The law of proportionality of the lengthening of a spring to 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 the most acute disputes about the priority in the discovery of certain laws with leading scientists (Huygens, Newton, etc.). However, Hooke's law has been so convincingly substantiated by numerous experiments that Hooke's priority here has never been disputed.
Robert Hooke's spring theory:
This is Hooke's Law!
PROBLEM SOLVING
Determine the stiffness of the spring, which, under the action of a force of 10 N, lengthened by 5 cm.
Given:
g = 10 H/kg
F=10H
X=5cm=0.05m
To find:
k = ?
The load is in balance.
Answer: spring stiffness k = 200H/m.
TASK FOR "5"
(we hand over on a piece of paper).
Explain why it is safe for an acrobat to jump onto a trampoline net with high altitude? (we call on the help of Robert Hooke)
Looking forward to an answer!
LITTLE EXPERIENCE
Place the rubber tube vertically, on which the metal ring is first tightly put on, and stretch the tube. What will happen to the ring?
Dynamics - Cool physics
This force arises as a result of deformation (changes in the initial state of matter). For example, when we stretch a spring, we increase the distance between the molecules of the spring material. When we compress the spring, we decrease it. When we twist or shift. In all these examples, a force arises that prevents deformation - the elastic force.
Hooke's law
The elastic force is directed opposite to the deformation.
Since the body is represented as a material point, the force can be depicted from the center
When connected in series, for example, springs, the stiffness is calculated by the formula
When connected in parallel, the stiffness
Sample stiffness. Young's modulus.
Young's modulus characterizes the elastic properties of a substance. This is a constant value that depends only on the material, its physical state. Characterizes the ability of a material to resist tensile or compressive deformation. The value of Young's modulus is tabular.
Body weight
Body weight is the force with which an object acts on a support. You say it's gravity! The confusion occurs in the following: indeed often body weight equal to strength gravity, but these are completely different forces. Gravity is the force that results from interaction with the Earth. Weight is the result of interaction with the support. The force of gravity is applied at the center of gravity of the object, while the weight is the force that is applied to the support (not to the object)!
There is no formula for determining weight. This force is denoted by the letter .
The support reaction force or elastic force arises in response to the impact of an object on a suspension or support, therefore the body weight is always numerically the same as the elastic force, but has the opposite direction.
The reaction force of the support and the weight are forces of the same nature, according to Newton's 3rd law they are equal and oppositely directed. Weight is a force that acts on a support, not on a body. The force of gravity acts on the body.
Body weight may not be equal to gravity. It can be either more or less, or it can be such that the weight is zero. This state is called weightlessness. Weightlessness is a state when an object does not interact with a support, for example, the state of flight: there is gravity, but the weight is zero!
It is possible to determine the direction of acceleration if we determine where the resultant force is directed.
Note that weight is a force, measured in Newtons. How to correctly answer the question: "How much do you weigh"? We answer 50 kg, naming not weight, but our mass! In this example, our weight is equal to gravity, which is approximately 500N!
Overload- the ratio of weight to gravity
Strength of Archimedes
Force arises as a result of the interaction of a body with a liquid (gas), when it is immersed in a liquid (or gas). This force pushes the body out of the water (gas). Therefore, it is directed vertically upwards (pushes). Determined by the formula:
In the air, we neglect the force of Archimedes.
If the Archimedes force is equal to the force of gravity, the body floats. If the Archimedes force is greater, then it rises to the surface of the liquid, if it is less, it sinks.
electrical forces
There are forces of electrical origin. Occur in the presence of an electric charge. These forces, such as the Coulomb force, the Ampère force, the Lorentz force.
Newton's laws
Newton's I law
There are such systems of reference, which are called inertial, with respect to which the bodies keep their speed unchanged, if they are not affected by other bodies or the action of other forces is compensated.
Newton's II law
The acceleration of a body is directly proportional to the resultant of the forces applied to the body and inversely proportional to its mass:
Newton's third law
The forces with which two bodies act on each other are equal in magnitude and opposite in direction. 
Local frame of reference - this is a frame of reference, which can be considered inertial, but only in an infinitely small neighborhood of any one point of space-time, or only along any one open world line.
Galilean transformations. The principle of relativity in classical mechanics.
Galilean transformations. Consider two frames of reference moving relative to each other and with a constant speed v 0. One of these frames will be denoted by the letter K. We will consider it stationary. Then the second system K will move rectilinearly and uniformly. Let's choose coordinate axes x,y,z systems K and x",y",z" of the K" system so that the x and x" axes coincide, and the y and y" , z and z" axes are parallel to each other. Find the relationship between the x,y,z coordinates of some point P in the K system and x",y",z" coordinates of the same point in the K system". that y=y", z=z". Let us add to these relations the assumption accepted in classical mechanics that time in both systems flows in the same way, that is, t=t". We obtain a set of four equations: x=x"+v 0 t;y=y";z=z"; t=t", called Galilean transformations. Mechanical principle of relativity. The position that all mechanical phenomena in different inertial reference frames proceed in the same way, as a result of which it is impossible to establish by any mechanical experiments whether the system is at rest or moves uniformly and rectilinearly is called the principle of relativity of Galileo. Violation of the classical law of addition of velocities. Based general principle relativity (no physical experience can distinguish one inertial system from another) formulated by Albert Einstein, Lawrence changed Galileo's transformations and got: x"=(x-vt)/(1-v 2 /c 2); y"=y; z "= z; t" \u003d (t-vx / c 2) / (1-v 2 / c 2). These transformations are called Lawrence transformations.
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 action 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 - Force of elasticity.
k - rigidity of the body (Proportionality factor, which depends on the material of the body and its shape).
x - Deformation of the body (lengthening or compression of the body).
This law was discovered by Robert Hooke in 1660. He conducted an experiment which consisted in the fact that. A thin steel string was fixed at one end, and a different force was applied to the other end. Simply put, the string was suspended from the ceiling, and a load of various masses was applied to it.
Figure 1 - Stretching of a string under the action of gravity.
As a result of the experiment, Hooke found out that in small aisles, the dependence of the stretching of the body is linear with respect to the force of elasticity. That is, when a unit of force is applied, the body lengthens by one unit of length.
Figure 2 - Graph of the dependence of the elastic force on the elongation of the body.
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 negative meaning. That is, she strives to return the body to its original state. Accordingly, it is directed opposite to the deforming force. Everything on the left is body compression. The force of elasticity is positive.
The stretching of the string of envy is not only from an external force, but also from the section of the string. A thin string will still somehow stretch from a small weight. But if you take a string of the same length, but let's say 1 m in diameter, it's hard to imagine how much weight it will take to stretch it.
To assess how a force acts on a body of a certain 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 relative elongation of the body. Relative elongation is the ratio of the increment in the length of the body to its total length. And the coefficient of proportionality is called Young's modulus. Module because the value of body elongation is taken modulo, without taking into account the sign. It is not taken 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 the normal tension of the body.
How many of us have thought about how amazingly objects behave when exposed to them?
For example, why is the fabric, if we stretch it in different sides, can stretch for a long time, and at one moment suddenly break? And why is the same experiment so much harder to do with a pencil? What does the resistance of a material depend on? How can you determine to what extent it can be deformed or stretched?
All these and many other questions were asked by an English researcher more than 300 years ago and found answers, now united under the general name "Hooke's Law".
According to his research, every material has a so-called elasticity coefficient. This is a property that allows the material to stretch within certain limits. The coefficient of elasticity is a constant value. This means that each material can only withstand a certain level of resistance, after which it reaches a level of irreversible deformation.
In general, Hooke's Law can be expressed by the formula:
where F is the elastic force, k is the already mentioned coefficient of elasticity, and /x/ is the change in the length of the material. What is meant by this change? Under the influence of force, a certain object under study, be it a string, rubber, or any other, changes, stretching or shrinking. In this case, the change in length is the difference between the initial and final length of the object under study. That is, how much the spring stretched / compressed (rubber, string, etc.)
Hence, knowing the length and constant coefficient of elasticity for this material, you can find the force with which the material is stretched, or elastic force, as is often called Hooke's Law.
There are also special cases in which this law in its standard form cannot be used. It's about about measuring the strain force under shear conditions, that is, in situations where the deformation is produced by a certain force acting on the material at an angle. Hooke's law in shear can be expressed as follows:
where τ is the desired force, G is a constant factor known as the shear modulus, y is the shear angle, the amount by which the angle of the object has changed.
Types of deformations
deformation called a change in the shape, size or volume of the body. Deformation can be caused by the action of external forces applied to the body. Deformations that completely disappear after the cessation of the action of external forces on the body are called elastic, and the deformations that persist even after the external forces have ceased to act on the body, - plastic. Distinguish tensile strain or compression(one-sided or all-sided), bending, torsion and shear.
elastic forces
With deformations solid body its particles (atoms, molecules, ions) located at the nodes crystal lattice, are displaced from their equilibrium positions. This displacement is counteracted by the forces of interaction between the 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 the body during its elastic deformation and directed against the direction of displacement of the particles of the body 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 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 the body to deform, 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 appearance of elastic forces during unilateral tension and compression of a solid body.
Hooke's law
The relationship between the elastic force and the elastic deformation of a body (for small deformations) was experimentally established by Newton's contemporary, the English physicist Hooke. mathematical expression Hooke's law for the deformation of one-sided 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 stiffness. The SI unit of stiffness is newton per meter (N/m).
Hooke's law for unilateral tension (compression) formulate as follows: the elastic force that occurs when a body is deformed is proportional to the elongation of this body.
Consider an experiment illustrating Hooke's law. Let the axis of symmetry of the cylindrical spring coincide with the 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 x coordinate, 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 > 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, the coordinate of which is x
It can be seen from the figure that the projection of the elastic force of the spring on the axis Ax 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. 20b shows a graph of Hooke's law. On the abscissa axis, the values of the elongation x of the spring are plotted, and on the ordinate axis, the values of 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 on a bracket, and a load is suspended from the other end, the weight of which is the 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 modulus of force F, but also on the cross-sectional area of the wire S.
Under the action of an external force applied to it, the wire is deformed and stretched. With not too much stretching, this deformation is elastic. In the elastically deformed wire there is an elastic force f y. According to Newton's third law, the elastic force is equal in absolute value and opposite in direction to the external force acting on the body, i.e.
f yn = -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). The normal stress s is equal to the ratio of the elastic modulus to the cross-sectional area of the body:
s = f y /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 elongation of the wire. The value e = DL/L 0 (2.12) is called relative elongation of the body. 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 ways of writing Hooke's law for one-sided tension (compression). In this formula, the elongation is taken modulo, since it can be both positive and negative. The coefficient of proportionality E in Hooke's law is called the modulus of longitudinal elasticity (Young's modulus).
Let's install physical meaning 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. Therefore, Young's modulus is numerically equal to such a normal stress that should have arisen in the body with an increase in its length by 2 times. (if for such a large deformation Hooke's law was fulfilled). From formula (2.13) it is also seen that in SI Young's modulus is expressed in pascals (1 Pa = 1 N/m2).
