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 strain 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.
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 rate... 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, 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 \u003d 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 (unilateral or comprehensive), bending, torsion and shift.
Elastic forces
With deformations solid its particles (atoms, molecules, ions) located in the nodes crystal latticeare shifted out of 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 a 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 stretching 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 tension (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 that illustrates 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 coordinate x, 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\u003e 0: At this point, the spring acts on the body M with an elastic force
We 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 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. On the abscissa axis, the values \u200b\u200bof the elongation x of the spring are plotted, and on the ordinate, the values \u200b\u200bof the elastic force. 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 to a bracket, and a load is suspended to 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 \u200b\u200bthe 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 \u003d -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 \u200b\u200bthe body:
s \u003d f pack / S (2.11)
Let the initial length of the unstretched wire be L 0. After applying the force F, the wire stretched and its length became equal to L. The value DL \u003d L-L 0 is called absolute wire elongation... The quantity
called body lengthening... For tensile strain e\u003e 0, for compression strain 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 \u003d 1 and L \u003d 2L 0 at DL \u003d L 0. It follows from formula (2.13) that in this case s \u003d 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 2 times. (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 \u003d 1 N / m 2).
Stretch diagram
Using formula (2.13), from the experimental values \u200b\u200bof the relative elongation e, we can calculate the corresponding values \u200b\u200bof 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 value of stress, 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 no permanent deformation occurs, 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 in formula (2.13) the values \u200b\u200bof s and e from formulas (2.11) and (2.12), we obtain
f pack / S \u003d 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 \u003d 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 \u003d A def. (2.15)
As can be seen from formula (2.14), the modulus of elastic force can change. It increases in proportion to body deformation. 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 \u003d
A def \u003d 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 \u003d ES | DL | 2 / 2L 0. (2.17)
For an elastically deformed spring ES / L 0 \u003d k - spring stiffness; x - spring elongation. Therefore, formula (2.17) can be written in the form
W \u003d 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 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 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.
Ministry of Education of the Autonomous Republic of Crimea
Tavrichesky National University named after Vernadsky
Physical Law Research
Hooke'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 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 characteristic of deformability 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 counteracting 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
then Hooke's law will be written like this
In this form, it is valid for any small volume of matter.
In general, stresses and strains are tensors of the second rank in three-dimensional space (they have 9 components). 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 was the law 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, as Hooke argued in his work "De potentia restitutiva", published in 1678, was made by him 18 years earlier, and in 1676 it was placed in his other book 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.
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 kdistance Δ l, then their product will be equal in magnitude to the force stretching the body (wire). Such a ratio will hold, 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 \u200b\u200bof 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. textbook on physics by V.A. Kasyanov grade 10
4. lectures on mechanics Ryabushkin D.S.
You and I know that if some force acts on the body, then the body will move under the influence of this force. For example, a snowflake falls to the ground because it is attracted by the Earth. And the gravity of the Earth acts constantly, but the snowflake, having reached the roof, does not continue to fall, but stops, keeping our house dry.
From the point of view of cleanliness and order in the house, everything is correct and logical, but from the point of view of physics, there should be an explanation for everything. And if the snowflake suddenly stops moving, it means that a force should have appeared that opposes its movement. This force acts in the direction opposite to the gravity of the Earth, and is equal to it in magnitude. In physics, this force opposing the force of gravity is called the elastic force and is studied in the seventh grade course. Let's figure out what it is.
What is elastic force?
For an example, explaining what the elastic force is, remember or imagine a simple clothesline on which we hang wet laundry. When we hang something wet, the rope, previously stretched horizontally, bends under the weight of the laundry and stretches slightly. Our little thing, for example, a wet towel, first moves to the ground along with the rope, then stops. And this happens when each new thing is added to the rope. That is, it is obvious that with an increase in the force of impact on the rope, it deforms until the moment when the forces of counteraction of this deformation become equal to the weight of all things. And then the downward movement stops. In simple terms, the work of the elastic force is to maintain the integrity of objects that we affect with other objects. And if the elastic force fails, then the body is deformed irrevocably. The rope breaks, the roof falls under too much snow, and so on. When does elastic force arise? At the moment of the beginning of the impact on the body. When we hang our laundry. And disappears when we take off our linen. That is, when the impact stops. The point of application of the elastic force is the point at which the impact occurs. If we are trying to break the stick on the knee, then the point of application of the elastic force will be the point at which we press on the stick with our knee. This is understandable.
How to find the elastic force: Hooke's law
To learn how to find the elastic force, we must get acquainted with Hooke's law. English physicist Robert Hooke was the first to establish the dependence of the magnitude of the elastic force on the deformation of the body. This dependence is directly proportional. The more deformation occurs, the greater the elastic force. I.e the formula for the elastic force is as follows:
F_control \u003d k * ∆l,
where ∆l is the amount of deformation,
and k is the stiffness coefficient.
The stiffness coefficient is naturally different for different bodies and substances. There are special tables to find it. The elastic force is measured in N / m (newtons per meter).
Elastic force in nature
Elastic force in nature - this is a flock of sparrows on a tree branch, bunches of berries on bushes or caps of snow on spruce legs. At the same time, the bending, but not yielding branches heroically and completely free of charge demonstrate to us the strength of elasticity.
You and I know that if some force acts on the body, then the body will move under the influence of this force. For example, a leaf falls to the ground because it is attracted by the Earth. But if a leaf falls on the bench, it does not continue to fall, and does not fall through the bench, but is at rest.
And if the leaf suddenly stops moving, it means that a force must have appeared that counteracts its movement. This force acts in the direction opposite to the gravity of the Earth, and is equal to it in magnitude. In physics, this force opposing gravity is called the elastic force.
What is elastic force?
Puppy Antoshka loves to watch birds.
For an example that explains what the elastic force is, let us remember the birds and the rope. When the bird sits down on the rope, the support, previously stretched horizontally, bends under the weight of the bird and slightly stretches. The bird first moves to the ground along with the rope, then stops. And this happens when you add another bird to the rope. And then another one. That is, it is obvious that with an increase in the force of impact on the rope, it deforms until the moment when the forces of counteraction to this deformation become equal to the weight of all the birds. And then the downward movement stops.
When the suspension is stretched, the elastic force will be equal to the gravity force, then the stretching stops.
In simple terms, the work of the elastic force is to maintain the integrity of objects that we affect with other objects. And if the elastic force fails, then the body is deformed irrevocably. The rope breaks under the abundance of snow, the handles of the package break if you overload it with food, with large harvests, apple tree branches break, and so on.
When does elastic force arise? At the moment of the beginning of the impact on the body. When the bird sat on the rope. And disappears when the bird takes off. That is, when the impact stops. The point of application of the elastic force is the point at which the impact occurs.
Deformation
The elastic force occurs only when the bodies are deformed. If the deformation of the body disappears, then the elastic force also disappears.
Deformations are of different types: tension, compression, shear, bending and torsion.
Stretching - we weigh the body on a spring scale, or an ordinary elastic band, which stretches under the weight of the body
Compression - we put a heavy object on the spring
Shear is the work of scissors or a saw, a loose chair, where the floor can be taken as the base, and the seat can be taken as the plane of application of the load.
Bend - our birds sat on a branch, a horizontal bar with students in a physical education lesson
Nature, being a macroscopic manifestation of intermolecular interaction. In the simplest case of tension / compression of a body, the elastic force is directed opposite to the displacement of body particles, perpendicular to the surface.
The force vector is opposite to the direction of body deformation (displacement of its molecules).
Hooke's law
In the simplest case of one-dimensional small elastic deformations, the formula for the elastic force has the form:
,where is the rigidity of the body, is the amount of deformation.
In a verbal formulation, Hooke's law reads as follows:
The force of elasticity arising from the deformation of the body is directly proportional to the elongation of the body and is directed opposite to the direction of movement of the body particles relative to other particles during deformation.
Non-linear deformations
With an increase in the amount of deformation, Hooke's law ceases to operate, the elastic force begins to depend in a complex way on the amount of tension or compression.
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See what "Elastic Force" is in other dictionaries:
elastic force - elastic energy - Topics oil and gas industry Synonyms elastic energy EN elastic energy ... Technical translator's guide
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elastic force - tamprumo jėga statusas T sritis fizika atitikmenys: angl. elastic force vok. elastische Kraft, f rus. elastic force, f; elastic force, f pranc. force élastique, f ... Fizikos terminų žodynas
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