body moves, and therefore, with an arbitrary choice of this FR, it can have an arbitrary value, respectively.

Relative acceleration is quite real in a non-inertial CO, since the difference between two real values ​​according to (11) cannot but be real.

We multiply both parts of equation (11) by the body weight and get:

In accordance with Newton's second law, formulated for inertial frames, the term on the left is the result of multiplying the mass by the vector defined in the inertial frame, and therefore a real force can be associated with it:

This is the force acting on the body in the first (inertial) CO, which will be called here " absolute power". It continues to act on the body with the same direction and magnitude in any coordinate system.

The next force, defined as:

according to the rules adopted for naming ongoing movements, it should be called "portable".

It is important that acceleration, in the general case, has nothing to do with the body under study, since it is caused by those forces that act only on the body chosen as a non-inertial frame of reference. But the mass included in the expression is the mass of the body under study. In view of the artificiality of the introduction of such a force, it must be considered a fictitious force.

Transferring the expressions for the absolute and portable forces to the left side of the equation:

and applying the introduced notation, we get:

It can be seen from this that due to the acceleration in new system reference on the body is not full force, but only part of it, remaining after subtracting the portable force from it so that:

then from (15) we get:

according to the accepted names for the ongoing movements, this force should be called "relative". It is this force that causes the body to move in a non-inertial coordinate system.

The result obtained in the difference between the "absolute" and "relative" forces is explained by the fact that in a non-inertial frame, in addition to the force, a certain force additionally acted on the body in such a way that:

This force is the force of inertia, as applied to the motion of bodies in non-inertial FR. It has nothing to do with the action of real forces on the body.

Then from (17) and (18) we get:

That is, the force of inertia in non-inertial CO equal in magnitude and opposite in direction to the force causing the accelerated motion of this system. She is attached to the accelerated body.

This force is not, in its origin, the result of the action of surrounding bodies and fields, and arises solely due to the accelerated movement of the second frame of reference relative to the first.

All quantities included in expression (18) can be measured independently of each other, and therefore the equal sign put here means nothing more than the recognition of the possibility of spreading Newtonian axiomatics, taking into account such “fictitious forces” (inertial forces) and on motion in non-inertial reference systems, and therefore requires experimental confirmation. Within the framework of classical physics, this is true and is confirmed.

The difference between the forces and consists only in the fact that the second is observed during the accelerated motion of the body in a non-inertial coordinate system, and the first corresponds to its immobility in this system. Since immobility is only the limiting case of motion at low speed, there is no fundamental difference between these fictitious forces of inertia.

Example 2

Let the second CO move at a constant speed or just be stationary in the inertial CO. Then the force of inertia is absent. A moving body experiences an acceleration caused by real forces acting on it.

Example 3

Let the second CO move with acceleration, that is, this CO is actually aligned with the moving body. Then in this non-inertial frame the body is motionless due to the fact that the force acting on it is completely compensated by the force of inertia:

Example 4

A passenger travels in a car at a constant speed. The passenger is a body, the car is its frame of reference (so far inertial), that is.

The car starts to slow down, and turns for the passenger into the second non-inertial system considered above, to which the braking force is applied towards its movement. Immediately there is an inertia force applied to the passenger, directed in the opposite direction (that is, along the movement): . This force causes the occupant's body to move involuntarily towards the windshield.

In a non-inertial system (for an observer standing on the Earth's surface), the following forces act on the body: the centrifugal force of inertia (blue vector), the gravitational force (red), in total giving the real gravity , which is balanced by the reaction of the support (black).

Example

When a body moves in a circle under the action of a centripetal force, which is the result of a connection superimposed on the movement of the body, the force acting on this connection will be both a reaction force and a "centrifugal force of inertia"

General approach to finding the forces of inertia

Comparing the motion of a body in inertial and non-inertial CO, one can come to the following conclusion:

Let there be the sum of all forces acting on the body in a fixed (first) coordinate system, which causes its acceleration. This sum is found by measuring the acceleration of the body in this system, if its mass is known.

Similarly, there is a sum of forces measured in a non-inertial coordinate system (the second one), causing an acceleration , which in the general case differs from due to the accelerated movement of the second CO relative to the first one.

Then the force of inertia in a non-inertial coordinate system will be determined by the difference:

In particular, if the body is at rest in a non-inertial frame, that is, then

If in expression (20) we assume that the acceleration is measured not in absolute, but in another non-inertial coordinate system, then the found force of inertia will be a force corresponding to the relative motion of two non-inertial RMs. If we take into account that all bodies in the Universe interact with each other due to all-penetrating gravity, and therefore there are no inertial FRs in principle, then this particular case is really realizable in practice.

Motion of a body along an arbitrary trajectory in a non-inertial CO

Position material body in a conditionally immobile and inertial frame is given here by the vector , and in a non-inertial frame - by the vector . The distance between the origins is determined by the vector . The angular velocity of rotation of the system is given by the vector , the direction of which is set along the axis of rotation according to the rule of the right screw . Line speed of the body with respect to the rotating CO is given by the vector .

AT this case inertial acceleration, in accordance with (11), will be equal to the sum:

The first term is the portable acceleration of the second system relative to the first; the second term is the acceleration arising from the uneven rotation of the system around its axis; the third term is the Coriolis acceleration caused by that component of the velocity vector that is not parallel to the axis of rotation of the non-inertial system; the last term, taken without sign, is a vector directed in the opposite direction from the vector, which can be obtained by expanding the double vector product, when we find that this term is equal to () and therefore represents the centripetal acceleration of the body in the reference frame of a stationary observer, taken for ISO, in which there can be no inertia forces by definition. However, formula (22) refers to accelerations observed in a non-inertial (rotating) frame of reference, and the last three terms in (11) represent the relative acceleration, that is, the acceleration experienced by a body in a non-inertial frame under the action of the centrifugal force of inertia (see blue arrow in the picture). The last term must represent (together with the sign) the centrifugal acceleration, and therefore must be preceded by a minus sign.

The work of fictitious inertia forces

In classical physics, the forces of inertia occur in three different situations depending on the frame of reference in which the observation is made. This is the force applied to the connection when observed in an inertial reference frame or to a moving body when observed in a non-inertial reference frame. Both of these forces are real and can do work. So, an example of the work done by the Coriolis force on a planetary scale is the Baer effect.

When solving problems on paper, when the dynamic problem of movement is artificially reduced to the problem of statics, a third type of force is introduced, called the d’Alembert forces, which do not perform work, since the work and immobility of bodies, despite the action of forces on it, are incompatible concepts in physics.

Equivalence of inertial and gravitational forces

Applications

1. V. Aircraft. Physics. Dictionary reference. Publishing house "Peter", 2005. S. 315.
2. inertia force- article from the Great Soviet Encyclopedia
3. Example: In history, as in nature, the force of inertia is great, from P. Gvozdev. Learning and Literary Mores in Roman Society in the Time of Pliny the Younger. // Journal of the Ministry of Public Education. T. 169. Ministry of Public Education, 1873. S. 119.
4. Walter Greiner Klassische Mehanik II. Wissenschaftlicher VerlagHarri Deutsch GmbH. Frankfurt am Main.2008 ISBN 978-3-8171-1828-1
5. ^Richard Phillips Feynman, Leighton R. B. & Sands M. L.(2006). The Feynman Lectures on Physics. San Francisco: Pearson/Addison-Wesley. Vol. I, section 12-5.

When studying the question of what is the force of inertia (SI), misunderstandings often occur, leading to pseudoscientific discoveries and paradoxes. Let's look into this issue, applying a scientific approach and substantiating everything that has been said with supporting formulas.

The force of inertia surrounds us everywhere. People noticed its manifestations in antiquity, but could not explain it. Galileo was seriously studying it, and then the famous one. It was because of his lengthy interpretation that erroneous hypotheses became possible. This is quite natural, because the scientist made an assumption, and the baggage of knowledge accumulated by science in this area did not yet exist.

Newton argued that the natural property of all material objects is the ability to be in a state in a straight line or at rest, provided that there is no external influence.

Let's "expand" this assumption on the basis of modern knowledge. Even Galileo Galilei drew attention to the fact that the force of inertia is directly related to gravity (attraction). And the natural attracting objects, the impact of which is obvious, are the planets and stars (due to their mass). And since they have the shape of a ball, this is what Galileo pointed out. However, Newton this moment completely ignored.

It is now known that the entire Universe is permeated with gravitational lines of varying intensity. Indirectly confirmed, although not mathematically proven, the existence of gravitational radiation. Therefore, the force of inertia always arises with the participation of gravity. Newton, in his assumption of a "natural property", also did not take this into account.

It is more correct to proceed from another definition - the indicated force is the value of which is the product of the mass (m) of the moving body and its acceleration (a). The vector is directed opposite to the acceleration, that is:

where F, a are the values ​​of the force vectors and the resulting acceleration; m is the mass of the moving body (or mathematical

Physics and mechanics offer two names for such an effect: Coriolis and portable inertia force (PSI). Both terms are equivalent. The difference is that the first option is generally recognized and is used in the course of mechanics. In other words, the equality is true:

F kor \u003d F per \u003d m * (-a kor) \u003d m * (-a per),

where F is the Coriolis force; F per - portable force of inertia; a kor and a per are the corresponding acceleration vectors.

PSI includes three components: inertia, translational SI and rotational. If there are usually no difficulties with the first, then the other two require explanation. The translational force of inertia is determined by the acceleration of the entire system as a whole relative to any inertial system in the translational type of motion. Accordingly, the third component arises due to the acceleration that appears during the rotation of the body. At the same time, these three forces can exist independently, without being part of the PSI. All of them are represented by the same basic formula F = m * a, and the differences are only in the type of acceleration, which, in turn, depends on the type of movement. Thus, they are a special case of inertia. Each of them is involved in the calculation of the theoretical absolute acceleration of a material body (point) in a fixed frame of reference (invisible for observation from a non-inertial frame).

PSI is necessary when studying the issue of relative motion, since in order to create formulas for the motion of a body in a non-inertial system, it is necessary to take into account not only other known forces, but also it (F kor or F per).