Ministry of Education of the Autonomous Republic of Crimea
Tauride National University named after. Vernadsky
Study of physical law
HOOKE'S LAW
Completed by: 1st year student
Faculty of Physics gr. F-111
Potapov Evgeniy
Simferopol-2010
Plan:
The connection between what phenomena or quantities is expressed by the law.
Statement of the law
Mathematical expression of the law.
How was the law discovered: based on experimental data or theoretically?
Experienced 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 effect of the law in practice.
Literature.
The relationship between what phenomena or quantities is expressed by the law:
Hooke's law relates phenomena such as stress and deformation of a solid, elastic modulus and elongation. The modulus of the elastic force arising during deformation of a body is proportional to its elongation. Elongation is a characteristic of the deformability of a material, assessed by the increase in the length of a sample of this material when stretched. Elastic force is a force that arises during deformation of a body and counteracts this deformation. Stress is a measure of internal forces that arise in a deformable body under the influence of external influences. Deformation is a change in the relative position of particles of a body 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.
Statement 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 elongation (compression), and k called elasticity coefficient(or rigidity). The minus in the equation indicates that the tension force is always directed in the direction opposite to the deformation.
If you enter the relative elongation
and normal stress in the cross section
then Hooke's law will be written like this
In this form it is valid for any small volumes of matter.
In the general case, stress and strain are tensors of the second rank in three-dimensional space (they have 9 components each). The tensor of elastic constants connecting them is a tensor of the fourth rank 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 was the law discovered: based on experimental data or theoretically:
The law was discovered in 1660 by the English scientist Robert Hooke (Hook) based on observations and experiments. The discovery, as stated by Hooke in his essay “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 above law of proportionality applies not only to metals, but also to wood, stones, horn, bones, glass, silk, hair, etc.
Experienced 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 to a distance Δ l, then their product will be equal in magnitude to the force stretching the body (wire). This relationship will hold true, however, not for all deformations, but for small ones. With large deformations, Hooke's law ceases to apply and the body collapses.
Examples of using the law and taking into account the effect of the law in practice:
As follows from Hooke's law, the elongation of a spring can be used to judge the force acting on it. This fact is used to measure forces using a dynamometer - a spring with a linear scale calibrated for different force values.
Literature.
1. Internet resources: - Wikipedia website (http://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83 %D0%BA%D0%B0).
2. textbook on physics Peryshkin A.V. 9th grade
3. textbook on physics V.A. Kasyanov 10th grade
4. lectures on mechanics Ryabushkin D.S.
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.
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:
F x= 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.
CONTROL QUESTIONS
1) What is called deformation? What types of deformations do you know?
Deformation- a change in the relative position of body particles associated with their movement. Deformation is the result of changes in interatomic distances and rearrangement of blocks of atoms. Typically, deformation is accompanied by a change in the magnitude of interatomic forces, the measure of which is elastic stress.
Types of deformations:
Tension-compression- in the resistance of materials - a type of longitudinal deformation of a rod or beam that occurs if a load is applied to it along its longitudinal axis (the resultant of the forces acting on it is normal to the cross section of the rod and passes through its center of mass).
Tension causes elongation of the rod (rupture and residual deformation are also possible), compression causes shortening of the rod (loss of stability and longitudinal bending are possible).
Bend- a type of deformation in which there is a curvature of the axes of straight bars or a change in the curvature of the axes of curved bars. Bending is associated with the occurrence of bending moments in the cross sections of the beam. Direct bending occurs when the bending moment in a given cross-section of a beam acts in a plane passing through one of the main central axes of inertia of this section. In the case when the plane of action of the bending moment in a given cross section of the beam does not pass through any of the main axes of inertia of this section, it is called oblique.
If, during direct or oblique bending, only a bending moment acts in the cross section of the beam, then, accordingly, there is a pure straight or pure oblique bend. If a transverse force also acts in the cross section, then there is a transverse straight or transverse oblique bend.
Torsion- one of the types of body deformation. Occurs when a load is applied to a body in the form of a pair of forces (moment) in its transverse plane. In this case, only one internal force factor appears in the cross sections of the body - torque. Tension-compression springs and shafts work for torsion.
Types of deformation of a solid body. Deformation is elastic and plastic.
Deformation solid body can be a consequence of phase transformations associated with changes in volume, thermal expansion, magnetization (magnetostrictive effect), the appearance of an electric charge (piezoelectric effect) or the result of the action of external forces.
A deformation is called elastic if it disappears after the load that caused it is removed, and plastic if it does not disappear (at least completely) after the load is removed. All real solids, when deformed, have plastic properties to a greater or lesser extent. Under certain conditions, the plastic properties of bodies can be neglected, as is done in the theory of elasticity. With sufficient accuracy, a solid body can be considered elastic, that is, it does not exhibit noticeable plastic deformations until the load exceeds a certain limit.
The nature of plastic deformation can vary depending on temperature, duration of load or strain rate. With a constant load applied to the body, the deformation changes with time; this phenomenon is called creep. As temperature increases, the creep rate increases. Special cases of creep are relaxation and elastic aftereffect. One of the theories explaining the mechanism of plastic deformation is the theory of dislocations in crystals.
Derivation of Hooke's law for various types of deformation.
Net shift:
Pure torsion: 
4) What is called the shear modulus and torsional modulus, what is their physical meaning?
Shear modulus or stiffness modulus (G or μ) characterizes the ability of a material to resist changes in shape while maintaining its volume; it is defined as the ratio of shear stress to shear strain, defined as the change in the right angle between the planes along which shear stresses act). The shear modulus is one of the components of the viscosity phenomenon.
Shear modulus:
Torsion modulus: 
5) What is the mathematical expression of Hooke's law? In what units are elastic modulus and stress measured?
Measured in Pa, - 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.
