Hooke's law is a mathematical notation. Hooke's law. Formula. Description of the experience

How many of us have ever wondered how amazingly objects behave when acted upon?

For example, why can fabric, if we stretch it in different directions, stretch for a long time, and then suddenly tear at one moment? And why is the same experiment much more difficult to carry out with a pencil? What does the resistance of a material depend on? How can you determine to what extent it can be deformed or stretched?

An English researcher asked himself all these and many other questions more than 300 years ago and found the answers, now united under common name"Hooke's Law".

According to his research, every material has a so-called elasticity coefficient. This is a property that allows a material to stretch within certain limits. The elasticity coefficient is a constant value. This means that each material can only withstand a certain level of resistance, after which it reaches a level of irreversible deformation.

In general, Hooke's Law can be expressed by the formula:

where F is the elastic force, k is the already mentioned elasticity coefficient, and /x/ is the change in the length of the material. What is meant by a change in this indicator? Under the influence of force, a certain object under study, be it a string, rubber or any other, changes, stretching or compressing. Changing the length in in this case The difference between the initial and final length of the object being studied is calculated. That is, how much the spring (rubber, string, etc.) has stretched/compressed.

Hence, knowing the length and constant coefficient of elasticity for of this material, you can find the force with which the material is stretched, or elastic force, as Hooke's Law is often called.

There are also special cases, in which this law in its standard form cannot be used. It's about about measuring the force of deformation under shear conditions, that is, in situations where the deformation is produced by a certain force acting on the material at an angle. Hooke's law under shear can be expressed as follows:

where τ is the desired force, G is a constant coefficient known as the shear modulus of elasticity, y is the shear angle, the amount by which the angle of inclination of the object has changed.

We continue our review of some topics from the “Mechanics” section. Our meeting today is dedicated to the force of elasticity.

It is this force that underlies the operation of mechanical watches; towing ropes and cables of cranes, shock absorbers of cars and railways are exposed to it. She is tested by a ball and a tennis ball, a racket and other sports equipment. How does this force arise, and what laws does it obey?

How is elastic force generated?

A meteorite falls to the ground under the influence of gravity and... freezes. Why? Does gravity disappear? No. Power cannot just disappear. At the moment of contact with the ground is balanced by another force equal in magnitude and opposite in direction. And the meteorite, like other bodies on the surface of the earth, remains at rest.

This balancing force is the elastic force.

The same elastic forces appear in the body during all types of deformation:

• sprains;
• compression;
• shift;
• bending;
• torsion.

The forces resulting from deformation are called elastic.

The nature of elastic force

The mechanism of the emergence of elastic forces was explained only in the 20th century, when the nature of the forces of intermolecular interaction was established. Physicists called them "a giant with short arms" What is the meaning of this witty comparison?

There are forces of attraction and repulsion between the molecules and atoms of a substance. This interaction is due to their constituents. tiny particles, carrying positive and negative charges. These forces are quite strong(hence the word giant), but appear only at very short distances(with short arms). At distances equal to three times the diameter of the molecule, these particles are attracted, “joyfully” rushing towards each other.

But, having touched, they begin to actively push away from each other.

With tensile deformation, the distance between the molecules increases. Intermolecular forces tend to reduce it. When compressed, the molecules come closer together, which generates repulsion between the molecules.

And, since all types of deformations can be reduced to compression and tension, the appearance of elastic forces under any deformations can be explained by these considerations.

Law established by Hooke

The study of elastic forces and their relationship with others physical quantities was engaged in by a compatriot and contemporary. He is considered the founder of experimental physics.

Scientist continued his experiments for about 20 years. He conducted experiments on the deformation of tension springs, hanging various loads from them. The suspended load caused the spring to stretch until the elastic force that arose in it balanced the weight of the load.

As a result of numerous experiments, the scientist concludes: an applied external force causes the appearance of an elastic force equal in magnitude, acting in the opposite direction.

The law he formulated (Hooke’s law) sounds like this:

The elastic force that arises during deformation of a body is directly proportional to the magnitude of the deformation and is directed in the direction opposite to the movement of particles.

The formula for Hooke's law is:

• F is the modulus, i.e. the numerical value of the elastic force;
• x - change in body length;
• k is the stiffness coefficient, depending on the shape, size and material of the body.

The minus sign indicates that the elastic force is directed in the direction opposite to the displacement of the particles.

Each physical law has its own limits of application. The law established by Hooke can only be applied to elastic deformations, when, after removing the load, the shape and size of the body are completely restored.

In plastic bodies (plasticine, wet clay) such restoration does not occur.

All solids have elasticity to one degree or another. Rubber takes first place in terms of elasticity, second place -. Even very elastic materials can exhibit plastic properties under certain loads. This is used for making wire and cutting out parts of complex shapes with special stamps.

If you have a manual kitchen scale (steelyard), then the maximum weight for which it is designed is probably written on it. Let's say 2 kg. When hanging a heavier load, the steel spring located in them will never regain its shape.

Work of elastic force

Like any force, the force of elasticity, capable of doing work. And very useful. She protects the deformable body from destruction. If she fails to cope with this, the destruction of the body occurs. For example, a cable breaks crane, a string on a guitar, an elastic band on a slingshot, a spring on a scale. This work always has a minus sign, since the elastic force itself is also negative.

Instead of an afterword

Armed with some information about elastic forces and deformations, we can easily answer some questions. For example, why do large human bones have a tubular structure?

Bend a metal or wooden ruler. Its convex part will experience tensile deformation, and its concave part will experience compression deformation. The middle part does not bear the load. Nature took advantage of this circumstance, providing humans and animals with tubular bones. During movement, bones, muscles and tendons experience all types of deformation. The tubular structure of bones significantly lightens their weight without affecting their strength at all.

Stems cereal crops have the same structure. Gusts of wind bend them to the ground, and elastic forces help them straighten. By the way, the bicycle frame is also made of tubes, not rods: the weight is much less and metal is saved.

The law established by Robert Hooke served as the basis for the creation of the theory of elasticity. Calculations performed using the formulas of this theory allow ensure the durability of high-rise buildings and other structures.

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Hooke's law is formulated as follows: the elastic force that occurs when a body is deformed due to the application of external forces is proportional to its elongation. Deformation, in turn, is a change in the interatomic or intermolecular distance of a substance under the influence of external forces. The elastic force is the force that tends to return these atoms or molecules to a state of equilibrium.

Formula 1 - Hooke's Law.

F - Elastic force.

k - body rigidity (Proportionality coefficient, which depends on the material of the body and its shape).

x - Body deformation (elongation or compression of the body).

This law was discovered by Robert Hooke in 1660. He conducted an experiment, which consisted of the following. Thin steel string was secured at one end, and different forces were applied to the other end. Simply put, a string was suspended from the ceiling and a load of varying mass was applied to it.

Figure 1 - String stretching under the influence of gravity.

As a result of the experiment, Hooke found out that in small aisles the dependence of the stretching of a body is linear with respect to the elastic force. That is, when a unit of force is applied, the body lengthens by one unit of length.

Figure 2 - Graph of the dependence of elastic force on body elongation.

Zero on the graph is the original length of the body. Everything on the right is an increase in body length. In this case, the elastic force has a negative value. That is, she strives to return the body to its original state. Accordingly, it is directed counter to the deforming force. Everything on the left is body compression. The elastic force is positive.

The stretching of the string depends not only on the external force, but also on the cross-section of the string. A thin string will somehow stretch due to its light weight. But if you take a string of the same length, but with a diameter of, say, 1 m, it is difficult to imagine how much weight will be required to stretch it.

To assess how a force acts on a body of a certain cross-section, the concept of normal mechanical stress is introduced.

Formula 2 - normal mechanical stress.

S-Cross-sectional area.

This stress is ultimately proportional to the elongation of the body. Relative elongation is the ratio of the increment in the length of a body to its total length. And the proportionality coefficient is called Young's modulus. Modulus because the value of the elongation of the body is taken modulo, without taking into account the sign. It does not take into account whether the body is shortened or lengthened. It is important to change its length.

Formula 3 - Young's modulus.

|e| - Relative elongation of the body.

s is normal body tension.

The force of resistance of an elastic substance to linear stretching or compression is directly proportional to the relative increase or decrease in length.

Imagine that you grabbed one end of an elastic spring, the other end of which is fixed motionless, and began to stretch or compress it. The more you compress or stretch a spring, the more it resists this. It is on this principle that any spring scale is designed - be it a steelyard (in which the spring is stretched) or a platform spring scale (the spring is compressed). In any case, the spring resists deformation under the influence of the weight of the load, and the force of gravitational attraction of the weighed mass to the Earth is balanced by the elastic force of the spring. Thanks to this, we can measure the mass of the object being weighed by the deviation of the end of the spring from its normal position.

First for real Scientific research The process of elastic stretching and compression of matter was undertaken by Robert Hooke. Initially, in his experiment, he did not even use a spring, but a string, measuring how much it extended under the influence of various forces applied to one end, while the other end was rigidly fixed. He managed to find out that up to a certain limit, the string stretches strictly proportionally to the magnitude of the applied force, until it reaches the limit of elastic stretching (elasticity) and begins to undergo irreversible nonlinear deformation ( cm. below). In equation form, Hooke's law is written in the following form:

Where F— elastic resistance force of the string, x- linear tension or compression, and k- so-called elasticity coefficient. The higher k, the stiffer the string and the harder it is to stretch or compress. The minus sign in the formula indicates that the string is resisting deformation: when stretched, it tends to shorten, and when compressed, it tends to straighten.

Hooke's law formed the basis of a branch of mechanics called theory elasticity. It turned out that it has much wider applications, since atoms in a solid behave as if they were connected to each other by strings, that is, elastically fixed in a three-dimensional crystal lattice. Thus, with slight elastic deformation of an elastic material active forces are also described by Hooke's law, but slightly more complex form. In the theory of elasticity, Hooke's law takes the following form:

σ /η = E

Where σ — mechanical stress(specific force applied to the cross-sectional area of ​​the body), η - relative elongation or compression of the string, and E - so-called Young's modulus, or elastic modulus, playing the same role as the elasticity coefficient k. It depends on the properties of the material and determines how much the body will stretch or contract during elastic deformation under the influence of a single mechanical stress.

In fact, Thomas Young is much better known in science as one of the proponents of the theory of the wave nature of light, who developed a convincing experiment with splitting a light beam into two beams to confirm it ( cm. The principle of complementarity and interference), after which doubts about fidelity wave theory no one had any light left (although Jung was never able to fully put his ideas into a strict mathematical form). Generally speaking, Young's modulus is one of three quantities that describe the response of a solid material to an external force applied to it. The second is displacement modulus(describes how much a substance is displaced under the influence of a force applied tangentially to a surface), and the third - Poisson's ratio(describes how much a solid thins when stretched). The latter is named after the French mathematician Simeon-Denis Poisson (1781-1840).

Of course, Hooke's law, even in the form improved by Jung, does not describe everything that happens to a solid under the influence of external forces. Imagine a rubber band. If you do not stretch it too much, a return force of elastic tension will arise from the rubber band, and as soon as you release it, it will immediately come together and take its previous shape. If you stretch the rubber band further, sooner or later it will lose its elasticity, and you will feel that the tensile strength has weakened. So you have crossed the so-called elastic limit material. If you pull the rubber further, after some time it will completely break and the resistance will disappear completely - you have crossed the so-called breaking point.

In other words, Hooke's law only applies to relatively small compressions or stretches. While the substance retains its elastic properties, the deformation forces are directly proportional to its magnitude, and you are dealing with linear system— each equal increment of applied force corresponds to an equal increment of deformation. It's worth re-tightening the tires elastic limit, and the interatomic bonds-springs inside the substance first weaken and then break - and a simple linear equation Guka stops describing what is happening. In this case, it is customary to say that the system has become nonlinear. Today, the study of nonlinear systems and processes is one of the main directions in the development of physics.

Robert Hooke, 1635—1703

English physicist. Born in Freshwater on the Isle of Wight, the son of a priest, he graduated from Oxford University. While still at the university, he worked as an assistant in the laboratory of Robert Boyle, helping the latter build a vacuum pump for the installation in which the Boyle-Mariotte law was discovered. Being a contemporary of Isaac Newton, he actively participated with him in the work of the Royal Society, and in 1677 he took up the post of scientific secretary there. Like many other scientists of his time, Robert Hooke was interested in a wide variety of fields. natural sciences and contributed to the development of many of them. In his monograph “Micrography” ( Micrographia) he published many sketches of the microscopic structure of living tissues and other biological specimens and was the first to introduce modern concept « living cell" In geology, he was the first to recognize the importance of geological strata and the first in history to engage in the scientific study of natural disasters ( cm. Uniformitarianism). He was one of the first to hypothesize that the force of gravitational attraction between bodies decreases in proportion to the square of the distance between them, and this is a key component of Newton’s Law of Universal Gravitation, and the two compatriots and contemporaries disputed each other’s right to be called its discoverer until the end of their lives. Finally, Hooke developed and personally built a number of important scientific measuring instruments - and many are inclined to see this as his main contribution to the development of science. In particular, he was the first to think of placing a crosshair made of two thin threads in the eyepiece of a microscope, the first to propose taking the freezing temperature of water as zero on the temperature scale, and also invented a universal joint (gimbal joint).

Types of deformations

Deformation called a change in the shape, size or volume of the body. Deformation can be caused by external forces applied to the body. Deformations that completely disappear after the action of external forces on the body ceases are called elastic, and deformations that persist even after external forces have ceased to act on the body - plastic. Distinguish tensile strain or compression(unilateral or comprehensive), bending, torsion And shift.

Elastic forces

For deformities solid its particles (atoms, molecules, ions), located at the nodes of the crystal lattice, are displaced from their equilibrium positions. This displacement is counteracted by the interaction forces between particles of a solid body, which keep these particles at a certain distance from each other. Therefore, with any type of elastic deformation in the body, internal forces, preventing its deformation.

The forces that arise in a body during its elastic deformation and are directed against the direction of displacement of the particles of the body caused by the deformation are called elastic forces. Elastic forces act in any section of a deformed body, as well as at the point of its contact with the body causing deformation. In the case of unilateral tension or compression, the elastic force is directed along the straight line along which the external force acts, causing deformation of the body, opposite to the direction of this force and perpendicular to the surface of the body. The nature of elastic forces is electrical.

We will consider the case of the occurrence of elastic forces during unilateral tension and compression of a solid body.

Hooke's law

The connection between the elastic force and the elastic deformation of a body (at small deformations) was experimentally established by Newton's contemporary, the English physicist Hooke. Mathematical expression Hooke's law for unilateral tension (compression) deformation has the form:

where f is the elastic force; x - elongation (deformation) of the body; k is a proportionality coefficient depending on the size and material of the body, called rigidity. The SI unit of stiffness is newton per meter (N/m).

Hooke's law for one-sided tension (compression) is formulated as follows: The elastic force arising during deformation of a body is proportional to the elongation of this body.

Let's consider an experiment illustrating Hooke's law. Let the axis of symmetry of the cylindrical spring coincide with the straight line Ax (Fig. 20, a). One end of the spring is fixed in the support at point A, and the second is free and the body M is attached to it. When the spring is not deformed, its free end is located at point C. This point will be taken as the origin of the coordinate x, which determines the position of the free end of the spring.

Let's stretch the spring so that its free end is at point D, the coordinate of which is x > 0: At this point the spring acts on the body M with an elastic force

Let us now compress the spring so that its free end is at point B, whose coordinate is x

It can be seen from the figure that the projection of the elastic force of the spring onto the Ax axis always has a sign opposite to the sign of the x coordinate, since the elastic force is always directed towards the equilibrium position C. In Fig. 20, b shows a graph of Hooke's law. The values ​​of elongation x of the spring are plotted on the abscissa axis, and the elastic force values ​​are plotted on the ordinate axis. The dependence of fx on x is linear, so the graph is a straight line passing through the origin of coordinates.

Let's consider another experiment.

Let one end of a thin steel wire be fixed to a bracket, and a load suspended from the other end, the weight of which is an external tensile force F acting on the wire perpendicular to its cross section (Fig. 21).

The action of this force on the wire depends not only on the force modulus F, but also on the cross-sectional area of ​​the wire S.

Under the influence of an external force applied to it, the wire is deformed and stretched. If the stretch is not too great, this deformation is elastic. In an elastically deformed wire, an elastic force f unit arises. According to Newton's third law, the elastic force is equal in magnitude and opposite in direction to the external force acting on the body, i.e.

f up = -F (2.10)

The state of an elastically deformed body is characterized by the value s, called normal mechanical stress(or, for short, just normal voltage). Normal stress s is equal to the ratio of the modulus of the elastic force to the cross-sectional area of ​​the body:

s = f up /S (2.11)

Let the initial length of the unstretched wire be L 0 . After applying force F, the wire stretched and its length became equal to L. The quantity DL = L - L 0 is called absolute wire elongation. The quantity e = DL/L 0 (2.12) is called relative body elongation. For tensile strain e>0, for compressive strain e< 0.

Observations show that for small deformations the normal stress s is proportional to the relative elongation e:

s = E|e|. (2.13)

Formula (2.13) is one of the types of writing Hooke’s law for unilateral tension (compression). In this formula, the relative elongation is taken modulo, since it can be both positive and negative. The proportionality coefficient E in Hooke's law is called the longitudinal modulus of elasticity (Young's modulus).

Let's install physical meaning Young's modulus. As can be seen from formula (2.12), e = 1 and L = 2L 0 for DL ​​= L 0 . From formula (2.13) it follows that in this case s = E. Consequently, Young’s modulus is numerically equal to the normal stress that should arise in the body if its length is doubled. (if Hooke's law were true for such a large deformation). From formula (2.13) it is also clear that in the SI Young’s modulus is expressed in pascals (1 Pa = 1 N/m2).