Inertia force

The deviation of the seats on the ride can be explained using centrifugal inertia

The force of inertia (also the inertial force ) is a multi-valued concept used in mechanics in relation to three different physical quantities . One of them is the “ d'Alembert force of inertia”- is introduced in inertial reference systems to obtain the formal possibility of writing dynamics equations in the form of simpler static equations. The other is the “ Euler inertia force” - used when considering the motion of bodies in non-inertial reference systems . Finally, the third - " Newtonian force of inertia" is the counteraction force considered in connection with Newton’s third law ^[1] ^[2] ^[3] .

Common to all three quantities is their vector character and dimension of force . In addition, the first two quantities are united by the possibility of their use in equations of motion, in the form of coinciding with the equation of Newton’s second law ^[1] ^[4] ^[5] , as well as their proportionality to the mass of bodies ^[6] ^[4] ^[5] .

Terminology

The Russian term "force of inertia" came from the French phrase fr. force d'inertie . The term is used to describe three different vector physical quantities having a dimension of force:

the value that is introduced when describing the motion of bodies in a non-inertial reference system - "Euler's force of inertia";
the value used in the D'Alembert principle - “d'Alembert force of inertia”;
force-counteraction from the third law of Newton - "Newtonian force of inertia."

The definitions of "Euler", "d'Alembert" and "Newton" were proposed by academician A. Yu. Ishlinsky ^[7] ^[8] . They are used in the literature, although they have not yet received widespread distribution. In the future, we will adhere to this terminology, as it allows us to make the presentation more concise and clear.

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

In other languages, the used names of inertia forces more clearly indicate their special properties: in German it. Scheinkraft ^[9] ("imaginary", "apparent", "visible", "false", "fictitious" force), in English English. pseudo force ^[10] (" pseudo-force ") or English. fictitious force. Less commonly used in English are the names “ d'Alembert force ” ( English d'Alembert force ^[11] ) and “inertial force” ( English inertial force ^[12] ). The literature published in Russian also uses similar characteristics with respect to the Euler and d'Alembert forces, calling these forces “fictitious” ^[13] , “apparent” ^[14] , “imaginary” ^[8] or “pseudo-forces” ^[15] .

At the same time, the literature sometimes emphasizes the reality of inertia forces ^[16] ^[17] , contrasting the meaning of this term with the meaning of the term fictitiousness . At the same time, however, various authors put different meanings in these words, and the forces of inertia turn out to be real or fictitious not because of differences in understanding their basic properties, but depending on the chosen definitions. Some authors consider this use of terminology unsuccessful and recommend simply avoiding it in the educational process ^[18] ^[19] .

Although the discussion of terminology is not yet complete, the differences do not affect the mathematical formulation of the equations of motion involving inertia forces and do not lead to any misunderstandings when using the equations in practice.

Forces in Classical Mechanics

In classical mechanics, ideas about forces and their properties are based on Newton’s laws and are inextricably linked with the concept of “ inertial reference frame ”.

Indeed, a physical quantity called force is introduced into Newton’s second law, while the law itself is formulated only for inertial reference systems ^[20] . Accordingly, the concept of force is determined only for such reference frames ^[21] .

The equation of Newton’s second law relating acceleration ${\vec {a}}$ ${\ displaystyle {\ vec {a}}}$ and $m$ ${\ displaystyle m}$ mass of material point with force acting on it ${\vec {F}}$ ${\ displaystyle {\ vec {F}}}$ is written as

{\vec {a}}={\frac {\vec {F}}{m}}.

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

It directly follows from the equation that only the forces are the cause of the acceleration of bodies, and vice versa: the action on the body of uncompensated forces necessarily causes its acceleration.

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

Taking into account the content of all Newton's laws leads to the conclusion that the forces referred to in classical mechanics have inherent properties:

force is a measure of mechanical action on a given material body of other bodies. ^[22]

in accordance with Newton’s third law, forces can exist only in pairs, while the nature of the forces in each such pair is the same ^[23] ^[24] .

any force acting on the body has a source of origin in the form of another body. In other words, forces are necessarily the result of the interaction of bodies ^[25] .

No other forces in classical mechanics are introduced or used ^[21] ^[26] . The possibility of the existence of forces that arose independently, without interacting bodies, is not allowed by mechanics ^[25] ^[27] .

Although in the names of the Euler and d'Alembert forces of inertia the word force is contained, these physical quantities are not forces in the sense accepted in mechanics ^[28] ^[15] .

Equations of Motion

Consider the motion of a material point $M$ ${\ displaystyle M}$ in an inertial reference system $K$ ${\ displaystyle K}$ starting at point $O$ ${\ displaystyle O}$ and in an arbitrary non-inertial reference frame $K^{'}$ ${\ displaystyle K '}$ starting at point $O^{'}$ ${\ displaystyle O '}$ that moves relative to the system $K$ ${\ displaystyle K}$ progressively accelerated ${\vec {W_{O'}}}$ ${\ displaystyle {\ vec {W_ {O '}}}}$ and rotates at instant speed ${\vec {\omega }}$ ${\ displaystyle {\ vec {\ omega}}}$ . Arbitrary point radius vectors $M$ ${\ displaystyle M}$ relative to these reference systems are related by the relations: ${\vec {r}}={\vec {r'}}+{\vec {R_{O'}}}(t)$ ${\ displaystyle {\ vec {r}} = {\ vec {r '}} + {\ vec {R_ {O'}}} (t)}$ . It can be shown that the equations of motion in a non-inertial reference frame have the form:

m{\frac {d'{\vec {v'}}}{dt}}=-m{\vec {W_{O'}}}-m\left[{\vec {\omega }}\left[{\vec {\omega }}{\vec {r'}}\right]\right]-m\left[{\vec {\dot {\omega }}}{\vec {r'}}\right]-2m\left[{\vec {\omega }}{\vec {v'}}\right]+{\vec {F}}

{\ displaystyle m {\ frac {d '{\ vec {v'}}} {dt}} = - m {\ vec {W_ {O '}}} - m \ left [{\ vec {\ omega}} \ left [{\ vec {\ omega}} {\ vec {r '}} \ right] \ right] -m \ left [{\ vec {\ dot {\ omega}}} {\ vec {r'}} \ right] -2m \ left [{\ vec {\ omega}} {\ vec {v '}} \ right] + {\ vec {F}}}

.

We introduce the following notation:

{\vec {J^{(p)}}}=-m{\vec {W_{p}}}=-m{\vec {W_{O'}}}-m\left[{\vec {\dot {\omega }}}{\vec {r'}}\right]-m\left[{\vec {\omega }}\left[{\vec {\omega }}{\vec {r'}}\right]\right]

{\ displaystyle {\ vec {J ^ {(p)}}} = - m {\ vec {W_ {p}}} = - m {\ vec {W_ {O '}}} - m \ left [{\ vec {\ dot {\ omega}}} {\ vec {r '}} \ right] -m \ left [{\ vec {\ omega}} \ left [{\ vec {\ omega}} {\ vec {r '}} \ right] \ right]}

{\vec {J^{(k)}}}=-m{\vec {W_{k}}}=2m\left[{\vec {\omega }}{\vec {v'}}\right]

{\ displaystyle {\ vec {J ^ {(k)}}} = - m {\ vec {W_ {k}}} = 2m \ left [{\ vec {\ omega}} {\ vec {v '}} \ right]}

That is, the equation of motion relative to an arbitrary non-inertial reference frame can be represented as:

m{\vec {w'}}={\vec {F}}+{\vec {J^{(p)}}}+{\vec {J^{(k)}}}

{\ displaystyle m {\ vec {w '}} = {\ vec {F}} + {\ vec {J ^ {(p)}}} {{\ vec {J ^ {(k)}}}}

Vector ${\vec {J^{(p)}}}$ ${\ displaystyle {\ vec {J ^ {(p)}}}}$ called the inertia force of the particle’s transport . It is the sum of two components: inertia forces $-m{\vec {W_{O'}}}$ ${\ displaystyle -m {\ vec {W_ {O '}}}}$ from portable translational motion of a particle and inertia $-m\left[{\vec {\dot {\omega }}}{\vec {r'}}\right]-m\left[{\vec {\omega }}\left[{\vec {\omega }}{\vec {r'}}\right]\right]$ ${\ displaystyle -m \ left [{\ vec {\ dot {\ omega}}} {\ vec {r '}} \ right] -m \ left [{\ vec {\ omega}} \ left [{\ vec {\ omega}} {\ vec {r '}} \ right] \ right]}$ from its figurative rotational motion, in turn, consisting of inertia $-m\left[{\vec {\dot {\omega }}}{\vec {r'}}\right]$ ${\ displaystyle -m \ left [{\ vec {\ dot {\ omega}}} {\ vec {r '}} \ right]}$ due to uneven rotation of the reference system $K^{'}$ ${\ displaystyle K '}$ and centrifugal inertia $-m\left[{\vec {\omega }}\left[{\vec {\omega }}{\vec {r'}}\right]\right]$ ${\ displaystyle -m \ left [{\ vec {\ omega}} \ left [{\ vec {\ omega}} {\ vec {r '}} \ right] \ right]}$ lying in the plane of vectors ${\vec {r'}}$ ${\ displaystyle {\ vec {r '}}}$ and ${\vec {\omega }}$ ${\ displaystyle {\ vec {\ omega}}}$ and directed perpendicular to the instantaneous axis of rotation.

Vector ${\vec {J^{(k)}}}$ ${\ displaystyle {\ vec {J ^ {(k)}}}}$ called the Coriolis force of inertia . It is perpendicular to the plane of vectors ${\vec {r'}}$ ${\ displaystyle {\ vec {r '}}}$ and ${\vec {\omega }}$ ${\ displaystyle {\ vec {\ omega}}}$ .

Newtonian inertia forces

Some authors use the term “inertia force” to refer to the force-reaction of Newton’s third law . The concept was introduced by Newton in his "Mathematical Principles of Natural Philosophy" ^[30] : "The innate force of matter is its inherent ability of resistance, according to which each separately taken body, since it is left to itself, maintains its state of rest or uniform rectilinear motion. It follows from the inertia of matter that every body is only difficult to get out of its rest or movement. Therefore, an innate force could be very intelligently 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“ inertia force ”itself, according to Euler , was first used in this meaning by Kepler ( ^[30] , with reference to E. L. Nikolai )

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

An echo of Newton’s choice of the word “resistance” for describing inertia is also an idea of a certain force that supposedly realizes this property in the form of resistance to changes in motion parameters. In this regard, Maxwell noted that with the same success it could be said that coffee resists to become sweet, since it does not become sweet by itself, but only after adding sugar ^[30] .

Euler inertia forces

Non-inertial JI Movement

Describing the motion in inertial CO and non-inertial CO, we will have ${\vec {r}}={\vec {R}}+{\vec {r}}{^{\prime }}={\vec {R}}+\mathbf {A} {\vec {\rho }}$ ${\ displaystyle {\ vec {r}} = {\ vec {R}} + {\ vec {r}} {^ {\ prime}} = {\ vec {R}} + \ mathbf {A} {\ vec {\ rho}}}$ where $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ there is a transition matrix from a moving reference frame to a fixed one. Differentiating, we obtain:

${\vec {v_{r}}}={\dot {\vec {R}}}+{\dot {\mathbf {A} }}{\vec {\rho }}+\mathbf {A} {\dot {\vec {\rho }}}={\vec {v_{e}}}+{\vec {v_{r^{\prime }}}}$ ${\ displaystyle {\ vec {v_ {r}}} = {\ dot {\ vec {R}}} + {\ dot {\ mathbf {A}}} {\ vec {\ rho}} + \ mathbf {A } {\ dot {\ vec {\ rho}}} = {\ vec {v_ {e}}} + {\ vec {v_ {r ^ {\ prime}}}}}$ (ten),

Where:

{\vec {v_{r}}}={\dot {\vec {r}}}

{\ displaystyle {\ vec {v_ {r}}} = {\ dot {\ vec {r}}}}

is the velocity of the body in inertial CO, hereinafter referred to as absolute velocity.

{\vec {v_{e}}}={\dot {\vec {R}}}+{\dot {\mathbf {A} }}{\vec {\rho }}={\dot {\vec {R}}}+{\vec {\omega }}\times \mathbf {A} {\vec {\rho }}={\dot {\vec {R}}}+{\vec {\omega }}\times {\vec {r}}{^{\prime }}

{\ displaystyle {\ vec {v_ {e}}} = {\ dot {\ vec {R}}} + {\ dot {\ mathbf {A}}} {\ vec {\ rho}} = {\ dot { \ vec {R}}} + {\ vec {\ omega}} \ times \ mathbf {A} {\ vec {\ rho}} = {\ dot {\ vec {R}}} + {\ vec {\ omega }} \ times {\ vec {r}} {^ {\ prime}}}

is the velocity of a non-inertial CO point with a radius vector

{\vec {r}}{^{\prime }}=\mathbf {A} {\vec {\rho }}

{\ displaystyle {\ vec {r}} {^ {\ prime}} = \ mathbf {A} {\ vec {\ rho}}}

in inertial CO, hereinafter referred to as transport speed, consisting of transport speed

{\dot {\vec {R}}}

{\ displaystyle {\ dot {\ vec {R}}}}

translational motion of non-inertial RM relative to inertial RM and carrying speed

{\dot {\mathbf {A} }}{\vec {\rho }}={\vec {\omega }}\times {\vec {r}}{^{\prime }}

{\ displaystyle {\ dot {\ mathbf {A}}} {\ vec {\ rho}} = {\ vec {\ omega}} \ times {\ vec {r}} {^ {\ prime}}}

associated with the rotation of a non-inertial system with an angular velocity

{\vec {\omega }}

{\ displaystyle {\ vec {\ omega}}}

{\vec {v_{r^{\prime }}}}=\mathbf {A} {\dot {\vec {\rho }}}

{\ displaystyle {\ vec {v_ {r ^ {\ prime}}} = \ mathbf {A} {\ dot {\ vec {\ rho}}}}

is the velocity of the body in non-inertial CO, hereinafter referred to as relative velocity.

Differentiating the sum (1), we obtain ${\dot {\vec {v_{r}}}}={\dot {\vec {v_{e}}}}+{\dot {\vec {v_{r^{\prime }}}}}$ ${\ displaystyle {\ dot {\ vec {v_ {r}}}} = {\ dot {\ vec {v_ {e}}}} + {\ dot {\ vec {v_ {r ^ {\ prime}}} }}}$ . Insofar as

{\dot {\vec {v_{e}}}}={\ddot {\vec {R}}}+{\dot {\vec {\omega }}}\times \mathbf {A} {\vec {\rho }}+{\vec {\omega }}\times {\dot {\mathbf {A} }}{\vec {\rho }}+{\vec {\omega }}\times \mathbf {A} {\dot {\vec {\rho }}}={\ddot {\vec {R}}}+{\dot {\vec {\omega }}}\times {\vec {r}}'+{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}')+{\vec {\omega }}\times \mathbf {A} {\dot {\vec {\rho }}}

{\ displaystyle {\ dot {\ vec {v_ {e}}}} = {\ ddot {\ vec {R}}} + {\ dot {\ vec {\ omega}}} times \ mathbf {A} { \ vec {\ rho}} + {\ vec {\ omega}} \ times {\ dot {\ mathbf {A}}} {\ vec {\ rho}} + {\ vec {\ omega}} \ times \ mathbf {A} {\ dot {\ vec {\ rho}}} = {\ ddot {\ vec {R}}} + {\ dot {\ vec {\ omega}}} times {\ vec {r}} ' + {\ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} ') + {\ vec {\ omega}} \ times \ mathbf {A} {\ dot {\ vec {\ rho}}}}

,

{\dot {\vec {v_{r^{\prime }}}}}={\dot {\mathbf {A} }}{\dot {\vec {\rho }}}+\mathbf {A} {\vec {\ddot {\rho }}}={\vec {\omega }}\times \mathbf {A} {\dot {\vec {\rho }}}+\mathbf {A} {\vec {\ddot {\rho }}}

{\ displaystyle {\ dot {\ vec {v_ {r ^ {\ prime}}}} = {\ dot {\ mathbf {A}}} {\ dot {\ vec {\ rho}}} + \ mathbf { A} {\ vec {\ ddot {\ rho}}} = {\ vec {\ omega}} \ times \ mathbf {A} {\ dot {\ vec {\ rho}}} + \ mathbf {A} {\ vec {\ ddot {\ rho}}}}

,

we have:

${\vec {a_{r}}}={\vec {a_{e}}}+{\vec {a_{c}}}+{\vec {a_{r^{\prime }}}}$ ${\ displaystyle {\ vec {a_ {r}}} = {\ vec {a_ {e}}} + {\ vec {a_ {c}}} + {\ vec {a_ {r ^ {\ prime}}} }}$ (11) where:

{\vec {a_{r}}}={\ddot {\vec {r}}}

{\ displaystyle {\ vec {a_ {r}}} = {\ ddot {\ vec {r}}}}

is the acceleration of the body in inertial CO, hereinafter called absolute acceleration.

{\vec {a_{e}}}={\ddot {\vec {R}}}+{\dot {\vec {\omega }}}\times {\vec {r}}'+{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}')

{\ displaystyle {\ vec {a_ {e}}} = {\ ddot {\ vec {R}}} + {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} '+ { \ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} ')}

there is an acceleration of a non-inertial CO point with a radius vector

{\vec {r}}{^{\prime }}=\mathbf {A} {\vec {\rho }}

{\ displaystyle {\ vec {r}} {^ {\ prime}} = \ mathbf {A} {\ vec {\ rho}}}

in inertial CO, hereinafter called portable acceleration, consisting of portable acceleration

{\ddot {\vec {R}}}

{\ displaystyle {\ ddot {\ vec {R}}}}

translational motion of non-inertial RM relative to inertial RM, portable acceleration

{\dot {\vec {\omega }}}\times {\vec {r}}{^{\prime }}

{\ displaystyle {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} {^ {\ prime}}}

related to the non-uniform rotation of the non-inertial system and centripetal portable acceleration

{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}')

{\ displaystyle {\ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} ')}

.

{\vec {a_{c}}}=2({\vec {\omega }}\times \mathbf {A} {\dot {\vec {\rho }}})=2({\vec {\omega }}\times {\vec {v_{r^{\prime }}}})

{\ displaystyle {\ vec {a_ {c}}} = 2 ({\ vec {\ omega}} \ times \ mathbf {A} {\ dot {\ vec {\ rho}}}) = 2 ({\ vec {\ omega}} \ times {\ vec {v_ {r ^ {\ prime}}}}}

there is the so-called Coriolis acceleration of non-inertial CO in inertial CO, hereinafter called portable acceleration.

{\vec {a_{r^{\prime }}}}=\mathbf {A} {\vec {\ddot {\rho }}}

{\ displaystyle {\ vec {a_ {r ^ {\ prime}}} = \ mathbf {A} {\ vec {\ ddot {\ rho}}}}

is the acceleration of the body in non-inertial CO, hereinafter referred to as relative acceleration.

Multiply both sides of equation (11) by body weight $m$ ${\ displaystyle m}$ and get:

$m{\vec {a_{r}}}=m{\vec {a_{e}}}+m{\vec {a_{c}}}+m{\vec {a_{r^{\prime }}}}$ ${\ displaystyle m {\ vec {a_ {r}}} = m {\ vec {a_ {e}}} + m {\ vec {a_ {c}}} + m {\ vec {a_ {r ^ {\ prime}}}}}$ (12)

In accordance with Newton’s second law formulated for inertial systems, the term on the left is the result of multiplying the mass by a vector defined in the inertial system, and therefore it is possible to relate the real force to it:

$m{\vec {a_{r}}}={\vec {F_{r}}}$ ${\ displaystyle m {\ vec {a_ {r}}} = {\ vec {F_ {r}}}}$ . This is the force acting on the body in the first (inertial) CO, which will be called "absolute force" here. It continues to act on the body with constant direction and magnitude in any coordinate system.

Desiring to bring the law of motion in a non-inertial reference frame to a form similar to Newton's law, all terms from the right-hand side of (12), except the last, are transferred to the left-hand side. The result is

$m{\vec {a_{r^{\prime }}}}={\vec {F_{r}}}-m{\vec {a_{e}}}-m{\vec {a_{c}}}={\vec {F_{r}}}+{\vec {I_{e}}}+{\vec {I_{c}}}$ ${\ displaystyle m {\ vec {a_ {r ^ {\ prime}}} = {\ vec {F_ {r}}} - m {\ vec {a_ {e}}} - m {\ vec {a_ { c}}} = {\ vec {F_ {r}}} + {\ vec {I_ {e}}} + {\ vec {I_ {c}}}}$ (13)

Where:

{\vec {I_{e}}}=-m{\vec {a_{e}}}

{\ displaystyle {\ vec {I_ {e}}} = - m {\ vec {a_ {e}}}}

- portable inertia force,

{\vec {I_{c}}}=-m{\vec {a_{c}}}

{\ displaystyle {\ vec {I_ {c}}} = - m {\ vec {a_ {c}}}}

- Coriolis force of inertia,

The next force, defined as:

Important that acceleration ${\vec {a_{e}}}$ ${\ displaystyle {\ vec {a_ {e}}}}$ and ${\vec {a_{c}}}$ ${\ displaystyle {\ vec {a_ {c}}}}$ in the general case, it has nothing to do with the body under study, since it is caused by those forces that act only on the body selected as a non-inertial reference frame. But the mass included in the expressions ${\vec {I_{e}}}=-m{\vec {a_{e}}}$ ${\ displaystyle {\ vec {I_ {e}}} = - m {\ vec {a_ {e}}}}$ and ${\vec {I_{c}}}=-m{\vec {a_{c}}}$ ${\ displaystyle {\ vec {I_ {c}}} = - m {\ vec {a_ {c}}}}$ , there is a mass of the studied body. In view of the artificial nature of the introduction of such forces, they should be considered fictitious. These forces are not in their origin the result of the action of surrounding bodies and fields, and arise solely due to the accelerated motion of the second reference system relative to the first.

From (13) it is clear that due to acceleration in the new, non-inertial, reference frame, the force no longer acts on the body ${\vec {F_{r}}}$ ${\ displaystyle {\ vec {F_ {r}}}}$ , but ${\vec {F^{\prime }}}={\vec {F_{r}}}+{\vec {I_{e}}}+{\vec {I_{c}}}$ ${\ displaystyle {\ vec {F ^ {\ prime}}} = {\ vec {F_ {r}}} + {\ vec {I_ {e}}} + {\ vec {I_ {c}}}}$ , and the law of motion is the same as in inertial CO:

${\vec {F^{\prime }}}=m{\vec {a_{r^{\prime }}}}$ ${\ displaystyle {\ vec {F ^ {\ prime}}} = m {\ vec {a_ {r ^ {\ prime}}}}$ . (14)

Example 1

Let the second RM move at a constant speed or simply motionless in inertial RM. Then ${\vec {a_{e}}}={\vec {a_{e}}}=0$ ${\ displaystyle {\ vec {a_ {e}}} = {\ vec {a_ {e}}} = 0}$ and inertia forces are absent. A moving body experiences acceleration caused by real forces acting on it.

Example 2

Let some external force act on the body, although the body is motionless in the second SD. Then ${\vec {a_{e}}}={\vec {a_{r}}}$ ${\ displaystyle {\ vec {a_ {e}}} = {\ vec {a_ {r}}}}$ , that is, this CO is actually combined with a 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 portable inertia force:

${\vec {I_{e}}}=-{\vec {F_{r}}}$ ${\ displaystyle {\ vec {I_ {e}}} = - {\ vec {F_ {r}}}}$

Example 3

Passenger rides in a passenger car at a constant speed. A passenger is a body, a car is its reference system (while inertial), i.e. ${\vec {F_{r}}}=0$ ${\ displaystyle {\ vec {F_ {r}}} = 0}$ .

The car starts to brake and turns for the passenger into the second non-inertial system discussed above, to which the braking force is applied towards its movement ${\vec {F_{R}}}$ ${\ displaystyle {\ vec {F_ {R}}}}$ . In this non-inertial reference frame, an inertia force appears, applied to the passenger and directed opposite to the acceleration of the car (that is, by its speed): ${\vec {I_{e}}}$ ${\ displaystyle {\ vec {I_ {e}}}}$ . The force of inertia tends to cause in this reference frame the movement of the passenger’s body towards the windshield .

However, the seat belt prevents the passenger from moving: under the influence of the passenger’s body, the belt stretches and acts on the passenger with the corresponding force. This reaction of the belt balances the force of inertia and the passenger in the reference system associated with the car does not experience acceleration, remaining stationary relative to the car during all braking.

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

Use

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

the movement of the moving parts of the car is conveniently described in the coordinate system associated with the car. In the case of acceleration of the car, this system becomes non-inertial;
The motion of a body along a circular path is sometimes conveniently described in the coordinate system associated with this body. Such a coordinate system is non-inertial due to centripetal acceleration .

In non-inertial reference systems, standard formulations of Newton's laws are not applicable. So when accelerating a car, 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 the body moves in orbit, in a non-inertial coordinate system connected with the body, the body rests, although it is affected by an unbalanced gravitational force, acting as a centripetal in the inertial coordinate system in which rotation was observed in the orbit.

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

Using this fictitious force, it becomes possible to briefly describe the actually observed effects: “why does the passenger press against the seat back when accelerating the car?” - “the inertia force acts on the passenger’s body”. In the inertial coordinate system associated with the road, inertia is not required to explain what is happening: the passenger’s body accelerates in it (along with the car), and this acceleration produces the force with which the seat acts on the passenger .

Inertia on the Earth's surface

Conditionally combined pictures of the acting forces for terrestrial and extraterrestrial observers

In an inertial reference system (an observer outside the Earth), a body located on the surface of the Earth experiences centripetal acceleration $a_{c}$ ${\ displaystyle a_ {c}}$ , coinciding in magnitude with the acceleration of points on the surface of the Earth caused by its daily rotation . This acceleration, in accordance with Newton’s second law, is determined by the centripetal force acting on the body $c$ ${\ displaystyle c}$ (green vector). The latter consists of the force of gravitational attraction to the center of the earth $g_{0}$ ${\ displaystyle g_ {0}}$ (red vector) and support reaction forces $b$ ${\ displaystyle b}$ (black vector) ^[31] . Thus, the equation of Newton’s second law for the body under consideration in the case of an inertial reference frame has the form $ma_{c}=c$ ${\ displaystyle ma_ {c} = c}$ or, which is the same $ma_{c}=g_{0}+b$ ${\ displaystyle ma_ {c} = g_ {0} + b}$ .

For an observer rotating with the Earth, the body is motionless, although it is affected by exactly the same forces as in the previous case: the force of gravity $g_{0}$ ${\ displaystyle g_ {0}}$ and support reaction $b$ ${\ displaystyle b}$ . There is no contradiction here, since in the non-inertial reference frame, which is the rotating Earth, it is unlawful to apply Newton’s second law in the usual form. At the same time, in a non-inertial reference frame it is possible to introduce inertia forces into consideration. In this case, the only force of inertia is centrifugal force. $a$ ${\ displaystyle a}$ (blue vector), equal to the product of the mass of the body and its acceleration in the inertial reference frame, taken with a minus sign, that is $-ma_{c}$ ${\ displaystyle -ma_ {c}}$ . After the introduction of this force, the equation of motion of the body given above is converted into the equation of equilibrium of the body, which has the form $g_{0}+a+b=0$ ${\ displaystyle g_ {0} + a + b = 0}$ .

The sum of the forces of gravity $g_{0}$ ${\ displaystyle g_ {0}}$ and centrifugal inertia $a$ ${\ displaystyle a}$ called gravity $g$ ${\ displaystyle g}$ (yellow vector) ^[32] . With this in mind, the last equation can be written as $g+b=0$ ${\ displaystyle g + b = 0}$ and to argue that the effects of gravity and the reaction forces of the supports cancel each other out. We also note that the relative value of centrifugal force is small: at the equator, where this value is maximum, its contribution to gravity is ~ 0.3% ^[33] . Accordingly, the deviations of vectors are also small. $g$ ${\ displaystyle g}$ and $b$ ${\ displaystyle b}$ from the radial direction.

General approach to finding the forces of inertia

Comparing the motion of the body in inertial and non-inertial CO, we can come to the following conclusion ^[30] :

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

Similarly ${\vec {F_{2}}}$ ${\ displaystyle {\ vec {F_ {2}}}}$ is the sum of forces measured in a non-inertial coordinate system (second), causing acceleration ${\vec {a_{2}}}$ ${\ displaystyle {\ vec {a_ {2}}}}$ generally different from ${\vec {a_{1}}}$ ${\ displaystyle {\ vec {a_ {1}}}}$ due to the accelerated movement of the second WITH relative to the first.

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

${\vec {F_{i_{2}}}}={\vec {F_{2}}}-{\vec {F_{1}}}$ ${\ displaystyle {\ vec {F_ {i_ {2}}}} = {\ vec {F_ {2}}} - {\ vec {F_ {1}}}}$ (nineteen)

or:

${\vec {F_{i_{2}}}}=m({\vec {a_{2}}}-{\vec {a_{1}}})$ ${\ displaystyle {\ vec {F_ {i_ {2}}}} = m ({\ vec {a_ {2}}} - {\ vec {a_ {1}}})}$ (20) ^[30]

In particular, if the body is resting in a non-inertial system, i.e. ${\vec {a_{2}}}=0$ ${\ displaystyle {\ vec {a_ {2}}} = 0}$ then

${\vec {F_{i_{2}}}}=-{\vec {F_{1}}}$ ${\ displaystyle {\ vec {F_ {i_ {2}}}} - - {\ vec {F_ {1}}}}$ (21) ^[30] .

Body motion along an arbitrary trajectory in a non-inertial RM

The position of the material body in a conditionally fixed and inertial system is set here by the vector ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ , and in a non-inertial system, by a vector ${\vec {r^{\prime }}}$ ${\ displaystyle {\ vec {r ^ {\ prime}}}}$ . The distance between the origin is determined by the vector ${\vec {R}}$ ${\ displaystyle {\ vec {R}}}$ . The angular velocity of rotation of the system is given by the vector ${\vec {\omega }}$ ${\ displaystyle {\ vec {\ omega}}}$ whose direction is set along the axis of rotation according to the rule of the right screw . The linear velocity of the body with respect to the rotating CO is given by the vector ${\vec {v}}$ ${\ displaystyle {\ vec {v}}}$ .

In this case, the acceleration, in accordance with (11), will be equal to the sum ^[9] :

${\vec {a_{r}}}={\frac {d^{2}{\vec {R}}}{dt^{2}}}+{\frac {d{\vec {\omega }}}{dt}}\times {\vec {r}}'+{2{\vec {\omega }}\times {\vec {v}}}+{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}'),\qquad (22)$ ${\vec {a_{r}}}={\frac {d^{2}{\vec {R}}}{dt^{2}}}+{\frac {d{\vec {\omega }}}{dt}}\times {\vec {r}}'+{2{\vec {\omega }}\times {\vec {v}}}+{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}'),\qquad (22)$

Where:

первый член — переносное ускорение второй системы относительно первой;

второй член — ускорение, возникающее из-за неравномерности вращения системы вокруг своей оси;

третий член — Кориолисово ускорение , вызванное той составляющей вектора скорости, которая не параллельна оси вращения неинерциальной системы;
последний член, взятый без знака, представляет собой вектор, направленный в противоположную сторону от вектора ${\vec {r}}'$ ${\vec {r}}'$ , что можно получить, раскрывая двойное векторное произведение, когда получаем, что этот член равен $-{{\vec {r}}'}\omega ^{2}$ $-{{\vec {r}}'}\omega ^{2}$ и потому представляет собой центростремительное ускорение тела в системе отсчёта неподвижного наблюдателя, принимаемой за ИСО, в которой сил инерции быть не может по определению.

Работа сил инерции

В классической физике силы инерции встречаются в двух различных ситуациях в зависимости от системы отсчёта, в которой производится наблюдение ^[30] . Это — сила, приложенная к связи при наблюдении в инерциальной СО, или сила, приложенная к рассматриваемому телу, при наблюдении в неинерциальной системе отсчёта. Обе эти силы могут совершать работу. Исключением является сила Кориолиса, которая работы не совершает, поскольку всегда направлена перпендикулярно вектору скорости. В то же время сила Кориолиса может изменить траекторию движения тела и, тем самым, способствовать совершению работы другими силами (такими, как сила трения). Примером этому может служить эффект Бэра .

Кроме того, в некоторых случаях бывает целесообразно разделить действующую силу Кориолиса на две составляющие, каждая из которых совершает работу. Суммарная работа, производимая этими составляющими, равна нулю, но такое представление может оказаться полезным при анализе процессов перераспределения энергии в рассматриваемой системе ^[34] .

При теоретическом рассмотрении, когда искусственно сводят динамическую задачу движения к задаче статики, вводят третий вид сил, называемый силами Даламбера, которые работы не совершают ввиду неподвижности тел, на которые эти силы действуют.

Эквивалентность сил инерции и гравитации

Согласно принципу эквивалентности сил гравитации и инерции локально невозможно отличить, какая сила действует на данное тело — гравитационная сила или сила инерции. Различие между силами гравитации и силами инерции классической механики заключается в невозможности устранения сил гравитации в конечной области пространства-времени переходом к какой-либо системе отсчёта. В этом смысле глобальные или даже конечные инерциальные системы отсчёта в общей теории относительности в общем случае отсутствуют.

Д'Аламберовы силы инерции

В принципе д'Аламбера в рассмотрение вводятся подлинно отсутствующие в природе силы инерции, которые невозможно измерить никакой физической аппаратурой. Эти силы вводятся ради использования искусственного математического приёма, основанного на применении принципа Д'Аламбера в формулировке Лагранжа , где задача на движение с помощью введения сил инерции формально сводится к проблеме равновесия ^[30] .

Приложения

↑ ¹ ² Тарг С. М. Сила инерции // Физическая энциклопедия / Гл. ed. А. М. Прохоров . — М. : Большая Российская энциклопедия , 1994. — Т. 4. Пойнтинга—Робертсона эффект — Стримеры. — С. 494-495. - 704 s. - 40,000 copies. — ISBN 5-85270-087-8 .
↑ Сила инерции — статья из Большой советской энциклопедии .
↑ Ишлинский А. Ю. Классическая механика и силы инерции. — М. : «Наука», 1987. — С. 15. — 320 с.
↑ ¹ ² Савельев И. В. Курс общей физики. Том 1. Механика. Молекулярная физика. - М., Наука, 1987. - Тираж 233000 экз. - с. 119-120
↑ ¹ ² Ландсберг Г. С. Элементарный учебник физики. Том 1. Механика. Теплота. Молекулярная физика. - М., Наука, 1975. - Тираж 350000 экз. - с. 291-292
↑ Кошкин Н. И., Ширкевич М. Г. Справочник по элементарной физике.- М., Наука , 1988. - Тираж 300000 экз. - с. 33
↑ Ишлинский А. Ю. Классическая механика и силы инерции. — М. : «Наука», 1987. — С. 14-18. — 320 с.
↑ ¹ ² Ишлинский А. Ю. К вопросу об абсолютных силах и силах инерции в классической механике // Теоретическая механика. Сборник научно-методических статей. — 2000. — № 23 . — С. 3-8 .
↑ ¹ ² Walter Greiner Klassische Mechanik II. Wissenschaftlicher Verlag Harri Deutsch GmbH. Frankfurt am Main. 2008 ISBN 978-3-8171-1828-1
↑ ^ Richard Phillips Feynman, Leighton RB & Sands ML (2006). The Feynman Lectures on Physics. San Francisco: Pearson/Addison-Wesley. Vol. I, section 12-5. ISBN 0-8053-9049-9 . https://books.google.com/books?id=zUt7AAAACAAJ& <=intitle:Feynman+intitle:Lectures+intitle: on+intitle:Physics&lr=&as_brr=0.
↑ ^ Cornelius Lanczos (1986). The Variational Principles of Mechanics. New York: Courier Dover Publications. p. 100. ISBN 0-486-65067-7 . https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103&dq=%22Euler+force%22&lr=&as_brr=0&sig=UV46Q9NIrYWwn5EmYpPv-LPuZd0#PPA100,M1 .
↑ ^ Max Born & Günther Leibfried (1962). Einstein's Theory of Relativity. New York: Courier Dover Publications. pp. 76-78. ISBN 0-486-60769-0 . https://books.google.com/books?id=Afeff9XNwgoC&pg=PA76&dq=%22inertial+forces%22&lr=&as_brr=0&sig=0kiN27BqUqHaZ9CkPdqLIjr-Nnw#PPA77,M1 .
↑ Зоммерфельд А. Механика. — Ижевск: НИЦ «Регулярная и хаотическая динамика», 2001. — С. 82. — 368 с. — ISBN 5-93972-051-X .
↑ Борн М. Эйнштейновская теория относительности. — М. : «Мир», 1972. — С. 81. — 368 с.
↑ ¹ ² Фейнман Р. , Лейтон Р., Сэндс М. Выпуск 1. Современная наука о природе. Законы механики // Фейнмановские лекции по физике. — М. : «Мир», 1965. — С. 225.
↑ Седов Л. И. Об основных моделях механики. М.: МГУ, 1992. Стр 17.; Седов Л. И. Очерки, связанные с основами механики и физики. М.: Знание, 1983. Стр 19.
↑ Матвеев А. Н. Механика и теория относительности. М.: Высшая школа, 1979. Стр 393. (в 3-е изд. 2003. Стр.393)
↑ [1] . Вестник высшей школы. Советская наука, 1987. С. 248.
↑ А. Ишлинский при переиздании своей работы удалил эти термины («Классическая механика и силы инерции», 1987, с. 279): ... термин «реальная сила» и «фиктивная сила» понимались по-разному. Считаю, что лучше не спорить на эту тему и от упомянутых слов вообще отказаться .
↑ Тарг С. М. Краткий курс теоретической механики. — М. : Высшая школа, 1995. — С. 182. — 416 с. — ISBN 5-06-003117-9 .
↑ ¹ ² Журавлёв В. Ф. Основания механики. Методические аспекты. — М. : ИПМ АН СССР , 1985. — С. 19. — 46 с.
↑ Тарг С. М. Сила // Физическая энциклопедия / Гл. ed. А. М. Прохоров . — М. : Большая Российская энциклопедия , 1994. — Т. 4. Пойнтинга—Робертсона эффект — Стримеры. — С. 494. — 704 с. - 40,000 copies. — ISBN 5-85270-087-8 .
↑ Зоммерфельд А. Механика. — Ижевск: НИЦ «Регулярная и хаотическая динамика», 2001. — С. 16. — 368 с. — ISBN 5-93972-051-X .
↑ Сивухин Д. В. Общий курс физики. — М. : Физматлит; Изд-во МФТИ, 2005. — Т. I. Механика. — С. 84. — 560 с. — ISBN 5-9221-0225-7 .
↑ ¹ ² Kleppner D., Kolenkow RJ An Introduction to Mechanics . — McGraw-Hill, 1973. — P. 59-60. — 546 p. — ISBN 0-07-035048-5 .
↑ Встречается утверждение, что применительно к силе Лоренца сказанное не верно и требует дополнительного уточнения ( Матвеев А.Н. Механика и теория относительности. — 3-е изд. — М. Высшая школа 1976. — С. 132). Согласно другой точке зрения, «в электродинамике силы противодействия силам Лоренца приложены к электромагнитному полю (подстрочное примечание: Стоит отметить, что ещё недавно некоторые видные учёные считали, что сила Лоренца вообще не удовлетворяет закону действия и противодействия…) как к физическому объекту, претерпевающему соответствующее влияние» (Седов, Очерки, с. 17).
↑ Ишлинский А. Ю. Классическая механика и силы инерции. — М. : «Наука», 1987. — С. 8. — 320 с.
↑ «"Силы инерции" — не силы». Журавлёв В. Ф. Основания механики. Методические аспекты. — М. : ИПМ АН СССР , 1985. — С. 21. — 46 с.
↑ Жирнов Н. И. Классическая механика. — Серия: учебное пособие для студентов физико-математических факультетов педагогических институтов. — М., Просвещение , 1980. — Тираж 28 000 экз. - with. 256
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ Хайкин, Семён Эммануилович. Силы инерции и невесомость. М.,1967 г. Издательство «Наука». Главная редакция физико-математической литературы.
↑ Китайгородский А. И. Введение в физику. М:Изд.-во «Наука», гл.ред.физико-математической литературы.1973
↑ Тарг С. М. Сила тяжести // Физическая энциклопедия / Гл. ed. А. М. Прохоров . — М. : Большая Российская энциклопедия , 1994. — Т. 4. Пойнтинга—Робертсона эффект — Стримеры. — С. 496. — 704 с. - 40,000 copies. — ISBN 5-85270-087-8 .
↑ Грушинский Н. П. Основы гравиметрии. — М. : «Наука», 1983. — С. 34. — 351 с.
↑ Krigel AM The theory of the index cycle in the general circulation of the atmosphere // Geophys. Astrophys. Fluid Dynamics.— 1980.— 16 .— p. 1-18.

Literature

Ishlinsky A. Yu . Classical mechanics and inertia forces (inaccessible link) . Science, 1987.
Khaikin S.E. Inertia forces and weightlessness (inaccessible link) . Publishing House "Science".The main edition of the physical and mathematical literature. M., 1967
Sedov L. I. On the basic models of mechanics. M .: Moscow State University, 1992 .-- 151s. ISBN 5-211-01570-3 Chapter 1 "On the forces of inertia" p.6-17.
Sedov L.I. Essays related to the foundations of mechanics and physics. M .: Knowledge, 1983.- 64s. Section "On the forces of inertia" p. 5-18.

[ФЭ-СИнерции-1] ¹ ² Тарг С. М. Сила инерции // Физическая энциклопедия / Гл. ed. А. М. Прохоров . — М. : Большая Российская энциклопедия , 1994. — Т. 4. Пойнтинга—Робертсона эффект — Стримеры. — С. 494-495. - 704 s. - 40,000 copies. — ISBN 5-85270-087-8 .

[2] Сила инерции — статья из Большой советской энциклопедии .

[Ишлинский-15-3] Ишлинский А. Ю. Классическая механика и силы инерции. — М. : «Наука», 1987. — С. 15. — 320 с.

[Sav-4] ¹ ² Савельев И. В. Курс общей физики. Том 1. Механика. Молекулярная физика. - М., Наука, 1987. - Тираж 233000 экз. - с. 119-120

[Land-5] ¹ ² Ландсберг Г. С. Элементарный учебник физики. Том 1. Механика. Теплота. Молекулярная физика. - М., Наука, 1975. - Тираж 350000 экз. - с. 291-292

[6] Кошкин Н. И., Ширкевич М. Г. Справочник по элементарной физике.- М., Наука , 1988. - Тираж 300000 экз. - с. 33

[Ишлинский-14-18-7] Ишлинский А. Ю. Классическая механика и силы инерции. — М. : «Наука», 1987. — С. 14-18. — 320 с.

[Ишлинский-Сб-8] ¹ ² Ишлинский А. Ю. К вопросу об абсолютных силах и силах инерции в классической механике // Теоретическая механика. Сборник научно-методических статей. — 2000. — № 23 . — С. 3-8 .

[WG-9] ¹ ² Walter Greiner Klassische Mechanik II. Wissenschaftlicher Verlag Harri Deutsch GmbH. Frankfurt am Main. 2008 ISBN 978-3-8171-1828-1

[RPF-10] Richard Phillips Feynman, Leighton RB & Sands ML (2006). The Feynman Lectures on Physics. San Francisco: Pearson/Addison-Wesley. Vol. I, section 12-5. ISBN 0-8053-9049-9 . https://books.google.com/books?id=zUt7AAAACAAJ& <=intitle:Feynman+intitle:Lectures+intitle: on+intitle:Physics&lr=&as_brr=0.

[CL-11] Cornelius Lanczos (1986). The Variational Principles of Mechanics. New York: Courier Dover Publications. p. 100. ISBN 0-486-65067-7 . https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103&dq=%22Euler+force%22&lr=&as_brr=0&sig=UV46Q9NIrYWwn5EmYpPv-LPuZd0#PPA100,M1 .

[MB-12] Max Born & Günther Leibfried (1962). Einstein's Theory of Relativity. New York: Courier Dover Publications. pp. 76-78. ISBN 0-486-60769-0 . https://books.google.com/books?id=Afeff9XNwgoC&pg=PA76&dq=%22inertial+forces%22&lr=&as_brr=0&sig=0kiN27BqUqHaZ9CkPdqLIjr-Nnw#PPA77,M1 .

[Зоммерфельд2-13] Зоммерфельд А. Механика. — Ижевск: НИЦ «Регулярная и хаотическая динамика», 2001. — С. 82. — 368 с. — ISBN 5-93972-051-X .

[Борн-14] Борн М. Эйнштейновская теория относительности. — М. : «Мир», 1972. — С. 81. — 368 с.

[Фейнман-15] ¹ ² Фейнман Р. , Лейтон Р., Сэндс М. Выпуск 1. Современная наука о природе. Законы механики // Фейнмановские лекции по физике. — М. : «Мир», 1965. — С. 225.

[16] Седов Л. И. Об основных моделях механики. М.: МГУ, 1992. Стр 17.; Седов Л. И. Очерки, связанные с основами механики и физики. М.: Знание, 1983. Стр 19.

[17] Матвеев А. Н. Механика и теория относительности. М.: Высшая школа, 1979. Стр 393. (в 3-е изд. 2003. Стр.393)

[18] [1] . Вестник высшей школы. Советская наука, 1987. С. 248.

[19] А. Ишлинский при переиздании своей работы удалил эти термины («Классическая механика и силы инерции», 1987, с. 279): ... термин «реальная сила» и «фиктивная сила» понимались по-разному. Считаю, что лучше не спорить на эту тему и от упомянутых слов вообще отказаться .

[Тарг-20] Тарг С. М. Краткий курс теоретической механики. — М. : Высшая школа, 1995. — С. 182. — 416 с. — ISBN 5-06-003117-9 .

[Журавлёв-19-21] ¹ ² Журавлёв В. Ф. Основания механики. Методические аспекты. — М. : ИПМ АН СССР , 1985. — С. 19. — 46 с.

[ФЭ4-Сила-22] Тарг С. М. Сила // Физическая энциклопедия / Гл. ed. А. М. Прохоров . — М. : Большая Российская энциклопедия , 1994. — Т. 4. Пойнтинга—Робертсона эффект — Стримеры. — С. 494. — 704 с. - 40,000 copies. — ISBN 5-85270-087-8 .

[Зоммерфельд-23] Зоммерфельд А. Механика. — Ижевск: НИЦ «Регулярная и хаотическая динамика», 2001. — С. 16. — 368 с. — ISBN 5-93972-051-X .

[24] Сивухин Д. В. Общий курс физики. — М. : Физматлит; Изд-во МФТИ, 2005. — Т. I. Механика. — С. 84. — 560 с. — ISBN 5-9221-0225-7 .

[Kleppner-25] ¹ ² Kleppner D., Kolenkow RJ An Introduction to Mechanics . — McGraw-Hill, 1973. — P. 59-60. — 546 p. — ISBN 0-07-035048-5 .

[26] Встречается утверждение, что применительно к силе Лоренца сказанное не верно и требует дополнительного уточнения ( Матвеев А.Н. Механика и теория относительности. — 3-е изд. — М. Высшая школа 1976. — С. 132). Согласно другой точке зрения, «в электродинамике силы противодействия силам Лоренца приложены к электромагнитному полю (подстрочное примечание: Стоит отметить, что ещё недавно некоторые видные учёные считали, что сила Лоренца вообще не удовлетворяет закону действия и противодействия…) как к физическому объекту, претерпевающему соответствующее влияние» (Седов, Очерки, с. 17).

[Ишлинский-8-27] Ишлинский А. Ю. Классическая механика и силы инерции. — М. : «Наука», 1987. — С. 8. — 320 с.

[28] «"Силы инерции" — не силы». Журавлёв В. Ф. Основания механики. Методические аспекты. — М. : ИПМ АН СССР , 1985. — С. 21. — 46 с.

[29] Жирнов Н. И. Классическая механика. — Серия: учебное пособие для студентов физико-математических факультетов педагогических институтов. — М., Просвещение , 1980. — Тираж 28 000 экз. - with. 256

[СЭХ-30] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ Хайкин, Семён Эммануилович. Силы инерции и невесомость. М.,1967 г. Издательство «Наука». Главная редакция физико-математической литературы.

[КАИ-31] Китайгородский А. И. Введение в физику. М:Изд.-во «Наука», гл.ред.физико-математической литературы.1973

[32] Тарг С. М. Сила тяжести // Физическая энциклопедия / Гл. ed. А. М. Прохоров . — М. : Большая Российская энциклопедия , 1994. — Т. 4. Пойнтинга—Робертсона эффект — Стримеры. — С. 496. — 704 с. - 40,000 copies. — ISBN 5-85270-087-8 .

[33] Грушинский Н. П. Основы гравиметрии. — М. : «Наука», 1983. — С. 34. — 351 с.

[34] Krigel AM The theory of the index cycle in the general circulation of the atmosphere // Geophys. Astrophys. Fluid Dynamics.— 1980.— 16 .— p. 1-18.

Inertia force

Terminology

Forces in Classical Mechanics

Equations of Motion

Newtonian inertia forces

Euler inertia forces

Non-inertial JI Movement

Example 1

Example 2

Example 3

Use

Inertia on the Earth's surface

General approach to finding the forces of inertia

Body motion along an arbitrary trajectory in a non-inertial RM

Работа сил инерции

Эквивалентность сил инерции и гравитации

Д'Аламберовы силы инерции

See also

Приложения

Literature

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