EntranceDefine Hooke's law. Hooke's law
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. A thin steel string was fixed at one end, and varying amounts of force 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.
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 that physics is not able to understand, it still plays a primary role, and every smallest 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 study methodology. Mechanics deals with the movement of bodies and the interaction of moving bodies, thermodynamics deals with thermal processes, electrodynamics deals with electrical processes.
Why should mechanics study deformation?
When talking about compression or tension, you should ask yourself 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, should hydrodynamics understand deformation? Or molecular kinetic theory?
It all depends on the force of deformation, on its degree. If the deformable medium (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 related, it means that the mechanics will deal with it.
Hooke's law and the condition for its fulfillment
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 satisfied, Let's limit ourselves to two parameters:
- Wednesday;
- force.
There are 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. Conversely, there are environments in which, with sufficiently large forces, the mechanics stop “working.”
Important! To the question: “Under what conditions is Hooke’s law true?”, a definite answer can be given: “At 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 stretching is small;
- elastic object;
- it consists of a material in which there are no nonlinear processes as a result of compression or tension.
Hooke's Law in Mathematical Form
Hooke's formulation, which we cited 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 stretching, F is the force applied to the body and causes deformation (elastic force), k is the elasticity coefficient, 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 appearance when stretched and compressed. Considering that 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 relative to which you are measuring will be directed.
What is the fundamental difference between compression and extension? Nothing if it is insignificant.
The degree of applicability can be considered as follows:
Let's pay attention to the graph. As we 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 relationship (dotted line) becomes nonlinear, and the law ceases to be true. In practice, this is reflected by such strong stretching that the spring stops returning to its original position and loses its properties. With even more stretching a fracture occurs and the structure collapses material.
With small compressions (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 relationship (dotted line) becomes nonlinear, and everything stops working again. In practice, this results in such strong compression that heat begins to be released 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 the length of the body, or, knowing the elastic force, measure the change in length:
Also, in some cases, you can find the elasticity coefficient. To understand how this is done, consider an example task:
A dynamometer is connected to the spring. It was stretched by applying a force of 20, due to which it became 1 meter long. Then they released her, 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.
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 indicator corresponds to spring steel.
Trouble with elasticity coefficient
Physics, as we know, is a very precise science; moreover, it is so precise that it has created entire applied sciences that measure errors. A model of unwavering precision, she cannot afford to be clumsy.
Practice shows that the linear dependence we 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 a variable value that 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, because otherwise, the formula:
can be called nothing more than a dependence between three variables.
Young's modulus
Let's try to figure out the elasticity coefficient. 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 area S from the coefficient, then we will obtain 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 greater the body length, the lower the coefficient k, and the dependence is inversely proportional.
This means that we can write the elasticity coefficient in this way:
where E is a new coefficient, which now precisely depends solely on the type of material.
Let us introduce the concept of “relative elongation”:
.
Conclusion
Let us formulate Hooke's law for tension and compression: For small compressions, normal stress is directly proportional to elongation.
The coefficient E is called Young's modulus and depends solely on the material.
Everything that happens in nature is based on the action of various forces - Hooke’s law is proof of this. This is one of the fundamental phenomena of science.
This process is the determining element in the processes of compression, bending, stretching and other modifications of materials of various structures.
Let's figure out what this law is, how Hooke's rule can be applied in practice, and whether it is always true.
Definition and formula of Hooke's law
For a long time people have tried to explain the origin of the phenomena of compression and tension. Lack of knowledge was the reason for the accumulation of experimental data. Actually, the English tester Hooke discovered his theorem from his observations and experiments. Only later, after the death of the scientist, will his contemporaries call the axiom he derived - Hooke's law.
The researcher noticed that with each elastic impact on an object, a force appears that returns it to its original shape. This was the beginning of the experiments.
Hooke's axiom states:
With very small elastic influences, a force is created that is proportional to the change in the object, but of the opposite sign in terms of the absolute value of the movement of its particles.
Mathematically, this definition can be written as follows:
Fx= Fcontrol= — k*x,
where on the left side it is indicated:
force acting on a body;
x– body movement (m);
k– deformation coefficient, depending on the properties of the object.
The unit of measurement, like any other force, is Newton.
By the way, k also called body stiffness, it is measured in N/m. Rigidity is not determined by the external parameters of the object, but depends on its material.
However, it is worth considering that his law is valid only for elastic deformations.
Elastic force
The formulation is based on the definition of elastic force. What is its difference from other effects on the body?
In fact, elastic force can arise at any point of the body during elastic deformation. What is meant by such influence? This is a change in body shape in which an object returns to its original form after a certain period of time.
And this, in turn, occurs due to the molecular effect of particles: with any deformation, the distance between the molecules of the object changes, and the Coulomb forces of attraction or repulsion tend to return the body to its original position.
The simplest model demonstrating the action of elastic forces is a spring pendulum.
What formula expresses the axiom established by the scientist in this case?
Here Hooke’s axiom will be written in the form:
ε = α * S,
where ε is the relative elongation of the body (its value is equal to the ratio of elongation to displacement);
α – proportionality coefficient (inversely proportional to Young’s modulus E);
S is the mechanical stress of the object (its value is equal to the ratio of the elastic force to the cross-sectional area of the body).
Considering the above, the equation can be written as follows:
Δx/ x= Fcontrol/ E*S,
where Δx is the maximum shear during deformation.
It is worth transforming this expression, then we get the following:
Fcontrol = (E*S/ x) Δx= k * Δx.
Since the elastic force is opposite to the external influence, the law can be briefly read as follows:
Fcontrol= — k * Δx.
It is not for nothing that small deformations are mentioned in it: for them Δx ̴ x, therefore, F control = - k * x.
Under what conditions does Hooke's law hold?
Now let's see what the limits of applicability of this expression are, and under what conditions it is generally true.
You should know that the main condition is:
s= E*e,
where on the left side of the equation is the stress arising during deformation, and on the right side is the Young’s modulus and elongation.
Moreover, E depends on the characteristics of the particles of the object, but not on its shape parameters, and the second factor is taken modulo.
In general, Hooke's axiom is valid for many situations.
So, with elastic bending of a spring lying on two supports, the mathematical representation of the theorem looks like this:
Fcontrol= — m*g
Fcontrol= — k*x
In other situations (with torsion, various pendulums and other deforming processes), the effect of forces on an object is recorded in a similar way.
How to apply the law of elastic deformation in practice
This law (generalized for many situations) is basic in the dynamics and statics of bodies, therefore its applicability is carried out in areas where it is necessary to calculate the rigidity and deformation stress of objects.
First of all, Hooke's rule must be applied in construction and technology. Thus, workers must know exactly what maximum load a tower crane can lift or what load the foundation of a future building can withstand.
None of the trains can do without tension and compression deformation, so Hooke's law is also valid for these situations. In addition, the mechanism and principle of operation of any dynamometers, which are equipped with some parts of technical equipment, are also based on this wonderful law.
Hooke's law is satisfied in all objects that are analogues of the “spring pendulum” model.
In everyday life, at home, you can see the applicability of this law in the springs of some mechanisms.
Thus, Hooke's law is applicable in many areas of human life. It is one of the basic phenomena on which the existence of all life on the planet rests.
Conclusion
To summarize, it should be noted that Hooke’s law is a universal assistant in problems with solutions to the deformation of objects, not only in student books on strength of materials, but also in various engineering fields.
It is these simple tasks that help scientists and craftsmen create new technical models necessary in the conditions of modern technological progress.
If a certain force is applied to a body, its size and (or) shape changes. This process is called body deformation. In bodies undergoing deformation, elastic forces arise that balance external forces.
Types of deformation
All deformations can be divided into two types: elastic deformation And plastic.
Definition
Elastic deformation is called if, after removing the load, the previous dimensions of the body and its shape are completely restored.
Definition
Plastic consider deformation in which changes in the size and shape of the body that appeared due to deformation are partially restored after removing the load.
The nature of the deformation depends on
- magnitude and time of exposure to external load;
- body material;
- body condition (temperature, processing methods, etc.).
There is no sharp boundary between elastic and plastic deformation. In a large number of cases, small and short-term deformations can be considered elastic.
Statements of Hooke's law
It has been empirically found that the greater the deformation necessary to obtain, the greater the deforming force should be applied to the body. By the magnitude of the deformation ($\Delta l$) one can judge the magnitude of the force:
\[\Delta l=\frac(F)(k)\left(1\right),\]
expression (1) means that the absolute value of elastic deformation is directly proportional to the applied force. This statement is the content of Hooke's law.
When deforming elongation (compression) of a body, the following equality holds:
where $F$ is the deforming force; $l_0$ - initial body length; $l$ is the length of the body after deformation; $k$ - elasticity coefficient (stiffness coefficient, stiffness), $ \left=\frac(N)(m)$. The elasticity coefficient depends on the material of the body, its size and shape.
Since elastic forces ($F_u$) arise in a deformed body, which tend to restore the previous size and shape of the body, Hooke’s law is often formulated in relation to elastic forces:
Hooke's law works well for deformations that occur in rods made of steel, cast iron, and other solid substances, in springs. Hooke's law is valid for tensile and compressive deformations.
Hooke's law for small deformations
The elastic force depends on the change in the distance between parts of the same body. It should be remembered that Hooke's law is valid only for small deformations. With large deformations, the elastic force is not proportional to the length measurement; with a further increase in the deforming effect, the body can collapse.
If the deformations of the body are small, then the elastic forces can be determined by the acceleration that these forces impart to the bodies. If the body is motionless, then the modulus of the elastic force is found from the equality to zero of the vector sum of the forces that act on the body.
Hooke's law can be written not only in relation to forces, but it is often formulated for such a quantity as stress ($\sigma =\frac(F)(S)$ is the force that acts on a unit cross-sectional area of a body), then for small deformations:
\[\sigma =E\frac(\Delta l)(l)\ \left(4\right),\]
where $E$ is Young's modulus;$\ \frac(\Delta l)(l)$ is the relative elongation of the body.
Examples of problems with solutions
Example 1
Exercise. A load of mass $m$ is suspended from a steel cable of length $l$ and diameter $d$. What is the tension in the cable ($\sigma $), as well as its absolute elongation ($\Delta l$)?
Solution. Let's make a drawing.
In order to find the elastic force, consider the forces that act on a body suspended from a cable, since the elastic force will be equal in magnitude to the tension force ($\overline(N)$). According to Newton's second law we have:
In the projection onto the Y axis of equation (1.1) we obtain:
According to Newton's third law, a body acts on a cable with a force equal in magnitude to the force $\overline(N)$, the cable acts on a body with a force $\overline(F)$ equal to $\overline(\N,)$ but opposite direction, so the cable deforming force ($\overline(F)$) is equal to:
\[\overline(F)=-\overline(N\ )\left(1.3\right).\]
Under the influence of a deforming force, an elastic force arises in the cable, which is equal in magnitude to:
We find the voltage in the cable ($\sigma $) as:
\[\sigma =\frac(F_u)(S)=\frac(mg)(S)\left(1.5\right).\]
Area S is the cross-sectional area of the cable:
\[\sigma =\frac(4mg\ )((\pi d)^2)\left(1.7\right).\]
According to Hooke's law:
\[\sigma =E\frac(\Delta l)(l)\left(1.8\right),\]
\[\frac(\Delta l)(l)=\frac(\sigma )(E)\to \Delta l=\frac(\sigma l)(E)\to \Delta l=\frac(4mgl\ ) ((\pi d)^2E).\]
Answer.$\sigma =\frac(4mg\ )((\pi d)^2);\ \Delta l=\frac(4mgl\ )((\pi d)^2E)$
Example 2
Exercise. What is the absolute deformation of the first spring of two springs connected in series (Fig. 2), if the spring stiffness coefficients are equal: $k_1\ and\ k_2$, and the elongation of the second spring is $\Delta x_2$?
Solution. If a system of series-connected springs is in a state of equilibrium, then the tension forces of these springs are the same:
According to Hooke's law:
According to (2.1) and (2.2) we have:
Let us express from (2.3) the elongation of the first spring:
\[\Delta x_1=\frac(k_2\Delta x_2)(k_1).\]
Answer.$\Delta x_1=\frac(k_2\Delta x_2)(k_1)$.
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. For different materials, Young's modulus varies widely. 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, when bending deformation the elastic force is proportional to the deflection of the rod, the ends of which lie on two supports (Fig. 1.12.2).
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Figure 1.12.2. Bend deformation.  |
The elastic force acting on the body from the side of the support (or suspension) is called ground reaction force. When the bodies come into contact, the support reaction force is directed perpendicular contact surfaces. That's why it's often called strength normal pressure. If a body lies on a horizontal stationary table, the support reaction force is directed vertically upward and balances the force of gravity: The force with which the body acts on the table is called body weight.
In technology, spiral-shaped springs(Fig. 1.12.3). When springs are stretched or compressed, elastic forces arise, which also obey Hooke's law. The coefficient k is called spring stiffness. Within the limits of applicability of Hooke's law, springs are capable of greatly changing their length. Therefore, they are often used to measure forces. A spring whose tension is measured in units of force is called dynamometer. It should be borne in mind that when a spring is stretched or compressed, complex torsional and bending deformations occur in its coils.
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| Figure 1.12.3. Spring extension deformation. |
Unlike springs and some elastic materials (for example, rubber), the tensile or compressive deformation of elastic rods (or wires) obeys Hooke's linear law within very narrow limits. For metals, the relative deformation ε = x / l should not exceed 1%. With large deformations, irreversible phenomena (fluidity) and destruction of the material occur.
§ 10. Elastic force. Hooke's law
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
When a solid body is deformed, 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, internal forces arise in the body that prevent 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. The mathematical expression of 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<0. В этой точке пружина действует на тело М упругой силой
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 value DL=L-L 0 is called absolute wire elongation. Size
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:
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 us establish the physical meaning of Young's modulus. As can be seen from formula (2.12), e=1 and L=2L 0 with 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).
Tension diagram
Using formula (2.13), from the experimental values of the relative elongation e, one can calculate the corresponding values of the normal stress s arising in the deformed body and construct a graph of 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 satisfied, i.e., the normal stress is proportional to the relative elongation. The maximum value of normal stress s p, at which Hooke’s law is still satisfied, is called limit of proportionality.
With a further increase in load, the dependence of stress on relative elongation becomes nonlinear (section 1-2), although the elastic properties of the body are still preserved. The maximum value s of normal stress, at which residual deformation does not yet occur, is called elastic limit. (The elastic limit exceeds the proportionality limit by only hundredths of a percent.) Increasing the load above the elastic limit (section 2-3) leads to the fact that the deformation becomes residual.
Then the sample begins to elongate at almost constant stress (section 3-4 of the graph). This phenomenon is called material fluidity. The normal stress s t at which the residual deformation reaches a given value is called yield strength.
At stresses exceeding the yield strength, 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 normal stress spr, above which the sample ruptures, is called tensile 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 up /S=E|DL|/L 0 .
whence it follows that the elastic force fуn, arising during deformation of the body, is determined by the formula
f up =ES|DL|/L 0 . (2.14)
Let us determine the work A def performed during 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 the 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
Then determined by the formula A def =
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 spring stiffness; x is the extension of the spring. 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 formulated and written for unilateral tension (compression)?
What is rigidity? What is the SI unit of hardness?
Draw a diagram and explain an experiment that illustrates Hooke's law. Draw a graph of this law.
After making an explanatory drawing, describe the process of stretching a metal wire under load.
What is normal mechanical stress? What formula expresses the meaning of this concept?
What is called absolute elongation? relative elongation? What formulas express the meaning of 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 stress-strain diagram of a metal specimen.
What is called the limit of proportionality? elasticity? turnover? strength?
Obtain formulas that determine the work of deformation and potential energy of an elastically deformed body.