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Newton's laws
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Newton's laws classical mechanics
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We will talk about two of Newton's laws - the first and the second
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Since measurement is always a comparison with a standard (with a unit of measurement), Newton's second law also predetermines the choice of the unit of force. Since the units of length, mass and time have already been established, this equation forces us to take as a unit of force such a force that imparts acceleration to a unit of mass, equal to one... In the SI system, newton (N) is taken as a unit of force. Newton is a force that imparts an acceleration of 1 m / s 2 to a mass of one kilogram.
By their nature, the forces of elastic interaction, friction forces, gravitational and electromagnetic forces are distinguished.
Above was an example of elastic forces. Friction forces depend on the speed of the relative motion of the contacting surfaces and the state of the surface. Gravitational and electromagnetic forces are due to the presence fields or field interaction and operate from a distance. Accordingly, the task of measuring forces is divided into two separate tasks: 1) measuring the fields that arise in a particular case, and 2) measuring the forces acting on a given body from the side of a given field.
To measure forces, first, a force standard must be established, and secondly, a way of comparing other forces with this standard.
Let's take some well-defined spring (for example, from steel wire, having the shape of a cylindrical spiral), stretched to a known length. The standard of force, we will consider the force with which this spring at a fixed tension acts on a body attached to any of its ends. Comparison of other forces with the standard is the measurement.
Having a way of measuring forces, it is possible to establish under what conditions forces arise, and to find their values in any specific cases. For example, by studying elastic forces, it can be established that a stretched cylindrical spring creates a force that, when the spring is not too large stretches, is proportional to the amount of tension (Hooke's law). Such a device for measuring forces is called a dynamometer (in accordance with the dimension of force in the CGS system - dyne). This law simplifies the calibration of dynamometers, since it is sufficient to note only the tension corresponding to greatest strength(not exceeding the above limits), and divide the entire dynamometer scale into equal parts. In the same way, for any other types of deformation, it is possible to establish the dependence of the magnitude of the emerging elastic force on the nature and magnitude of the deformation.
Friction forces can be measured in the same way. If a dynamometer is attached to a moving body and the extension of the dynamometer is set, at which the body will move rectilinearly and uniformly, then the friction force will be equal in magnitude and opposite in direction to the force acting from the dynamometer (of course, provided that no other forces on the body do not work).
For example, the well-known method of weighing bodies on a spring balance allows you to measure the attraction of these bodies by the Earth (albeit only approximately, since the Earth, on which the body rests during weighing, moves relative to the chosen "stationary" coordinate system and this somewhat distorts the measurement results).
After checking Newton's second law experimentally, we can, on the basis of this law, find the acceleration of the body for a given body by the known forces, or, conversely, by the known accelerations, find the sum of the forces acting on it, if at least once for this body we simultaneously determine both the acting force and the communicated force acceleration.
Since the same second Newton's law is used to establish a method for measuring body mass (the value of mass is determined by the simultaneous measurement of force and acceleration), Newton's second law contains, on the one hand, the statement that acceleration is proportional to the force, and on the other, - the definition of body mass as the ratio of the force acting on the body to the acceleration imparted by this force.
It should be emphasized here that Newton formulated a law for elastic forces, gravitational forces, but knew almost nothing about the nature of more complex forces, for example, about the forces between atoms. However, he discovered one rule, one common property of all forces, which constitutes his third law:
"The force of action is equal to the force of reaction".
Now, relying on the accumulated experience of cognition, we can note what Newton did not notice, we can generalize his formulation, taking into account all types of interactions known to science today. In accordance with Newton's third law, any two bodies, say two particles, will be " push»Each other in opposite directions with equal strength. Newton had in mind, only known at that time, interactions: the forces of gravity and elasticity. However, now we can argue that the law is also valid for other types of interactions established by science today.
What else is interesting about Newton's third law? Let the interacting particles have different mass... What follows from this? According to the Second Law, the force is equal to the rate of change in momentum with time, so that the rate of change in the momentum of particle 1, in accordance with the Third Law, will be equal to the rate of change in momentum of particle two, i.e.
d p 1 / dt = - d p 2 / dt.
That is, complete change momentum of particle 1 is equal and opposite complete change momentum of particle 2. That is, the rate of change of the sum of the total momentum of both particles is equal to zero
d (p 1 + p 2) / dt = 0.
It is necessary, however, to remember that in our problem of a system consisting of two interacting bodies, we assumed the absence of any other forces, except for internal ones. Thus, we have obtained that if there are only one internal forces in a system of interacting bodies, the total momentum of the system of interacting particles remains unchanged. This statement expresses itself momentum conservation law... It follows from it that if we measure or calculate the value m 1v 1 + m 2v 2 + m 3v 3 +… , that is, the sum of the impulses of all particles, then for any forces acting between them, no matter how complex they are, we should get the same result, both before the action of the forces and after, since the total impulse remains constant.
Thus, the total momentum conservation law in the absence external forces can be written as
m 1v 1 + m 2v 2 + m 3v 3 +… = const.
Since for each particle Newton's second law has the form
f =d (m v) / dt,
then for any component of the total force, in any given direction, for example NS,
f x= d (mv x) / dt.
Exactly the same formulas can be written for y, z component.
If, however, there are external forces, forces external to an isolated system of particles, then the sum of all these external forces will be equal to the rate of change of the total momentum of all particles of the system.
Friction
So, to truly understand Newton's laws, we must discuss the properties of forces; the purpose of this chapter is to start this discussion and provide a kind of supplement to Newton's laws. We are already familiar with the properties of acceleration and with other similar concepts, but now we have to deal with the properties of forces. Because of their complexity, in this chapter (unlike the previous ones) we will not chase precise wording. To start with a specific force, consider the resistance that air has to a flying plane. What is the law of this force? (We must find it; after all, the law exists for every force!) It is unlikely that it will be simple. It is worth imagining the air braking of an aircraft - the whistling of the wind in the wings, whirlwinds, gusts, fuselage tremors and many other difficulties - to understand that this law is unlikely to come out simple and convenient. All the more remarkable is the fact that strength has a very simple pattern: F ≈ s υ 2 (constant times the square of the speed).
What is the position of this law among others? Is he like the law F = ma ? Not at all. First, it is empirical, and it was obtained by rough measurements in a wind tunnel. But you will object: “Well, the law F = ma could also be empirical. " But is that really the point? The difference is not in empiricality, but in the fact that, as far as we understand, this law of friction is the result of many influences and is fundamentally not at all simple. The more we study it, the more accurately we measure it, the more harder(but not simpler) he will introduce himself to us. In other words, as we delve deeper into the law of aircraft deceleration, we will more and more clearly understand its “falsehood”. The deeper the look, the more accurate the measurements, the more complicated the truth becomes; it will not appear before us as a result of simple fundamental processes (however, we guessed about it from the very beginning). At very low speeds (for example, they are not even available to an airplane), the law changes: braking already depends on the speed almost linearly: Or, for example, the braking of a ball (or an air bubble or something else) due to friction against a viscous liquid (like honey ), - at low speeds it is also proportional to the speed, but large ones, when vortices are formed (not in honey, of course, but in water or air), again there is an approximate proportionality to the square of the speed (F = c u 2 ); with a further increase in speed, this rule does not work either. You can, of course, say: "Well, here the coefficient changes slightly." But this is just a gimmick.
Secondly, there are other difficulties: is it possible, say, to divide this force into parts - by the friction force of the wings, fuselage, tail, etc.? Of course, when you need to find out rotational moments acting on a part of the aircraft, then you can do this, but then you already need to have a special law of friction for the wings, etc. And it turns out that amazing fact, that the force acting on the wing depends on. the other wing, that is, if the plane is removed and one wing is left in the air; then the force will not be at all what it would be if the entire plane were in the air; The reason, of course, is that the wind blowing into the nose of the aircraft flows down onto the wings and changes the braking force. And although it seems a miracle that there is such a simple, crude empirical law suitable for creating airplanes, it is not one of those laws of physics that are called the main : as it deepens, it becomes more and more complex. Any study of the coefficient dependence with from the shape of the aircraft's nose immediately destroys its simplicity. No simple dependency remains. Whether the matter is the law of gravitation: it is simple, and its further deepening only emphasizes this.
We just talked about two types of friction resulting from fast movement in the air or slow in honey. But there is also a type of friction - dry, or sliding friction: they talk about it when one solid slides differently. Such a body needs strength to keep moving. It is called frictional force. Its origin is a very confusing question. Both surfaces in contact are uneven when viewed at the atomic level. At the points of contact, the atoms are linked; when you press on the body, the hitch breaks and vibrations occur (in any case, something similar happens). Previously, it was thought that the friction mechanism is simple: the surface is covered with irregularities and friction is the result of lifting the sliding parts to these irregularities; but this is wrong, because then there would be no energy loss, but in fact the energy is spent on friction. The mechanism of losses is different: irregularities are crumpled during sliding, vibrations and motion of atoms arise, and heat spreads over both bodies. And here it turns out to be extremely unexpected that this friction can be approximately described empirically by a simple law. The force required to overcome friction and drag one object along the surface of another depends on the force directed along the normal (perpendicular) to the contact surfaces. In a fairly good approximation, we can assume that the friction force is proportional to the normal force with a more or less constant coefficient:
F = μ N, (12.1)
where μ - coefficient of friction(Fig. 12.1).
Although the μ coefficient is not very constant, this formula turns out to be a good rule of thumb for estimating how much force will be needed in any given practical or engineering circumstance. Only when the normal force or speed of movement is too great does the law fail: too much heat is emitted. It is important to understand that any of these empirical laws have limitations outside of which they do not work.
The approximate validity of the formula F = μ N can be attested simple experience... We put a bar of weight W on a plane inclined at an angle θ. Raise the plane steeper until the bar slides off it under the weight of its own weight. The downward weight component along the plane W sin θ is equal to the friction force F, once the bar slides evenly. The component of the weight normal to the plane is W · cosθ; she is normal force N. The formula becomes W sin θ = μ· W cosθ, whence μ = sin θ / cosθ = tgθ. According to this law, at a certain inclination of the plane, the bar begins to slide. If the bar is loaded with additional weight, then all forces in the formula will increase in the same proportion, and W will drop out of the formula. If the value μ has not changed, then the loaded bar will slide off again at the same inclination. Having determined the angle θ from experience, we will make sure that with a greater weight of the bar, sliding still starts at the same angle of inclination. Even if the weight has increased many times, this rule is respected. We come to the conclusion that the coefficient of friction does not depend on weight.
When doing this experiment, it is easy to see that when right angle tilt into the bar slides not continuously, but with stops: in one place it will get stuck, and in another it will rush forward. This behavior is a sign that the coefficient of friction can only be roughly considered constant: it varies from place to place. The same uncertain behavior is observed when the load on the bar changes. Differences in friction arise from different smoothness or hardness of parts of the surface, from dirt, rust and other extraneous influences. The tables that list the friction coefficients of "steel on steel", "copper on copper" and so on are all sheer cheating, because they neglect these trifles, and in fact they determine the value μ ... Friction "copper on copper", etc., is actually friction "on contaminants adhering to copper."
In experiments of the described type, friction is almost independent of speed. Many people believe that the friction that must be overcome to set the object in motion (static) is greater than the force required to maintain the motion that has already arisen (sliding friction). But on dry metals it is difficult to notice any difference. This opinion was probably generated by experiments in which traces of oil or grease were present, or maybe the bars were fixed there with a spring or something flexible, as if tied to a support.
It is very difficult to achieve accuracy in quantitative friction experiments, and until now friction has not been very well analyzed, despite the enormous importance of such analysis for technology. Although the law F = μ N for standard surfaces is almost accurate, the reason for this kind of law is not really understood. To show that μ does not depend much on speed, especially delicate experiments are needed, because the visible friction greatly decreases from the rapid vibrations of the lower surface. In experiments at high speeds, care must be taken that the bodies do not tremble, otherwise the visible friction immediately decreases. In any case, this law of friction belongs to those semi-experienced laws that are not fully understood and do not become clearer, despite enormous efforts. Almost no one can now estimate the coefficient of friction between two substances.
Earlier it was said that attempts to measure μ sliding of pure substances (copper on copper) leads to questionable results, because the contacting surfaces are not pure copper, but mixtures of oxides and other contaminants. If we want to get completely pure copper, if we clean and polish the surfaces, degass the substance in a vacuum and observe all the necessary precautions, then all the same μ we will not receive. Because two pieces of copper stick together, and then at least put the plane upside down! Coefficient μ , for moderately hard surfaces, usually less than a unit, here it grows to several units! The reason for this unexpected behavior is as follows: when atoms of the same kind come into contact, they cannot "know" that they belong to different pieces of copper. If there were other atoms between them (atoms, oxides, lubricants, thin surface layers of impurities), then it would be “clear” to the copper atoms whether they are on the same piece or on different ones. Remember now that it is precisely because of the forces of attraction between the copper atoms, copper is an solid matter, and you will understand why it is impossible to correctly determine the coefficient of friction for pure metals.
The same phenomenon is observed in a simple home experience with a glass plate and a glass. Place the glass on a plate, put a loop on it and pull; it glides well and the coefficient of friction is felt; of course this ratio is slightly irregular, but it is still a ratio. Now moisten the plate and stem of the glass and pull; you will feel that they are stuck together. If you look closely, you can even find scratches. The fact is that water can remove grease and other substances that clog the surface; a clean glass-to-glass contact remains. This contact is so good that breaking it is not so easy: breaking it is more difficult than tearing out pieces of glass, and that is why scratches appear.
1) Newton's first law: There are such frames of reference, called inertial, relative to which free bodies move uniformly and rectilinearly.
The first law of mechanics, or the law of inertia, as it is often called, ball, was essentially established by Galileo, but Newton gave it a general formulation.
Free body - is called a body that is not affected by any other bodies or fields. When solving some problems, the body can be considered free if external influences are balanced.
Reference frames in which a free material point is at rest or moving rectilinearly and uniformly are called inertial reference frames. The rectilinear and uniform motion of a free material point in an inertial reference frame is called coasting. With such a motion, the velocity vector of the material point remains constant (= const). The rest of the point is a special case of inertial motion (= 0).
In inertial reference frames, rest or uniform motion is a natural state, and dynamics must explain the change in this state (i.e., the appearance of the body's acceleration under the action of forces). There are no free bodies that are not influenced 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.
Reference frames in which a free body does not keep the speed of movement unchanged are called non-inertial. A non-inertial frame of reference is one that moves with acceleration relative to any inertial frame of reference. In a non-inertial frame of reference, even a free body can move with acceleration.
Uniform and straight-line movement of the reference frame does not affect the stroke mechanical phenomena flowing 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 applies to electromagnetic and optical phenomena.
2) Mass, density, strength.
The property of a body to maintain its speed in the absence of interaction with other bodies is called inertia. A physical quantity that is a measure of the inertness of a body in translational motion is called inert mass... Body weight is measured in kilograms:. Mass also characterizes the body's ability to interact with other bodies in accordance with the law universal gravitation... In these cases, mass acts as a measure of gravity and is called gravitational mass.
In modern physics, the identity of the values of the inert and gravitational masses of a given body has been proven with a high degree of accuracy. Therefore, they just talk about body weight(m).
In Newtonian mechanics, it is believed that
a) the body mass 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, mass conservation law: for any processes occurring in a system of bodies, its mass remains unchanged.
The density of a homogeneous body is equal to. Density unit 1 kg / m 3.
By force called vector physical quantity, which is a measure of the mechanical effect on the body from other bodies or fields. A force is fully defined if its modulus, direction, and point of application are given. The straight line along which the force is directed is called line of action of force.
As a result of the action of the force, the body changes the speed of movement (acquires acceleration) or deforms. On the basis of these experimental facts, the forces are measured.
The force is not the cause of the speed, but the acceleration of the body. In all cases, the direction of the force coincides with the direction of the acceleration, but not the speed.
In the tasks of mechanics, gravitational forces (gravitational forces) and two types of electromagnetic forces - elastic forces and frictional 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 connection between the acceleration of a particle, its mass and the force with which these bodies act on it:
If the surrounding bodies act on a particle with mass m with 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:
On the basis of this law, the unit of force is established - 1 N (newton). 1 N is the force with which it is necessary to act on a body weighing 1 kg in order to impart an acceleration of 1 m / s 2 to it.
If the strength , with which the bodies act on a given particle, is known, then the equation of Newton's second law written for this particle calls 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 basic 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, expressing each of these actions in the form of a corresponding force, it is necessary to compose the equation of motion of this particle. The particle acceleration is found from the equation of motion (for a known mass). Knowing
the acceleration can be determined by its speed, and after the speed - and the position of a given particle at any time.
Practice shows that solving the main 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 are directed along one straight line in opposite directions.
This means that if the body A from the side of the body V the force acts, then at the same time on the body V from the side of the body A force will act , and = -.
Using Newton's second law, you can write:
Hence it follows that
i.e., the ratio of the modules of acceleration and bodies interacting with each other is determined by the inverse ratio of their masses and does not depend at all on the nature of the forces acting between them. A more massive body gets less acceleration, and a lighter one gets more.
It is important to understand that the forces about which in question in Newton's third law, 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 laws:
1. 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.
Attention! The task is as follows: find and correct the errors in the article, fill in the blanks. Expand the article necessary materials... Be careful: there may be several spaces and several errors in one sentence.Cart experience
We roll the trolley on wheels from an inclined plane to the floor, where a pile of sand is poured. Having reached it, the cart will get bogged down in the sand and stop. We smooth out the sand and again let the cart slide down the hill. Now the speed of the cart will decrease much more slowly. If the sand is removed, then the decrease in the speed of the cart will be barely noticeable at all.
1.V= 0, the reason is sand on the plane.
2.V decreases more slowly, because the force of friction affects.
3. Movement of the cart by inertia, V approximately does not change.
If the resultant of all forces applied to a material point (body) is equal to zero, then the speed of the point (body) does not change either in magnitude or in direction.
The wording of the law: Material point (body) Isolated from external forces, it maintains a state of rest or uniform rectilinear motion until the applied forces force him (her) to change this state.
In other words, the body remains at rest or moves uniformly in a straight line, since all the resultant of all the forces acting on it is zero. As long as this is maintained, the speed of the body is either zero (at rest) or constant (with rectilinear uniform motion).
Inertia- this is a phenomenon in which a body maintains its speed in the absence of other bodies acting on it. The motion of a free body is called motion by inertia, and the preservation of speed by it is called the phenomenon of inertia.
Newton's first law is valid for inertial reference frames. For such reference systems can be taken: heliocentric system, train movement, trolley movement. A non-inertial frame of reference is also considered to be such a frame of reference that moves with acceleration relative to the non-inertial frame. Reference systems moving with acceleration relative to the Earth or to some bodies are called inertial.
Experimental confirmation of Newton's first law.
A coin lying on the plexiglass covering the neck of the bottle falls into the bottle with a sharp click on the plexiglass in the horizontal plane.
If the vehicle is suddenly braked, passengers who are not wearing their seat belts continue to coast forward, which can lead to injury.
Output: thus, it follows from Newton's first law that a body can move both in the presence and in the absence of external influence.
Inertia is the phenomenon of keeping the speed of bodies constant. A body that is not subject to external influences (called free) is at rest or moves evenly and rectilinearly.
There are such frames of reference, relative to which a translationally moving body keeps its speed constant, if other bodies do not act on it or the actions of bodies are compensated.
Inertial frames of reference are a frame of reference with the origin at the center of the Earth, a frame of reference associated with the center of the Sun. Any system moving relative to inertial reference frames is inertial.
Systems moving with acceleration are non-inertial and Newton's laws are not fulfilled in them.
The main laws of classical mechanics are the three laws of Newton. Now we will look at them in more detail.
Newton's first law
Observations and experience show that bodies receive acceleration relative to the Earth, that is, they change their speed relative to the Earth, only when other bodies act on them.
Let us imagine that the plug of the air "pistol" comes in motion under the action of the gas compressed by the extended piston, i.e. we get such a consistent chain of forces:
Force driving the piston => Force of the piston compressing the gas in the cylinder => Force of the gas driving the plug.
In this and other similar cases, the change in speed, i.e. the emergence of acceleration is the result of the action of forces on a given body of other bodies.
If forces do not act on the body (or the forces are compensated, i.e.), then the body will remain at rest (relative to the Earth), or move uniformly and rectilinearly, i.e. without acceleration.
Based on this, it was possible to establish Newton's first law, which is more often called the law of inertia:
There are such inertial frames of reference, relative to which, the body is at rest (a special case of motion) or moves uniformly and rectilinearly, if the body is not acted upon by forces or the actions of these forces are compensated.
It is practically impossible to verify this law by simple experiments, because it is impossible to completely eliminate the action of all surrounding forces, especially the action of friction.
Meticulous experiments on the study of the movement of bodies were first carried out by the Italian physicist Galilei Galileo at the end of XVI and early XVII centuries. Later, this law was described in more detail by Isaac Newton, therefore this law was named after him.
Such manifestations of inertia of bodies are widely used in everyday life and technology. Shaking a dusty rag, "dropping" a column of mercury in a thermometer.
Newton's second law
Various experiments show that the acceleration coincides with the direction of the force causing this acceleration. Therefore, it is possible to formulate the law of the dependence of the forces applied to the body on the acceleration:
In the inertial reference system, the product of mass and acceleration is equal to the resultant force (the resultant force is the geometric sum of all forces applied to the body).
Body weight is the coefficient of proportionality of this relationship.By definition of acceleration () write the law in a different form, andfurther it turns out that in the numerators of the right-hand side of the equality is the change in momentum Δp since Δ p = mΔv
Hence, the second law can be written as follows:
In this form, Newton wrote down his second law.
This law is valid only for speeds much less than the speed of light and in inertial frames of reference.
Newton's third law
When two bodies collide, their speed changes, i.e. both bodies get acceleration. The earth attracts the moon and makes it move along a curved path; in turn, the Moon also attracts the Earth (the force of gravity).
These examples show that forces always arise in pairs: if one body acts with force on another, then the second body acts on the first with the same force. All forces are mutual.
Then we can formulate Newton's third law:
Bodies act in pairs on each other with forces directed along a straight line, equal in magnitude and opposite in direction.
This law is often called a difficult law, because do not understand the meaning of this law. For ease of understanding of the law, you can reformulate thislaw ( "Action is equal to reaction") on « The opposing force is equal to the acting force ", since these forces are applied to different bodies.
Even the fall of bodies strictly obeys the law of opposition. The apple falls on the Earth because it is attracted Earth; but exactly with the same force and the apple attracts our entire planet.
For the Lorentz force, Newton's third law does not hold.
Newton formulated the basic laws of mechanics in his book "Mathematical Principles of Natural Philosophy".
So, we can conclude that all these three Newton's laws are fundamental to classical mechanics; and each of the laws follows in the other.