Inertia force of a point. Inertia forces and the basic law of mechanics

They are used in the literature, although they have not yet become widespread. In the future, we will adhere to this terminology, as it allows us to make the presentation more concise and clear.

Euler's inertial force in general case consists of several components of various origins, which are also given special names (“portable”, “Coriolis”, etc.). This is discussed in more detail in the corresponding section below.

In other languages, the names used for inertial forces more clearly indicate their special properties: in German German Scheinkräfte ("imaginary", "apparent", "apparent", "false", "fictitious" force), in English English pseudo force (“pseudo-force”) or English fictitious force (“fictitious force”). Less commonly used in English are the names "strength" d'Alembert » ( English d’Alembert force) and “inertial force” ( English inertial force). In literature published in Russian, similar characteristics are also used in relation to Euler and d’Alembert forces, calling these forces “fictitious”, “apparent”, “imaginary” or “pseudo-forces”

At the same time, the literature sometimes emphasizes reality inertial forces, opposing the meaning this term meaning of the term fictitiousness. At the same time, however, different authors put different meanings into these words, and the forces of inertia turn out to be real or fictitious not due to differences in the understanding of their basic properties, but depending on the chosen definitions. Some authors consider this use of terminology to be unfortunate and recommend simply avoiding it in educational process.

Although the debate over terminology is not yet over, the existing disagreements do not affect mathematical formulation equations of motion involving inertial forces and do not lead to any misunderstandings when using the equations in practice.

Forces in classical mechanics

Indeed, a physical quantity called force is introduced into consideration by Newton’s second law, while the law itself is formulated only for inertial reference systems. Accordingly, the concept of force turns out to be defined only for such reference systems.

Newton's second law equation relating acceleration a → (\displaystyle (\vec (a))) And m (\displaystyle m) mass material point with the force acting on it F → (\displaystyle (\vec (F))), is written in the form

It immediately follows from the equation that the acceleration of bodies is caused only by forces, and vice versa: the action of uncompensated forces on a body necessarily causes its acceleration.

Newton's third law complements and develops what was said about forces in the second law.

No other forces are introduced or used in classical mechanics. The possibility of the existence of forces that arise independently, without interacting bodies, is not allowed by mechanics.

Although the names of Euler and d’Alembertian inertia forces contain the word strength, these physical quantities are not forces in the sense accepted in mechanics.

Newtonian inertial forces

Some authors use the term “inertial force” to denote the reaction force from Newton's third law. The concept was introduced Newton in his “Mathematical principles of natural philosophy”: “The innate force of matter is the ability of resistance inherent in it, by which every single body, in so far as it is left to itself, maintains its state of rest or uniform rectilinear movement. From the inertia of matter it happens that every body is only with difficulty removed from its rest or movement. Therefore, the innate force could very sensibly be called the force of inertia. This force is manifested by the body only when another force applied to it produces a change in its state. The manifestation of this force can be considered in two ways - both as resistance and as pressure.” And the term “force of inertia” itself was, according to Euler, first used in this meaning Kepler(, with reference to E. L. Nikolai).

To denote this reaction force, some authors propose to use the term “Newtonian inertial force” to avoid confusion with fictitious forces used in calculations in non-inertial reference frames and when using d’Alembert’s principle.

An echo of Newton’s choice of the word “resistance” to describe inertia is also the idea of ​​a certain force that supposedly realizes this property in the form resistance changes in movement parameters. Due to this Maxwell noticed that it could just as well be said that coffee resists becoming sweet, since it does not become sweet on its own, but only after adding sugar.

Existence of inertial reference systems

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, which fills all space). Further development physics showed that there is no such system, but this led to the need to go beyond classical physics.

Motion in inertial FR

Having done 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 impeccable notation:

F 1 → + F 2 → = 0 (\displaystyle (\vec (F_(1)))+(\vec (F_(2)))=0)(6)

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

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

The fact is that the concept of the resultant arises only in the case of estimating joint action several forces on same thing body. In this case, although the forces are equal in magnitude and opposite in direction, they are applied to different bodies and therefore, regarding each of the bodies under consideration 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 mutual neutralization of their action for each of the bodies; it speaks about the system as a whole.

The equation expressing Newton's second law in an inertial reference frame is used everywhere:

F r → = m a r → (\displaystyle (\vec (F_(r)))=m(\vec (a_(r)))) (7)

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

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

F r → − m a r → = 0 (\displaystyle (\vec (F_(r)))-m(\vec (a_(r)))=0) (8)

For an outside observer who is in an inertial frame and analyzes the acceleration of a body, based on the above, such an entry has physical meaning only if the terms on the left side of the equality refer to forces that arise simultaneously, but relate to different bodies. And in (8) the second term on the left represents a force of the same magnitude, but directed towards the opposite side and applied to another body, namely force, that is

F i 1 → = − m a r → (\displaystyle (\vec (F_(i_(1))))=-m(\vec (a_(r)))) (9)

In the case when it turns out to be appropriate to divide interacting bodies into accelerated and accelerating and, in order to distinguish the forces acting then on the basis of the Third Law, those of them that act from the accelerated body on the accelerating body are called inertial forces F → i 1 (\displaystyle (\vec (F))_(i_(1))) or " Newtonian forces inertia”, which corresponds to writing expression (5) for the Third Law in new notation:

F r → = − F i 1 → (\displaystyle (\vec (F_(r)))=-(\vec (F_(i_(1))))) (10)

It is important 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 special name The “force of inertia” is only a consequence of the agreement reached. It is as conditional as the division of forces into action and reaction itself.

The situation is different when the masses of interacting bodies are incomparable with each other (a person and the hard floor, pushing off from which he walks). In this case, the division of bodies into accelerating and accelerated becomes quite clear, and the accelerating body can be considered as mechanical connection, accelerating the body, but not accelerating itself.

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

Euler inertia forces

Motion in non-inertial FR

Differentiating both sides of the equality twice with respect to time r = R + r ′ (\displaystyle r=R+r(^(\prime ))), we get:

A r → = a R → + a r ′ → (\displaystyle (\vec (a_(r)))=(\vec (a_(R)))+(\vec (a_(r^(\prime ))) ))(11), where:

It is important 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 a correspondingly arbitrary value.

Let's multiply both sides of equation (11) by body mass m (\displaystyle m) and we get:

M a r → = m a R → + m a r ′ → (\displaystyle m(\vec (a_(r)))=m(\vec (a_(R)))+m(\vec (a_(r^(\prime ))))) (12)

According to 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:

M a r → = F r → (\displaystyle m(\vec (a_(r)))=(\vec (F_(r)))). This is the force acting on the body in the first (inertial) CO, which will be called here “absolute force”. It continues to act on the body with unchanged direction and magnitude in any coordinate system.

The following force is defined as:

M a R → = F R → (\displaystyle m(\vec (a_(R)))=(\vec (F_(R)))) (13)

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

It is important that acceleration a R → (\displaystyle (\vec (a_(R)))) in the general case, it has nothing to do with the body being studied, since it is caused by those forces that act only on the body chosen as not inertial system countdown. But the mass included in the expression is the mass of the body being studied. Due to the artificiality of introducing such force, it must be considered a fictitious force.

Transferring the expressions for absolute and figurative force to left side equality:

M a r → − m a R → = m a r ′ → (\displaystyle m(\vec (a_(r)))-m(\vec (a_(R)))=m(\vec (a_(r^(\prime ))))) (14)

and applying the introduced notations, we obtain:

F r → − F R → = m a r ′ → (\displaystyle (\vec (F_(r)))-(\vec (F_(R)))=m(\vec (a_(r^(\prime ))) )) (15)

From this it is clear that due to acceleration in new system reference does not affect the body full strength, but only part of it F ′ → (\displaystyle (\vec (F^(\prime )))), remaining after subtracting the transfer force from it F R → (\displaystyle (\vec (F_(R)))) So:

F ′ → = m a r ′ → (\displaystyle (\vec (F^(\prime )))=m(\vec (a_(r^(\prime ))))) (16)

then from (15) we obtain:

F r → − F R → = F ′ → (\displaystyle (\vec (F_(r)))-(\vec (F_(R)))=(\vec (F^(\prime )))) (17)

According to the conventions for naming the movements occurring, 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 system, in addition to the force F → r (\displaystyle (\vec (F))_(r)), a certain force additionally acted on the body F → i 2 (\displaystyle (\vec (F))_(i_(2))) in such a way that:

F r → + F i 2 → = F ′ → (\displaystyle (\vec (F_(r)))+(\vec (F_(i_(2))))=(\vec (F^(\prime ) ))) (18)

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

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

F i 2 → = − F R → (\displaystyle (\vec (F_(i_(2))))=-(\vec (F_(R)))) (19)

That is, the force of inertia in non-inertial FR equal in magnitude and opposite in direction to the force causing the accelerated movement 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 recognition of the possibility of extending Newton’s axiomatics, taking into account such “fictitious forces” (forces of inertia) to motion in non-inertial reference systems, and therefore requires experimental confirmation. Within the framework of classical physics, this is indeed confirmed.

Difference between forces F i 1 → (\displaystyle (\vec (F_(i_(1))))) and consists only in the fact that the second is observed during the accelerated movement of a body in a non-inertial coordinate system, and the first corresponds to its immobility in this system. Since immobility is only an extreme case of motion at low speed, there is no fundamental difference between these fictitious inertial forces.

Example 2

Let the second CO move with constant speed or simply motionless in the inertial CO. Then a R → = 0 (\displaystyle (\vec (a_(R)))=0) and there is no inertial force. A moving body experiences acceleration caused by real forces acting on it.

Example 3

Let the second CO move with acceleration a R → = a r → (\displaystyle (\vec (a_(R)))=(\vec (a_(r)))), that is, this CO is actually combined with the moving body. Then in this non-inertial CO the body is motionless due to the fact that the force acting on it is completely compensated by the force of inertia:

F i 2 → = − F r → = F i 1 → (\displaystyle (\vec (F_(i_(2)))=-(\vec (F_(r)))=(\vec (F_(i_ (1)))))

Example 4

A passenger travels in a car at a constant speed. The passenger is the body, the car is its reference system (inertial for now), that is F r → = 0 (\displaystyle (\vec (F_(r)))=0).

The car begins to slow down and turns for the passenger into the second non-inertial system discussed above, to which a braking force is applied towards its movement F R → (\displaystyle (\vec (F_(R)))). In this non-inertial reference frame, an inertial force appears, applied to the passenger and directed oppositely to the acceleration of the car (that is, its speed): F i 2 → (\displaystyle (\vec (F_(i_(2))))). The force of inertia tends to cause, in a given frame of reference, the movement of the passenger’s body towards windshield.

However, the passenger's movement is impeded seat belt: Under the influence of the passenger's body, the belt stretches and exerts a corresponding force on the passenger. This reaction of the belt balances the force of inertia and the passenger in the reference frame associated with the car does not experience acceleration, remaining motionless relative to the car during the entire braking process.

From the point of view of an observer located in an arbitrary inertial frame of reference (for example, associated with the road), the passenger loses speed as a result of the force exerted on him by the belt. Thanks to this force, acceleration (negative) of the passenger occurs, its work causes a decrease kinetic energy passenger. It is clear that no inertial forces arise in the inertial reference frame, and they are not used to describe the passenger’s movement.

Examples of use

In some cases, it is convenient to use a non-inertial reference system in calculations, for example:

• It is convenient to describe the movement of moving parts of a car in a coordinate system associated with the car. If the car accelerates, this system becomes non-inertial;
• It is sometimes convenient to describe the movement of a body along a circular path in a coordinate system associated with this body. Such a coordinate system is non-inertial due to centripetal acceleration.

In non-inertial reference systems, standard formulations Newton's laws not applicable. Thus, when a car accelerates, in the coordinate system associated with the car body, loose objects inside receive acceleration in the absence of any force applied directly to them; and when a body moves in orbit, in the non-inertial coordinate system associated with the body, the body is at rest, although it is acted upon by an unbalanced gravitational force, which acts as centripetal in the inertial coordinate system in which the orbital rotation was observed.

To restore the possibility of applying in these cases the usual formulations of Newton’s laws and related equations of motion for each body under consideration it turns out to be convenient to introduce a fictitious force - inertia force- proportional to the mass of this body and the magnitude of the acceleration of the coordinate system, and opposite to the vector of this acceleration.

With the use of this fictitious power, it becomes possible brief description actually observed effects: “why is the passenger pressed against the back of the seat when accelerating a car?” - “the force of inertia acts on the passenger’s body.” In an inertial coordinate system associated with the road, inertial force is not required to explain what is happening: the passenger’s body accelerates in it (together with the car), and this acceleration is produced by a force with which the seat acts on the passenger.

Inertia force on the Earth's surface

Let F 1 → (\displaystyle (\vec (F_(1)))) is the sum of all forces acting on a body in a fixed (first) coordinate system, which causes its acceleration. This sum is found by measuring the acceleration of a body in this system if its mass is known.

Likewise, F 2 → (\displaystyle (\vec (F_(2)))) is the sum of forces, measured in a non-inertial coordinate system (second), causing acceleration a 2 → (\displaystyle (\vec (a_(2)))), which in general differs from a 1 → (\displaystyle (\vec (a_(1)))) due to the accelerated movement of the second CO relative to the first.

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

F i 2 → = F 2 → − F 1 → (\displaystyle (\vec (F_(i_(2))))=(\vec (F_(2)))-(\vec (F_(1))) ) (19)

F i 2 → = m (a 2 → − a 1 →) (\displaystyle (\vec (F_(i_(2))))=m((\vec (a_(2)))-(\vec (a_ (1))))) (20)

In particular, if the body is at rest in a non-inertial frame, that is a 2 → = 0 (\displaystyle (\vec (a_(2)))=0), That

F i 2 → = − F 1 → (\displaystyle (\vec (F_(i_(2))))=-(\vec (F_(1)))) (21) .

Movement of a body along an arbitrary trajectory in a non-inertial reference frame

The position of a material body in a conditionally stationary and inertial system is given here by the vector r → (\displaystyle (\vec (r))), and in a non-inertial system - by the vector r ′ → (\displaystyle (\vec (r^(\prime )))). The distance between the origins is determined by the vector R → (\displaystyle (\vec (R))). The angular speed of rotation of the system is specified by the vector ω → (\displaystyle (\vec (\omega ))), the direction of which is set along the axis of rotation along right screw rule. Linear speed body in relation to the rotating reference frame is given by the vector v → (\displaystyle (\vec (v))).

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

A r → = d 2 R → d t 2 + d ω → d t × r ′ → + 2 ω → × v → + ω → × [ ω → × r ′ → ] , (22) (\displaystyle (\vec (a_ (r)))=(\frac (d^(2)(\vec (R)))(dt^(2)))+(\frac (d(\vec (\omega )))(dt)) \times (\vec (r"))+(2(\vec (\omega ))\times (\vec (v)))+(\vec (\omega ))\times \left[(\vec (\ omega ))\times (\vec (r"))\right],\qquad (22))

• the first term is the portable acceleration of the second system relative to the first;

• the second term is the acceleration arising due to the uneven rotation of the system around its axis;

Work of inertia forces

In classical physics, inertial forces occur in two different situations depending on the reference system in which the observation is made. This is the force applied to the connection when observed in an inertial reference frame, or the force applied to the body in question when observed in a non-inertial reference frame. Both of these forces can do work. The exception is the Coriolis force, which does no work, since it is always directed perpendicular to the velocity vector. At the same time, the Coriolis force can change the trajectory of a body and, thereby, contribute to the performance of work by other forces (such as friction). An example of this would be Beer effect.

In addition, in some cases it may be advisable to divide the acting Coriolis force into two components, each of which does work. The total work performed by these components is zero, but such a representation may be useful in analyzing the processes of energy redistribution in the system under consideration.

At theoretical consideration, when the dynamic problem of motion is artificially reduced to a static problem, a third type of force is introduced, called d'Alembert's forces, which do not perform work due to the immobility of the bodies on which these forces act.

Inertia - the ability to maintain one's state unchanged is intrinsic property all material bodies.

Inertia force - a force that occurs during acceleration or deceleration of a body (material point) and is directed in the opposite direction from acceleration. The force of inertia can be measured; it is applied to “links” - bodies connected to an accelerating or decelerating body.

It is calculated that the inertial force is equal to

F in = | m*a|

Thus, the forces acting on material points m 1 And m 2(Fig. 14.1), when overclocking the platforms are respectively equal

F in1 = m 1 *a ; F in2 = m 2 *a

Accelerating body (platform with mass T(Fig. 14.1)) does not perceive the force of inertia, otherwise acceleration of the platform would be impossible at all.

During rotational motion (curvilinear), the resulting acceleration is usually represented in the form of two components: normal a p and tangent a t(Fig. 14.2).

Therefore, when considering curvilinear motion, two components of the inertia force may arise: normal and tangential

a = a t + a n ;

With uniform motion along an arc, normal acceleration always occurs; tangential acceleration is zero, therefore only the normal component of the inertial force, directed radially from the center of the arc, acts (Fig. 14.3).

The principle of kinetostatics (D'Alembert's principle)

The principle of kinetostatics is used to simplify the solution of a number of technical problems.

In reality, inertial forces are applied to bodies connected to the accelerating body (to connections).

d'Alembert suggested conditionally apply the force of inertia to an actively accelerating body. Then the system of forces applied to a material point becomes balanced, and it is possible to use statics equations when solving problems of dynamics.

D'Alembert's principle:

A material point under the influence of active forces, coupling reactions and a conditionally applied inertial force is in equilibrium;

End of work -

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Let solid under the influence of external forces rotates around the Oz axis with angular velocity

Voltages
The section method makes it possible to determine the value of the internal force factor in the section, but does not make it possible to establish the distribution law internal forces by section. To assess the strength of n

Internal force factors, tensions. Construction of diagrams
Have an idea of ​​longitudinal forces and normal stresses in cross sections. Know the rules for constructing diagrams of longitudinal forces and normal stresses, the distribution law

Longitudinal forces
Let us consider a beam loaded with external forces along its axis. The beam is fixed in the wall (fastening “fixing”) (Fig. 20.2a). We divide the beam into loading areas. Loading area with

Geometric characteristics of flat sections
Have an idea about physical sense and the procedure for determining the axial, centrifugal and polar moments of inertia, about the main central axes and the main central moments inertia.

Static moment of sectional area
Let's consider an arbitrary section (Fig. 25.1). If we divide the section into infinitesimal areas dA and multiply each area by the distance to the coordinate axis and integrate the resulting

Centrifugal moment of inertia
The centrifugal moment of inertia of a section is the sum of the products of elementary areas taken over both coordinates:

Axial moments of inertia
The axial moment of inertia of a section relative to a certain yard lying in the same plane is called the sum of the products of elementary areas taken over the entire area by the square of their distance

Polar moment of inertia of the section
The polar moment of inertia of a section relative to a certain point (pole) is the sum of the products of elementary areas taken over the entire area by the square of their distance to this point:

Moments of inertia of the simplest sections
Axial moments of inertia of a rectangle (Fig. 25.2) Imagine directly

Polar moment of inertia of a circle
For a circle, first calculate the polar moment of inertia, then the axial ones. Let's imagine a circle as a collection of infinitely thin rings (Fig. 25.3).

Torsional Deformation
Torsion of a round beam occurs when it is loaded with pairs of forces with moments in planes perpendicular to the longitudinal axis. In this case, the generatrices of the beam are bent and rotated through an angle γ,

Hypotheses for torsion
1. The hypothesis is fulfilled flat sections: the cross section of the beam, flat and perpendicular to the longitudinal axis, after deformation remains flat and perpendicular to the longitudinal axis.

Internal force factors during torsion
Torsion is a loading in which only one internal force factor appears in the cross section of the beam - torque. External loads are also two

Torque diagrams
Torque moments can vary along the axis of the beam. After determining the values ​​of the moments along the sections, we construct a graph of the torques along the axis of the beam.

Torsional stress
We draw a grid of longitudinal and transverse lines on the surface of the beam and consider the pattern formed on the surface after Fig. 27.1a deformation (Fig. 27.1a). Pop

Maximum torsional stresses
From the formula for determining stresses and the diagram of the distribution of tangential stresses during torsion, it is clear that the maximum stresses occur on the surface. Let's determine the maximum voltage

Types of strength calculations
There are two types of strength calculations: 1. Design calculation - the diameter of the beam (shaft) in the dangerous section is determined:

Stiffness calculation
When calculating rigidity, the deformation is determined and compared with the permissible one. Let us consider the deformation of a round beam under the action of an external pair of forces with a moment t (Fig. 27.4).

Basic definitions
Bending is a type of loading in which an internal force factor—a bending moment—appears in the cross section of the beam. Timber working on

Internal force factors during bending
Example 1. Consider a beam that is acted upon by a pair of forces with a moment m and an external force F (Fig. 29.3a). To determine internal force factors, we use the method with

Bending moments
A transverse force in a section is considered positive if it tends to rotate it

Differential dependencies for direct transverse bending
The construction of diagrams of shear forces and bending moments is greatly simplified by using differential relationships between the bending moment, shear force and uniform intensity

Using the section method The resulting expression can be generalized
The transverse force in the section under consideration is equal to algebraic sum of all forces acting on the beam up to the section under consideration: Q = ΣFi Since we are talking

Voltages
Let us consider the bending of a beam clamped to the right and loaded with a concentrated force F (Fig. 33.1).

Stress state at a point
The stress state at a point is characterized by normal and tangential stresses that arise on all areas (sections) passing through this point. Usually it is enough to determine for example

The concept of a complex deformed state
A set of deformations arising in various directions and in different planes, passing through a point, determine the deformed state at this point. Complex deformation

Calculation of a round beam for bending with torsion
In the case of calculating a round beam under the action of bending and torsion (Fig. 34.3), it is necessary to take into account normal and shear stresses, because maximum values stresses in both cases arose

The concept of stable and unstable equilibrium
Relatively short and massive rods are designed for compression, because they fail as a result of destruction or residual deformations. Long rods of a small cross section under the day

Stability calculation
The stability calculation consists of determining the permissible compressive force and, in comparison with it, the acting force:

Calculation using Euler's formula
The problem of determining the critical force was mathematically solved by L. Euler in 1744. For a rod hinged on both sides (Fig. 36.2), Euler’s formula has the form

Critical stresses
Critical stress is the compressive stress corresponding to the critical force. The stress from the compressive force is determined by the formula

Limits of applicability of Euler's formula
Euler's formula is valid only within the limits of elastic deformations. Thus, the critical stress must be less than the elastic limit of the material. Prev

Inertial and non-inertial reference systems

Newton's laws are satisfied only in inertial frames of reference. Relative to all inertial systems, this body moves with the same acceleration $w$. Any non-inertial frame of reference moves relative to the inertial frames with some acceleration, therefore the acceleration of the body in the non-inertial frame of reference $w"$ will be different from $w$. Let us denote the difference in the acceleration of the body in both the inertial and non-inertial frames by the symbol $a$:

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

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

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

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

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 it were acted upon by a force equal to $-ma$.

This means that when describing motion in non-inertial frames of reference, one can use Newton’s equations if, along with the forces caused by the influence of bodies on each other, one takes into account the so-called inertial forces $F_(in) $, which should be assumed equal to the product mass of a body by the difference of its accelerations taken with the opposite sign in relation to the inertial and non-inertial reference systems:

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

Let us clarify our statement following example. Let's consider a cart with a bracket attached to it, from which a ball is suspended by a thread.

Figure 1.

While the cart is at rest or moving without acceleration, the thread is located vertically and the force of gravity $P$ is balanced by the reaction of the thread $F_(r)$. Now let's put the cart in translational motion with acceleration $a$. The thread will deviate from the vertical at such an angle that the resulting forces $P$ and $F_(r)$ impart an acceleration to the ball equal to $a$. With respect 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 absence of acceleration of the ball with respect to this reference frame 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 acted upon by the inertial force $F_(in) = -ma$.

Inertia forces and their properties

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

Note 1

It should be clearly understood that inertial forces cannot be put on a par with forces such as elastic, gravitational forces and frictional forces, i.e. forces caused by the influence of other bodies on the body. Inertial forces are determined by the properties of the reference system in which they are considered. mechanical phenomena. In this sense, they can be called fictitious forces.

The introduction of inertial forces into consideration is not fundamentally necessary. In principle, any movement can always be considered in relation to an inertial reference frame. However, in practice, it is often the motion of bodies with respect to non-inertial reference systems, 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 in relation to such a reference system, which often turns out to be much simpler than considering motion in an inertial frame.

A characteristic property of inertial forces is their proportionality to the mass of the body. Thanks to this property, the forces of inertia turn out to be similar to the forces of gravity. Let's imagine that we are in a closed cabin remote from all external bodies, which moves with acceleration g in the direction that we will call “top”.

Figure 2.

Then all bodies located 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 inertia force $-mg$. However, the same phenomena would have been observed if the cabin had been stationary and located near the surface of the Earth. Without the opportunity to “look” outside the cabin, no experiments carried out inside the cabin would allow us to establish whether the $-mg$ force is due to the accelerated movement of the cabin or the action gravitational field Earth. On this basis they speak of the equivalence of the forces of inertia and gravity. This equivalence underlies general theory Einstein's relativity.

Example 1

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

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

Find: $l-$?

Solution: B earth system reference point, the Coriolis inertial force acts on a freely falling body:

\, \]

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 speed of the body’s movement relative to Earth.

The Coriolis inertial force is many times less than the gravitational force of a body towards the Earth. Therefore, to a first approximation, when determining $F_(k) $, we can assume that the speed $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.

From the figure you can see 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 tangential to the Earth’s surface, $l$ is the displacement of the 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 body's fall $t=0,v=0,l=0$, therefore the integration constants are equal to zero and then we have:

Duration free fall bodies from height $h$:

so the desired deviation of the body to the east is:

$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.

Perhaps this unusual question will cause confusion among the average person who is new to the basic postulates classical mechanics. The expressions “inertia” and “by inertia” are firmly entrenched in the everyday lexicon, and it would seem that their essence is clear to everyone. But what is inertia, and not everyone can explain why bodies can move by inertia.

Let's try to understand this issue using the basic postulates of mechanics and more or less scientific knowledge about the world around us.

First, we will conduct virtual experiments, the results of which can be presented by everyone.
Let a heavy cast-iron ball (for example, a large cannonball) rest in front of us on a smooth horizontal floor, and one of the “experimenters” tries to roll it in any direction, resting his feet on the floor and pushing with his hands.
First, we will have to make a significant effort to move the ball from its place, after which it will begin to confidently roll in the direction you have chosen, and if we stop pushing it, it will continue to roll (for the purity of the experiment, we will leave the forces of friction and aerodynamic resistance without virtual attention for now ).

Now, on the contrary, try to stop this ball by grabbing it with your hands and using your legs as a brake. Do you feel resistance?.. I think so.
At the same time, no one will deny that the more massive the ball, the more difficult it is to change its mechanical state, that is, to move or stop.
So, the conclusion is that moving a stationary ball or stopping it while moving is quite difficult - you need to make a noticeable effort. From a mechanical point of view, in this case we are making an effort to overcome some incomprehensible force.

Let's take a closer look at our core resting on the floor. From the point of view of classical mechanics, again, only two forces are applied to it - the force of gravity, which attracts the ball to the center of our planet, and also the force of the floor reaction, which counteracts the force of gravity, i.e., directed opposite to it.
When our ball rolls on a smooth floor at a constant speed, it is also acted upon only by the two forces described above - attraction to the Earth and the reaction of the supporting surface. Both of these forces balance each other, and the ball is in equilibrium state. And what force prevents an attempt to move the ball from its place or stop it during a straight and uniform movement?
I think that the smartest ones have already guessed - of course, this is the force of inertia.
Where did she come from? After all, in fact, we applied only one force to the ball, trying to move or stop the ball. Where has the force of inertia been hiding until now and when did it “awaken”?

Textbooks on mechanics claim that the force of inertia, as such, does not exist in nature. The concept of this force was introduced into scientific use by the Frenchman Jean Leron d'Alembert (D'Alembert) in 1743, when he proposed using it to balance bodies moving with acceleration. The method was called d'Alembert's principle, and was used to transform problems of dynamics into problems of statics, thereby simplifying their solution.
But this solution to the problem was not explained and even came into conflict with other postulates of mechanics, in particular, with the laws described somewhat earlier by the great Englishman, Isaac Newton.

When in 1686 I. Newton published his work “Mathematical Principles of Natural Philosophy” and opened mankind’s eyes to the basic laws of mechanics, including the law describing the movement of bodies under the influence of any force ( F = ma), he expanded somewhat as measures of a certain property of material bodies - inertia.
In accordance with the conclusions of the genius, all material bodies around us have a certain property of “laziness” - they strive for eternal peace, trying to get rid of accelerated movement. Newton called this “laziness” of material bodies inertia.
That is, inertia is not a force, but a certain property of all bodies that form the environment around us material world, expressed in opposition to attempts to change their mechanical state (to give any acceleration).
However, it would not be entirely fair to attribute the merits of explaining the nature of inertia to Newton alone. The fundamental conclusions on this issue were made by the Italian G. Galileo and the Frenchman R. Descartes, and I. Newton only generalized them and used them in the description of the laws of mechanics.

According to the thoughts of medieval geniuses, material bodies(i.e. bodies with mass) are extremely reluctant to allow their mechanical state to be changed, agreeing to this only under the influence external force. At the same time, the same Newton, describing the laws of interaction of bodies, argued that forces in nature do not appear alone - they, as a result of the interaction of two bodies, appear only in pairs, and both forces of such a pair are equal in magnitude and directed along the same straight line towards each other , i.e. compensate each other in pairs.

Based on this, in the case of a cast iron ball there should also be two forces - the effort of the experimenter and the force counteracting this effort, due to the above-mentioned property of inertia of this ball.
But the strength general concepts classical mechanics is the result of the interaction of bodies. And no property of the body, in accordance with this postulate, can be the cause of the appearance of any force.

The contradiction with Newton's laws led to the emergence of the concepts in the scientific community inertial and non-inertial reference systems.
Inertial began to be called a frame of reference in which all bodies, in the absence external influences are at rest, and non-inertial - all other reference systems relative to which bodies move with acceleration. At the same time, in an inertial frame of reference, the laws of mechanics described by Newton are observed unconditionally, but in a non-inertial frame they are not observed.
However, all the laws of classical mechanics can be applied to non-inertial frames of reference, if, along with the real active forces(loads and reactions) use the force of inertia – virtual power, due to the same unfortunate property of inertia of bodies.

Thus, it was possible to get rid of the contradiction arising from the nature of the emergence of forces described by Newton, and to achieve conditional equilibrium of bodies under any accelerated motion, using d'Alembert's principle.
The force of inertia gained the right to exist, and physicists began to study it more closely, without fear of being ridiculed by their colleagues.

The emergence of inertial forces is directly related to the acceleration of the body - in a state of rest (immobility or rectilinear uniform motion bodies) these forces do not arise and appear only in non-inertial frames of reference. In this case, the magnitude of the inertial force is equal in magnitude and oppositely directed to the force causing the acceleration of the body, so they mutually balance each other.

IN real world any body is affected by inertial forces, i.e. the concept of an inertial frame of reference is abstract. But in many practical situations, one can conditionally accept the reference system as inertial, which makes it possible to simplify the solution of problems related to mechanical movement material bodies.

Relationship between inertia and gravity

Even G. Galileo pointed out some connection between the concepts of inertia and gravity.

The inertial forces acting on bodies in a non-inertial frame of reference are proportional to their masses and other equal conditions give these bodies the same accelerations. Therefore, under the same conditions in the “field of inertial forces,” these bodies move in exactly the same way. And the same property is possessed by bodies under the influence of the forces of the gravitational field.

For this reason, in some conditions, inertial forces are associated with gravitational forces. For example, the movement of bodies in a uniformly accelerated elevator occurs in exactly the same way as in a stationary elevator hanging in a uniform field of gravity. No experiment performed inside an elevator can separate a uniform gravitational field from uniform field inertial forces.

The analogy between gravitational forces and inertial forces underlies the principle of equivalence of gravitational forces and inertial forces (Einstein’s equivalence principle): all physical phenomena in the gravitational field occur in exactly the same way as in the corresponding field of inertial forces, if the strengths of both fields at the corresponding points in space coincide, and the rest initial conditions for the bodies under consideration are the same.
This principle forms the basis of the general theory of relativity.

What are the types of inertial forces?

Inertial forces are caused by the accelerated movement of the reference system relative to the measured system, therefore, in the general case, the following cases of manifestation of these forces must be taken into account:

• inertia forces at accelerated forward movement reference systems (determined by translational acceleration);
• inertial forces acting on a body at rest in a rotating reference frame (due to centrifugal acceleration);
• inertial forces acting on a body moving in a rotating frame of reference (due to translational and centrifugal acceleration, as well as Coriolis acceleration).

By the way, the term “inertia” has Latin origin- word " inertia" means inactivity.