Classification of forces of inertia. §7

Let on a material point M some system of forces is at work.

Among the forces there can be active forces and reactions of connections.

Based on the axiom of independence of the action of forces, the point M under the action of these forces, it will receive the same acceleration as if it were acted upon by only one force equal to the geometric sum of the given forces,

where but- point acceleration M; m- point mass M F ; - resultant system of forces.

We transfer the vector on the left side of the equation to right side. After that, we get the sum of vectors equal to zero,

We introduce the notation then the above equation can be written as:

Thus, all forces, including the force , must be balanced, since the forces and F are equal to each other and directed along one straight line in opposite directions. A force equal to the product of the mass of a point and its acceleration, but directed in the direction opposite to the acceleration, is called the force of inertia.

From the last equation it follows that in each this moment time, the forces applied to a material point are balanced by the forces of inertia. The above conclusion is called the beginning of D "Alembert. It can be applied not only to a material point, but also to a solid body or to a system of bodies. In the latter case, it is formulated as follows: if all the acting forces applied to a moving body or system of bodies, apply inertial forces, then the resulting system of forces can be considered as being in equilibrium.

It should be emphasized that the forces of inertia do exist, but are applied not to a moving body, but to those bodies that cause accelerated motion.

The use of the beginning of D "Alembert allows you to use equilibrium equations when solving dynamic problems. This technique for solving problems of dynamics is called kinetostatic method.

Let us consider how the force of inertia of a material point is determined in various cases of its motion.

1. Point M weight m moves rectilinearly with acceleration (Fig. a, b).

In rectilinear motion, the direction of acceleration coincides with the trajectory. The force of inertia is directed in the direction opposite to the acceleration, and numerical value it is determined by the formula:

With accelerated movement (Fig. a), the directions of acceleration and speed coincide and the force of inertia is directed in the direction opposite to the movement. In slow motion (Fig. b), when the acceleration is directed in the direction opposite to the speed, the inertia force acts in the direction of motion.

2. Point M moves curvilinearly and unevenly (Fig. c).

In this case, as is known from the previous one, its acceleration can be decomposed into a normal a n and tangent a t components. Similarly, the force of inertia of a point also consists of two components: normal and tangential.

The normal component of the inertial force is equal to the product of the mass of the point and the normal acceleration and is directed opposite to this acceleration:

The tangential component of the inertial force is equal to the product of the mass of the point and the tangential acceleration and is directed opposite to this acceleration:

Obviously, the total force of inertia of the point M is equal to the geometric sum of the normal and tangent components, i.e.

Considering that the tangential and normal components are mutually perpendicular, the total inertia force is:

3.3 The work of a constant force on rectilinear movement

Let us define the work for the case when acting force is constant in magnitude and direction, and the point of its application moves along a rectilinear trajectory. Consider a material point C, to which a force F, constant in value and direction, is applied.

For some period of time t dot FROM moved to position From 1 along a straight path to a distance s.

Work A constant strength F during rectilinear motion of the point of its application is equal to the product of the modulus of force F at a distance s and by the cosine of the angle between the direction of the force and the direction of displacement, i.e.

The angle α between the direction of the force and the direction of motion can vary from 0 to 180°. For α< 90° работа положительна, при α>90° - negative, at α = 90° A=0(work is zero).

If the force is equal to the direction of motion sharp corner, it is called driving force her work is always positive. If the angle between the directions of the force and the movement is obtuse, the force resists the movement, performs negative work and is called the resistance force. Examples of resistance forces are the forces of cutting, friction, air resistance, and others, which are always directed in the direction opposite to the movement.

When α = 0, i.e. when the direction of the force coincides with the direction of the velocity, A = Fs , because cosα = 1. Product Fcosα is the force projection F on the direction of movement of the material point. Therefore, the work of a force can be defined as the product of displacement s and force projections F on the direction of movement of the point.

The unit of work in the International System of Units (SI) is the joule (J), equal to the work of a force of one newton (N) on a one meter (m) long coinciding with it in the direction of motion:. A larger unit of work is also used - kilojoule (kJ), 1 kJ \u003d 1000 J \u003d 10 3 J. In technical system(MKGSS) a kilogram-force meter (kgf m) is accepted as a unit of work.

Having established that the individual points in Newtonian absolute space are not a physical reality, we must now ask ourselves what remains within

this concept at all? The following remains: the resistance of all bodies to acceleration must be interpreted in the Newtonian sense as an action of absolute space. The locomotive that sets the train in motion overcomes the resistance of inertia. A projectile that takes down a wall draws its destructive force from inertia. The action of inertia is manifested whenever accelerations take place, and the latter are nothing more than changes in velocity in absolute space (we can use the latter expression, since the change in velocity has the same magnitude in all inertial frames). Thus, frames of reference, which themselves move with acceleration relative to inertial frames, are not equivalent to the latter or to each other. It is possible, of course, to determine the laws of mechanics in such systems, but they will become more complex shape. Even the trajectory of a free body turns out to be no longer uniform and rectilinear in an accelerated system (see Chap. p. 59). The latter can be expressed in the form of a statement that in an accelerated system, in addition to real forces, there are apparent, or inertial, forces. The body, which is not affected by real forces, is still subject to the action of these inertial forces, so its movement in the general case turns out to be uneven and non-rectilinear. For example, a car that starts moving or slows down is such an accelerated system. Everyone knows the push of a moving or stopping train; it is nothing but the action of the inertial force we are talking about.

Let us consider this phenomenon in detail using the example of a system moving rectilinearly with acceleration. If we measure the acceleration of a body relative to such a moving system, then its acceleration relative to absolute space will obviously be greater by. Therefore, the fundamental law of mechanics in this space has the form

If we write it in the form

then we can say that the law of motion in the Newtonian form is satisfied in the accelerated system, namely

except that now the force must be set to K, which is equal to

where K is the real force, and is the apparent force, or the force of inertia.

So, this force acts on a free body. Its action can be illustrated by the following reasoning: we know that gravity on Earth - the force of gravity - is determined by the formula G = mg, where is a constant acceleration due to gravity. The force of inertia acts in this case like gravity; the minus sign means that the force of inertia is directed opposite to the acceleration of the frame of reference, which is used as a basis. The magnitude of the apparent gravitational acceleration y coincides with the acceleration of the frame of reference. Thus, the motion of a free body in the frame is simply motion of the type we know as the fall or motion of a thrown body.

This relationship between inertial forces in accelerated systems and the force of gravity still seems somewhat artificial here. In fact, it went unnoticed for two hundred years. However, already at this stage we must point out that it forms the basis 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) retains 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 only be valid 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, i.e. 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 has been 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 of them that act from the accelerated body on the accelerating are called inertia forces or " Newtonian forces inertia”, which corresponds to the expression (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 frame of reference 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, in the general case, acceleration 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 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 SO move with constant speed or simply immobile in 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 surface of the Earth), the following forces act on the body: the centrifugal force of inertia (blue vector), the force of gravity (red), in total giving the real force of 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 inertia force will be a force corresponding to the relative motion of two non-inertial FRs. If we take into account that all bodies in the Universe interact with each other due to the 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 .

IN 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 frame of reference 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 of reference 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 motion 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.

From everyday experience, we can confirm the following conclusion: the speed and direction of a body can change only during its interaction with another body. This gives rise to the phenomenon of inertia, which we will discuss in this article.

What is inertia? An example of life observations

Let us consider the cases when some body on initial stage experiment is already in motion. Later we will see that a decrease in speed and a stop of a body cannot occur arbitrarily, because the reason for this is the action of another body on it.

You have probably seen more than once how passengers who travel in transport suddenly lean forward during braking or press on their side on sharp turn. Why? Let's explain further. When, for example, athletes run a certain distance, they try to develop maximum speed. Having run through the finish line, you can no longer run, but you can’t stop abruptly, and therefore the athlete runs a few more meters, that is, he moves by inertia.

From the above examples, we can conclude that all bodies have the peculiarity of maintaining the speed and direction of movement, while not being able to instantly change them after the action of another body. It can be assumed that in the absence of an external action, the body will retain both the speed and the direction of motion for an arbitrarily long time. So what is inertia? This is the phenomenon of maintaining the speed of a body in the absence of the influence of other bodies on it.

Discovery of inertia

This property of bodies was discovered by the Italian scientist Galileo Galilei. On the basis of his experiments and reasoning, he argued: if the body does not interact with other bodies, then it either remains in a state of calm, or moves in a straight line and uniformly. His discoveries entered science as the Law of Inertia, but Rene Descartes formulated it in more detail, and Isaac Newton introduced it into his system of laws.

An interesting fact: inertia, the definition of which Galileo gave us, was considered in Ancient Greece Aristotle, but due to the insufficient development of science, the exact formulation was not given. says: there are
frames of reference, relative to which a body that moves forward, keeps its speed constant, if it is not acted upon by other bodies. There is no formula for inertia in a single and generalized form, but below we present many other formulas that reveal its features.

Body inertia

We all know that a car, train, ship or other body increases gradually when they start moving. You've all seen rocket launches on TV or planes taking off at an airport - they increase speed not in jerks, but gradually. Observations, as well as everyday practice, show that all bodies have common feature: the speed of the bodies in the process of their interaction changes gradually, and therefore it takes some time to change them. This feature of bodies is called inertia.

All bodies are inert, but not all have the same inertia. Of the two interacting bodies, it will be higher for the one that acquires less acceleration. So, for example, when fired, a gun acquires less acceleration than a cartridge. With the mutual repulsion of an adult skater and a child, the adult receives less acceleration than the child. This indicates that the inertia of an adult is greater.

To characterize the inertia of bodies, a special value was introduced - the mass of the body, it is usually denoted by the letter m. In order to be able to compare the masses of different bodies, the mass of one of them must be taken into account as a unit. Its choice can be arbitrary, but it should be convenient for practical use. In the SI system, the mass of a special standard made of a hard alloy of platinum and iridium was taken as a unit. She wears all of us famous name- kilogram. It should be noted that the inertia of a rigid body can be of 2 types: translational and rotational. In the first case, the measure of inertia is the mass, in the second, the moment of inertia, which we will discuss later.

Moment of inertia

This is the name of a scalar physical quantity. In the SI system, the unit of moment of inertia is kg * m 2. The generalized formula is the following:

Here m i is the mass of body points, r i is the distance from the points of the body to the axis z in the spatial coordinate system. In verbal interpretation, we can say this: the moment of inertia is determined by the sum of the products of elementary masses, multiplied by the square of the distance to the base set.

There is another formula that characterizes the definition of the moment of inertia:

Here dm is the mass of the element, r- distance from the element dm to the axis z. It can be verbally formulated as follows: the moment of inertia of a system of material points or a body relative to a pole (point) is algebraic sum the product of the masses of the material points that make up the body by the square of their distance to the pole 0.

It is worth mentioning that there are 2 types of moments of inertia - axial and centrifugal. There is also such a thing as the main moments of inertia (GMI) (relative to the main axes). As a rule, they are always different from each other. Now it is possible to calculate the moments of inertia for many bodies (cylinder, disk, ball, cone, sphere, etc.), but we will not delve into the refinement of all formulas.

reference systems

Newton's 1st law dealt with uniform rectilinear motion, which can only be considered in a certain frame of reference. Even an approximate analysis of mechanical phenomena shows that the law of inertia is far from being fulfilled in all frames of reference.

Let's consider a simple experiment: let's put a ball on a horizontal table in a car and watch it move. If the train is in a state of calm relative to the Earth, then the ball will also remain calm until we act on it with another body (for example, a hand). Therefore, in the reference frame that is connected with the Earth, the law of inertia is fulfilled.

Let's imagine that the train will travel uniformly and rectilinearly relative to the Earth. Then in the frame of reference, which is connected with the train, the ball will remain in a state of calm, and in the one connected with the Earth, it will remain in a state of uniform and rectilinear motion. Consequently, the law of inertia is satisfied not only in the reference frame associated with the Earth, but also in all others moving uniformly and rectilinearly relative to the Earth.

Now imagine that the train picks up speed quickly or turns sharply (in all cases, it is moving with acceleration relative to the Earth). Then, as before, the ball retains the uniform and which it had before the start of the train's acceleration. However, with respect to the train, the ball itself leaves the state of calmness, although there are no bodies that would take it out of it. This means that in the frame of reference associated with the acceleration of the train relative to the Earth, the law of inertia is violated.

So, frames of reference in which the law of inertia is fulfilled are called inertial. And those in which it is not fulfilled are non-inertial. It is easy to determine them: if the body moves uniformly and rectilinearly (in some cases it is calm), then the system is inertial; if the movement is uneven - non-inertial.

inertia force

This is a rather ambiguous concept, and therefore we will try to consider it in as much detail as possible. Let's take an example. You are standing quietly on the bus. Suddenly, it starts to move, which means it picks up speed. You unwillingly lean back. But why? Who pulled you? From the point of view of an observer on Earth, you remain in place, while Newton's 1st law is fulfilled. From the point of view of an observer on the bus itself, you begin to move backwards, as if under some kind of force. In fact, your legs, which are connected by friction with the floor of the bus, went forward with it, and you,
losing balance, had to fall back. Thus, to describe the motion of a body in a non-inertial frame of reference, it is necessary to introduce and take into account additional forces that act from the body's connections with such a frame. These forces are the forces of inertia.

It must be taken into account that they are fictitious, because there is not a single body or field, under the influence of which you began to move in the bus. Newton's laws do not apply to the forces of inertia, but their use along with "real" forces makes it possible to describe the motion of arbitrary non-inertial frames of reference using various tools. This is the whole point of introducing inertial forces.

So, now you know what inertia is, the moment of inertia and inertial systems, the forces of inertia. We move on.

Translational motion of systems

Let some body, located in a non-inertial frame of reference, moving with acceleration a 0 relative to the inertial one, the force F acts. For such a non-inertial system, the equation-analogue of Newton's second law has the form:

Where a 0 is the acceleration of a body with mass m, which is caused by the action of the force F relative to the non-inertial frame of reference; F ін - force of inertia. The force F on the right side is “real” in the sense that it is the resultant of the interaction of bodies, depending only on the difference in coordinates and velocities of the interacting material points, which do not change when moving from one frame of reference to another, moving forward. Therefore, the force F does not change either. It is invariant under such a transition. But F іn arises not because of but because of the accelerated motion of the reference frame, because of which it changes when moving to another accelerated frame, therefore it is not invariant.

Centrifugal force of inertia

Consider the behavior of bodies in a non-inertial frame of reference. XOY rotates relative to the inertial frame, which we will consider the Earth, with a constant angular velocity ω. An example is the system in the figure below.

Above is a disk, where a radially directed rod is fixed, and a blue ball is put on, "tied" to the axis of the disk with an elastic rope. As long as the disk does not rotate, the rope does not deform. However, when the disk is untwisted, the ball gradually stretches the rope until the elastic force F cp becomes such that it is equal to the product of the mass of the ball m to its normal acceleration a p \u003d -ω 2 R, i.e F cf \u003d -mω 2 R , where R is the radius of the circle that describes the ball as it rotates around the system.

If the angular velocity ω disk remains constant, then the ball will stop moving about the OX axis. In this case, relative to the XOY reference system, which is associated with the disk, the ball will be in a state of calm. This can be explained by the fact that in this system, in addition to the force F cf, the force of inertia acts on the ball Fcf, which is directed along the radius from the axis of rotation of the disk. A force that looks like the formula below is called inertia. It can arise only in rotating frames of reference.

Coriolis force

It turns out that when bodies move relative to rotating frames of reference, in addition to the centrifugal force of inertia, another force acts on them - Coriolis. It is always perpendicular to the body's velocity vector V, which means that it does not do any work on this body. We emphasize that the Coriolis force manifests itself only when the body moves relative to a non-inertial frame of reference, which performs rotation. Its formula looks like this:

Since the expression (v*ω) is the cross product of the vectors given in brackets, then we can conclude that the direction of the Coriolis force is determined by the gimlet rule in relation to them. Its modulus is:

Here Ө is the angle between the vectors v And ω .

Finally

Inertia is an amazing phenomenon that haunts each person hundreds of times every day, even if we ourselves do not notice it. We think that the article has given you important answers to questions about what is inertia, what is force and moments of inertia, who discovered the phenomenon of inertia. We are sure you were interested.

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 of reference $w"$ 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 reference frame associated with the trolley, 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 such forces 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 a 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 $v$ is the numerical value of the body's velocity component 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.