As you already know from the basic physics course, elastic forces are associated with the deformation of bodies, that is, a change in their shape and (or) size.

The deformation of bodies associated with elastic forces is not always noticeable (we will dwell on this below). For this reason, the properties of elastic forces are usually studied using springs for clarity: their deformation is clearly visible to the eye.

Put the experience

We hang the load to the spring (Fig. 15.1, a). (We assume that the mass of the spring can be neglected.) The spring will stretch, that is, be deformed.

The suspended load is affected by the gravity t and the elastic force exerted on the side of the stretched spring (Fig. 15.1, b). It is caused by deformation of the spring.

According to Newton’s third law, the spring on the load side is affected by the same modulus but opposite directional force (Fig. 15.1, c). This force is the weight of the load: after all, this is the force with which the body stretches a vertical tray (spring).

The control forces with which the load and the spring interact with each other are related by Newton’s third law and therefore have the same physical nature. Therefore, weight is also an elastic force. (The elastic force acting on the spring side of the load (the weight of the load) is caused by the deformation of the load. This deformation is invisible if the load is a weight or bar. To make the load deformation also noticeable, we can take a massive spring as a load: we will see that it stretches. ) Acting on the spring, the weight of the load stretches it, that is, is the cause of its deformation. (In order to avoid misunderstandings, we emphasize once again that the spring to which the load is suspended is not stretched by the load’s gravity force, but by the elastic force applied to the spring from the load side (load weight).)

In this example, we see that the elastic forces are both a consequence and a cause of the elastic deformation of bodies:
  - if the body is deformed, then elastic forces act on the side of this body (for example, the control force in Figure 15.1, b);
  - if elastic forces are applied to the body (for example, the force in Figure 15.1, c), then this body is deformed.

1. Which of the forces depicted in Figure 15.1
  a) balance each other if the load is at rest?
  b) have the same physical nature?
  c) are related by Newton’s third law?
  d) cease to be equal in absolute value if the load moves with acceleration directed up or down?

Is body deformation always noticeable? As we have already said, the “insidious” feature of the elastic forces is that the deformation of bodies associated with them is far from always noticeable.

Put the experience

The deformation of the table, due to the weight of the apple lying on it, is invisible to the eye (Fig. 15.2).

Nevertheless, it is: only thanks to the strength of elasticity arising from the deformation of the table, it holds the apple! Deformation of the table can be detected with the help of witty experience. In Figure 15.2, the white lines schematically indicate the course of the ray of light when there is no apple on the table, and the yellow lines indicate the course of the ray of light when the apple is on the table.

2. Consider Figure 15.2 and explain why the deformation of the table was made noticeable.

Some danger lies in the fact that, without noticing the deformation, you can not notice the elastic force associated with it!

So, under the conditions of some tasks, an “inextensible thread” appears. These words mean that it is possible to neglect only the amount of deformation of the thread (increase in its length), but not to neglect the elastic forces applied to the thread or acting on the side of the thread. Actually, there are no “absolutely inextensible threads”: accurate measurements show that any thread is at least a little stretched.

For example, if in the experiment described above with a load suspended from a spring (see Fig. 15.1), the spring is replaced with an “inextensible thread”, then under the weight of the load the thread will stretch, although its deformation will be invisible. And consequently, all considered elastic forces will be present. The role of the spring elastic force will be played by the thread tension force directed along the thread.

3. Draw the drawings corresponding to Figure 15.1 (a, b, c), replacing the spring with an inextensible thread. Indicate in the drawings the forces acting on the thread and the load.

4. Two people pull the rope in opposite directions with a force of 100 N each.
  a) What is the tension of the rope?
  b) Will the tension force of the rope change if one end is tied to a tree and the other end is pulled with a force of 100 N?

The nature of elastic forces

The elastic forces are due to the forces of interaction of the particles that make up the body (molecules or atoms). When a body is deformed (its size or shape is changed), the distances between particles change. As a result of this, forces arise between the particles that tend to return the body to an undeformed state. This is the force of elasticity.

2. Hooke's Law

Put the experience

We will hang identical weights to the spring. We will notice that the elongation of the spring is proportional to the number of weights (Fig. 15.3).

It means that spring deformation is directly proportional to the elastic force.

Denote the deformation (elongation) of the spring

x \u003d l - l 0, (1)

where l is the length of the deformed spring, and l 0 is the length of the undeformed spring (Fig. 15.4). When the spring is stretched, x\u003e 0, and the projection of the elastic force F x acting on the side of the spring< 0. Следовательно,

F x \u003d –kx. (2)

The minus sign in this formula recalls that the elastic force applied from the side of the deformed body is opposite to the deformation of this body: an extended spring tends to compress, and a compressed spring tends to stretch.

Coefficient k is called spring stiffness. Rigidity depends on the material of the spring, its size and shape. The stiffness unit is 1 N / m.

The relation (2) is called hooke's law in honor of the English physicist Robert Hook, who discovered this pattern. Hooke's law is valid when the deformation is not too large (the magnitude of the allowable deformation depends on the material from which the body is made).

Formula (2) shows that the modulus of elasticity F is related to the modulus of deformation x by the relation

From this formula it follows that the graph of the dependence F (x) is a straight line segment passing through the origin.

5. Figure 15.5 shows graphs of the dependence of the elastic modulus on the deformation modulus for three springs.
  a) Which spring has the greatest stiffness?
  b) What is the stiffness of the softest spring?

6. What mass should be suspended from a spring with a stiffness of 500 N / m so that the spring extension becomes 3 cm?

It is important to distinguish the spring extension x from its length l. The difference between them is shown by formula (1).

7. When a load weighing 2 kg is suspended from a spring, its length is 14 cm, and when a load weighing 4 kg is suspended, the length of the spring is 16 cm.
  a) What is the spring stiffness?
  b) What is the length of an undeformed spring?

3. Spring connection

Serial connection

We take one spring with rigidity k (Fig. 15.6, a). If you stretch it by force (Fig. 15.6, b), its elongation is expressed by the formula

  Now take the second spring of the same type and connect the springs, as shown in Figure 15.6, c. In this case, they say that the springs are connected in series.

Find the stiffness k after the system of two series-connected springs.

If you stretch the spring system by force, then the spring force of each spring will be equal to modulo F. The total elongation of the spring system will be 2x, because each spring will be extended by x (Fig. 15.6, d).

Consequently,

k last \u003d F / (2x) \u003d ½ F / x \u003d k / 2,

where k is the stiffness of one spring.

So, the rigidity of a system of two identical series-connected springs is 2 times less than the rigidity of each of them.

If the springs with different stiffness are connected in series, then the elastic forces of the springs will be the same. And the total elongation of the spring system is equal to the sum of the elongations of the springs, each of which can be calculated using Hooke's law.

8. Prove that when connecting two springs in series
  1 / k last \u003d 1 / k 1 + 1 / k 2, (4)
  where k 1 and k 2 - spring stiffness.

9. What is the stiffness of a system of two springs connected in series with a stiffness of 200 N / m and 50 N / m?

In this example, the stiffness of the system of two series-connected springs was less than the stiffness of each spring. Is this always the case?

10. Prove that the stiffness of the system of two series-connected springs is less than the stiffness of any of the springs that make up the system.

Parallel connection

Figure 15.7 on the left shows parallel connected identical springs.

  Denote the stiffness of one spring k, and the stiffness of the spring system k pairs.

11. Prove that k pairs \u003d 2k.

Hint. See figure 15.7.

So, the rigidity of a system of two identical springs connected in parallel is 2 times greater than the rigidity of each of them.

12. Prove that with a parallel connection of two springs with rigidity k 1 and k 2

k pairs \u003d k 1 + k 2. (5)

Hint. When the springs are connected in parallel, their elongation is the same, and the elastic force acting on the side of the spring system is equal to the sum of their elastic forces.

13. Two springs with a stiffness of 200 N / m and 50 N / m are connected in parallel. What is the rigidity of a system of two springs?

14. Prove that the stiffness of the system of two springs connected in parallel is greater than the stiffness of any of the springs that make up the system.

Additional questions and tasks

15. Build a graph of the dependence of the modulus of elasticity on elongation for a spring with a stiffness of 200 N / m.

16. A cart weighing 500 g is pulled across the table with a spring of 300 N / m stiffness, applying force horizontally. The friction between the wheels of the cart and the table can be neglected. What is the extension of the spring if the trolley moves with an acceleration of 3 m / s 2?

17. A load of mass m is suspended from a spring of stiffness k. What is the extension of the spring when the load is at rest?

18. A spring of rigidity k was cut in half. What is the stiffness of each spring formed?

19. A spring of rigidity k was cut into three equal parts and connected in parallel. What is the stiffness of the resulting spring system?

20. Prove that the stiffness of the same springs connected in series is n times less than the stiffness of one spring.

21. Prove that the stiffness of n parallel connected identical springs is n times greater than the stiffness of one spring.

22. If two springs are connected in parallel, then the spring system stiffness is 500 N / m, and if the same springs are connected in series, the spring system stiffness is 120 N / m. What is the stiffness of each spring?

23. A block located on a smooth table is attached to vertical stops by springs with a stiffness of 100 N / m and 400 N / m (Fig. 15.8). In the initial state, the springs are not deformed. What will be the elastic force acting on the bar if it is shifted 2 cm to the right? 3 cm to the left?

1 . What type of deformation is experienced under load:

a) the leg of the bench;

b) bench seat;

c) a tense string of a guitar;

d) a meat grinder screw;

e) drill;

2 . What kind of deformation (elastic or plastic) do you have when sculpting figures from clay, plasticine?

3 . The 5.40 m long wire under load has lengthened to 5.42 m. Determine the absolute elongation of the wire.

4 . With an absolute elongation of 3 cm, the length of the spring became equal to 27 cm. Determine its initial length if the spring:

a) stretched;

5 . Absolute elongation of a wire 40 cm long is 2.0 mm. Determine the elongation of the wire.

6 . Absolute and relative elongation of the rod are 1 mm and 0.1%, respectively. Determine the length of the undeformed rod?

7 . When the rod is deformed with a cross section of 4.0 cm 2, the elastic force is 20 kN. Determine the mechanical stress of the material.

8 . Determine the modulus of elasticity in a deformed rod with an area of \u200b\u200b4.0 cm 2 if a mechanical stress of 1.5 · 10 8 Pa arises.

9 . Find the mechanical stress that occurs in a steel cable with its elongation of 0.001.

10 . When tensile aluminum wire, a mechanical stress of 35 MPa arose in it. Find elongation.

11 . What is the coefficient of spring stiffness, which is extended by 10 cm with an elastic force of 5.0 H?

12 . How long is the spring with a stiffness of 100 N / m, if the elastic force is 20 N?

13 . Determine the maximum force a steel wire can withstand with a cross-sectional area of \u200b\u200b5.0 mm 2.

14 . The human tibia can withstand a compression force of 50 kN. Considering the human bone strength to be 170 MPa, estimate the average cross-sectional area of \u200b\u200bthe tibia.

Level B

1 . Which bulb can withstand more pressure from the outside - round or flat-bottomed?

2 . Why is the bicycle frame made of hollow tubes, not solid rods?

3 . When stamping parts are sometimes preheated (hot stamping). Why are they doing this?

4 . Indicate the direction of the elastic forces acting on the bodies at the indicated points (Fig. 1).

Fig. 1

5 . Why there are no tables for the body stiffness coefficient k, like tables for substance density?

6 . At what brick laying (Fig. 2) will the bottom of the bricks be under a lot of stress?

7 . The elastic force is a variable force: it changes from point to point as it elongates. And how does the acceleration caused by this force behave?

8 . A weight of 10 kg is suspended from a wire with a diameter of 2.0 mm fixed at one end. Find the mechanical stress in the wire.

9 . On two vertical wires, the diameters of which differ by 3 times, the same weights were attached. Compare the stresses arising in them.

10 . In fig. Figure 3 shows a graph of the dependence of the stress occurring in a concrete pile on its relative compression. Find the modulus of elasticity of concrete.

11 . A wire 10 m long with a cross-sectional area of \u200b\u200b0.75 mm 2, when stretched by a force of 100 N, was extended by 1.0 cm. Determine the Young's modulus for the material of the wire.

12 . With what force do you need to stretch a fixed steel wire 1 m long with a cross-sectional area of \u200b\u200b0.5 mm 2 to extend it by 3 mm?

13 . Determine the diameter of the steel wire 4.2 m long, so that under the action of a longitudinal tensile force of 10 kN, its absolute elongation is 0.6 cm?

14 . Determine the body stiffness coefficient from the graph (Fig. 4).

15 . From the graph of the dependence of the change in the length of the rubber bundle on the force applied to it, find the stiffness of the bundle (Fig. 5).

16 . Build a graph of the dependence of the elastic force arising in a deformed spring F  control \u003d f(Δ l), from its extension, if the spring stiffness is 200 N / m.

17 . Build a graph of the dependence of the spring extension on the applied force Δ l = f(F) if the spring rate of 400 N / m.

18 . Hooke's law for the projection of the spring elastic force has the form F x = –200 x. What is the projection of the elastic force if, when extending the spring from an undeformed state, the projection of the movement of the end of the spring onto the axis X  is 10 cm?

19 . Two boys stretch a rubber band, fastening dynamometers to its ends. When the harness was extended by 2 cm, dynamometers showed forces of 20 N each. What do dynamometers show when pulling a tow rope 6 cm?

20 . Two springs of equal length, connected in series, are stretched by the free ends by hand. A spring with a stiffness of 100 N / m was extended by 5 cm. What is the stiffness of the second spring if its elongation is 1 cm?

21 . The spring changed its length by 6 cm when a load of 4 kg was suspended from it. How much would she change her length under the influence of a cargo weighing 6 kg?

22 . On two wires of the same stiffness, 1 and 2 m long, the same loads are suspended. Compare absolute wire extensions.

23 . The diameter of the nylon fishing line is 0.12 mm, and the breaking load is 7.5 N. Find the tensile strength of this grade of nylon.

24 . At what largest cross-sectional diameter does a steel wire break under a force of 7850 N?

25 . A chandelier weighing 10 kg must be suspended on a wire with a cross-section of not more than 5.0 mm 2. What material should I take the wire from if I need to provide a five-fold safety margin?

Level FROM

1.   If a wooden bar weighing 200 g is attached to a vertically located dynamometer, then the dynamometer reading will be as shown in Figure 1. Determine the acceleration with which the same block will start to move if it is pulled so that the spring lengthens another 2 cm, and then release the bar.

We have repeatedly used a dynamometer - a device for measuring forces. Let us now get acquainted with the law that allows us to measure forces with a dynamometer and determines the uniformity of its scale.

It is known that under the influence of forces arises deformation of bodies  - change in their shape and / or size. For example, from clay or clay, you can fashion an object whose shape and size will remain after we remove our hands. This deformation is called plastic. However, if our hands deform the spring, then when we remove them, two options are possible: the spring will completely restore its shape and dimensions, or the spring will retain permanent deformation.

  If the body restores the shape and / or dimensions that were before the deformation, then elastic deformation.   The force arising in this case in the body is   elastic force obeying   Hooke's law:

Since the elongation of the body is included in the Hooke law modulo, this law will be valid not only in tension, but also in compression of bodies.

Experiments show:   if the elongation of the body is small compared with its length, then the deformation is always elastic;  if the elongation of the body is large compared to its length, then deformation will usually be plastic  or even destructive. However, some bodies, for example, gum and springs are deformed elastically even with significant changes in their length. The figure shows more than twofold extension of the dynamometer spring.

To clarify the physical meaning of the stiffness coefficient, we express it from the formula of the law. We obtain the ratio of the modulus of elasticity to the modulus of elongation of the body. Recall: any ratio shows how many units of the numerator are per unit of the denominator. therefore   the stiffness coefficient shows the force arising in an elastically deformed body when its length is changed by 1 m.

1. A dynamometer is ...
2. Thanks to Hooke's law, a dynamometer is observed ...
3. The phenomenon of deformation of bodies is called ...
4. We call a plastic deformed body ...
5. Depending on the module and / or the direction of the force applied to the spring, ...
6. The deformation is called elastic and is considered to obey Hooke's law, ...
7. Hooke's law is scalar in nature, since it can only be used to determine ...
8. Hooke's law is valid not only in tension, but also in compression of bodies, ...
9. Observations and experiments on the deformation of various bodies show that ...
10. Ever since the time of the children's games, we well know that ...
11. Compared to the zero stroke of the scale, that is, the undeformed initial state, on the right ...
12. To understand the physical meaning of the stiffness coefficient, ...
13. As a result of the expression of the quantity "k" we ...
14. From the mathematics of elementary school, we know that ...
15. The physical meaning of the stiffness coefficient is that it ...

The initial spring extension is A /. How to change
  potential energy of the spring, if its extension
  will be twice as much?
  1) increase by 2 times
  2) will increase 4 times
3) decrease by 2 times
  4) will decrease by 4 times
  Two bodies move along mutually perpendicular
a straight line, as shown. Module
  momentum of the first body p \\ \u003d  8 kg-m / s, and the second body
p 2 \u003d  6 kg-m / s. What is the modulus of body impulse
  resulting from their absolutely inelastic impact?
  At
R \\
  1) 2 kg - m / s
  2) 48 kg - m / s
  3) 10 kg * m / s
  4) 14 kg - m / s
156

In the study of the dependence of the sliding friction force
A5
Fjp  steel bar on the horizontal surface of the table
  by mass t  the bar received the graph presented on
  figure. According to the schedule, in this study the coefficient
  friction coefficient is approximately equal
2) 0,02
3) 1,00
4) 0,20
  Car moving on a horizontal road, sover
A6
  makes a turn along an arc of a circle. What is the minimum
  the radius of this circle with the coefficient of friction of the car
  mobile tires on road 0.4 and car speed
  10 m / s?
  1) 25 m
  2) 50 m
  3) 100 m
  4) 250 m
  For 2 s of rectilinear uniformly accelerated body movement
A7
  20 m passed, increasing its speed by 3 times. Define
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  1) 5 m / s
  2) 10 m / s
  3) 15 m / s
  4) 30 m / s
157

The figure shows a graph of the process carried out on 1
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  the heater was evenly heated from
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  temperature bridge T  substance
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1) 5-6
2) 2-3
3) 3-4
4) 4-5
  The gas in the heat engine received an amount of heat of 300 J
A P
  and completed the work7 36 J. How the internal energy has changed
  gas?
  1) decreased by 264 J
  2) decreased by 336 J
  3) increased by 264 J
  4) increased by 336 J
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  The ideal gas was first heated at a constant pressure
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A13
  Two point electric charges act on each other
friend with forces of 9 microns. What will be the forces of interaction?
vii between them, if, without changing the distance between the dawn
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  1) 1 μN
  2) 3 μN
  3) 27 μN
  4) 81 μN
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  A direct current flows through the conductor. Know
  --- - the charge passing through the conductor increases with
  over time according to the schedule presented on
figure. The current strength in the conductor is
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  2) 4 A
  3) 6 A
  4) 24 A
  Using the basic law of electromagnetic
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