Does leverage give a gain in work? Inclined plane

Bibliographic description: Shumeiko A. V., Vetashenko O. G. A modern view of the simple mechanism “block” studied from physics textbooks for grade 7 // Young scientist. - 2016. - No. 2. — S. 106-113..07.2019).

Physics textbooks for grade 7, when studying a simple block mechanism, interpret getting a win in different ways. force when lifting a load using this mechanism, for example: Peryshkin's textbook BUT. B. win in strength achieved with using the wheel of the block, on which the forces of the lever act, and in Gendenstein's textbook L. E. the same gain is obtained with using a rope, which is subjected to the tension of the rope. Different textbooks, different subjects and different forces - in order to win force when lifting a load. Therefore, the purpose of this article is to search for objects and strength, with through which gains are made in force, when lifting a load with a simple block mechanism.

Keywords:

First, let's get acquainted and compare how they get a gain in strength, when lifting a load with a simple block mechanism, in physics textbooks for grade 7, for this we will place excerpts from the texts of textbooks, with the same concepts, for clarity, we will place in the table.

Let's sum up the acquaintance and comparison of texts and figures in textbooks:

Evidence of obtaining a gain in strength in the textbook of A. V. Peryshkin is carried out on the wheel of the block and acting force- leverage force; when lifting a load, a fixed block does not give a gain in strength, and a movable block gives a gain in strength by 2 times. There is no mention of a cable on which a load hangs on a fixed block and a movable block with a load.

On the other hand, in L. E. Gendenshtein's textbook, evidence of a gain in strength is carried out on a cable on which a load or a movable block with a load hangs and the acting force is the cable tension force; when lifting a load, a fixed block can give a 2-fold gain in strength, and there is no mention of a lever on the block wheel in the text.

A search for literature describing obtaining a gain in strength with a block and a cable led to the "Elementary Textbook of Physics" edited by Academician G. S. Landsberg, in §84. Simple machines on pages 168-175 are descriptions of: "simple pulley, double pulley, gate, chain hoist and differential pulley". Indeed, by its design, “a double block gives a gain in strength, when lifting a load, due to the difference in the length of the radii of the blocks”, with the help of which the load is lifted, and “chain hoist - gives a gain in strength, when lifting a load, due to the rope , on several parts of which, a load hangs. Thus, it was possible to find out why they give a gain in strength, when lifting a load, separately a block and a cable (rope), but it was not possible to find out how the block and cable interact with each other and transfer the weight of the load to each other, since the load can be suspended on a cable , and the cable is thrown over the block or the load can hang on the block, and the block hangs on the cable. It turned out that the tension force of the cable is constant and acts along the entire length of the cable, so the transfer of the weight of the load by the cable to the block will be at each point of contact between the cable and the block, as well as the transfer of the weight of the load suspended on the block to the cable. To clarify the interaction of the block with the cable, we will conduct experiments on obtaining a gain in strength by a movable block, when lifting a load, using the equipment of a school physics classroom: dynamometers, laboratory blocks and a set of loads in 1N (102 g). Let's start the experiments with a mobile block, because we have three different versions of getting a gain in strength by this block. The first version is “Fig.180. A movable block as a lever with unequal shoulders "- A. V. Peryshkin's textbook, the second" Fig. 24.5 ... two identical cable tension forces F "- according to L. E. Gendenstein's textbook and finally the third" Fig. 145. Polyspast " . Lifting a load with a movable clip of a chain hoist on several parts of one rope - according to the textbook by Landsberg G.S.

Experience number 1. "Fig.183"

To conduct experiment No. 1, obtaining a gain in strength on the movable block “lever with unequal shoulders OAB fig. 180” according to the textbook by A. V. Peryshkin, on the movable block “fig. 183” position 1, draw a lever with unequal shoulders OAB, as on "Fig. 180", and start lifting the load from position 1 to position 2. At the same moment, the block begins to rotate, counterclockwise, around its axis at point A, and point B - the end of the lever, beyond which the lifting takes place, goes beyond the semicircle along which the cable goes around the movable block from below. Point O - the fulcrum of the lever, which should be fixed, goes down, see "Fig. 183" - position 2, i.e. the lever with unequal arms OAB changes as a lever with equal arms (points O and B pass the same paths).

Based on the data obtained in experiment No. 1 on changes in the position of the OAB lever on the movable block when lifting the load from position 1 to position 2, we can conclude that the representation of the movable block as a lever with unequal arms in “Fig. 180”, when lifting load, with the rotation of the block around its axis, corresponds to a lever with equal arms, which does not give a gain in strength when lifting the load.

Let's start experiment No. 2 by attaching dynamometers to the ends of the cable, on which we will hang a movable block with a load weighing 102 g, which corresponds to a gravity force of 1 N. We will fix one of the ends of the cable on a suspension, and for the second end of the cable we will lift the load on the movable block. Before lifting, the readings of both dynamometers of 0.5 N each, at the beginning of the lifting, the readings of the dynamometer, for which the lifting takes place, changed to 0.6 N, and remained so during the lifting, at the end of the lifting, the readings returned to 0.5 N. The readings of the dynamometer, fixed for a fixed suspension did not change during the ascent and remained equal to 0.5 N. Let's analyze the results of the experiment:

1. Before lifting, when a load of 1 N (102 g) hangs on a movable block, the weight of the load is distributed over the entire wheel and transferred to the cable, which goes around the block from below, with the entire semicircle of the wheel.
2. Before lifting, the readings of both dynamometers are 0.5 N each, which indicates the distribution of the weight of the load of 1 N (102 g) on ​​two parts of the cable (before and after the block) or that the cable tension force is 0.5 N, and is the same along the entire length of the cable (which is at the beginning, the same is at the end of the cable) - both of these statements are true.

Let's compare the analysis of experiment No. 2 with the versions of textbooks on obtaining a gain in strength by 2 times with a moving block. Let's start with the statement in Gendenstein's textbook L. E. "... that three forces are applied to the block: the weight of the load P, directed downward, and two identical cable tension forces directed upward (Fig. 24.5)". It would be more accurate to say that the weight of the load in “Fig. 14.5" was distributed into two parts of the cable, before and after the block, since the cable tension force is one. It remains to analyze the signature under “Fig. 181” from the textbook by A. V. Peryshkin “A combination of movable and fixed blocks - a chain hoist”. A description of the device and obtaining a gain in strength, when lifting a load, with a chain hoist is given in the Elementary Textbook of Physics, ed. Lansberg G. S. where it is said: “Each piece of rope between the blocks will act on a moving load with a force T, and all pieces of the rope will act with a force nT, where n is the number of individual sections of the rope connecting both parts of the block.” It turns out that if to “Fig. 181” we apply the gain in strength by the “rope connecting both parts” of the chain hoist from the Elementary Textbook of Physics by G. Landsberg, then the description of the gain in strength by the movable block in “Fig. 179 and, accordingly, Fig. 180" would be an error.

After analyzing four physics textbooks, we can conclude that the existing description of obtaining a gain in strength by a simple block mechanism does not correspond to the real state of affairs and therefore requires a new description of the operation of a simple block mechanism.

Simple lifting mechanism consists of a block and a cable (rope or chain).

The blocks of this lifting mechanism are divided into:

by design into simple and complex;

according to the method of lifting the load on mobile and stationary.

Let's start our acquaintance with the construction of blocks with simple block, which is a wheel rotating around its axis, with a groove around the circumference for a cable (rope, chain) Fig. 1 and it can be considered as an equal-arm lever, in which the arms of forces are equal to the radius of the wheel: OA \u003d OB \u003d r. Such a block does not give a gain in strength, but allows you to change the direction of movement of the cable (rope, chain).

double block consists of two blocks of different radii, rigidly fastened together and mounted on a common axis Fig.2. The radii of the blocks r1 and r2 are different and when lifting the load they act as a lever with unequal arms, and the gain in strength will be equal to the ratio of the lengths of the radii of a block of a larger diameter to a block of a smaller diameter F = Р·r1/r2.

gate consists of a cylinder (drum) and a handle attached to it, which acts as a block of large diameter. The gain in strength given by the collar is determined by the ratio of the radius of the circle R described by the handle to the radius of the cylinder r, on which the rope is wound F = Р r / R.

Let's move on to the method of lifting the load in blocks. From the design description, all blocks have an axis around which they rotate. If the axis of the block is fixed and does not rise or fall when lifting loads, then such a block is called fixed block, simple block, double block, gate.

At rolling block the axle rises and falls together with the load (Fig. 10) and it is intended mainly to eliminate the kink of the cable at the place of suspension of the load.

Let's get acquainted with the device and method of lifting the load. The second part of a simple lifting mechanism is a cable, rope or chain. The cable is made of steel wires, the rope is made of threads or strands, and the chain consists of links connected to each other.

Ways of suspension of the load and obtaining a gain in strength, when lifting the load, with a cable:

On fig. 4, the load is fixed at one end of the cable, and if you lift the load at the other end of the cable, then to lift this load, a force slightly more than the weight of the load will be required, since a simple block of gain in strength does not give F = P.

In Fig. 5, the load is lifted by the worker himself by the cable, which goes around a simple block from above, at one end of the first part of the cable there is a seat on which the worker sits, and by the second part of the cable the worker lifts himself with a force 2 times less than his weight, because the weight of the worker was distributed into two parts of the cable, the first - from the seat to the block, and the second - from the block to the hands of the worker F \u003d P / 2.

In Fig. 6, the load is lifted by two workers for two cables and the weight of the load is distributed equally between the cables and therefore each worker will lift the load with the force of half the weight of the load F = P / 2.

In Fig. 7, workers lift a load that hangs on two parts of one cable and the weight of the load is distributed equally between the parts of this cable (as between two cables) and each worker will lift the load with a force equal to half the weight of the load F = P / 2.

In Fig. 8, the end of the cable, for which one of the workers lifted the load, was fixed on a fixed suspension, and the weight of the load was distributed into two parts of the cable, and when the worker lifts the load by the second end of the cable, the force with which the worker will lift the load is doubled less than the weight of the load F = P / 2 and the lifting of the load will be 2 times slower.

In Fig. 9, the load hangs on 3 parts of one cable, one end of which is fixed and the gain in strength, when lifting the load, will be equal to 3, since the weight of the load will be distributed over three parts of the cable F = P / 3.

To eliminate the inflection and reduce the friction force, a simple block is installed in the place of suspension of the load and the force required to lift the load has not changed, since a simple block does not give a gain in strength in Fig. 10 and Fig. 11, and the block itself will be called moving block, since the axis of this block rises and falls along with the load.

Theoretically, the load can be hung on an unlimited number of parts of one cable, but in practice they are limited to six parts and such a lifting mechanism is called chain hoist, which consists of a fixed and movable clips with simple blocks, which are alternately bent around by a cable, fixed at one end to a fixed clip, and the load is lifted by the second end of the cable. The gain in strength depends on the number of parts of the rope between the fixed and movable clips, as a rule it is 6 parts of the rope and the gain in strength is 6 times.

The article considers the real-life interactions between the blocks and the cable when lifting the load. The current practice in determining that “a fixed block does not give a gain in strength, and a movable block gives a gain in strength by 2 times” erroneously interpreted the interaction of a cable and a block in a lifting mechanism and did not reflect the whole variety of block designs, which led to the development of one-sided erroneous ideas about block. Compared with the existing volumes of material for studying a simple block mechanism, the volume of the article has increased by 2 times, but this made it possible to clearly and intelligibly explain the processes occurring in a simple lifting mechanism not only to students, but also to teachers.

Literature:

1. Peryshkin, A. V. Physics, 7th grade: textbook / A. V. Peryshkin. - 3rd ed. ISBN 978-5-358-14436-1. § 61. Application of the lever balance rule to the block, pp. 181–183.
2. Gendenstein, L. E. Physics. 7th grade. At 2 pm Part 1. Textbook for educational institutions / L. E. Gendenshten, A. B. Kaydalov, V. B. Kozhevnikov; ed. V. A. Orlova, I. I. Roizen. - 2nd ed., corrected. - M.: Mnemosyne, 2010.-254 p.: ill. ISBN 978-5-346-01453-9. § 24. Simple mechanisms, pp. 188–196.
3. Elementary textbook of physics, edited by Academician G. S. Landsberg Volume 1. Mechanics. Heat. Molecular physics. - 10th ed. - M.: Nauka, 1985. § 84. Simple machines, pp. 168–175.
4. Gromov, S. V. Physics: Proc. for 7 cells. general education institutions / S. V. Gromov, N. A. Rodina. - 3rd ed. - M.: Enlightenment, 2001.-158 s,: ill. ISBN-5-09-010349-6. §22. Block, pp. 55-57.

Keywords: block, double block, fixed block, movable block, chain hoist..

Annotation: Physics textbooks for the 7th grade, when studying a simple mechanism, the block interprets the gain in strength when lifting a load using this mechanism in different ways, for example: in the textbook by A. V. Peryshkin, the gain in strength is achieved using the block wheel, which is acted upon by lever forces, and in L. E. Gendenshtein's textbook, the same gain is obtained with the help of a cable, on which the cable tension force acts. Different textbooks, different objects and different forces - to get a gain in strength when lifting a load. Therefore, the purpose of this article is to search for objects and forces with the help of which a gain in strength is obtained when lifting a load with a simple block mechanism.

Blocks are classified as simple mechanisms. In addition to the blocks, the group of these devices, which serve to convert forces, includes a lever, an inclined plane.

DEFINITION

Block- a rigid body that has the ability to rotate around a fixed axis.

Blocks are made in the form of discs (wheels, low cylinders, etc.) with a groove through which a rope (torso, rope, chain) is passed.

A block is called fixed, with a fixed axis (Fig. 1). It does not move when lifting a load. A fixed block can be considered as a lever that has equal leverage.

The equilibrium condition for a block is the equilibrium condition for the moments of forces applied to it:

The block in Fig. 1 will be in equilibrium if the tension forces of the threads are equal:

since the shoulders of these forces are the same (OA = OB). A fixed block does not give a gain in strength, but it allows you to change the direction of the force. Pulling on a rope that comes from above is often more comfortable than pulling on a rope that comes from below.

If the mass of the load tied to one of the ends of the rope thrown over the fixed block is equal to m, then in order to lift it, a force F should be applied to the other end of the rope, equal to:

provided that we do not take into account the friction force in the block. If it is necessary to take into account the friction in the block, then the drag coefficient (k) is introduced, then:

A smooth fixed support can serve as a replacement for the block. A rope (rope) is thrown through such a support, which slides along the support, but the friction force increases.

The fixed block does not give a gain in work. The paths that pass through the points of application of forces are the same, the forces are equal, therefore, the work is equal.

In order to get a gain in strength, using fixed blocks, a combination of blocks is used, for example, a double block. When the blocks must have different diameters. They are fixedly connected to each other and mounted on a single axis. A rope is attached to each block so that it can be wound on or off the block without slipping. Shoulders of forces in this case will be unequal. The double block acts as a lever with arms of different lengths. Figure 2 shows a diagram of a double block.

The equilibrium condition for the lever in Fig. 2 will become the formula:

Double block can transform force. By applying a smaller force to a rope wrapped around a block of large radius, a force is obtained that acts from the side of the rope wound onto a block of smaller radius.

A movable block is a block whose axis moves together with the load. On fig. 2 the movable block can be considered as a lever with arms of different sizes. In this case, point O is the fulcrum of the lever. OA - shoulder strength; OB - shoulder of strength. Consider Fig. 3. The arm of the force is twice as large as the arm of the force, therefore, for balance it is necessary that the magnitude of the force F be half as much as the modulus of the force P:

It can be concluded that with the help of a movable block, we get a twofold gain in strength. The equilibrium condition of the moving block without taking into account the friction force can be written as:

If we try to take into account the friction force in the block, then we introduce the coefficient of resistance of the block (k) and get:

Sometimes a combination of a movable and a fixed block is used. In this combination, a fixed block is used for convenience. It does not give a gain in strength, but allows you to change the direction of the force. The movable block is used to change the magnitude of the applied force. If the ends of the rope enclosing the block make the same angles with the horizon, then the ratio of the force acting on the load to the weight of the body is equal to the ratio of the radius of the block to the chord of the arc that the rope covers. In the case of parallel ropes, the force required to lift the load will be required two times less than the weight of the load being lifted.

The golden rule of mechanics

Simple mechanisms of gain in work do not give. How much we gain in strength, how many times we lose in distance. Since the work is equal to the scalar product of the force and the displacement, therefore, it will not change when using movable (as well as fixed) blocks.

In the form of a formula "golden rule" can be written as follows:

where - the path that passes the point of application of force - the path passed by the point of application of force.

Golden Rule is the simplest formulation of the law of conservation of energy. This rule applies to cases of uniform or almost uniform movement of mechanisms. The translational distances of the ends of the ropes are related to the radii of the blocks ( and ) as:

We get that in order to fulfill the "golden rule" for a double block, it is necessary that:

If the forces and are balanced, then the block is at rest or moves uniformly.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2

In modern technology for the transfer of goods at construction sites and enterprises, hoisting mechanisms are widely used, indispensable constituent parts which can be called simple mechanisms. Among them are the most ancient inventions of mankind: block and lever. The ancient Greek scientist Archimedes facilitated the work of man, giving him a gain in strength when using his invention, and taught him to change the direction of the force.

A block is a wheel with a groove around the circumference for a rope or chain, the axis of which is rigidly attached to a wall or ceiling beam.

Lifting devices usually use not one, but several blocks. The system of blocks and cables, designed to increase the carrying capacity, is called a chain hoist.

The movable and fixed block are the same ancient simple mechanisms as the lever. Already in 212 BC, with the help of hooks and grabs connected to blocks, the Syracusans seized the means of siege from the Romans. The construction of military vehicles and the defense of the city was led by Archimedes.

Archimedes considered the fixed block as an equal-armed lever.

The moment of force acting on one side of the block is equal to the moment of force applied on the other side of the block. The forces that create these moments are also the same.

There is no gain in strength, but such a block allows you to change the direction of the force, which is sometimes necessary.

Archimedes took the movable block as an unequal lever, giving a gain in strength by 2 times. Moments of forces act relative to the center of rotation, which should be equal at equilibrium.

Archimedes studied the mechanical properties of the moving block and put it into practice. According to Athenaeus, "many methods were invented to launch the gigantic ship built by the Syracusan tyrant Hieron, but the mechanic Archimedes, using simple mechanisms, alone managed to move the ship with the help of a few people. Archimedes came up with a block and through it launched a huge ship" .

The block does not give a gain in work, confirming the golden rule of mechanics. It is easy to verify this by paying attention to the distances covered by the hand and the kettlebell.

Sports sailboats, like the sailboats of the past, cannot do without blocks when setting and managing sails. Modern ships need blocks for lifting signals, boats.

This combination of movable and fixed units on an electrified railway line to adjust the tension of the wires.

Such a system of blocks can be used by glider pilots to lift their vehicles into the air.

Most often, simple mechanisms are used to gain strength. That is, with less force to move a greater weight in comparison with it. At the same time, the gain in strength is not achieved “for free”. The price to pay for it is the loss in distance, that is, it is required to make a greater movement than without the use of a simple mechanism. However, when forces are limited, "trading" distance for strength is advantageous.

Movable and fixed blocks are one of the types of simple mechanisms. In addition, they are a modified lever, which is also a simple mechanism.

Fixed block does not give a gain in strength, it simply changes the direction of its application. Imagine that you need to lift a heavy load up with a rope. You will have to pull it up. But if you use a fixed block, then you will need to pull down, while the load will rise up. In this case, it will be easier for you, since the necessary strength will be the sum of muscle strength and your weight. Without the use of a fixed block, the same force would have to be applied, but it would be achieved solely due to muscle strength.

The fixed block is a wheel with a groove for the rope. The wheel is fixed, it can rotate around its axis, but cannot move. The ends of the rope (cable) hang down, a load is attached to one, and a force is applied to the other. If you pull the cable down, the load rises.

Since there is no gain in strength, there is no loss in distance. At what distance the load will rise, the rope must be lowered the same distance.

Usage rolling block gives a gain in strength twice (ideally). This means that if the weight of the load is F, then in order to lift it, a force F / 2 must be applied. The movable block consists of the same wheel with a cable groove. However, one end of the cable is fixed here, and the wheel is movable. The wheel moves with the load.

The weight of the load is the downward force. It is balanced by two upward forces. One is created by a support to which the cable is attached, and the other by pulling the cable. The cable tension is the same on both sides, which means that the weight of the load is equally distributed between them. Therefore, each of the forces is 2 times less than the weight of the load.

In real situations, the gain in strength is less than 2 times, since the lifting force is partially “spent” on the weight of the rope and block, as well as friction.

The movable block, giving almost a double gain in strength, gives a double loss in distance. To lift a load to a certain height h, it is necessary that the ropes on each side of the block decrease by this height, that is, a total of 2h is obtained.

Typically, combinations of fixed and movable blocks are used - chain hoists. They allow you to get a gain in strength and direction. The more moving blocks in the chain hoist, the greater the gain in strength.