Newton's 3 laws of inertia. Newton's laws of mechanics

DEFINITION

Statement of Newton's first law. There are such frames of reference, in relation to which the body maintains a state of rest or a state of uniform rectilinear motion, if other bodies do not act on it or the action of other bodies is compensated.

Description of Newton's first law

For example, the ball on the thread hangs at rest, because the force of gravity is compensated by the tension in the thread.

Newton's first law is valid only in . For example, bodies at rest in the cabin of an aircraft that is moving uniformly can begin to move without any influence from other bodies if the aircraft begins to maneuver. In vehicles, when braking hard, passengers fall, although no one pushes them.

Newton's first law shows that the state of rest and the state do not require external influences for their maintenance. The property of a free body to keep its speed constant is called inertia. Therefore, Newton's first law is also called law of inertia. Uniform rectilinear motion of a free body is called inertial motion.

Newton's first law contains two important statements:

1. all bodies have the property of inertia;
2. inertial reference systems exist.

It should be remembered that in Newton's first law we are talking about bodies that can be taken for.

The law of inertia is by no means obvious, as it might seem at first glance. With his discovery, one long-standing misconception was done away with. Prior to this, for centuries it was believed that in the absence of external influences on the body, it can only be in a state of rest, that rest is, as it were, the natural state of the body. For a body to move at a constant speed, another body must act on it. Everyday experience seemed to confirm this: in order for a wagon to move at a constant speed, it must be pulled all the time by a horse; for the table to move along the floor, it must be continuously pulled or pushed, etc. Galileo Galilei was the first to point out that this is not true, that in the absence of external influence, the body can not only rest, but also move rectilinearly and uniformly. Rectilinear and uniform motion is, therefore, the same "natural" state of bodies as rest. In fact, Newton's first law says that there is no difference between a body at rest and uniform rectilinear motion.

It is impossible to test the law of inertia empirically, because it is impossible to create such conditions under which the body would be free from external influences. However, the opposite can always be traced. Anyway. When a body changes the speed or direction of its movement, you can always find the cause - the force that caused this change.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2

Exercise

A light toy car stands on a table in a uniformly and rectilinearly moving train. When the train braked, the car rolled forward without any external influence. Is the law of inertia satisfied: a) in the frame of reference associated with the train during its rectilinear uniform motion? during braking? b) in the reference system connected with the Earth?

Answer

a) the law of inertia is satisfied in the reference frame associated with the train during its rectilinear motion: the toy car is at rest relative to the train, since the action from the Earth is compensated by the action from the table (reaction of the support). When braking, the law of inertia is not satisfied, since braking is movement with and the train in this case is not an inertial frame of reference. b) in the reference frame associated with the Earth, the law of inertia is satisfied in both cases - with a uniform train movement, the toy car moves relative to the Earth at a constant speed (train speed); When the train brakes, the car tries to keep its speed relative to the Earth unchanged, and therefore rolls forward.

Newton's laws of dynamics (classical dynamics) have a limited area of ​​applicability. They are valid for macroscopic bodies moving at speeds much lower than the speed of light in vacuum.

The formulation of Newton's first law (it is also known as the law of inertia):

Newton's first law There are such frames of reference, called inertial ones, relative to which the body moves in a straight line and uniformly, if no other bodies act on it or the action of these bodies is compensated.

In an inertial frame of reference, a body moves uniformly and rectilinearly in the absence of forces acting on it.

Inertia The phenomenon of maintaining the speed of a body in the absence of external influences or with their compensation is called inertia. Therefore, Newton's first law is called the law of inertia.

If the resultant of all forces acting on a given body is zero, then the body moves uniformly and rectilinearly or does not move at all. In reality, it is impossible to achieve equality to zero of the resultant force. But you can neglect some actions and choose a section of motion when the speed of the body does not change significantly.

The law of inertia was first formulated by Galileo Galilei (1632). Newton generalized the conclusions of Galileo and included them among the basic laws of motion.

ISO inertial frames of reference are frames of reference in which Newton's 1st law is fulfilled.

So, the reason for changing the speed of a body in an inertial frame of reference is always its interaction with other bodies. For a quantitative description of the motion of a body under the influence of other bodies, it is necessary to introduce two new physical quantities - inert body weight And force.

Weight

Mass is a property of a body that characterizes its inertia. With the same impact from the surrounding bodies, one body can quickly change its speed, and the other, under the same conditions, much more slowly. It is customary to say that the second of these two bodies has more inertia, or, in other words, the second body has more mass.

If two bodies interact with each other, then as a result, the speed of both bodies changes, i.e., in the process of interaction, both bodies acquire accelerations. The ratio of the accelerations of two given bodies is constant under any impact. It is accepted in physics that the masses of interacting bodies are inversely proportional to the accelerations acquired by the bodies as a result of their interaction.

Comparison of the masses of two bodies.

\[ \dfrac(m_1)(m_2) =-\dfrac(a_2)(a_1) \]

In this relation, the quantities \(a_1\) and \(a_2\) should be considered as projections of the vectors \(a_1\) and \(a_2\) on the OX axis. The minus sign on the right side of the formula means that the accelerations of the interacting bodies are directed in opposite directions.

In the International System of Units (SI), body weight is measured in kilograms (kg).

The mass of any body can be determined experimentally by comparison with standard mass (\(m_(\text(et)) = 1 \text(kg) \)). Let \(m_1 = m_(\text(et)) = 1 \text(kg) \). Then

\[ m_2=-\dfrac(a_1)(a_2) m_(\text(et)) \]

Body mass - scalar. Experience shows that if two bodies with masses \ (m_1 \) and \ (m_2 \) are combined into one, then the mass \ (m \) of the composite body turns out to be equal to the sum of the masses \ (m_1 \) and \ (m_2 \) of these bodies :

\[ M = m_1 + m_2 \]

This mass property is called additivity.

Force

Force is a quantitative measure of the interaction of bodies. Force is the cause of a change in the speed of a body. In Newtonian mechanics, forces can have a different physical nature: friction force, gravity force, elastic force, etc. The force is vector quantity, has modulus, direction and point of application.

The vector sum of all forces acting on a body is called resultant force.

To change the speed of a body, it is necessary to act on it with some force. Naturally, the result of the action of forces of the same magnitude on different bodies will be different.

There are 4 main types interactions:

• gravitational,
• electromagnetic,
• strong,
• weak.

All interactions are manifestations of these basic types.

Examples of forces: gravity, elastic force, body weight, friction force, buoyant (Archimedean) force, lifting force.

What is strength? Force is a measure of the influence of one body on another.

Force is a vector quantity. Strength is characterized by:

• module (absolute value);
• direction;
• application point.

To measure forces, you need to install strength standard And comparison method other forces with this standard.

As a force standard, you can take a spring stretched to some given length. Force module F 0, with which this spring acts on the body attached to it at a fixed tension, is called standard of strength. The way to compare other forces with the standard is as follows: if the body under the action of the measured force \(\vec(F) \) and the reference force \(\vec(F_0) \) remains at rest (or moves uniformly and rectilinearly), then the forces are modulo \(\vec(F) \) = \(\vec(F_0) \) .

Comparison of force \(\vec(F) \) with the standard. \(\vec(F) \) = \(\vec(F_0 ) \)

If the measured force \(\vec(F ) \) is greater (in modulus) than the reference force, then two reference springs can be connected in parallel. In this case, the measured force is \(\vec( 2 F_0 ) \) . The forces \(\vec( 3 F_0 ) \) , \(\vec( 4 F_0 ) \) etc. can be measured similarly.

Comparison of force \(\vec(F ) \) with the standard. \(\vec(F) \) = \(\vec(2 F_0) \)

Measurement of forces smaller than \(\vec(2 F_0) \)

Comparison of force \(\vec(F ) \) with the standard. \(\vec(F) \) = \(\vec(2 F_0) \cos (\alpha) \)

The reference force in the International System of Units is called Newton(N).

A force of 1 N gives a body with a mass of 1 kg an acceleration of 1 m/s2

Unit [N]

\[ 1\text(H) = 1\dfrac(\text(kg)\cdot \text(m))(\text(c)^2) \]

In practice, there is no need to compare all measured forces with the standard. To measure forces, use springs calibrated as described above. These calibrated springs are called dynamometers . The force is measured by stretching the dynamometer.

Javascript is disabled in your browser.
ActiveX controls must be enabled in order to make calculations!

Newton's laws- three laws that underlie classical mechanics and allow writing the equations of motion for any mechanical system, if the force interactions for its constituent bodies are known. First fully formulated by Isaac Newton in the book "Mathematical Principles of Natural Philosophy" (1687)

Newton's first law postulates the existence of inertial frames of reference. Therefore, it is also known as Law of inertia. Inertia is the phenomenon of the body maintaining the speed of movement (both in magnitude and in direction), when no forces act on the body. To change the speed of a body, it is necessary to act on it with some force. Naturally, the result of the action of forces of the same magnitude on different bodies will be different. Thus bodies are said to have inertia. Inertia is the property of bodies to resist a change in their speed. The value of inertia is characterized by body mass.

Modern wording

In modern physics, Newton's first law is usually formulated as follows:

There are such frames of reference, called inertial ones, relative to which a material point, in the absence of external influences, retains the magnitude and direction of its velocity indefinitely.

The law is also true in a situation where external influences are present, but mutually compensated (this follows from Newton's 2nd law, since the compensated forces impart zero total acceleration to the body).

Historical wording

Newton in his book "Mathematical Principles of Natural Philosophy" formulated the first law of mechanics in the following form:

Every body continues to be held in a state of rest, or uniform and rectilinear motion, until and insofar as it is compelled by applied forces to change this state.

From a modern point of view, such a formulation is unsatisfactory. First, the term "body" should be replaced by the term "material point", since a body of finite dimensions in the absence of external forces can also perform rotational motion. Secondly, and most importantly, Newton in his work relied on the existence of an absolute fixed frame of reference, that is, absolute space and time, and modern physics rejects this idea. On the other hand, in an arbitrary (say, rotating) frame of reference, the law of inertia is incorrect. Therefore, the Newtonian formulation needs to be clarified.

Newton's second law

Newton's second law is a differential law of motion that describes the relationship between the force applied to a material point and the resulting acceleration of this point. In fact, Newton's second law introduces mass as a measure of the manifestation of the inertia of a material point in a chosen inertial frame of reference (ISR).

In this case, the mass of a material point is assumed to be constant in time and independent of any features of its movement and interaction with other bodies.

Modern wording

In an inertial frame of reference, the acceleration that a material point receives with a constant mass is directly proportional to the resultant of all forces applied to it and inversely proportional to its mass.

With a suitable choice of units of measurement, this law can be written as a formula:

where is the acceleration of the material point;
is the force applied to the material point;
is the mass of a material point.

Newton's second law can also be formulated in an equivalent form using the concept of momentum:

In an inertial frame of reference, the rate of change in the momentum of a material point is equal to the resultant of all external forces applied to it.

where is the momentum of the point, is its velocity, and is the time. With this formulation, as with the previous one, it is believed that the mass of a material point is unchanged in time

Sometimes attempts are made to extend the scope of the equation to the case of bodies of variable mass. However, along with such a broad interpretation of the equation, it is necessary to significantly modify the previously accepted definitions and change the meaning of such fundamental concepts as material point, momentum and force.

When several forces act on a material point, taking into account the principle of superposition, Newton's second law is written as:

or, if the forces are independent of time,

Newton's second law is valid only for speeds much less than the speed of light and in inertial frames of reference. For speeds close to the speed of light, the laws of the theory of relativity are used.

It is impossible to consider a special case (for ) of the second law as an equivalent of the first, since the first law postulates the existence of the IFR, and the second is already formulated in the IFR.

Historical wording

Newton's original formulation:

The change in momentum is proportional to the applied driving force and occurs in the direction of the straight line along which this force acts.

Newton's third law

This law explains what happens to two material points. Take for example a closed system consisting of two material points. The first point can act on the second with some force , and the second - on the first with the force . How are the forces related? Newton's third law states that the action force is equal in magnitude and opposite in direction to the reaction force. We emphasize that these forces are applied to different material points, and therefore are not compensated at all.

Modern wording

Material points interact with each other by forces of the same nature, directed along the straight line connecting these points, equal in magnitude and opposite in direction:

The law reflects the principle of pair interaction.

Historical wording

Action always has an equal and opposite reaction, otherwise, the interactions of two bodies on each other are equal and directed in opposite directions.

For the Lorentz force, Newton's third law does not hold. Only by reformulating it as the law of conservation of momentum in a closed system of particles and an electromagnetic field, one can restore its validity.

conclusions

Some interesting conclusions immediately follow from Newton's laws. So, Newton's third law says that, no matter how the bodies interact, they cannot change their total momentum: there is law of conservation of momentum. Further, if we require that the interaction potential of two bodies depend only on the modulus of the difference in the coordinates of these bodies , then there arises law of conservation of total mechanical energy interacting bodies:

Newton's laws are the basic laws of mechanics. The equations of motion of mechanical systems can be derived from them. However, not all laws of mechanics can be derived from Newton's laws. For example, the law of universal gravitation or Hooke's law are not consequences of Newton's three laws.

1) Newton's first law: There are such frames of reference, called inertial ones, relative to which free bodies move uniformly and rectilinearly.

The first law of mechanics, or the law of inertia, as it is often called, was essentially established by Galileo, but it was Newton who gave it a general formulation.

free body - called a body that is not affected by any other bodies or fields. When solving some problems, a body can be considered free if external influences are balanced.

Reference systems in which a free material point is at rest or moves in a straight line and uniformly are called inertial reference systems. The rectilinear and uniform motion of a free material point in an inertial frame of reference is called inertia. With such a motion, the velocity vector of a material point remains constant ( = const). The rest of a point is a special case of motion by inertia (=0).

In inertial frames of reference, rest or uniform motion is a natural state, and the dynamics must explain the change in this state (ie, the appearance of body acceleration under the action of forces). There are no free bodies that are not affected by other bodies. However, due to the decrease of all: known interactions with increasing distance, such a body can be realized with any required accuracy.

Frames of reference in which a free body does not keep its speed unchanged are called non-inertial. Non-inertial is a frame of reference moving with acceleration relative to any inertial frame of reference. In a non-inertial frame of reference, even a free body can move with acceleration.

The uniform and rectilinear motion of the reference system does not affect the course of mechanical phenomena occurring in it. No mechanical experiments make it possible to distinguish the rest of an inertial frame of reference from its uniform rectilinear motion. For any mechanical phenomena, all initial frames of reference are equal. These statements express mechanical principle of relativity (Galileo's principle of relativity). The principle of relativity is one of the most general laws of nature; in the special theory of relativity, it extends to electromagnetic and optical phenomena.

2) Mass, density, force.

The property of a body to maintain its speed in the absence of interaction with other bodies is called inertia. The physical quantity, which is a measure of the inertia of a body in translational motion, is called inertial mass. Body weight is measured in kilograms: . Mass also characterizes the ability of a body to interact with other bodies in accordance with the law of universal gravitation. In these cases, the mass acts as a measure of gravity and is called gravitational mass.

In modern physics, the identity of the values ​​of the inertial and gravitational masses of a given body has been proven with a high degree of accuracy. So they just talk about body weight(m).

In Newtonian mechanics, it is believed that

a) the mass of a body is equal to the sum of the masses of all particles (or material points) of which it consists;

b) for a given set of bodies, law of conservation of mass: for any processes occurring in the system of bodies, its mass remains unchanged.

The density of a homogeneous body is . Density unit 1 kg / m 3.

By force called a vector physical quantity, which is a measure of the mechanical impact on the body from other bodies or fields. A force is completely defined if its modulus, direction, and point of application are given. The line along which the force is directed is called line of force.

As a result of the action of a force, the body changes the speed of movement (acquires acceleration) or deforms. Based on these experimental facts, the forces are measured.

The force is not the cause of the occurrence of speed, but of the acceleration of the body. In all cases, the direction of acceleration coincides with the direction of the force, but not the direction of velocity.

The problems of mechanics take into account gravitational forces (gravitational forces) and two types of electromagnetic forces - elastic forces And friction forces.

3) Newton's second law

Newton's second law describes the motion of a particle caused by the influence of surrounding bodies, and establishes a relationship between the particle's acceleration, its mass, and the force with which these bodies act on it:

If the surrounding bodies act on a particle with a mass m with a force , then this particle acquires such an acceleration that the product of its mass and acceleration will be equal to the acting force.

Mathematically, Newton's second law is written as:

Based on this law, the unit of force is established - 1 N (newton). 1 N is the force with which you need to act on a body with a mass of 1 kg in order to tell it an acceleration of 1 m / s 2.

If strength , with which the bodies act on a given particle, is known, then the equation of Newton's second law written for this particle is called it the equation of motion.

Newton's second law is often called the basic law of dynamics, since it is in it that the principle of causality finds the most complete mathematical expression, and it is he who, finally, allows solving the main problem of mechanics. To do this, it is necessary to find out which of the bodies surrounding the particle have a significant effect on it, and, having expressed each of these actions in the form of an appropriate force, one should draw up an equation of motion for this particle. From the equation of motion (with a known mass) is the acceleration of the particle. Knowing

same acceleration, you can determine its speed, and after the speed - and the position of the particle at any time.

Practice shows that the solution of the basic problem of mechanics with the help of Newton's second law always leads to correct results. This is an experimental confirmation of the validity of Newton's second law.

4) Newton's third law.

Newton's third law: The forces with which the bodies act on each other are equal in modules and directed along one straight line in opposite directions.

This means that if the body A from the side of the body IN force acts on the body at the same time IN from the side of the body A force will act , and = - .

Using Newton's second law, we can write:

Hence it follows that

i.e., the ratio of the acceleration modules and bodies interacting with each other is determined by the inverse ratio of their masses and is completely independent of the nature of the forces acting between them. A more massive body receives less acceleration, while a lighter body receives more.

It is important to understand that the forces referred to in Newton's third law are applied to different bodies and therefore they cannot balance each other.

5) Consequences from Newton's laws

Newton's laws are a system of interrelated laws that allow a deeper understanding of the essence of the concepts of force and mass. Consequences from the laws:

1. The force is a measure of the impact exerted on a given particle by other bodies, and decreases with increasing distance to them, tending to zero.

It is a question of the behavior of a body isolated from the influence of other bodies. The second law speaks of the exact opposite situation. It deals with cases where a body or several bodies act on a given one.

Both of these laws describe the behavior of one particular body. But at least two bodies always participate in the interaction. What will happen to both of these bodies? How to describe their interaction? Newton took up the analysis of this situation after formulating his first two laws. Let's do the same research.

Interaction of two bodies

We know that when interacting, both bodies act on each other. It does not happen that one body pushes another, and the second in response would not react in any way. This can happen among differently educated people, but not in nature.

We know that if we kick the ball, the ball kicks us back. Another thing is that the ball has a much smaller mass than the human body, and therefore its impact is practically not noticeable.

However, if you try to kick a heavy iron ball, you will vividly feel this response. In fact, we kick a very, very heavy ball to our planet many times every day. We push it with every step we take, only at the same time it is not she who flies away, but we. And all because the planet is millions of times larger than us in mass.

The ratio of forces in the interaction between bodies

So from these considerations it can be seen that when two bodies interact, not only the first acts on the second with some force, but the second in response acts on the first also with some force. The question arises: how are these forces related? Which one is bigger, which one is smaller?

To do this, you need to make some measurements. You will need two dynamometers, but at home I can easily replace them with two steelyards. They measure weight, and weight is also a force, only expressed in units of mass in the case of a steelyard. Therefore, if you have two steelyards, then do the following.

Put one of them with a ring on something motionless, for example, on a nail in the wall, and connect the second to the first with hooks. And pull the ring of the second steelyard. Follow the readings of both instruments. Each of them will show the force with which the other steelyard acts on it.

And although we are only pulling for one of them, it turns out that the testimony of both, as at a confrontation, will coincide. It turns out that the force with which we act on the first steelyard with the second is equal to the force with which the first steelyard acts on the second.

Newton's third law: definition and formula

The force of action is equal to the force of reaction. This is the essence of Newton's third law. Its definition is as follows: the forces with which two bodies act on each other are equal in magnitude and opposite in direction. Newton's third law can be written as a formula:

F_1 = - F_2,

Where F_1 and F_2 are the forces of action on each other, respectively, of the first and second bodies.

The validity of Newton's third law has been confirmed by numerous experiments. This law is valid both for the case when one body pulls another, and for the case when the bodies repel each other. All bodies in the universe interact with each other, obeying this law.