This force arises as a result of deformation (change in the initial state of matter). For example, when we stretch a spring, we increase the distance between the molecules of the spring material. When we compress the spring, we decrease it. When we twist or shift. In all these examples, a force arises that prevents deformation - the elastic force.

Hooke's law

The elastic force is directed opposite to the deformation.

Since the body is represented as a material point, the force can be depicted from the center

When connecting springs in series, for example, the stiffness is calculated by the formula

Parallel connection stiffness

The rigidity of the sample. Young's modulus.

Young's modulus characterizes the elastic properties of a substance. This is a constant value that depends only on the material, its physical state. Characterizes the ability of a material to resist tensile or compressive deformation. Young's modulus is tabular.

Body weight

Body weight is the force with which an object acts on a support. You say, it's gravity! The confusion is as follows: indeed, often the weight of the body is equal to the force of gravity, but these forces are completely different. Gravity is a force that results from interaction with the Earth. Weight is the result of interaction with the support. The force of gravity is applied at the center of gravity of the object, while the weight is the force that is applied to the support (not to the object)!

There is no formula for determining weight. This force is designated by a letter.

The reaction force of the support or the elastic force arises in response to the action of the object on the suspension or support, therefore the weight of the body is always numerically the same as the elastic force, but has the opposite direction.

The reaction force of the support and the weight are forces of the same nature, according to Newton's 3 law they are equal and oppositely directed. Weight is a force that acts on the support, not on the body. The force of gravity acts on the body.

Body weight may not be equal to gravity. It can be either more or less, or it can be such that the weight is zero. This state is called weightlessness... Weightlessness is a state when an object does not interact with a support, for example, a state of flight: there is gravity, and the weight is zero!

It is possible to determine the direction of acceleration if we determine where the resultant force is directed.

Note, weight is force, measured in Newtons. How to correctly answer the question: "How much do you weigh"? We answer 50 kg, naming not the weight, but our own mass! In this example, our weight is equal to gravity, which is approximately 500N!

Overload- the ratio of weight to gravity

Archimedes' strength

Force arises as a result of the interaction of a body with a liquid (gas), when it is immersed in a liquid (or gas). This force pushes the body out of the water (gas). Therefore, it is directed vertically upward (pushes). Determined by the formula:

We neglect the power of Archimedes in the air.

If the force of Archimedes is equal to the force of gravity, the body floats. If the force of Archimedes is greater, then it rises to the surface of the liquid, if less, it sinks.

Electrical forces

There are forces of electrical origin. Occur when there is an electrical charge. These forces, such as Coulomb Force, Ampere Force, Lorentz Force.

Newton's laws

Newton's I law

There are such frames of reference, which are called inertial, relative to which the bodies keep their speed unchanged, if other bodies do not act on them or the action of other forces is compensated.

II Newton's law

The acceleration of a body is directly proportional to the resultant forces applied to the body, and inversely proportional to its mass:

III Newton's law

The forces with which two bodies act on each other are equal in magnitude and opposite in direction.

Local frame of reference is a frame of reference that can be considered inertial, but only in an infinitely small neighborhood of some one point in space-time, or only along some one open world line.

Galileo's transformations. The principle of relativity in classical mechanics.

Galileo's transformations. Consider two frames of reference moving relative to each other and with constant velocity v 0. One of these frames will be denoted by the letter K. We will assume motionless. Then the second system K will move rectilinearly and uniformly. Let's choose the coordinate axes x, y, z of the system K and x ", y", z "of the system K" so that the axes x and x "coincide, and the axes y and y", z and z ", were parallel to each other. between the x, y, z coordinates of some point P in the K system and the x ", y", z "coordinates of the same point in the K system. If we start counting the time from the moment when the origin of the system coordinates coincided, then x = x "+ v 0, moreover, it is obvious that y = y", z = z ". Let us add to these relations the assumption accepted in classical mechanics that time in both systems flows in the same way, that is, t = t ". We obtain a set of four equations: x = x" + v 0 t; y = y "; z = z"; t = t ", called Galileo transformations. The mechanical principle of relativity. The proposition that all mechanical phenomena in different inertial reference systems proceed in the same way, as a result of which it is impossible to establish by any mechanical experiments whether the system is at rest or moves uniformly and in a straight line is called Galileo's principle of relativity. Violation of the classical law of addition of velocities. Based on the general principle of relativity (no physical experience can distinguish one inertial system from another), formulated by Albert Einstein, Lawrence changed Galileo's transformations and received: x "= (x-vt) /  (1-v 2 / c 2); y "= y; z "= z; t" = (t-vx / c 2) /  (1-v 2 / c 2). These transformations are called Lawrence transforms.

How many of us have wondered how amazingly objects behave when exposed to them?

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

All these and many other questions more than 300 years ago were asked by an English researcher And found answers, now united under the general name "Hooke's Law".

According to his research, each material has a so-called coefficient of elasticity... This is the property that allows the material to stretch within certain limits. The coefficient of elasticity is a constant value. This means that each material can only withstand a certain level of resistance, after which it reaches the 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 coefficient of elasticity, 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 studied object, be it a string, rubber or any other, changes, stretching or contracting. The change in length in this case is the difference between the initial and final length of the studied subject. That is, how much the spring stretched / compressed (rubber, string, etc.)

From here, knowing the length and constant coefficient of elasticity for a given material, you can find the force with which the material is pulled, or elastic force, as it is often called Hooke's Law.

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

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

Ministry of Education of the Autonomous Republic of Crimea

Tavrichesky National University named after Vernadsky

Physical Law Research

THE HOOK'S LAW

Completed: 1st year student

Faculty of Physics gr. F-111

Potapov Evgeniy

Simferopol-2010

Plan:

The relationship between what phenomena or quantities expresses the law.

The wording of the law

Mathematical expression of the law.

How the law was discovered: on the basis of experimental data or theoretically.

Experiential facts on the basis of which the law was formulated.

Experiments confirming the validity of the law formulated on the basis of the theory.

Examples of using the law and taking into account the operation of the law in practice.

Literature.

The relationship between what phenomena or quantities expresses the law:

Hooke's law connects phenomena such as stress and deformation of a rigid body, modulus of elasticity and elongation. The modulus of the elastic force arising from the deformation of the body is proportional to its elongation. Elongation is the deformability characteristic of a material, assessed by the increase in the length of a specimen from this material under tension. The force of elasticity is the force arising from the deformation of the body and opposing this deformation. Stress is a measure of the internal forces that arise in a deformable body under the influence of external influences. Deformation is a change in the relative position of body particles associated with their movement relative to each other. These concepts are related by the so-called stiffness coefficient. It depends on the elastic properties of the material and the size of the body.

The wording of the law:

Hooke's law is an equation of the theory of elasticity that relates stress and deformation of an elastic medium.

The formulation of the law is that the elastic force is directly proportional to the deformation.

Mathematical expression of the law:

For a thin tensile rod, Hooke's law has the form:

Here F rod tension force, Δ l- its lengthening (compression), and k called coefficient of elasticity(or rigidity). A minus in the equation indicates that the pulling force is always directed in the opposite direction to the deformation.

If we introduce the relative elongation

and the normal stress in the cross section

Hooke's law will be written like this

In this form, it is valid for any small volume of matter.

In the general case, stresses and strains are tensors of the second rank in three-dimensional space (they have 9 components each). The tensor of elastic constants connecting them is the fourth rank tensor C ijkl and contains 81 coefficients. Due to the symmetry of the tensor C ijkl, as well as stress and strain tensors, only 21 constants are independent. Hooke's law looks like this:

where σ ij- stress tensor, - strain tensor. For an isotropic material, the tensor C ijkl contains only two independent coefficients.

How the law was discovered: based on experimental data or theoretically:

The law was discovered in 1660 by the English scientist Robert Hooke (Hooke) on the basis of observations and experiments. The discovery, according to Hooke in his work "De potentia restitutiva", published in 1678, was made by him 18 years earlier, and in 1676 it was placed in another of his books under the guise of the anagram "ceiiinosssttuv", meaning "Ut tensio sic vis" ... According to the author's explanation, the aforementioned law of proportionality applies not only to metals, but also to wood, stones, horn, bones, glass, silk, hair, etc.

Experiential facts on the basis of which the law was formulated:

History is silent about this ..

Experiments confirming the validity of the law formulated on the basis of the theory:

The law is formulated on the basis of experimental data. Indeed, when stretching a body (wire) with a certain stiffness coefficient k distance Δ l, then their product will be equal in magnitude to the force stretching the body (wire). This ratio will be fulfilled, however, not for all deformations, but for small ones. With large deformations, Hooke's law ceases to operate, the body collapses.

Examples of using the law and taking into account the operation of the law in practice:

As follows from Hooke's law, the elongation of the spring can be judged on the force acting on it. This fact is used to measure forces using a dynamometer - a spring with a linear scale, graduated to different values ​​of the forces.

Literature.

1. Internet resources: - Wikipedia site (http://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83 % D0% BA% D0% B0).

2.A textbook on physics Peryshkin A.V. Grade 9

3. a textbook on physics by V.A. Kasyanov Grade 10

4. lectures on mechanics Ryabushkin D.S.

Elastic Coefficient

Elasticity coefficient(sometimes called the Hooke's coefficient, the coefficient of stiffness or the stiffness of a spring) is a coefficient connecting in Hooke's law the elongation of an elastic body and the elastic force resulting from this elongation. It is used in solid mechanics in the elastic section. Denoted by a letter k, sometimes D or c... Has the dimension N / m or kg / s2 (in SI), dyne / cm or g / s2 (in CGS).

The coefficient of elasticity is numerically equal to the force that must be applied to the spring in order for its length to change per unit distance.

Definition and properties

The coefficient of elasticity is by definition equal to the elastic force divided by the change in the length of the spring: k = F e / Δ l. (\ displaystyle k = F _ (\ mathrm (e)) / \ Delta l.) The coefficient of elasticity depends on both the properties of the material and the dimensions of the elastic body. So, for an elastic bar, you can isolate the dependence on the bar dimensions (cross-sectional area S (\ displaystyle S) and length L (\ displaystyle L)) by writing the coefficient of elasticity as k = E ⋅ S / L. (\ displaystyle k = E \ cdot S / L.) The quantity E (\ displaystyle E) is called Young's modulus and, unlike the elastic coefficient, depends only on the material properties of the bar.

The stiffness of deformable bodies when they are connected

When several elastically deformable bodies are connected (hereinafter, for brevity - springs), the overall rigidity of the system will change. With a parallel connection, the rigidity increases, with a serial connection, it decreases.

Parallel connection

When n (\ displaystyle n) springs are connected in parallel with stiffnesses k 1, k 2, k 3,. ... ... , kn, (\ displaystyle k_ (1), k_ (2), k_ (3), ..., k_ (n),) the stiffness of the system is equal to the sum of the stiffnesses, that is, k = k 1 + k 2 + k 3 + ... ... ... + k n. (\ displaystyle k = k_ (1) + k_ (2) + k_ (3) + ... + k_ (n).)

Proof

In a parallel connection, there are n (\ displaystyle n) springs with stiffness k 1, k 2,. ... ... , k n. (\ displaystyle k_ (1), k_ (2), ..., k_ (n).) From III of Newton's Law, F = F 1 + F 2 +. ... ... + F n. (\ displaystyle F = F_ (1) + F_ (2) + ... + F_ (n).) (A force F (\ displaystyle F) is applied to them, and a force F 1 is applied to spring 1, (\ displaystyle F_ (1),) to spring 2 force F 2, (\ displaystyle F_ (2),)…, to spring n (\ displaystyle n) force F n. (\ Displaystyle F_ (n).))

Now, from Hooke's law (F = - k x (\ displaystyle F = -kx), where x is the aspect ratio) we deduce: F = k x; F 1 = k 1 x; F 2 = k 2 x; ... ... ... ; F n = k n x. (\ displaystyle F = kx; F_ (1) = k_ (1) x; F_ (2) = k_ (2) x; ...; F_ (n) = k_ (n) x.) Substitute these expressions into equality (1): kx = k 1 x + k 2 x +. ... ... + k n x; (\ displaystyle kx = k_ (1) x + k_ (2) x + ... + k_ (n) x;) can be reduced by x, (\ displaystyle x,) we get: k = k 1 + k 2 +. ... ... + k n, (\ displaystyle k = k_ (1) + k_ (2) + ... + k_ (n),) as required.

Serial connection

When n (\ displaystyle n) springs are connected in series with stiffnesses k 1, k 2, k 3,. ... ... , kn, (\ displaystyle k_ (1), k_ (2), k_ (3), ..., k_ (n),) the total stiffness is determined from the equation: 1 / k = (1 / k 1 + 1 / k 2 + 1 / k 3 +.. + 1 / kn). (\ displaystyle 1 / k = (1 / k_ (1) + 1 / k_ (2) + 1 / k_ (3) + ... + 1 / k_ (n)).)

Proof

In series connection there are n (\ displaystyle n) springs with stiffness k 1, k 2,. ... ... , k n. (\ displaystyle k_ (1), k_ (2), ..., k_ (n).) From Hooke's law (F = - kl (\ displaystyle F = -kl), where l is the aspect ratio) it follows that F = k ⋅ l. (\ displaystyle F = k \ cdot l.) The sum of the elongations of each spring equals the total elongation of the entire connection l 1 + l 2 +. ... ... + l n = l. (\ displaystyle l_ (1) + l_ (2) + ... + l_ (n) = l.)

The same force F acts on each spring. (\ displaystyle F.) According to Hooke's Law, F = l 1 ⋅ k 1 = l 2 ⋅ k 2 =. ... ... = l n ⋅ k n. (\ displaystyle F = l_ (1) \ cdot k_ (1) = l_ (2) \ cdot k_ (2) = ... = l_ (n) \ cdot k_ (n).) From the previous expressions, we deduce: l = F / k, l 1 = F / k 1, l 2 = F / k 2,. ... ... , l n = F / k n. (\ displaystyle l = F / k, \ quad l_ (1) = F / k_ (1), \ quad l_ (2) = F / k_ (2), \ quad ..., \ quad l_ (n) = F / k_ (n).) Substituting these expressions in (2) and dividing by F, (\ displaystyle F,) we get 1 / k = 1 / k 1 + 1 / k 2 +. ... ... + 1 / k n, (\ displaystyle 1 / k = 1 / k_ (1) + 1 / k_ (2) + ... + 1 / k_ (n),) as required.

Rigidity of some deformable bodies

Constant section bar

A uniform bar of constant cross-section, elastically deformable along the axis, has a stiffness coefficient

K = E S L 0, (\ displaystyle k = (\ frac (E \, S) (L_ (0))),) E- Young's modulus, which depends only on the material from which the rod is made; S- cross-sectional area; L 0 is the length of the rod.

Coiled coil spring

Coiled cylindrical compression or tension spring, wound from a cylindrical wire and elastically deformable along the axis, has a stiffness coefficient

K = G ⋅ d D 4 8 ⋅ d F 3 ⋅ n, (\ displaystyle k = (\ frac (G \ cdot d _ (\ mathrm (D)) ^ (4)) (8 \ cdot d _ (\ mathrm (F )) ^ (3) \ cdot n)),) d- wire diameter; d F is the diameter of the winding (measured from the axis of the wire); n- number of turns; G- shear modulus (for ordinary steel G≈ 80 GPa, for spring steel G≈ 78.5 GPa, for copper ~ 45 GPa).

Sources and Notes

1. Elastic deformation (Russian). Archived June 30, 2012.
2. Dieter Meschede, Christian Gerthsen. Physik. - Springer, 2004. - P. 181 ..
3. Bruno Assmann. Technische Mechanik: Kinematik und Kinetik. - Oldenbourg, 2004. - P. 11 ..
4. Dynamics, Elastic Force (Russian). Archived June 30, 2012.
5. Mechanical properties of bodies (Russian). Archived June 30, 2012.

10. Hooke's law in tension-compression. Elastic modulus (Young's modulus).

Under axial tension or compression up to the proportionality limit σ pr Hooke's law is valid, i.e. law of directly proportional relationship between normal stresses  and longitudinal relative deformations :

(3.10)

or

(3.11)

Here E is the coefficient of proportionality in Hooke's law has the dimension of voltage and is called modulus of elasticity of the first kind characterizing the elastic properties of the material, or Young's modulus.

The relative longitudinal deformation is the ratio of the absolute longitudinal deformation of the area

rod to the length of this section before deformation:

(3.12)

The relative transverse deformation will be equal to:  "= = b / b, where b = b 1 - b.

The ratio of the relative transverse deformation  "to the relative longitudinal deformation , taken in modulus, is a constant value for each material and is called Poisson's ratio:

Determination of the absolute deformation of a section of a bar

In formula (3.11), instead of and substitute expressions (3.1) and (3.12):

From here, we obtain the formula for determining the absolute elongation (or shortening) of a section of a rod with a length:

(3.13)

In formula (3.13), the product ЕА is called the stiffness of the timber in tension or compression, which is measured in kN, or in MN.

According to this formula, the absolute deformation is determined if the longitudinal force is constant in the section. In the case when the longitudinal force is variable in the section, it is determined by the formula:

(3.14)

where N (x) is a function of the longitudinal force along the length of the section.

11.The ratio of lateral deformation (Poisson's ratio

12. Determination of displacements in tension-compression. Hooke's law for a section of a bar. Determination of displacements of sections of a bar

Determine the horizontal movement of the point a the axis of the bar (Figure 3.5) - u a: it is equal to the absolute deformation of a part of the bar ad, enclosed between the embedment and the section drawn through the point, i.e.

In turn, the lengthening of the section ad consists of extensions of individual cargo sections 1, 2 and 3:

Longitudinal forces in the areas under consideration:

Hence,

Then

Similarly, you can determine the displacement of any section of the bar and formulate the following rule:

displacement of any section jbar under tension-compression is defined as the sum of absolute deformations ncargo sections enclosed between the considered and fixed (fixed) sections, i.e.

(3.16)

The stiffness condition for the timber will be written as follows:

, (3.17)

where

- the largest value of the section displacement, taken in modulus from the displacement diagram; u - the permissible value of the section displacement for a given structure or its element, set in the norms.

13. Determination of the mechanical characteristics of materials. Tensile test. Compression test.

To quantify the basic properties of materials like

As a rule, the tension diagram is experimentally determined in the coordinates  and  (Fig. 2.9). The characteristic points are marked on the diagram. Let's give their definition.

The greatest stress to which a material follows Hooke's law is called proportional limit  NS... Within the limits of Hooke's law, the tangent of the slope of the straight line  = f() to the  axis is determined by the value E.

The elastic properties of the material are retained up to stress  Have called elastic limit... Elastic limit  Have is understood to be the highest stress up to which the material does not receive permanent deformations, i.e. after complete unloading, the last point of the diagram coincides with the starting point 0.

The quantity  T called yield point material. The yield point is understood as the stress at which an increase in deformations occurs without a noticeable increase in the load. If it is necessary to distinguish between tensile and compressive yield strength  T is respectively replaced by  TR and  TS... At voltages large  T plastic deformations develop in the body of the structure  NS that do not disappear when the load is removed.

The ratio of the maximum force that the sample can withstand to its initial cross-sectional area is called the ultimate strength, or ultimate resistance, and is denoted by,  BP(when compressing  Sun).

When performing practical calculations, the real diagram (Fig. 2.9) is simplified, and for this purpose, various approximating diagrams are used. To solve problems taking into account resiliently plastic properties of structural materials is most often used Prandtl diagram... According to this diagram, the stress changes from zero to the yield point according to Hooke's law  = E, and further with the growth of ,  =  T(fig. 2.10).

The ability of materials to receive permanent deformations is called plasticity... In fig. 2.9 was presented a characteristic diagram for plastic materials.

Rice. 2.10 Fig. 2.11

The opposite property of plasticity is the property fragility, i.e. the ability of the material to break down without the formation of noticeable permanent deformations. A material with this property is called fragile... Brittle materials include cast iron, high-carbon steel, glass, brick, concrete, natural stones. A typical diagram of deformation of brittle materials is shown in Fig. 2.11.

1. What is called body deformation? How is Hooke's Law formulated?

Vakhit shavaliev

Deformations are any changes in the shape, size and volume of the body. Deformation determines the end result of the movement of body parts relative to each other.
Elastic deformations are deformations that completely disappear after the elimination of external forces.
Plastic deformations are deformations that are fully or partially preserved after the cessation of the action of external forces.
Elastic forces are forces that arise in a body during its elastic deformation and are directed in the direction opposite to the displacement of particles during deformation.
Hooke's law
Small and short-term deformations with a sufficient degree of accuracy can be considered elastic. For such deformations, Hooke's law is valid:
The elastic force arising from the deformation of the body is directly proportional to the absolute elongation of the body and is directed in the direction opposite to the displacement of the body particles:
\
where F_x is the projection of the force on the x axis, k is the stiffness of the body, depending on the size of the body and the material from which it is made, the unit of stiffness in the SI system N / m.
http://ru.solverbook.com/spravochnik/mexanika/dinamika/deformacii-sily-uprugosti/

Varya Guseva

Deformation is a change in the shape or volume of a body. Types of deformation - stretching or compression (examples: stretch an elastic band or squeeze, accordion), bending (the board bent under a person, bent a sheet of paper), twisting (working with a screwdriver, wringing out clothes with your hands), shear (when braking a car, tires are deformed due to friction force ).
Hooke's Law: The elastic force arising in a body during its deformation is directly proportional to the magnitude of this deformation
or
The force of elasticity arising in the body during its deformation is directly proportional to the magnitude of this deformation.
Hooke's law formula: Fcont = kx

Hooke's Law. Can be expressed by the formula F = -kx or F = kx?

⚓ Otter ☸

Hooke's law is an equation of the theory of elasticity that relates stress and deformation of an elastic medium. Discovered in 1660 by the English scientist Robert Hooke. Since Hooke's law is written for low stresses and strains, it has the form of simple proportionality.

For a thin tensile rod, Hooke's law has the form:
Here F is the tensile force of the rod, Δl is its elongation (compression), and k is called the coefficient of elasticity (or stiffness). A minus in the equation indicates that the pulling force is always directed in the opposite direction to the deformation.

The coefficient of elasticity depends both on the properties of the material and on the dimensions of the bar. The dependence on the bar dimensions (cross-sectional area S and length L) can be distinguished explicitly by writing the coefficient of elasticity as
The quantity E is called Young's modulus and depends only on the properties of the body.

If we introduce the relative elongation
and the normal stress in the cross section
then Hooke's law will be written as
In this form, it is valid for any small volume of matter.
[edit]
Generalized Hooke's Law

In the general case, stresses and strains are tensors of the second rank in three-dimensional space (they have 9 components each). The tensor of elastic constants connecting them is the fourth-rank tensor Cijkl and contains 81 coefficients. Due to the symmetry of the tensor Cijkl, as well as the stress and strain tensors, only 21 constants are independent. Hooke's law looks like this:
For an isotropic material, the tensor Cijkl contains only two independent coefficients.

It should be borne in mind that Hooke's law is fulfilled only for small deformations. When the proportionality limit is exceeded, the relationship between stresses and strains becomes nonlinear. For many media, Hooke's law is inapplicable even for small deformations.
[edit]

in short, you can do this and that, depending on what you want to indicate in the end: just the module of Hooke's force or also the direction of this force. Strictly speaking, of course, -kx, since Hooke's force is directed against the positive increment of the coordinate of the end of the spring.

As you know, physics studies all the laws of nature: from the simplest to the most general principles of natural science. Even in those areas where, it would seem, physics is not able to figure it out, it still plays a primary role, and every slightest law, every principle - nothing escapes it.

In contact with

It is physics that is the basis of the foundations, it is this that lies at the origins of all sciences.

Physics studies the interaction of all bodies, both paradoxically small and incredibly large. Modern physics is actively studying not just small, but hypothetical bodies, and even this sheds light on the essence of the universe.

Physics is divided into sections, this simplifies not only the science itself and its understanding, but also the methodology of study. Mechanics deals with the movement of bodies and the interaction of moving bodies, thermodynamics - thermal processes, electrodynamics - electrical.

Why deformation should be studied by mechanics

Speaking about compressions or stretches, one should ask oneself a question: which branch of physics should study this process? With strong distortions, heat can be released, maybe these processes should be dealt with by thermodynamics? Sometimes, when liquids are compressed, it begins to boil, and when gases are compressed, liquids are formed? So what, deformation should be learned by hydrodynamics? Or molecular kinetic theory?

It all depends from the force of deformation, from its degree. If a deformable medium (a material that is compressed or stretched) allows, and the compression is small, it makes sense to consider this process as the movement of some points of the body relative to others.

And since the question concerns purely, it means that the mechanic will deal with it.

Hooke's law and the condition for its implementation

In 1660, the famous English scientist Robert Hooke discovered a phenomenon that can mechanically describe the process of deformation.

In order to understand under what conditions Hooke's law is fulfilled, let's restrict ourselves to two parameters:

• Wednesday;
• force.

There are such media (for example, gases, liquids, especially viscous liquids, close to solid states, or, conversely, very fluid liquids) for which the process cannot be described mechanically. And vice versa, there are such environments in which mechanics ceases to "work" at sufficiently large forces.

Important! To the question: "Under what conditions is Hooke's law fulfilled?", One can give a definite answer: "At small deformations."

Hooke's law, definition: the deformation that occurs in the body is directly proportional to the force that causes this deformation.

Naturally, this definition implies that:

• little compression or expansion;
• the subject is elastic;
• it consists of a material in which there are no non-linear processes as a result of compression or tension.

Hooke's law in mathematical form

Hooke's formulation, which we gave above, makes it possible to write it in the following form:

where is the change in body length due to compression or tension, F is the force applied to the body and causing deformation (elastic force), k is the coefficient of elasticity, measured in N / m.

It should be remembered that Hooke's law is valid only for small stretches.

Also note that it has the same appearance when stretched and compressed. Considering that the force is a vector quantity and has a direction, in the case of compression, the following formula will be more accurate:

But again, it all depends on where the axis, relative to which you are measuring, will be directed.

What's the cardinal difference between squeezing and stretching? Nothing if it is insignificant.

The degree of applicability can be considered as follows:

Let's pay attention to the chart. As you can see, with small stretches (the first quarter of the coordinates) for a long time the force with the coordinate has a linear relationship (red line), but then the real dependence (dotted line) becomes nonlinear, and the law ceases to be fulfilled. In practice, this is reflected in such a strong tension that the spring stops returning to its original position and loses its properties. With even greater stretch a break occurs and the structure collapses material.

With small compressions (the third quarter of the coordinates) for a long time the force with the coordinate also has a linear relationship (red line), but then the real dependence (dotted line) becomes nonlinear, and everything again ceases to be fulfilled. In practice, this is reflected in such strong compression that heat starts to build up and the spring loses its properties. With even greater compression, the coils of the spring "stick together" and it begins to deform vertically, and then completely melt.

As you can see, the formula expressing the law allows you to find the force, knowing the change in body length, or, knowing the elastic force, measure the change in length:

Also, in some cases, you can find the coefficient of elasticity. In order to understand how this is done, consider an example task:

A dynamometer is connected to the spring. It was stretched with a force of 20, which made it 1 meter long. Then they let her go, waited until the hesitation stopped and she returned to her normal state. In its normal state, its length was 87.5 centimeters. Let's try to find out what material the spring is made of.

Let's find the numerical value of the spring deformation:

From here we can express the value of the coefficient:

Looking at the table, we can find that this figure corresponds to spring steel.

Trouble with the coefficient of elasticity

Physics, as you know, is a very precise science; moreover, it is so accurate that it has created whole applied sciences that measure errors. As a benchmark of unshakable precision, it cannot afford to be awkward.

Practice shows that the linear dependence we have considered is nothing more than Hooke's law for a thin and tensile rod. It can only be used as an exception for springs, but even that is undesirable.

It turns out that the coefficient k is a variable quantity that depends not only on the material of the body, but also on the diameter and its linear dimensions.

For this reason, our conclusions require clarification and development, because otherwise, the formula:

cannot be called anything other than a relationship between three variables.

Young's modulus

Let's try to figure out the coefficient of elasticity. This parameter, as we found out, depends on three quantities:

• material (which suits us perfectly);
• length L (which indicates its dependence on);
• S.

Important! Thus, if we manage to somehow "separate" the length L and the area S from the coefficient, then we will get a coefficient that completely depends on the material.

What we know:

• the larger the cross-sectional area of ​​the body, the greater the coefficient k, and the dependence is linear;
• the longer the body length, the lower the coefficient k, and the dependence is inversely proportional.

So, we can write the coefficient of elasticity in this way:

moreover, E is a new coefficient, which now exactly depends solely on the type of material.

Let's introduce the concept of "relative elongation":

. 

Output

Let us formulate Hooke's law in tension and compression: at low compressions, the normal stress is directly proportional to the elongation.

The E factor is called Young's modulus and depends solely on the material.

The coefficient E in this formula is called Young's modulus... Young's modulus depends only on the properties of the material and does not depend on the size and shape of the body. Young's modulus varies widely for different materials. For steel, for example, E ≈ 2 · 10 11 N / m 2, and for rubber E ≈ 2 · 10 6 N / m 2, that is, five orders of magnitude less.

Hooke's law can be generalized to the case of more complex deformations. For example, for bending deformations the elastic force is proportional to the deflection of the rod, the ends of which lie on two supports (Fig. 1.12.2).

Figure 1.12.2. Bending deformation.

The elastic force acting on the body from the side of the support (or suspension) is called support reaction force... When the bodies touch, the reaction force of the support is directed perpendicular contact surfaces. Therefore, it is often called strength. normal pressure... If the body lies on a horizontal stationary table, the reaction force of the support is directed vertically upward and balances the force of gravity: The force with which the body acts on the table is called body weight.

The technique often uses spiral springs(fig. 1.12.3). When the springs are stretched or compressed, elastic forces arise, which also obey Hooke's law. The coefficient k is called spring rate... Within the limits of applicability of Hooke's law, springs are capable of greatly varying their length. Therefore, they are often used to measure forces. A spring, the tension of which is graduated in units of force, is called dynamometer... It should be borne in mind that when the spring is stretched or compressed, complex torsion and bending deformations occur in its coils.

Figure 1.12.3. Spring tension deformation.

Unlike springs and some elastic materials (for example, rubber), the deformation of tension or compression of elastic rods (or wires) obeys the linear Hooke's law within very narrow limits. For metals, the relative deformation ε = x / l should not exceed 1%. With large deformations, irreversible phenomena (fluidity) and material destruction occur.

§ 10. The force of elasticity. Hooke's law

Types of deformations

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

Elastic forces

During deformations of a 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 forces of interaction between the particles of the solid, which keep these particles at a certain distance from each other. Therefore, for any type of elastic deformation, internal forces arise in the body that prevent its deformation.

The forces arising in the body during its elastic deformation and directed against the direction of displacement of the body particles caused by deformation are called elastic forces. Elastic forces act in any section of the deformed body, as well as in the place of its contact with the body, causing deformations. In the case of unilateral tension or compression, the elastic force is directed along a straight line along which an 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 emergence of elastic forces during unilateral tension and compression of a rigid 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. The mathematical expression of Hooke's law for the deformation of unilateral tension (compression) has the form

where f is the elastic force; x - elongation (deformation) of the body; k - coefficient of proportionality, 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 unilateral stretching (compression) is formulated as follows: the elastic force arising from the deformation of a body is proportional to the elongation of this body.

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 other is free and body M. is attached to it. When the spring is not deformed, its free end is at point C. This point will be taken as the origin of the x coordinate, which determines the position of the free end of the spring.

We 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

We will now compress the spring so that its free end is at point B, the coordinate of which is x<0. В этой точке пружина действует на тело М упругой силой

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

Consider another experience.
Let one end of a thin steel wire be fixed on a bracket, and a load is 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 effect of this force on the wire depends not only on the modulus of force F, but also on the cross-sectional area of ​​the wire S.

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

f pack = -F (2.10)

The state of an elastically deformed body is characterized by the quantity s, called normal mechanical stress(or, for short, just normal voltage). The 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 pack / S (2.11)

Let the initial length of the unstretched wire be L 0. After the application of force F, the wire stretched and its length became equal to L. The value DL = L-L 0 is called absolute wire elongation... The value

are called relative lengthening of the body... For tensile deformation e> 0, for compression deformation e<0.

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

Formula (2.13) is one of the types of recording 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 modulus of longitudinal elasticity (Young's modulus).

Let us establish the physical meaning of Young's modulus. As can be seen from formula (2.12), e = 1 and L = 2L 0 at 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 have arisen in the body with an increase in its length by a factor of 2. (if Hooke's law was satisfied for such a large deformation). It is also seen from formula (2.13) that in SI Young's modulus is expressed in pascals (1 Pa = 1 N / m 2).

Stretch diagram

Using formula (2.13), from the experimental values ​​of the relative elongation e, we can calculate the corresponding values ​​of the normal stress s arising in the deformed body, and plot the dependence of s on e. This graph is called stretch diagram... A similar graph for a metal sample is shown in Fig. 22. In section 0-1, the graph looks like a straight line passing through the origin. This means that up to a certain stress value, the deformation is elastic and Hooke's law is fulfilled, i.e., the normal stress is proportional to the relative elongation. The maximum value of the normal stress s p, at which Hooke's law still holds, is called proportional limit.

With a further increase in the load, the dependence of the stress on the relative elongation becomes nonlinear (section 1-2), although the elastic properties of the body are still preserved. The maximum value of s at normal stress, at which permanent deformation does not yet occur, is called elastic limit... (The elastic limit is only hundredths of a percent higher than the proportional limit.) An increase in the load above the elastic limit (section 2-3) leads to the fact that the deformation becomes permanent.

Then the sample begins to elongate at almost constant stress (section 3-4 of the graph). This phenomenon is called material flow. The normal stress s t, at which the permanent deformation reaches a given value, is called yield point.

At stresses exceeding the yield point, the elastic properties of the body are restored to a certain extent, and it again begins to resist deformation (section 4-5 of the graph). The maximum value of the normal stress s pr, above which the sample ruptures, is called ultimate strength.

Energy of an elastically deformed body

Substituting the values ​​of s and e from formulas (2.11) and (2.12) into formula (2.13), we obtain

f pack / S = E | DL | / L 0.

whence it follows that the elastic force f yn, arising during the deformation of the body, is determined by the formula

f pack = ES | DL | / L 0. (2.14)

Let us determine the work A def, performed during the deformation of the body, and the potential energy W of the elastically deformed body. According to the law of conservation of energy,

W = A def. (2.15)

As can be seen from formula (2.14), the modulus of elastic force can change. It increases in proportion to the deformation of the body. Therefore, to calculate the work of deformation, it is necessary to take the average value of the elastic force equal to half of its maximum value:

= ES | DL | / 2L 0. (2.16)

Then determined by the formula A def = | DL | deformation work

A def = ES | DL | 2 / 2L 0.

Substituting this expression into formula (2.15), we find the value of the potential energy of an elastically deformed body:

W = ES | DL | 2 / 2L 0. (2.17)

For an elastically deformed spring ES / L 0 = k is the stiffness of the spring; x - spring elongation. Therefore, formula (2.17) can be written in the form

W = kx 2/2. (2.18)

Formula (2.18) determines the potential energy of an elastically deformed spring.

Questions for self-control:

 What is deformation?

 What deformation is called elastic? plastic?

 Name the types of deformations.

 What is elastic force? How is it directed? What is the nature of this force?

 How is Hooke's law for unilateral tension (compression) formulated and written?

 What is stiffness? What is the SI unit of stiffness?

 Draw a diagram and explain the experience that illustrates Hooke's Law. Plot this law.

 After drawing an explanatory drawing, describe the process of stretching a metal wire under load.

 What is called normal mechanical stress? What formula expresses the meaning of this concept?

 What is called absolute elongation? relative lengthening? What formulas express these concepts?

• What is the form of Hooke's law in a record containing normal mechanical stress?

 What is called Young's modulus? What is its physical meaning? What is the SI unit of Young's modulus?

• Draw and explain the tensile diagram of a metal specimen.

 What is called the proportional limit? elasticity? fluidity? strength?

 Obtain formulas by which the work of deformation and potential energy of an elastically deformed body are determined.