Definition and formula of the spring constant

The elastic force (), which arises as a result of the deformation of the body, in particular the spring, directed in the direction opposite to the movement of the particles of the deformable body, is proportional to the elongation of the spring:

It depends on the shape of the body, its dimensions, the material from which the body is made (spring).

Sometimes the stiffness coefficient is denoted by the letters D and c.

The value of the coefficient of rigidity of the spring indicates its resistance to the action of loads and how great its resistance is when exposed.

Coefficient of rigidity of spring connections

If a certain number of springs are connected in series, then the total stiffness of such a system can be calculated as:

In the event that we are dealing with n springs that are connected in parallel, then the resulting stiffness is obtained as:

Coil spring constant

Consider a spring in the form of a spiral, which is made of wire with a circle cross section. If we consider the deformation of a spring as a set of elementary shifts in its volume under the influence of elastic forces, then the stiffness coefficient can be calculated using the formula:

where is the radius of the spring, is the number of turns in the spring, is the radius of the wire, is the shear modulus (a constant that depends on the material).

Units

The basic unit of measure for the stiffness coefficient in the SI system is:

Examples of problem solving

en.solverbook.com

Elastic Coefficient - Chemist's Handbook 21

Rice. 61. The coefficient of elastic expansion of coke obtained in a cube from the cracked residue of sour Devonian oil and calcined at 1300 ° C for 5 hours

mylink" data-url="http://chem21.info/info/392465/">chem21.info Elements of the theory of elasticity | world of welding Introduction Under the influence of external forces, any solid body changes its shape - it is deformed. A deformation that disappears with the cessation of the action of forces is called elastic. When the body is elastically deformed, internal elastic forces arise, tending to return the body to its original shape. The magnitude of these forces is proportional to the deformation of the body. Tensile and Compressive Deformation The resulting elongation of the sample (Δl) under the action external force(F) proportional to the value operating force, initial length (l) and inversely proportional to the cross-sectional area (S) - Hooke's law: The value E is called the modulus of elasticity of the first kind or Young's modulus and characterizes the elastic properties of the material. The value F / S \u003d p is called voltage. Deformation of rods of any lengths and sections (samples) is characterized by a value called relative longitudinal deformation, ε = Δl/l. Hooke's law for samples of any shape: 2) Young's modulus is numerically equal to the voltage that doubles the sample length. However, sample rupture occurs at much lower stresses. Figure 1 graphically shows the experimental dependence of p on ε, where pmax is the ultimate strength, i.e. stress at which a local narrowing (neck) is obtained on the rod, ptech is the yield point, i.e. the stress at which fluidity appears (i.e., an increase in deformation without an increase in the deforming force), pupr is the elastic limit, i.e. voltage below which Hooke's law is valid (meaning the short-term action of a force). Materials are divided into brittle and ductile. Brittle substances break down at very low relative elongations. Brittle materials usually withstand more compression than tension without breaking. Together with tensile deformation, a decrease in the sample diameter is observed. If Δd is the change in the sample diameter, then ε1 = Δd/d is usually called the relative transverse strain. Experience shows that |ε1/ε| The absolute value μ = |ε1/ε| is called the transverse strain ratio or Poisson's ratio. A shear is a deformation in which all layers of the body parallel to a certain plane are displaced relative to each other. During shear, the volume of the deformed sample does not change. The segment AA1 (Fig. 2), on which one plane has shifted relative to the other, is called an absolute shift. At small shear angles, the angle α ≈ tg α = AA1/AD characterizes the relative deformation and is called the relative shear. where the coefficient G is called the shear modulus. Compressibility of matter Comprehensive compression of the body leads to a decrease in the volume of the body by ΔV and the emergence of elastic forces that tend to return the body to its original volume. Compressibility (β) is a value that is numerically equal to the relative change in the volume of the body ΔV / V when the stress (p) acting along the normal to the surface changes by one. The reciprocal of compressibility is called the bulk modulus (K). The change in body volume ΔV with a comprehensive increase in pressure by ΔP is calculated by the formula Relations between elastic constants Young's modulus, Poisson's ratio, bulk modulus and shear modulus are related by the equations: which, according to two known elastic characteristics, allow, in a first approximation, to calculate the rest. The potential energy of elastic deformation is determined by the formula Units of elasticity modulus: N/m2 (SI), dyne/cm2 (CGS), kgf/m2 (MKGSS) and kgf/mm2. 1 kgf/mm2 = 9.8 106 N/m2 = 9.8 107 dynes/cm2 = 10-6 kgf/m2 Application Table 1 - Tensile strengths of some materials (kg/mm2) Material Tensile strengthin tension in compression Amino layered 8 20 Bakelite 2–3 8–10 Concrete - 0,5–3,5 Viniplast 4 8 Getinax 15–17 15–18 Granite 0,3 15–26 Graphite 0,5–1,0 1,6–3,8 Oak (at 15% humidity) along the grain 9,5 5 Oak (at 15% humidity) across the grain - 1,5 Brick - 0,74–3 brass, bronze 22–50 - Ice (0 °C) 0,1 0,1–0,2 Polyfoam tiled 0,06 - Polyacrylate (plexiglass) 5 7 Polystyrene 4 10 Pine (at 15% moisture) along the grain 8 4 Pine (at 15% moisture) across the grain - 0,5 Structural steel 38–42 - Silicon-chromium-manganese steel 155 - Carbon steel 32–80 - Rail steel 70–80 - Textolite PTK 10 15–25 Phenoplast textolite 8–10 10–26 Fluoroplast-4 2 - Cellon 4 16 Celluloid 5–7 - Cast iron white - up to 175 Cast iron gray fine-grained 21–25 up to 140 Cast iron gray ordinary 14–18 60–100 Table 2 - Elastic moduli and Poisson's ratios Name of material Young's modulus E,107 N/m2 Shear modulus G,107 N/m2 Poisson's ratioμ Aluminum 6300–7000 2500–2600 0,32–0,36 Concrete 1500–4000 700–1700 0,1–0,15 Bismuth 3200 1200 0,33 Bronze aluminum, casting 10300 4100 0,25 Bronze phosphorous rolled 11300 4100 0,32–0,35 Granite, marble 3500–5000 1400–4400 0,1–0,15 Duralumin rolled 7000 2600 0,31 Limestone is dense 3500 1500 0,2 Invar 13500 5500 0,25 Cadmium 5000 1900 0,3 Rubber 0,79 0,27 0,46 Quartz filament (fused) 7300 3100 0,17 Constantan 16000 6100 0,33 Ship rolled brass 9800 3600 0,36 Manganin 12300 4600 0,33 Copper rolled 10800 3900 0,31–0,34 Cold drawn copper 12700 4800 0,33 Nickel 20400 7900 0,28 Plexiglass 525 148 0,35 Rubber soft vulcanized 0,15–0,5 0,05–0,15 0,46–0,49 Silver 8270 3030 0,37 Alloy steels 20600 8000 0,25–0,30 Carbon steels 19500–20500 800 0,24–0,28 Glass 4900–7800 1750–2900 0,2–0,3 Titanium 11600 4400 0,32 Celluloid 170–190 65 0,39 Zinc rolled 8200 3100 0,27 Cast iron white, gray 11300–11600 4400 0,23–0,27 Table 3 - Compressibility of liquids at different temperatures Substance Temperature, °C In the pressure range, atm Compressibility β, 10-6 atm-1 Acetone 14,2 9–36 111 0 100–500 82 0 500–1000 59 0 1000–1500 47 0 1500–2000 40 Benzene 16 8–37 90 20 99–296 78,7 20 296–494 67,5 Water 20 1–2 46 Glycerol 14,8 1–10 22,1 Castor oil 14,8 1–10 47,2 Kerosene 1 1–15 67,91 16,1 1–15 76,77 35,1 1–15 82,83 52,2 1–15 92,21 72,1 1–15 100,16 94 1–15 108,8 Sulfuric acid 0 1–16 302,5 Acetic acid 25 92,5 81,4 Kerosene 10 1–5,25 74 100 1–5,25 132 Nitrobenzene 25 192 43,0 Olive oil 14,8 1–10 56,3 20,5 1–10 63,3 Paraffin (with a melting point of 55 ° C) 64 20–100 83 100 20–400 24 185 20–400 137 Mercury 20 1–10 3,91 Ethanol 20 1–50 112 20 50–100 102 20 100–200 95 20 200–300 86 20 300–400 80 100 900–1000 73 Toluene 10 1–5,25 79 20 1–2 91,5 weldworld.com Elastic coefficient - WiKi en-wiki.org Elasticity coefficient - Wikipedia. There are n(\displaystyle n) springs in series connection with stiffnesses k1,k2,...,kn.(\displaystyle k_(1),k_(2),...,k_(n).) From Hooke's law ( F=−kl(\displaystyle F=-kl) , where l is extension) it follows that F=k⋅l.(\displaystyle F=k\cdot l.) The sum of the extensions of each spring is equal to the total extension of the entire connection l1+l2+ ...+ln=l.(\displaystyle l_(1)+l_(2)+...+l_(n)=l.) Each spring is subject to the same force F.(\displaystyle F.) According to Hooke's law, F=l1⋅k1=l2⋅k2=...=ln⋅kn.(\displaystyle F=l_(1)\cdot k_(1)=l_(2)\cdot k_(2)=...=l_(n)\cdot k_(n).) We derive from the previous expressions: l=F/k,l1=F/k1,l2 =F/k2,...,ln=F/kn.(\displaystyle l=F/k,\quad l_(1)=F/k_(1),\quad l_(2)=F/k_(2 ),\quad ...,\quad l_(n)=F/k_(n).) Substituting these expressions into (2) and dividing by F,(\displaystyle F,) we get 1/k=1/k1+ 1/k2+...+1/kn,(\displaystyle 1/k=1/k_(1)+1/k_(2)+...+1/k_(n)), which was to be proved. http-wikipedia.ru Poisson's ratio formula and examples Definition and formula of Poisson's ratio Let us turn to the consideration of deformation solid body. In the process under consideration, there is a change in the size, volume and often the shape of the body. So, the relative longitudinal stretching (compression) of an object occurs with its relative transverse narrowing (expansion). In this case, the longitudinal deformation is determined by the formula: where is the length of the sample before deformation, is the change in length under load. However, during tension (compression), not only does the length of the sample change, but the transverse dimensions of the body also change. Deformation in the transverse direction is characterized by the magnitude of the relative transverse narrowing (expansion): where is the diameter of the cylindrical part of the sample before deformation (transverse size of the sample). It has been empirically obtained that under elastic deformations the following equality holds: Poisson's ratio in conjunction with Young's modulus (E) is a characteristic of the elastic properties of the material. Poisson's ratio at volumetric strain If the coefficient of volumetric deformation () is taken equal to: where is the change in the volume of the body, is the initial volume of the body. Then, under elastic deformations, the relation is fulfilled: Often in formula (6) terms of small orders are discarded and used in the form: For isotropic materials, Poisson's ratio must be within: The existence of negative values ​​of the Poisson's ratio means that the transverse dimensions of the object could increase during stretching. This is possible in the presence of physical and chemical changes in the process of deformation of the body. Materials with a Poisson's ratio less than zero are called auxetics. The maximum value of Poisson's ratio is a characteristic of more elastic materials. Its minimum value refers to fragile substances. So steels have a Poisson's ratio of 0.27 to 0.32. Poisson's ratio for rubber varies between: 0.4 - 0.5. Poisson's ratio and plastic deformation Expression (4) is also valid for plastic deformations, however, in this case, the Poisson's ratio depends on the magnitude of the deformation: With an increase in deformation and the occurrence of significant plastic deformations It has been experimentally established that plastic deformation occurs without a change in the volume of the substance, since this type of deformation occurs due to shifts in the layers of the material. Units Poisson's ratio is physical quantity, which has no dimension. Examples of problem solving en.solverbook.com Poisson's Ratio - WiKi This article is about a parameter that characterizes the elastic properties of a material. For the concept in thermodynamics, see Adiabatic exponent. Poisson's ratio (denoted as ν(\displaystyle \nu ) or μ(\displaystyle \mu )) is the ratio of relative transverse compression to relative longitudinal tension. This coefficient does not depend on the size of the body, but on the nature of the material from which the sample is made. Poisson's ratio and Young's modulus fully characterize the elastic properties of an isotropic material. Dimensionless, but can be specified in relative units: mm/mm, m/m. A homogeneous rod before and after applying tensile forces to it. Let us apply tensile forces to a homogeneous rod. As a result of the action of such forces, the rod will generally be deformed both in the longitudinal and transverse directions. Let l(\displaystyle l) and d(\displaystyle d) be the length and transverse dimension of the sample before deformation, and let l′(\displaystyle l^(\prime )) and d′(\displaystyle d^(\prime )) be the length and transverse dimension of the specimen after deformation. Then the longitudinal elongation is called the value equal to (l′−l)(\displaystyle (l^(\prime )-l)) , and the transverse compression is the value equal to −(d′−d)(\displaystyle -(d^( \prime )-d)) . If (l′−l)(\displaystyle (l^(\prime )-l)) is denoted as Δl(\displaystyle \Delta l) , and (d′−d)(\displaystyle (d^(\prime )- d)) as Δd(\displaystyle \Delta d) , then the relative longitudinal elongation will be equal to the value Δll(\displaystyle (\frac (\Delta l)(l))) , and the relative transverse compression will be equal to the value −Δdd(\displaystyle - (\frac (\Delta d)(d))) . Then, in the accepted notation, the Poisson ratio μ(\displaystyle \mu ) has the form: μ=−ΔddlΔl.(\displaystyle \mu =-(\frac (\Delta d)(d))(\frac (l)(\Delta l)).) Usually, when tensile forces are applied to the rod, it lengthens in the longitudinal direction and contracts in the transverse directions. Thus, in such cases, Δll>0(\displaystyle (\frac (\Delta l)(l))>0) and Δdd<0{\displaystyle {\frac {\Delta d}{d}}<0} , так что коэффициент Пуассона положителен. Как показывает опыт, при сжатии коэффициент Пуассона имеет то же значение, что и при растяжении. For absolutely brittle materials, Poisson's ratio is 0, for absolutely incompressible materials - 0.5. For most steels, this coefficient lies in the region of 0.3; for rubber, it is approximately 0.5. There are also materials (mainly polymers) in which Poisson's ratio is negative, such materials are called auxetics. This means that when a tensile force is applied, the cross section of the body increases. For example, paper made from single-walled nanotubes has a positive Poisson's ratio, and as the proportion of multi-walled nanotubes increases, there is a sharp transition to a negative value of −0.20. Many anisotropic crystals have a negative Poisson's ratio, since the Poisson's ratio for such materials depends on the angle of orientation of the crystal structure relative to the tension axis. A negative coefficient is found in materials such as lithium (the minimum value is -0.54), sodium (-0.44), potassium (-0.42), calcium (-0.27), copper (-0.13) and others. 67% of cubic crystals from the periodic table have a negative Poisson's ratio.

6. Sound research methods in medicine: percussion, auscultation. Phonocardiography.

Auscultation

Percussion

Phonocardiography

7. Ultrasound. Obtaining and registration of ultrasound based on the reverse and direct piezoelectric effect.

8. Interaction of ultrasound of different frequency and intensity with matter. The use of ultrasound in medicine.

Electromagnetic oscillations and waves.

4. Scale of electromagnetic waves. Classification of frequency intervals adopted in medicine

5. The biological effect of electromagnetic radiation on the body. Electrical injury.

6. Diathermy. UHF therapy. Inductothermy. Microwave therapy.

7. Depth of penetration of non-ionizing electromagnetic radiation into the biological environment. Its dependence on frequency. Methods of protection against electromagnetic radiation.

Medical optics

1. The physical nature of light. Wave properties of light. The length of the light wave. Physical and psychophysical characteristics of light.

2. Reflection and refraction of light. total internal reflection. Fiber optics, its application in medicine.

5. Resolution and resolution limit of the microscope. Ways to improve resolution.

6. Special methods of microscopy. immersion microscope. Dark field microscope. polarizing microscope.

The quantum physics.

2. Line spectrum of radiation of atoms. Its explanation is in N. Bohr's theory.

3. Wave properties of particles. De Broglie's hypothesis, its experimental substantiation.

4. Electron microscope: principle of operation; resolution, application in medical research.

5. Quantum-mechanical explanation of the structure of atomic and molecular spectra.

6. Luminescence, its types. Photoluminescence. Stokes law. Chemiluminescence.

7. Application of luminescence in biomedical research.

8. Photoelectric effect. Einstein's equation for the external photoelectric effect. Photodiode. Photomultiplier.

9. Properties of laser radiation. Their connection with the quantum structure of radiation.

10. Coherent radiation. Principles of obtaining and restoring holographic images.

11. The principle of operation of a helium-neon laser. Inverse population of energy levels. The emergence and development of photon avalanches.

12. Application of lasers in medicine.

13. Electron paramagnetic resonance. EPR in medicine.

14. Nuclear magnetic resonance. The use of NMR in medicine.

ionizing radiation

1. X-ray radiation, its spectrum. Bremsstrahlung and characteristic radiation, their nature.

3. The use of x-rays in diagnostics. X-ray. Radiography. Fluorography. CT scan.

4. Interaction of X-rays with matter: photoabsorption, coherent scattering, Compton scattering, pair formation. The probabilities of these processes.

5. Radioactivity. Law of radioactive decay. Half life. Units of activity of radioactive preparations.

6 The law of attenuation of ionizing radiation. Linear attenuation coefficient. The thickness of the half attenuation layer. Mass attenuation factor.

8. Obtaining and using radioactive preparations for diagnosis and treatment.

9. Methods for registration of ionizing radiation: Geiger counter, scintillation sensor, ionization chamber.

10. Dosimetry. The concept of absorbed, exposure and equivalent dose and their power. Units of their measurement. The off-system unit is the roentgen.

Biomechanics.

1. Newton's second law. Protecting the body from excessive dynamic loads and injuries.

2. Types of deformation. Hooke's law. Stiffness coefficient. Elastic modulus. properties of bone tissue.

3. Muscle tissue. The structure and function of the muscle fiber. Energy conversion during muscle contraction. The efficiency of muscle contraction.

4. Isotonic mode of muscle work. Static muscle work.

5. General characteristics of the circulatory system. The speed of blood movement in the vessels. Stroke volume of blood. Work and power of the heart.

6. Poiseuille equation. The concept of the hydraulic resistance of blood vessels and how to influence it.

7. Laws of fluid motion. Continuity equation; its relationship with the features of the capillary system. Bernoulli equation; its connection with the blood supply to the brain and lower extremities.

8. Laminar and turbulent fluid motion. Reynolds number. Measurement of blood pressure by the Korotkov method.

9. Newton's equation. Viscosity coefficient. Blood is a non-Newtonian fluid. Blood viscosity in normal and pathological conditions.

Biophysics of cytomembranes and electrogenesis

1. The phenomenon of diffusion. Fick's equation.

2. Structure and models of cell membranes

3. Physical properties of biological membranes

4. The concentration element and the Nernst equation.

5. Ionic composition of the cytoplasm and intercellular fluid. The permeability of the cell membrane for various ions. Potential difference across the cell membrane.

6. Resting potential of the cell. Goldman-Hodgkin-Katz equation

7. Excitability of cells and tissues. Excitation methods. The all-or-nothing law.

8. Action potential: graphical view and characteristics, mechanisms of occurrence and development.

9. Potential-gated ion channels: structure, properties, functioning

10. Mechanism and rate of propagation of the action potential along the amyopiatic nerve fiber.

11. Mechanism and rate of propagation of the action potential along the myelinated nerve fiber.

Biophysics of reception.

1. Classification of receptors.

2. Structure of receptors.

3. General mechanisms of reception. receptor potentials.

4. Encoding information in the senses.

5. Features of light and sound perception. Weber-Fechner law.

6. Main characteristics of the auditory analyzer. Mechanisms of auditory reception.

7. Main characteristics of the visual analyzer. Mechanisms of visual reception.

Biophysical aspects of ecology.

1. Geomagnetic field. Nature, biotropic characteristics, role in the life of biosystems.

2. Physical factors of ecological significance. natural background levels.

Elements of probability theory and mathematical statistics.

Sample mean properties

2. Types of deformation. Hooke's law. Stiffness coefficient. Elastic modulus. properties of bone tissue.

Deformation- change in the size, shape and configuration of the body as a result of the action of external or internal forces. types of deformation:

tension-compression - a type of body deformation that occurs if a load is applied to it along its longitudinal axis

shear - deformation of the body caused by shear stresses

bending - deformation, characterized by the curvature of the axis or gray surface of the deformable object under the action of external forces.

torsion - occurs when a load is applied to a body in the form of a pair of forces in its transverse plane.

Hooke's law- the equation of the theory of elasticity, relating the stress and deformation of an elastic medium. In verbal form, the law reads as follows:

The elastic force that occurs in the body when it is deformed is directly proportional to the magnitude of this deformation

For a thin tensile rod, Hooke's law has the form:

Here F is the tension force of the rod, Δl is the absolute elongation (compression) of the rod, and k is called the coefficient of elasticity (or stiffness).

Elastic coefficient depends on both the properties of the material and the dimensions of the rod. It is possible to isolate the dependence on the dimensions of the rod (cross-sectional area S and length L) by writing the coefficient of elasticity as

The stiffness coefficient is equal to the force that causes a unit displacement at a characteristic point (most often at the point of force application).

Elastic modulus- the general name of several physical quantities characterizing the ability of a solid body (material, substance) to deform elastically when a force is applied to them.

There are no absolutely rigid bodies in nature, real solid bodies can "spring" a little - this is elastic deformation. Real solids have an elastic strain limit, i.e. such a limit after which a trace of pressure will already remain and will not disappear itself.

properties of bone tissue. Bone is a solid body, for which the main properties are strength and elasticity.

Bone strength is the ability to withstand an external destructive force. Quantitatively, strength is determined by the tensile strength and depends on the structure and composition of the bone tissue. Each bone has a specific shape and a complex internal structure that allows it to withstand the load in a certain part of the skeleton. A change in the tubular structure of the bone reduces its mechanical strength. The composition of the bone also significantly affects the strength. When mineral substances are removed, the bone becomes rubbery, and when organic substances are removed, it becomes brittle.

Bone elasticity is the property to acquire its original shape after the cessation of exposure to environmental factors. It, like strength, depends on the structure and chemical composition of the bone.

3. Muscle tissue. The structure and function of the muscle fiber. Energy conversion during muscle contraction. The efficiency of muscle contraction.

Muscle tissue called tissues that are different in structure and origin, but similar in ability to pronounced contractions. They provide movement in the space of the body as a whole, its parts and the movement of organs inside the body and consist of muscle fibers.

A muscle fiber is an elongated cell. The composition of the fiber includes its shell - sarcolemma, liquid contents - sarcoplasm, nucleus, mitochondria, ribosomes, contractile elements - myofibrils, and also containing Ca 2+ ions - the sarcoplasmic reticulum. The surface membrane of the cell forms transverse tubes at regular intervals, through which the action potential penetrates into the cell when it is excited.

The functional unit of a muscle fiber is the myofibril. The repeating structure in a myofibril is called a sarcomere. Myofibrils contain 2 types of contractile proteins: thin filaments of actin and twice as thick filaments of myosin. The contraction of the muscle fiber occurs due to the sliding of myosin filaments over actin filaments. In this case, the overlap of the filaments increases and the sarcomere shortens.

home muscle fiber function- ensuring muscle contraction.

Energy conversion during muscle contraction. For muscle contraction, the energy released during the hydrolysis of ATP by actomyosin is used, and the process of hydrolysis is closely associated with the contractile process. By the amount of heat released by the muscle, one can evaluate the efficiency of energy conversion during contraction. When the muscle is shortened, the rate of hydrolysis increases in accordance with the increase in the work performed. the energy released during hydrolysis is sufficient to provide only the work done, but not the full energy production of the muscle.

Efficiency(efficiency) of muscle work ( r) is the ratio of the magnitude of external mechanical work ( W) to the total amount released in the form of heat ( E) energy:

The highest value of the efficiency of an isolated muscle is observed with an external load that is about 50% of the maximum value of the external load. Work performance ( R) in a person is determined by the amount of oxygen consumption during the period of work and recovery according to the formula:

where 0.49 is the coefficient of proportionality between the volume of oxygen consumed and the mechanical work performed, i.e. at 100% efficiency to perform work equal to 1 kgf․m(9,81J), you need 0.49 ml oxygen.

Motor action / efficiency

Walking/23-33%; Running at an average speed / 22-30%; Cycling/22-28%; Rowing/15-30%;

Shot put/27%; Throwing/24%; Lifting the bar / 8-14%; Swimming / 3%.

"

Sooner or later, when studying a physics course, pupils and students are faced with problems on the elastic force and Hooke's law, in which the coefficient of spring stiffness appears. What is this quantity, and how is it related to the deformation of bodies and Hooke's law?

First, let's define the basic terms that will be used in this article. It is known that if you act on a body from the outside, it will either gain acceleration or deform. Deformation is a change in the size or shape of a body under the influence of external forces. If the object is fully restored after the termination of the load, then such a deformation is considered elastic; if the body remains in an altered state (for example, bent, stretched, compressed, etc.), then the deformation is plastic.

Examples of plastic deformations are:

• clay crafting;
• bent aluminum spoon.

In its turn, elastic deformations will be considered:

• elastic band (you can stretch it, after which it will return to its original state);
• spring (after compression, it straightens again).

As a result of elastic deformation of a body (in particular, a spring), an elastic force arises in it, equal in absolute value to the applied force, but directed in the opposite direction. The elastic force for a spring will be proportional to its elongation. Mathematically, this can be written like this:

where F is the elastic force, x is the distance by which the length of the body has changed as a result of stretching, k is the stiffness coefficient that we need. The above formula is also a special case of Hooke's law for a thin tensile rod. In general form, this law is formulated as follows: "The deformation that has arisen in an elastic body will be proportional to the force that is applied to this body." It is valid only in those cases when we are talking about small deformations (tension or compression is much less than the length of the original body).

Determination of the stiffness factor

Stiffness factor(it also has the names of the coefficient of elasticity or proportionality) is most often written with the letter k, but sometimes you can see the designation D or c. Numerically, the stiffness will be equal to the magnitude of the force that stretches the spring per unit length (in the case of SI, by 1 meter). The formula for finding the elasticity coefficient is derived from a special case of Hooke's law:

The greater the value of rigidity, the greater will be the resistance of the body to its deformation. The Hooke coefficient also shows how stable the body is to the action of an external load. This parameter depends on the geometric parameters (wire diameter, number of turns and winding diameter from the wire axis) and on the material from which it is made.

The unit of stiffness in SI is N/m.

System Stiffness Calculation

There are more complex tasks in which total stiffness calculation required. In such tasks, the springs are connected in series or in parallel.

Serial connection of the spring system

When connected in series, the overall rigidity of the system is reduced. The formula for calculating the coefficient of elasticity will be as follows:

1/k = 1/k1 + 1/k2 + … + 1/ki,

where k is the total stiffness of the system, k1, k2, …, ki are the individual stiffnesses of each element, i is the total number of all springs involved in the system.

Parallel connection of the spring system

When the springs are connected in parallel, the value of the total coefficient of elasticity of the system will increase. The calculation formula will look like this:

k = k1 + k2 + … + ki.

Measuring the stiffness of the spring empirically - in this video.

Calculation of the stiffness coefficient by experimental method

With the help of a simple experiment, you can independently calculate, what will be the Hooke coefficient. For the experiment you will need:

• ruler;
• spring;
• cargo with a known mass.

The sequence of actions for experience is as follows:

1. It is necessary to fix the spring vertically, hanging it from any convenient support. The bottom edge must remain free.
2. Using a ruler, its length is measured and written as x1.
3. At the free end, you need to hang a load with a known mass m.
4. The length of the spring is measured in the loaded state. Denoted by x2.
5. Absolute elongation is calculated: x = x2-x1. In order to get the result in the international system of units, it is better to immediately convert it from centimeters or millimeters to meters.
6. The force that caused the deformation is the force of gravity of the body. The formula for calculating it is F = mg, where m is the mass of the load used in the experiment (translated into kg), and g is the free acceleration value, which is approximately 9.8.
7. After the calculations, it remains to find only the stiffness coefficient itself, the formula of which was indicated above: k = F / x.

Examples of tasks for finding stiffness

Task 1

A force F = 100 N acts on a spring 10 cm long. The length of the stretched spring is 14 cm. Find the stiffness coefficient.

1. We calculate the length of the absolute elongation: x = 14-10 = 4 cm = 0.04 m.
2. According to the formula, we find the stiffness coefficient: k = F / x = 100 / 0.04 = 2500 N / m.

Answer: the spring stiffness will be 2500 N/m.

Task 2

A load of mass 10 kg, when suspended on a spring, stretched it by 4 cm. Calculate how long another load of mass 25 kg will stretch it.

1. Let's find the force of gravity that deforms the spring: F = mg = 10 9.8 = 98 N.
2. Let's determine the coefficient of elasticity: k = F/x = 98 / 0.04 = 2450 N/m.
3. Calculate the force with which the second load acts: F = mg = 25 9.8 = 245 N.
4. According to Hooke's law, we write the formula for absolute elongation: x = F/k.
5. For the second case, we calculate the stretching length: x = 245 / 2450 = 0.1 m.

Answer: in the second case, the spring will stretch by 10 cm.

Video

This video will show you how to determine the stiffness of a spring.

Springs are an elastic element, through which rotational motion is transmitted to the mechanisms, almost all mechanisms are equipped with them. The reliability of this product and its service depend on such a concept as spring rate. It is the rigidity that determines how reliable the operation of the mechanism will be in various operating conditions. "" is determined by the force necessary for its compression. Spreading the spring is a slightly different issue, which is directly dependent on the material from which the spring is made. By the way, not always the high stiffness of the spring determines its long service life. Rather, it depends on the mechanism that the spring drives.

Types of hardness:

Springs, according to their varieties, are divided into types. Each type is used in certain mechanisms. In general, coil springs, springs, conical springs, belleville and cylindrical springs are in demand. The “stiffness of the spring” also determines the factor of how it transfers its own deformation to the mechanism. So, springs have another important characteristic, deformation, which divides the springs into , and of course .
varied section. So, springs are obtained, which are then completed with various types of equipment, mechanisms, and cars.

How to calculate the spring constant?

In the production of springs, the coefficient of stiffness is necessarily taken into account, which actually serves as an indicator of the service life of the product. "" is calculated in accordance with the calculation formula.
So, for example, if we take a standard cylindrical coil spring made of ordinary cylindrical wire, then the coefficient can be calculated using the following formula:

In the formula, the designation G should be taken as the shear modulus. If the spring is copper, then it will be approximately 45 GPa, and if it is just steel, then the modulus will be approximately 80 GPa. The letter n indicates the number of turns that the spring has, and dF is the winding diameter. The designation dD remains, but it only indicates the diameter of the wire from which the spring is made. Actually, the arithmetic is quite simple, if you only take the appropriate measurements, and substitute digital equivalents for visible letters and values.

If, under the influence of external forces on a solid body, it is deformed, then displacements of particles of the nodes of the crystal lattice occur in it. This shift is resisted by the forces of particle interaction. This is how elastic forces arise, which are applied to a body that has undergone deformation. The modulus of the elastic force is proportional to the deformation:

where is the stress at elastic deformation, K is the modulus of elasticity, which is equal to the stress at a relative strain equal to unity. where - relative deformation, - absolute deformation, - the initial value of the quantity that characterized the shape or size of the body.

DEFINITION

elasticity coefficient called the physical quantity that connects in Hooke's law the elongation that occurs when an elastic body is deformed and the elastic force. The value equal is called the coefficient of elasticity. It shows the change in the size of a body under the influence of a load during elastic deformation.

The coefficient of elasticity depends on the material of the body, its dimensions. So with an increase in the length of the spring and a decrease in its thickness, the coefficient of elasticity decreases.

Young's modulus and elasticity coefficient

With longitudinal deformation, in unilateral tension (compression), the relative elongation, which is denoted by or , serves as a measure of deformation. In this case, the modulus of elasticity is determined as:

where is Young's modulus, which in the case under consideration is equal to the elastic modulus () and characterizes the elastic properties of the body; - initial body length; - change in length under load. When S is the cross-sectional area of ​​the sample.

Coefficient of elasticity of a stretched (compressed) spring

When a spring is stretched (compressed) along the X axis, Hooke's law is written as:

where is the modulus of the projection of the elastic force; - coefficient of elasticity of the spring, - elongation of the spring. Then the coefficient of elasticity is the force that should be applied to the spring in order to change its length by one.

Units

The basic unit of measure for the elasticity coefficient in the SI system is:

Examples of problem solving

EXAMPLE 1

Exercise	What is the work done when the spring is compressed by ? Assume that the elastic force is proportional to compression, the coefficient of elasticity of the spring is equal to k.
Solution	As the main formula, we use the definition of the work of the form: The force is proportional to the amount of compression, which can be represented mathematically as: Let us substitute expressions for force (1.2) into formula (1.1):
Answer

EXAMPLE 2

Exercise	The car was moving at a speed of . He hit the wall. Upon impact, each buffer of the car shrank by l m. There are two buffers. What are the elastic coefficients of the springs, if we assume that they are equal?
Solution	Let's make a drawing.