Buy a diploma of higher education inexpensively. Elastic forces, formulas

Strengthelasticity is that power which occurs when the body is deformed and which seeks to restore the former shape and dimensions of the body.

The elastic force arises as a result of the electromagnetic interaction between the molecules and atoms of a substance.

The simplest version of deformation can be considered using the example of compression and extension of a spring.

In this picture (x > 0) — tensile strain; (x< 0) — compression deformation. (FX) is an external force.

In the case when the deformation is the most insignificant, i.e. small, the elastic force is directed to the side, which is opposite in the direction of the moving particles of the body and is proportional to the deformation of the body:

Fx = Fcontrol = - kx

With the help of this ratio, Hooke's law is expressed, which was established by the experimental method. Coefficient k commonly referred to as the rigidity of the body. The stiffness of a body is measured in newtons per meter (N/m) and depends on the size and shape of the body, as well as on what materials the body is made of.

Hooke's law in physics for determining the compressive or tensile deformation of a body is written in a completely different form. In this case, the relative deformation is called

Robert Hooke

(18.07.1635 - 03.03.1703)

English naturalist, encyclopedist

attitude ε = x / l . At the same time, stress is the cross-sectional area of ​​the body after relative deformation:

σ = F / S = -Fcontrol / S

In this case, Hooke's law is formulated as follows: the stress σ is proportional to the relative strain ε . In this formula, the coefficient E called Young's modulus. This module does not depend on the shape of the body and its dimensions, but at the same time, it directly depends on the properties of the materials that make up the given body. For various materials Young's modulus fluctuates over a fairly wide range. For example, for rubber E ≈ 2 106 N/m2, and for steel E ≈ 2 1011 N/m2 (i.e. five orders of magnitude more).

It is quite possible to generalize Hooke's law in cases where more complex deformations are performed. For example, consider bending deformation. Consider a rod that rests on two supports and has a significant deflection.

From the side of the support (or suspension), an elastic force acts on this body, this is the reaction force of the support. The reaction force of the support at the contact of the bodies will be directed to the contact surface strictly perpendicular. This force is called the force of normal pressure.

Let's consider the second option. The path of the body lies on a fixed horizontal table. Then the reaction of the support balances the force of gravity and it is directed vertically upwards. Moreover, the weight of the body is considered the force with which the body acts on the table.

As you know, physics studies all the laws of nature: from the simplest to the most general principles natural sciences. 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.

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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 is concerned with the motion of bodies and the interaction of moving bodies, thermodynamics with thermal processes, and electrodynamics with electrical processes.

Why deformation should be studied by mechanics

Speaking of contractions or tensions, one should ask oneself the question: which branch of physics should study this process? With strong distortions, heat can be released, perhaps thermodynamics should deal with these processes? Sometimes, when liquids are compressed, it begins to boil, and when gases are compressed, liquids are formed? So what, the hydrodynamics should learn the deformation? Or molecular kinetic theory?

It all depends on the force of deformation, on its degree. If the 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 is purely concerned, it means that mechanics will deal with this.

Hooke's law and the condition for its implementation

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

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

• Wednesday;
• strength.

There are such media (for example, gases, liquids, especially viscous liquids close to solid states or, conversely, very fluid liquids) for which it is impossible to describe the process mechanically. And vice versa, there are such environments in which, with sufficiently large forces, the mechanics ceases to “work”.

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

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

Naturally, this definition implies that:

• compression or tension is small;
• the object 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 have given above, makes it possible to write it in the following form:

where is the change in the length of the body 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 valid only for small stretches.

We also note that it has the same form under tension and compression. Given that the force is a vector quantity and has a direction, then in the case of compression, the following formula will be more accurate:

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

What is the fundamental difference between compression and stretching? Nothing if it's insignificant.

The degree of applicability can be considered in the following form:

Let's take a look at the chart. As you can see, with small tensions (the first quarter of the coordinates) for a long time force with coordinate has linear connection(red line), but then the real dependence (dotted line) becomes non-linear, and the law ceases to hold. In practice, this is reflected by such a strong stretch that the spring stops returning to its original position and loses its properties. With more stretch fracture 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 (dashed line) becomes non-linear, and everything again ceases to be fulfilled. In practice, this is reflected by such strong compression that heat starts to radiate 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 melts.

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

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

A dynamometer is connected to the spring. She was stretched, applying a force of 20, because of which she began to have a length of 1 meter. Then they let her go, waited until the vibrations stopped, and she returned to her normal state. In normal condition, its length was 87.5 centimeters. Let's try to find out what material the spring is made of.

Find the numerical value of the spring deformation:

From here we can express the value of the coefficient:

After looking at the table, we can find that this indicator corresponds to spring steel.

Trouble with the coefficient of elasticity

Physics, as you know, is a very precise science, moreover, it is so precise that it has created entire applied sciences that measure errors. As the standard of unwavering precision, she cannot afford to be clumsy.

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

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

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

can not 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 quite well);
• length L (which indicates its dependence on);
• area 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:

• how more area section of the body, the greater the coefficient k, and the dependence is linear;
• the longer the body length, the smaller the coefficient k, and the dependence is inversely proportional.

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

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

Let us introduce the concept of “relative elongation”:

It should be recognized that this value is more meaningful than , since it reflects not just how much the spring has compressed or stretched, but how many times this has happened.

Since we have already “put into play” S, we will introduce the concept of normal stress, which is written as follows:

Important! The normal stress is the proportion of the deforming force per element of the cross-sectional area.

Hooke's law and elastic deformations

Conclusion

We formulate Hooke's law for tension and compression: at low compressions, the normal stress is directly proportional to the relative elongation.

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

All bodies near the Earth are affected by its attraction. Under the influence of gravity, raindrops, snowflakes, leaves torn off the branches fall to the Earth.

But when the same snow lies on the roof, it is still attracted by the Earth, but it does not fall through the roof, but remains at rest. What prevents it from falling? Roof. It acts on snow with a force equal to gravity, but directed in the opposite direction. What is this power?

Figure 34, a shows a board lying on two stands. If a weight is placed in its middle, then under the influence of gravity the weight will begin to move, but after a while, having bent the board, it will stop (Fig. 34, b). In this case, the force of gravity will be balanced by the force acting on the weight from the side of the curved board and directed vertically upwards. This force is called elastic force. The elastic force arises during deformation. Deformation is a change in the shape or size of the body. One type of deformation is bending. The more the support bends, the more power elasticity acting from this support on the body. Before the body (weight) was placed on the board, this force was absent. As the weight moved, which bent its support more and more, the elastic force also increased. At the moment the weight stops, the elastic force has reached the force of gravity and their resultant has become equal to zero.

If a sufficiently light object is placed on the support, then its deformation may turn out to be so insignificant that we will not notice any change in the shape of the support. But the deformation will still be! And along with it, the elastic force will also act, preventing the fall of the body located on this support. In such cases (when the deformation of the body is imperceptible and the change in the size of the support can be neglected), the elastic force is called support reaction force.

If some kind of suspension (thread, rope, wire, rod, etc.) is used instead of a support, then the object attached to it can also be held at rest. The force of gravity here will also be balanced by the oppositely directed force of elasticity. In this case, the elastic force arises due to the fact that the suspension is stretched under the action of the load attached to it. stretching another kind of distortion.

The elastic force also occurs when compression. It is she who makes the compressed spring straighten and push the body attached to it (see Fig. 27, b).

A great contribution to the study of the force of elasticity was made by the English scientist R. Hooke. In 1660, when he was 25 years old, he established a law that was later named after him. Hooke's law says:

The elastic force that occurs when a body is stretched or compressed is proportional to its elongation.

If the elongation of the body, i.e., the change in its length, is denoted by x, and the elastic force is denoted by F control, then Hooke's law can be given the following mathematical form:

F control \u003d kx,

where k is the proportionality factor, called rigidity body. Each body has its own rigidity. The greater the rigidity of a body (spring, wire, rod, etc.), the less it changes its length under the action of a given force.

The SI unit of stiffness is newton per meter(1 N/m).

Having done a series of experiments that confirmed this law, Hooke refused to publish it. Therefore, for a long time no one knew about his discovery. Even after 16 years, still not trusting his colleagues, Hooke in one of his books gave only an encrypted formulation (anagram) of his law. She looked

After waiting two years for competitors to claim their discoveries, he finally deciphered his law. The anagram was deciphered as follows:

ut tensio, sic vis

(which in Latin means: what is the tension, such is the force). “The strength of any spring,” Hooke wrote, “is proportional to its stretching.”

Hooke studied elastic deformations. This is the name of deformations that disappear after the cessation of external influence. If, for example, a spring is stretched a little and then released, it will return to its original shape. But the same spring can be stretched so much that, after it is released, it will remain stretched. Deformations that do not disappear after the cessation of external influence are called plastic.

Plastic deformations are used in modeling from plasticine and clay, in metal processing - forging, stamping, etc.

For plastic deformations, Hooke's law is not satisfied.

In ancient times, the elastic properties of some materials (in particular, a tree such as yew) allowed our ancestors to invent onion - hand weapon, designed for throwing arrows with the help of the elastic force of a stretched bowstring.

Having appeared about 12 thousand years ago, the bow has existed for many centuries as the main weapon of almost all tribes and peoples of the world. Before invention firearms the bow was the most effective combat weapon. English archers could shoot up to 14 arrows per minute, which, with the massive use of bows in battle, created a whole cloud of arrows. For example, the number of arrows fired at the Battle of Agincourt (during the Hundred Years' War) was approximately 6 million!

The widespread use of this formidable weapon in the Middle Ages caused a justified protest from certain circles society. In 1139, the Lateran (Church) Council, which met in Rome, banned the use of these weapons against Christians. However, the struggle for "bow disarmament" was not successful, and the bow as military weapon continued to be used by humans for another five hundred years.

The improvement of the design of the bow and the creation of crossbows (crossbows) led to the fact that the arrows fired from them began to pierce any armor. But military science did not stand still. And in the XVII century. the bow was supplanted by firearms.

Nowadays, archery is just one of the sports.

1. In what cases does the elastic force arise? 2. What is called deformation? Give examples of deformations. 3. Formulate Hooke's law. 4. What is hardness? 5. How do elastic deformations differ from plastic ones?

DEFINITION

Deformations any changes in the shape, size and volume of the body are called. Deformation determines the final result of the movement of body parts relative to each other.

DEFINITION

Elastic deformations are called deformations that completely disappear after the removal of external forces.

Plastic deformations are called deformations that are fully or partially preserved after the cessation of the action of external forces.

The ability to elastic and plastic deformation depends on the nature of the substance of which the body consists, the conditions in which it is located; ways of making it. For example, if you take different grades of iron or steel, then you can find completely different elastic and plastic properties in them. At ordinary room temperatures, iron is a very soft, ductile material; hardened steel, on the other hand, is a hard, resilient material. The plasticity of many materials is a condition for their processing, for the manufacture of the necessary parts from them. Therefore, it is considered one of the most important technical properties of a solid.

When a solid body is deformed, particles (atoms, molecules, or ions) are displaced from their original equilibrium positions to new positions. In this case, the force interactions between the individual particles of the body change. As a result, in the deformed body, internal forces preventing its deformation.

There are tensile (compression), shear, bending, and torsion strains.

elastic forces

DEFINITION

elastic forces are the forces that arise in the body during its elastic deformation and are directed in the direction opposite to the displacement of particles during deformation.

Elastic forces are of electromagnetic nature. They prevent deformations and are directed perpendicular to the contact surface of the interacting bodies, and if such bodies as springs and threads interact, then the elastic forces are directed along their axis.

The elastic force acting on the body from the side of the support is often called the reaction force of the support.

DEFINITION

Tensile deformation (linear deformation)- this is a deformation in which only one linear dimension of the body changes. Its quantitative characteristics are absolute and relative elongation.

Absolute elongation:

where and are the lengths of the body in the deformed and undeformed states, respectively.

Relative extension:

Hooke's Law

Small and short-term deformations with a sufficient degree of accuracy can be considered as elastic. For such deformations, Hooke's law is valid:

where the projection of the force on the axis is the rigidity of the body, depending on the dimensions of the body and the material from which it is made, the unit of stiffness in the SI system N/m.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2

Elastic Forces and Deformations

Definition 1

The force that arises in the body as a result of its deformation and tends to return it to its initial state is called the elastic force.

All bodies material world subject to various types of deformation. Deformations arise due to movement and, as a result, changes in the position of body particles relative to each other. According to the degree of reversibility, we can distinguish:

• elastic, or reversible deformations;
• plastic (residual), or irreversible deformations.

In cases where the body, upon completion of the action of forces leading to deformation, restores its original parameters, the deformation is called elastic.

It should be noted that during elastic deformation, the effect of an external force on the body does not exceed the elastic limit. Thus, the elastic forces compensate for the external influence on the body.

Otherwise, the deformation is plastic or permanent. A body that has been exposed to this kind of impact does not restore its original size and shape.

The elastic forces arising in the bodies are not able to fully balance the forces that cause plastic deformation.

In general, there are a number of simple deformations:

• stretching (compression);
• bend;
• shift;
• torsion.

As a rule, deformations are often a combination of several types of action presented, which makes it possible to reduce all deformations to the two most common types, namely tension and shear.

Characteristics of elastic forces

The modulus of elastic force acting per unit area is physical quantity, called stress (mechanical).

Mechanical stress, depending on the direction of force application, can be:

• normal (directed along the normal to the surface, $σ$);
• tangential (directed tangentially to the surface, $τ$).

Remark 1

The degree of deformation is characterized by a quantitative measure - relative deformation.

So, for example, the relative change in the length of the rod can be described by the formula:

$ε=\frac(\Delta l)(l)$,

and relative longitudinal tension (compression):

$ε’=\frac(\Delta d)(d)$, where:

$l$ is the length, and $d$ is the rod diameter.

The deformations $ε$ and $ε'$ proceed simultaneously and have opposite signs, due to the fact that during tension the change in the length of the body is positive, and the change in diameter is negative; in cases with compression of the body, the signs are reversed. Their relationship is described by the formula:

Here $μ$ is the Poisson's ratio, which depends on the properties of the material.

Hooke's Law

By their nature, elastic forces are electromagnetic, not fundamental forces, and, therefore, they are described by approximate formulas.

So, it is empirically established that for small deformations, the relative elongation and stress are proportional, or

Here $E$ is the coefficient of proportionality, also called Young's modulus. It takes on a value at which the relative elongation is equal to one. Young's modulus is measured in newtons per square meter(Pascals).

According to Hooke's law, the elongation of a rod under elastic deformation is proportional to the force acting on the rod, or:

$F=\frac(ES)(l)\Delta l=k\Delta l$

The value of $k$ is called the coefficient of elasticity.

Deformation solids is described by Hooke's law only until the limit of proportionality is reached. With increasing stress, the deformation ceases to be linear, but, up to the achievement of the elastic limit, residual deformations do not occur. Thus, Hooke's Law is valid only for elastic deformations.

Plastic deformation

With a further increase in the acting forces, residual deformations occur.

Definition 2

The value of mechanical stress at which a noticeable residual deformation occurs is called the yield strength ($σt$).

Further, the degree of deformation increases without increasing stress until the ultimate strength ($σr$) is reached, when the body is destroyed. If we graphically depict the return of the body to its original state, then the area between the points $σт$ and $σр$ will be called the yield region (plastic deformation region). Depending on the size of this area, all materials are divided into viscous, in which the yield area is significant, and brittle, in which the yield area is minimal.

Note that before we considered the effect of forces applied along the normal to the surface. If external forces were applied tangentially, shear deformation occurs. In this case, at each point of the body, a tangential stress occurs, determined by the modulus of force per unit area, or:

$τ=\frac(F)(S)$.

The relative shift, in turn, can be calculated by the formula:

$γ=\frac(1)(G)τ$, where $G$ is the shear modulus.

The shear modulus takes the value of the tangential stress at which the shear value is equal to one; $G$ is measured in the same way as voltage, in pascals.