Hooke's law for relative quantities. Derivation of Hooke's law for various types of deformation

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.

The law of proportionality between the elongation of a spring and the applied force was discovered by the English physicist Robert Hooke (1635-1703)

Hooke's scientific interests were so broad that he often did not have time to complete his research. This gave rise to heated disputes about the priority in the discovery of certain laws with major scientists (Huygens, Newton, etc.). However, Hooke's law was so convincingly substantiated by numerous experiments that Hooke's priority was never disputed.

Robert Hooke's spring theory:

This is Hooke's law!

PROBLEM SOLVING

Determine the stiffness of a spring that, under the action of a force of 10 N, lengthens by 5 cm.

Given:
g = 10 N/kg
F=10H
X = 5cm = 0.05m
Find:
k = ?

The load is in balance.

Answer: spring stiffness k = 200N/m.

TASK FOR "5"

(hand in on a piece of paper).

Explain why it is safe for an acrobat to jump onto a trampoline net from a great height? (we call on Robert Hooke for help)
I'm looking forward to your answer!

LITTLE EXPERIENCE

Place the rubber tube, on which the metal ring has been tightly placed, vertically, and stretch the tube. What will happen to the ring?

Dynamics - Cool physics

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

abnormal stress in 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 work “De potentia restitutiva”, published in 1678, was made by him 18 years earlier, and in 1676 it was placed in another of his books under the guise of the anagram “ceiiinosssttuv”, meaning “Ut tensio sic vis” . According to the author's explanation, the 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.

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 in the body, internal forces, preventing 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

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 fх 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 quantity DL = L - L 0 is called absolute wire elongation. The quantity e = DL/L 0 (2.12) is 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:

s = E|e|. (2.13)

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 for 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).

DEFINITION

Deformations are any changes in the shape, size and volume of the body. 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 remain fully or partially after the cessation of external forces.

The ability to elastic and plastic deformations depends on the nature of the substance of which the body is composed, the conditions in which it is located; methods of its manufacture. For example, if you take different types of iron or steel, you can find completely different elastic and plastic properties in them. At normal room temperatures, iron is a very soft, ductile material; hardened steel, on the contrary, is a hard, elastic material. The plasticity of many materials is a condition for their processing and 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 individual particles of the body change. As a result, internal forces arise in the deformed body, preventing its deformation.

There are tensile (compressive), shear, bending, and torsional deformations.

Elastic forces

DEFINITION

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

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

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

DEFINITION

Tensile strain (linear strain) 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 is the length of the body in the deformed and undeformed state, respectively.

Elongation:

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

Examples of problem solving

EXAMPLE 1

Exercise	A spring with stiffness N/m in an unloaded state has a length of 25 cm. What will be the length of the spring if a load weighing 2 kg is suspended from it?
Solution	Let's make a drawing. An elastic force also acts on a load suspended on a spring. Projecting this vector equality onto the coordinate axis, we get: According to Hooke's law, elastic force: so we can write: where does the length of the deformed spring come from: Let us convert the length of the undeformed spring, cm, to the SI system. Substituting numerical values ​​into the formula physical quantities, let's calculate:
Answer	The length of the deformed spring will be 29 cm.

EXAMPLE 2

Exercise	A body weighing 3 kg is moved along a horizontal surface using a spring with stiffness N/m. How much will the spring lengthen if under its action, with uniformly accelerated motion, the speed of the body changes from 0 to 20 m/s in 10 s? Ignore friction.
Solution	Let's make a drawing. The body is acted upon by the reaction force of the support and the elastic force of the spring.