Units of power. Dynamometer

Now that the properties of the force and the methods of its measurement have been determined, let us return to the second experimental result (§ 43) and determine the quantitative relationship between force and acceleration.

Roughly such a connection can be established on the already familiar experience with a cart that is set in motion by a load (Fig. 2.28). In order to determine the accelerations, we install a dropper on the trolley, which will allow us to mark the positions of the trolley at regular intervals.

To change the force acting on the entire moving system, we will make several identical weights. The entire system can be considered as a complex body consisting of several parts,

moving with accelerations of the same modulus (a trolley with a dropper and a load). In order for the inertial properties of the system to be the same in all experiments, we will place some of the loads on the cup, and the rest on the trolley.

If only one load is placed on the cup, then the entire system will be set in motion by a force equal to the force of gravity acting on it. If two or three such loads are placed on the cup, then the force causing the movement will increase two or three times, respectively. By measuring the distance between the marks left by the dropper during each such experiment, it is possible for all cases to calculate the accelerations that occur in the body under the action of different forces.

Having carried out such experiments, we will be convinced that the accelerations of the cart grow in direct proportion to the acting forces, i.e.

Of course, our experience is very crude, but similar experiments carried out with very accurate measurements forces and accelerations, invariably confirm the found result: accelerations in the motion of bodies are directly proportional to the forces acting on them:

the directions of the resulting accelerations coincide with the directions of the acting forces 1).

In our experiment, the trolley made a rectilinear motion. The force, causing a change in the velocity modulus, created only tangential acceleration. It can be seen from simple experiments that the same connection between force and acceleration is preserved for normal accelerations.

We place the ball in a chute mounted on the axis of a centrifugal machine, and connect it with a thread with a load (Fig. 2.29). Let's make the car rotate at a constant number of revolutions per second. In this case, the ball, if it is at a distance from the axis of rotation,

will acquire some speed and normal acceleration

In order to keep the ball on this circle, the thread must stretch and act on it with some force. The tension force will be created by a load that is tied to the end of the thread passed through the tube on the axis of the centrifugal machine. It is this force that will create a normal (centripetal) acceleration, forcing the ball to move in a circle. Set speed When a ball moves along a circle, a quite definite force will correspond. If you increase the number of revolutions, i.e., increase the normal acceleration, then to keep the ball on a given circle, you must correspondingly increase the thread tension force.

The accelerations of bodies are determined by the forces acting on them. After we have learned how to measure force and know in principle how to determine acceleration, we can answer main question: "How does the acceleration of a body depend on the forces acting on it?"
Experimental determination of the dependence of acceleration on force
It is impossible to establish by experience the connection between acceleration and force with absolute accuracy, since any measurement gives an approximate value of the measured quantity. But it is possible to notice the nature of the dependence of acceleration on force with the help of simple experiments. Already simple observations show that the greater the force, the faster the speed of the body changes, that is, the greater its acceleration. It is natural to assume that acceleration is directly proportional to force. In principle, of course, the dependence of acceleration on force can be more complex, but first we need to see if the simplest assumption is not true.
It is best to study the translational motion of a body, such as a metal bar, on the horizontal surface of the table, since only in translational motion the acceleration of all points is the same, and we can talk about a certain acceleration of the body as a whole. However, in this case, the force of friction against the table is large and, most importantly, it is difficult to accurately measure it.
Therefore, we take a cart with light wheels and install it on rails. Then the friction force is relatively small, and the mass
Rice. 2.14
X
Q
O
Rice. 2.13 soy wheels can be neglected in comparison with the mass of the trolley moving forward (Fig. 2.13).
Let a constant force act on the cart from the side of the thread, to the end of which the load is attached. The modulus of force is measured with a spring dynamometer. This force is constant, but not equal in motion to the force with which the Earth attracts a suspended load. It is very difficult to measure the acceleration of the trolley directly by determining the change in its speed over a short time interval. But it can be estimated by measuring the time t it takes the cart to cover the path s.
Given that under the action of a constant force, the acceleration is also constant, since it is uniquely determined by the force, it is possible to use the kinematic formulas of uniformly accelerated motion. At an initial speed of zero,
at ~2~ where and ¦ Hence
the start and end coordinates of the body. 2s
(2.5.1) It can be seen directly by eye that the trolley picks up speed the faster, the greater the force acting on it. Careful measurements of the modules of force and acceleration show a direct proportionality between them:
a ~ F.
There are other experiments confirming this connection. Here is one of them. A massive roller (Fig. 2.14) is installed on the platform. If you bring the platform into rotation, then the roller under the action of a stretched thread acquires centripetal acceleration, which is easy to determine by the radius of rotation R and the number of revolutions per second n:
a = 4 K2n2R.
We find the force from the readings of the dynamometer. By changing the number of revolutions and comparing F and a, we will make sure that F ~ a.
If several forces simultaneously act on the body, then the modulus of the acceleration of the body will be proportional to the modulus of the geometric sum of all these forces, equal to:
F = Fj + F2+ ... . (2.5.2)
->
The vectors a and F are directed along the same straight line in the same direction:
a ~ F. (2.5.3)
This can be seen in the experiment with the cart: the acceleration of the cart is directed along the thread tied to it.
What is inertia?
According to Newtonian mechanics, the force uniquely determines the acceleration of the body, but not its speed. This needs to be very clearly understood. Force does not determine speed, but how quickly it changes. Therefore, a body at rest will acquire a noticeable speed under the action of a force only for a certain time interval.
mm

Acceleration occurs immediately, simultaneously with the onset of the force, but the speed increases gradually. Even a very large force is not able to impart significant speed to the body at once. This takes time. To stop the body, again, it is necessary that the braking force, no matter how great it is, act for some time.
It is these facts that are meant when they say that bodies are inert. Let us give examples of simple experiments in which the inertia of bodies is very clearly manifested.
1. A massive ball is suspended on a thin thread, exactly the same thread is tied to it below (Fig. 2.15). If you slowly pull the bottom Fig. 2.15
thread, then, as expected, the upper thread breaks. After all, it is affected by the weight of the ball, and the force with which we pull the ball down. However, if you pull the lower thread very quickly, then it will break, which at first glance is rather strange. But it's easy to explain. When we pull the thread slowly, the ball gradually descends, stretching the upper thread until it breaks.
With a quick jerk with great force, the ball receives a large acceleration, but its speed does not have time to increase any significantly during that small period of time during which the lower thread is greatly stretched, and therefore it breaks, and the upper thread is stretched a little and remains whole.
An interesting experience is with a long stick suspended on paper rings (Fig. 2.16). If you hit the stick sharply with an iron rod, the stick breaks, and the paper rings remain intact. Try to explain this experience yourself.
An even simpler experiment can be done at home. The idea of ​​experience is clear from figure 2.17. The left side of the figure corresponds to the situation when v \u003d const or a \u003d 0. On the right side of the figure, v Ф const, i.e., a Ф 0.

Rice. 2.16
Rice. 2.17
Finally, perhaps the most spectacular experience. If you shoot at an empty plastic vessel, the bullet will leave holes in the walls, but the vessel will remain intact. If you shoot at the same vessel filled with water, the vessel will burst into small pieces. This result of the experiment is explained as follows. Water is very little compressible, and a small change in its volume leads to a sharp increase in pressure. When the bullet enters the water very quickly, breaking through the walls of the vessel, the pressure rises sharply. Due to the inertness of water, its level does not have time to rise and the increased pressure tears the vessel apart.
It is sometimes said that, thanks to inertia, a body "resists" attempts to change its speed. This is not entirely true. A body always changes speed under the action of a force, but changing speed takes time. As J. Maxwell emphasized, it is just as wrong to talk about the body's resistance to attempts to change its speed, as it is to say that tea "resists" becoming sweet. It just takes a while for the sugar to dissolve.
The laws of mechanics and everyday experience
The basic statement of mechanics is clear enough and not complicated. It easily fits into our minds. After all, from birth we live in the world of bodies, the movement of which obeys the laws of Newtonian mechanics.
But sometimes ideas acquired from life experience can fail. So, the idea that the speed of a body is directed in the same direction as the force applied to it is too rooted. In fact, the force determines not the speed, but the acceleration of the body, and the direction of the speed and force may not coincide. This is clearly seen in Figure 2.18.
When a body is thrown at an angle to the horizon, the force of gravity is always directed downward, and the velocity tangent to the trajectory forms an angle with the force, which changes during the flight of the body.
The direction of the force coincides with the direction of the velocity only in the particular case of rectilinear motion with an increasing velocity in absolute value.
The main fact for dynamics has been established: the acceleration of a body is directly proportional to the force acting on it.

1. The thread on which the ball is suspended is deflected at a certain angle and released. Where is the resultant of the forces acting on the ball directed at the moment when the thread is vertical?
2. Draw a small circle on the floor and have a competition. Each participant quickly walks in a straight line towards the circle, holding tennis ball. The task is to get the ball released from the hands into the circle. This competition will show which of you understands the essence of Newtonian mechanics better. Rice. 2.18

More on the topic § 2.5. RELATIONSHIP BETWEEN ACCELERATION AND FORCE:

1. The authors of the Declaration saw a close connection between the "natural and inalienable rights of man",
2. Researchers rightly note that feeding strengthened ties between rulers and their vassals and contributed to
3. § 6. Causal relationship between a socially dangerous action (inaction) and socially dangerous consequences that have occurred

The main task of mechanics is to find the laws of mechanical motion of a body under the action of forces applied to it. Empirically, it was found that at speeds vc, Where c is the speed of light in a vacuum, under the action of a force F a free body changes the speed of its translational motion, moving with acceleration a, and the connection of the force F and acceleration a linear:

a = k 1 F,

Where k 1 - positive coefficient of proportionality depending on the choice of units of measurement of force and acceleration, constant for each specific body, but different for different bodies.

The property of the inertia of the body is manifested in the fact that under the action of a force, the speed of translational motion does not change instantly, but gradually with a finite acceleration corresponding to the change a. As a measure of inertia, a scalar value is introduced m called body weight. The higher the inertia of the body, the less acceleration is acquired under the action of a certain force. It has been experimentally obtained that the acceleration depends on the mass inversely proportional to k 1 =k/m:

where is the coefficient of proportionality k depends only on the choice of the system of units of acceleration, force and mass and is the same for different bodies. If the units of measurement refer to the same system (for example, SI), then the coefficient k=1.

Thus, the acceleration of the body is directly proportional to the force causing it, coincides with it in direction and inversely proportional to the mass of the body:

The equation () is called the basic equation of dynamics. Body mass m is a constant value that does not depend on the state of motion of the body, nor on its position in space, therefore, to compare the masses, it is enough to compare the accelerations acquired by the bodies under the action of the same force:

m 2 / m 1 = a 1 / a 2 .

If the body is divided into N parts by weight m, then it has been experimentally established that under the action of the same force the whole body acquires acceleration in N times less than when the force acts on each part separately. Therefore, the mass of a body is an additive quantity - the mass of a body is equal to the sum of the masses of its parts. The mass of a system of bodies is equal to the sum of the masses of all the bodies included in the system. Often in dynamic calculations, the body is mentally divided into a system of material points with mass. The mass of the whole body will be equal to the sum of the masses of all its material points.

A balance scale can be used to measure body weight. The principle of their work is as follows. Since the free fall acceleration g in the same place on the surface of the Earth is the same for all bodies, then gravity will act on the body P satisfying the relation

For two different masses

When weighing a body balance scales measured mass m 1 balance with reference masses of weights m 2. In balance P 1 =P 2 , and hence m 1 =m 2 .

In the standard system of units, mass is measured in kilograms (kg).

Equation () describes the motion of a body only if it moves forward and does not deform. Otherwise, the acceleration of different points of the body will be different. A material point cannot be deformed or rotated, so equation () will always be valid for it.

If several forces act on a material point F i ( i=1, …, n) with the resulting F, then the acceleration of the material point will be:

Where a i - acceleration of a material point under the action of one force on it F i , that is, it acts principle of independence of action of forces- if several forces simultaneously act on a material point, then each of them imparts to the material point the same acceleration as if there were no other forces.

Like any vector, you can decompose the resultant force vector into two components: tangent to the point trajectory Fτ and normal to it F n:

F = F τ + F n.

Comparing with the expansion of the acceleration vector into a tangential and normal component and using the basic equation of dynamics () we get:

Normal strength F n only changes the direction of the vector v, is directed to the center of curvature of the trajectory of radius R and called centripetal force:

The tangential force changes the amount of speed v: positive value Fτ accelerates the body, and negative slows it down; at Fτ =0 the body moves uniformly at a constant speed. If during uniform motion the normal force is zero, then the trajectory will be rectilinear, if the normal force is constant and non-zero, then the trajectory will have a constant radius of curvature (that is, a circle on a plane or a helix in space):

Rice. 1 Experience in demonstrating inertia

Let's place a sheet of paper on the horizontal surface of the table, and place a body (for example, a glass) on it. At the beginning, the leaf and the glass are at rest. If you slowly pull a sheet of paper with force F, then the glass will remain motionless relative to the sheet, but will begin to move with acceleration relative to the table, that is, the glass will move with the same acceleration as the sheet of paper. If you sharply pull a sheet of paper, it will pull out from under the glass and the glass will practically not move relative to the table.

To move the glass, a force must act on it, and the only force that occurs in the horizontal direction is the friction force F tr arising between the sheet and the glass. If the acceleration of a sheet of paper is a, then a force arises in the direction of this acceleration F=ma, the friction force is directed in the opposite direction and is equal to F tr =- ma for low power F, that is, if you pull a sheet of paper slowly, then the forces are compensated and the glass will be stationary relative to the sheet of paper. With growing strength F the value of the friction force reaches maximum value F tr =μ mg called sliding friction force, where μ is the coefficient of friction between a sheet of paper and a glass. If you apply force F<μmg, then the friction force will no longer be able to compensate it completely and the glass will move relative to the sheet under the action of the force F-F tr with acceleration a 1 =a-μ g, and relative to the table with acceleration a 2 =a-a 1 =μ g. Since the time it takes for a sheet of paper to be pulled out from under the glass is short, the glass will travel a negligible distance.

Note that after pulling out a sheet of paper, the glass has an acceleration a 2 relative to the table and then it will stop due to the friction force between the glass and the table. If the same experiment is carried out not on a table, but, for example, on ice, where the coefficient of friction is much smaller (and hence the sliding friction force is much smaller), then the glass will move by inertia under the action of a smaller force and move a greater distance.

References

• A.A. Detlaf, B.M. Yavorsky, L.B. Milkovskaya. Physics course. M.: Higher school. 1973.

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Relationship between force and acceleration.

Newton's second lawIn accordance with Newton's first law in an inertial frame of reference, a free body has no acceleration. The acceleration of a body is due to its interaction with other bodies, i.e., the forces acting on the body. Since we can measure acceleration and force independently of each other, we can establish a relationship between them by experience. This relationship turns out to be very simple: in all cases, the acceleration of a body is proportional to the force that causes it. The proportionality between acceleration and force is valid for forces of any physical nature, and the proportionality coefficient is a constant value for a given body. The direction of the acceleration vector coincides with the direction of the force. Deviations from this fundamental regularity are found only for very fast movements occurring at speeds comparable to the speed of light c = 300,000 km/s. In the world of macroscopic bodies surrounding us, such speeds do not occur. Samos is the fastest movement known here - the movement of the Earth in orbit around the Sun - occurs at a speed of "only" 30 km / s. Only micro-objects move with relativistic speeds: particles in cosmic rays, electrons and protons in charged particle accelerators, etc.

The regularity can be illustrated in visual demonstration experiments. It is convenient to use the same air path as in the coasting demonstration. It is possible to ensure the constancy of the force acting on the cart in the direction of its movement as follows. We attach a dynamometer to it (Fig. 65), to the other end of the spring of which a thread with a load is tied, thrown over a fixed block at the end of the track. According to the dynamometer reading, one can judge the force acting on the trolley from the side of the thread. Suspending different weights to the end of the thread, you can give this force different values. The acceleration acquired by the trolley under the action of this force can be calculated using kinematic formulas, measuring, for example, the paths traveled by the trolley over certain periods of time. For this purpose, in particular, stroboscopic photography can be used, when the object is illuminated by short flashes of light at regular intervals (Fig. 66).

Experience shows that under the action of a constant force (which can be judged by the constant reading of the dynamometer during the movement of the cart), the movement does indeed occur with constant acceleration. If the experiment is repeated by changing the value of the acting force, then the acceleration of the cart will change by the same amount.

Inertia.

The coefficient of proportionality between acceleration and force, which is unchanged for a given body, turns out to be different for different bodies. Having coupled together two identical carts, we will see that a certain force F imparts to them an acceleration that is half that which it imparted to one cart. Thus, the coefficient of proportionality between acceleration and force is associated with a certain physical property of the body. This property is called inertia. The greater the inertia of the body, the less acceleration the acting force imparts to it. The physical quantity that quantitatively characterizes the property of the inertia of the body is the mass, or inertial mass. Using the concept of mass, the relationship between acceleration and force can be expressed as follows: Mass as a measure of inertia. The mass included in the formula is a measure of the inertia of the body. It does not depend not only on the force acting on the body, but also on other physical conditions in which this body is located - on the ambient temperature, the presence of an electric or gravitational field, etc. You can verify this if you do it with a given body similar experiments, using the power of a different physical nature, at different temperatures and humidity of the surrounding air, on the surface of the earth or in high mountains, etc. Mass properties. From experience, the following properties of mass are known: it is an additive scalar quantity that does not depend on the position of the body. The mass of a body does not depend on its speed, provided that this speed is much less than the speed of light. Additivity means that the mass of a compound body is equal to the sum of the masses of its parts. The property of additivity of mass is very accurately fulfilled for macroscopic bodies and is violated only when the energy of interaction of the constituent parts of the body is high, for example, when protons and neutrons combine to form an atomic nucleus. The fact that the mass is a scalar means that the inertial properties of the body are the same in all directions. The equality can be interpreted as follows. If one day a simultaneous measurement of the force acting on it and the acceleration acquired by it is made with a given body, then its mass will be found, and in the future it is possible to calculate the acceleration a of this body from a known force, or vice versa, calculate the acting force from a known acceleration a. We will further compare this so-called dynamic method of determining mass with the common method of measuring mass by weighing. Experience shows that with the simultaneous action of several forces on a body, acceleration a is proportional to the vector sum of these forces. Therefore, the equality is generalized as follows.

Newton's second law.

Equality expresses the content of Newton's second law: In an inertial frame of reference, the acceleration of a body is proportional to the vector sum of all forces acting on it and inversely proportional to the mass of the body. The relationship between acceleration and force expressed by Newton's second law is universal. It does not depend on the specific choice of inertial frame of reference. The law is valid for any direction of the acting force. When this force is directed along the velocity of the body, it changes the modulus of velocity, i.e., the acceleration imparted by such a force will be tangential. This is exactly what happened in the described experiments with the air track. When the force is directed perpendicular to the speed, it changes the direction of the speed, i.e. with. the acceleration imparted to the body will be normal (centripetal). For example, with an almost circular motion of the Earth around the Sun, the force of attraction to the Sun acting perpendicular to the orbital velocity imparts centripetal acceleration to the Earth. When all the forces acting on the body are balanced, their vector sum is zero, and there is no acceleration of the body relative to the inertial frame of reference. A body is either at rest or moving uniformly and in a straight line. Its motion in this case is indistinguishable from the motion by inertia, which was discussed in the discussion of Newton's first law. However, if there the motion in the absence of forces was used to introduce inertial frames of reference, then here the zero acceleration when the acting forces are compensated is a consequence of Newton's second law. Force and motion. The essence of Newton's second law, expressed by the formula, is very simple. However, often the results of its action are unexpected due to the peculiar manifestations of the inertia of bodies. The fact is that acceleration appears in the law itself, and movement is visually perceived through speed. Consider the following experiment. We hang a massive body on a thin thread, and from below we tie another similar thread to it (Fig. 67). If you slowly pull it down, gradually increasing the applied force, then at some point the upper thread will break.

This is easy to understand, since the pull on the upper thread is due to both the applied external force and the weight of the suspended body. However, if the bobbin thread is pulled down with a sharp movement, the bobbin thread will break. The explanation for this is as follows. Thread breakage occurs when its elongation reaches a certain value. For the upper thread to stretch, the load must move down the same distance. But this cannot happen instantly due to the inertia of a massive body, it takes some time to change its speed, which is exactly what is lacking with a sharp jerk for the bottom thread.

What is the property of inertia? What is inertial mass of a body?

What experiments testify to the adaptability of the mass?

What statements are contained in Newton's second law?

How should the force acting on the body be directed so that its speed changes only in direction? Give examples of such movements.

Can the acceleration of a body in an inertial frame of reference be equal to zero if forces act on it?

Let's depict the right side of the Hill curve, which establishes the relationship between the largest (record) values ​​​​of the developed strength and speed (quickness) of muscle movement, in a separate figure.

Figure 2.3

The ratio of strength and speed of muscle contractions in some sports

(according to V.L. Utkin, 1989., revised)

Any physical exercise in one way or another requires the manifestation of strength and speed of muscle contraction. Depending on the magnitude of the relationship between strength and speed, manifested in certain physical exercises, these exercises are usually divided into power, speed-strength and speed. So, the bench press in weightlifting refers to strength exercises, shot put, javelin throw - to speed-strength, and strikes in table tennis - to speed.

Pull-ups on the crossbar can be attributed to strength exercises, but you need to take into account that since pull-ups are to a large extent associated with the manifestation of endurance, and not strength itself, it is not characterized by the development of maximum efforts, especially in the initial period of the exercise. If the athlete, from the very beginning of the pull-ups, sought to show maximum strength in the phase of lifting the body, he would develop maximum speed and fly out over the bar to the chest (as happens when performing a “exit by force”). But since the athlete is required not to display maximum efforts for a short time, but to maintain efforts of a certain magnitude for a long time, the speed of movement of an athlete in the phase of lifting the body in the initial period of pull-ups is much less than the maximum possible (point A). With the development of fatigue processes during the performance of pull-ups, the athlete’s power capabilities decrease, the load on the muscles (equal to the athlete’s weight) becomes relatively higher, which, in accordance with the rule “the greater the load, the lower the speed”, leads to a decrease in the athlete’s movement speed in the lifting phase torso (point B).

2.4.3 Dependence of the limiting time of static work on the absolute and relative muscle strength.

Endurance during static work is determined by the time during which a constant pressure force is maintained or a certain load is held in a constant position.

The maximum time of static work is inversely related to the developed muscle efforts (Figure 1.8). When the required force is less than 20% of the maximum force, static work can be performed for a very long time. There is evidence in the literature that in the pressure (load) range of 20 - 80% of the maximum force, the limiting time of static work decreases with increasing pressure (load) according to the following relationship:

(2.1)

where: - limiting time of static work;

Constant;

Force of pressure (load);

Maximum strength;

n is an exponent equal to approximately 2.5.

It can be seen from the formula that even a slight decrease in the strength of a static contraction leads to a significant increase in the length of time during which this contraction can be maintained.

To establish specific parameters of the dependence described by formula (2.1), a special experiment was conducted in 2005, the essence of which was that after a standard warm-up and a standard procedure for processing the palms and the bar of the crossbar, the athlete performed a hang on one arm “to failure”. At the same time, the time of hanging and the magnitude of the load on the hand were recorded. After a short rest (5-10 minutes), the athlete performed the hang "to failure" on the other hand. After 30 minutes of rest, the athlete once again performed similar hangs, but with a different load on the hand - more or less (in accordance with the experiment plan). Similar hangs "to failure" were carried out every other day for a month. Weights placed directly on the athlete's belt were used as weights, and when performing hangings with relief, weights of the required size were fixed at the end of the cable thrown over the block and fixed at the other end on the belt. The sign of the end of the experiment with increasing load was the inability of the athlete to perform the hang for more than 10 seconds. At the same time, the total value of the load and the athlete's own weight, equal to 68 kg, was taken as the maximum strength of the muscles - the flexors of the fingers Fmax. In the experiment, it was 129 kg for the left hand and 117 kg for the right.

The results of the experiment are reflected in the graphs of Figures 2.4 and 2.5. At the same time, Figure 2.4 shows the dependence of the maximum hanging time on one hand on the absolute value of the load, and Figure 2.5 shows the dependence of the maximum hanging time on the left hand on the relative magnitude of the load.

The dependence curves of the hanging time on the absolute value of the load for the right and left hands (Figure 2.4) do not coincide, but run almost parallel at some distance from each other. This means that the flexor muscles of the fingers are not equal in their static strength abilities. In the experiment, the athlete's left hand turned out to be more enduring - the leading one. Indeed, in almost all approaches with the same load, the hanging time on the left hand turned out to be longer than on the right.

By the degree of closeness of the curves, one can judge the degree of difference in the static strength endurance of the athlete's hands. So, in order for the limiting time of hanging on the right and left hands to be the same in the above experiment, it was necessary to increase the load on the left hand (or reduce it on the right) by an average of 8 kg.

Weakening the grip of the less enduring - weaker - hand during pull-ups in competition often leads to premature falls off the bar. To avoid this, at least two ways of redistributing body weight between the arms in proportion to their strength abilities are used in practice. In the first case, the athlete, even before the start of the pull-ups, shifts the grip from the center of the crossbar so that the leading hand is a little closer to the vertical support. In this case, the leading hand is higher than the weakest and it accounts for most of the body weight of the athlete. In the second case, during a pause of rest in the hang, the athlete shifts his legs towards the leading arm, transferring part of the body weight to it and thereby somewhat unloading the weakest arm for its faster recovery and preventing "acidification" of the forearm muscles.

If we consider the dependence of the duration of the hang on one arm not on the absolute, but on the relative value of the load, it should be noted that although the qualitatively experimental dependence of the hang time “to failure” on the relative value of the load coincides with that described in the literature, but at the same time there are quantitative differences, associated, apparently, with the specifics of polyathlon. So, if, on the basis of the obtained experimental data, to build a graph of the dependence of the limiting time of static work on the relative load in the load range from 0.2 to 0.8 Fmax, it will have the form shown in Figure 2.5. The exponent in the formula for the approximation curve, equal to 1.6, is significantly less than in the previously given formula (2.1), therefore, the decrease in the time of static work with an increase in the load on the hand when hanging on the crossbar will be less sharp than it is described in the literature. Apparently, the curve shown in is obtained with the participation of subjects who do not specialize in pull-ups, while the dependence shown in Figure 2.5 is taken for an athlete who has been involved in polyathlon for many years. Naturally, the energy potential of the trained finger flexor muscles makes it possible to maintain a given static force for a longer time in their usual exercise. This is exactly what is manifested in the fact that the experimental dependence of the limiting time of static stress is flatter than the analogous classical curve.

But in any case, the dependence of the hang time on the load value is non-linear, which means that any change in the load value leads to a more significant change in the hang time.

Since the traction force of the flexor muscles of the fingers of each hand in the hang is equal to half the weight of the athlete's body minus the friction force acting in the grip area, even a slight increase in the friction force leads to a significant increase in the time of the hang. Therefore, in order to increase the time of a reliable grip, it is very important to facilitate the work of the finger flexor muscles as much as possible by increasing the friction force between the palms and the bar by applying magnesia.

An experiment conducted in St. Petersburg at the end of the 2005 season helps to assess the degree of influence of the quality of hand grip on the crossbar bar on the result in pull-ups. One well-known polyathlete, potentially capable of showing high results in pull-ups, constantly fell off the bar between the second and third minutes of the exercise, managing to pull himself up from 30 to 42 times. It must be said that before the prohibition of the use of adhesives, this athlete on rosin steadily pulled himself up in the region of 50 times. After switching to magnesium, despite intense training, the athlete could not get close to his best results. Therefore, it was interesting to see what result an athlete can show if pulling up according to the old rules - without taking into account time and using rosin. It turned out to be very high. Without visible effort, the athlete pulled himself up 77 times in 8 minutes 10 seconds. And this is after during the entire competitive season, at best, he could hardly manage to pull 42 times in 3 minutes.

In order to visualize the degree of influence of the grip of the palms with the bar on the result, Figure 2.6 shows two graphs of pull-ups of this polyathlete athlete. The first of them shows the dependence of the average time of the pull-up cycle during the exercise at the Cup of St. Petersburg 2005, and the second shows the same dependence, but with experimental pull-ups without taking into account time and using rosin. And if on the first curve, even with the naked eye, a characteristic rise is visible, indicating an insufficient level of development of static endurance, then on the second dependence, there are no obvious signs of static fatigue. Thus, the sticky properties of rosin reduced the static load on the athlete's finger flexor muscles to such an extent that he was able to perform pull-ups until he completely exhausted the reserves of dynamic endurance.