Simple mechanisms. Upward movement of a body on an inclined plane Inclined plane determination of application force

Topics of the Unified State Examination codifier: simple mechanisms, mechanism efficiency.

Mechanism - this is a device for converting force (increasing or decreasing it).
Simple mechanisms - a lever and an inclined plane.

Lever arm.

Lever arm is a rigid body that can rotate around a fixed axis. In Fig.

1) shows a lever with an axis of rotation. Forces and are applied to the ends of the lever (points and ). The shoulders of these forces are equal to and respectively.

The equilibrium condition of the lever is given by the rule of moments: , whence

Rice. 1. Lever

From this relationship it follows that the lever gives a gain in strength or distance (depending on the purpose for which it is used) as many times as the larger arm is longer than the smaller one.

For example, to lift a 700 N load with a force of 100 N, you need to take a lever with a 7:1 arm ratio and place the load on the short arm. We will gain 7 times in strength, but will lose the same amount of times in distance: the end of the long arm will describe a 7 times greater arc than the end of the short arm (that is, the load).

Examples of levers that provide a gain in strength are a shovel, scissors, and pliers. The rower's oar is the lever that gives the gain in distance. And ordinary lever scales are an equal-armed lever that does not provide any gain in either distance or strength (otherwise they can be used to weigh customers).

Fixed block. An important type of lever is block

- a wheel fixed in a cage with a groove through which a rope is passed. In most problems, a rope is considered to be a weightless, inextensible thread.

In Fig.

Figure 2 shows a stationary block, i.e. a block with a stationary axis of rotation (passing perpendicular to the plane of the drawing through the point ).

At the right end of the thread, a weight is attached to a point. Let us recall that body weight is the force with which the body presses on the support or stretches the suspension. In this case, the weight is applied to the point where the load is attached to the thread.

Why then do we need a fixed block at all? It is useful because it allows you to change the direction of the effort. Typically a fixed block is used as part of more complex mechanisms.

Movable block.

In Fig. 3 shown moving block

, the axis of which moves along with the load. We pull the thread with a force that is applied at a point and directed upward. The block rotates and at the same time also moves upward, lifting a load suspended on a thread.

At a given moment in time, the fixed point is the point, and it is around it that the block rotates (it would “roll” over the point). They also say that the instantaneous axis of rotation of the block passes through the point (this axis is directed perpendicular to the plane of the drawing).

The weight of the load is applied at the point where the load is attached to the thread. The leverage of force is equal to .

But the shoulder of the force with which we pull the thread turns out to be twice as large: it is equal to . Accordingly, the condition for equilibrium of the load is equality (which we see in Fig. 3: the vector is half as long as the vector).

Consequently, the movable block gives a double gain in strength. At the same time, however, we lose by the same two times in distance: in order to raise the load one meter, the point will have to be moved two meters (that is, pull out two meters of thread).

The block in Fig.

3 there is one drawback: pulling the thread up (beyond the point) is not the best idea. Agree that it is much more convenient to pull the thread down! This is where the stationary block comes to our rescue.

In Fig. Figure 4 shows a lifting mechanism, which is a combination of a moving block and a fixed one. A load is suspended from the movable block, and the cable is additionally thrown over the fixed block, which makes it possible to pull the cable down to lift the load up. The external force on the cable is again symbolized by the vector .

Fundamentally, this device is no different from a moving block: with its help we also get a double gain in strength.

Inclined plane. As we know, it is easier to roll a heavy barrel along inclined walkways than to lift it vertically. The bridges are thus a mechanism that provides gains in strength. In mechanics, such a mechanism is called an inclined plane.

Inclined plane

Let's select the axis as shown in the figure. Since the load moves without acceleration, the forces acting on it are balanced:

We project on the axis:

This is exactly the force that needs to be applied to move the load up an inclined plane.

To evenly lift the same load vertically, a force equal to . It can be seen that, since . An inclined plane actually gives a gain in strength, and the smaller the angle, the greater the gain.

Widely used types of inclined plane are wedge and screw.

The golden rule of mechanics.

A simple mechanism can give a gain in strength or distance, but cannot give a gain in work.

For example, a lever with a leverage ratio of 2:1 gives a double gain in strength. In order to lift a weight on the smaller shoulder, you need to apply force to the larger shoulder. But to raise the load to a height, the larger arm will have to be lowered by , and the work done will be equal to:

i.e. the same value as without using the lever.

In the case of an inclined plane, we gain in strength, since we apply a force to the load that is less than the force of gravity. However, in order to raise the load to a height above the initial position, we need to go along the inclined plane. At the same time we do work

i.e. the same as when lifting a load vertically.

These facts serve as manifestations of the so-called golden rule of mechanics.

The golden rule of mechanics. None of the simple mechanisms provide any gains in work. The number of times we win in strength, the same number of times we lose in distance, and vice versa.

The golden rule of mechanics is nothing more than a simple version of the law of conservation of energy.

Efficiency of the mechanism.

In practice, we have to distinguish between useful work A useful, which must be accomplished using the mechanism under ideal conditions without any losses, and complete work A full,
which is performed for the same purposes in a real situation.

The total work is equal to the sum:
-useful work;
-work done against frictional forces in various parts of the mechanism;
-work done to move the component elements of the mechanism.

So, when lifting a load with a lever, you have to additionally do work to overcome the frictional force in the axis of the lever and to move the lever itself, which has some weight.

Full work is always more useful. The ratio of useful work to total work is called the coefficient of performance (efficiency) of the mechanism:

=A useful/ A full

Efficiency is usually expressed as a percentage. The efficiency of real mechanisms is always less than 100%.

Let's calculate the efficiency of an inclined plane with an angle in the presence of friction. The coefficient of friction between the surface of the inclined plane and the load is equal to .

Let the mass load rise uniformly along the inclined plane under the action of force from point to point to a height (Fig. 6). In the direction opposite to the movement, the sliding friction force acts on the load.

There is no acceleration, so the forces acting on the load are balanced:

We project on the X axis:

. (1)

We project on the Y axis:

. (2)

Besides,

, (3)

From (2) we have:

Then from (3):

Substituting this into (1), we get:

The total work is equal to the product of the force F and the path traveled by the body along the surface of the inclined plane:

A full=.

The useful work is obviously equal to:

A useful=.

For the required efficiency we obtain:

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Simple machines - This name refers to the following mechanisms, a description and explanation of the operation of which can be found in all elementary courses in physics and mechanics: lever, blocks, pulleys, gates, inclined plane, wedge and screw. The blocks and gates are based on the lever principle, the wedge and screw are based on the inclined plane principle.

Lever arm- the simplest mechanical device, which is a solid body (crossbar) rotating around a fulcrum. The sides of the crossbar on either side of the fulcrum are called lever arms.

The lever is used to obtain more force on the short arm with less force on the long arm (or to obtain more movement on the long arm with less movement on the short arm). By making the lever arm long enough, theoretically, any force can be developed.

Two other simple mechanisms are also special cases of a lever: a gate and a block. The principle of operation of the lever is a direct consequence of the law of conservation of energy. For levers, as for other mechanisms, a characteristic is introduced that shows the mechanical effect that can be obtained due to the lever. This characteristic is the gear ratio; it shows how the load and the applied force relate:

There are levers of the 1st class, in which the fulcrum is located between the points of application of forces, and levers of the 2nd class, in which the points of application of forces are located on one side of the support.

Block- a simple mechanical device that allows you to regulate the force, the axis of which is fixed when lifting loads, does not rise or fall. It is a wheel with a groove around its circumference, rotating around its axis. The groove is intended for a rope, chain, belt, etc. The axis of the block is placed in cages attached to a beam or wall, such a block is called stationary; if a load is attached to these clips, and the block can move with them, then such a block is called movable.

A fixed block is used to lift small loads or to change the direction of force.

Block equilibrium condition:

F is the applied external force, m is the mass of the load, g is the acceleration of gravity, f is the resistance coefficient in the block (for chains approximately 1.05, and for ropes - 1.1). In the absence of friction, lifting requires a force equal to the weight of the load.

The moving block has a free axis and is designed to change the magnitude of the applied forces. If the ends of the rope clasping the block make equal angles with the horizon, then the force acting on the load is related to its weight, as the radius of the block is to the chord of the arc clasped by the rope; hence, if the ropes are parallel (that is, when the arc encircled by the rope is equal to a semicircle), then lifting the load will require a force half as much as the weight of the load, that is:

In this case, the load will travel a distance half as large as that traveled by the point of application of force F; accordingly, the gain in the force of the moving block is equal to 2.

In fact, any block is a lever, in the case of a fixed block - equal arms, in the case of a moving one - with a ratio of arms of 1 to 2. As for any other lever, the rule is true for a block: The number of times we win in an effort, the same number of times we lose in the distance. In other words, the work done when moving a load a certain distance without using a block is equal to the work expended when moving a load the same distance using a block, provided there is no friction. In a real block there is always some loss.

As we know, it is easier to roll a heavy barrel along inclined walkways than to lift it vertically. The bridges are thus a mechanism that provides gains in strength.- this is a flat surface installed at an angle other than straight and/or zero to a horizontal surface. An inclined plane allows you to overcome significant resistance by applying relatively little force over a greater distance than the load needs to be lifted.

The inclined plane is one of the well-known simple mechanisms. Examples of inclined planes are:

• ramps and ladders;
• tools: chisel, axe, hammer, plow, wedge and so on;

The most canonical example of an inclined plane is an inclined surface, such as the entrance to a bridge with a difference in height.

§ tr - where m is the mass of the body, is the acceleration vector, is the reaction force (impact) of the support, is the free fall acceleration vector, tr is the friction force.

§ a = g(sin α + μcos α) - when climbing an inclined plane and in the absence of additional forces;

§ a = g(sin α − μcos α) - when descending from an inclined plane and in the absence of additional forces;

here μ is the coefficient of friction of the body on the surface, α is the angle of inclination of the plane.

The limiting case is when the angle of inclination of the plane is 90 degrees, that is, the body falls, sliding along the wall. In this case: α = g, that is, the friction force does not affect the body in any way; it is in free fall. Another limiting case is the situation when the angle of inclination of the plane is zero, i.e. the plane is parallel to the ground; in this case, the body cannot move without the application of an external force. It should be noted that, following from the definition, in both situations the plane will no longer be inclined - the angle of inclination should not be equal to 90o or 0o.

The type of movement of the body depends on the critical angle. The body is at rest if the angle of inclination of the plane is less than the critical angle, is at rest or moves uniformly if the angle of inclination of the plane is equal to the critical angle, and moves uniformly accelerated, provided that the angle of inclination of the plane is greater than the critical angle.

§ or α< β - тело покоится;

§ or α = β - the body is at rest or moving uniformly;

§ or α > β - the body moves with uniform acceleration;

Wedge- a simple mechanism in the form of a prism, the working surfaces of which converge at an acute angle. Used for moving apart and dividing the object being processed into parts. The wedge is one of the varieties of the mechanism called "inclined plane". When a force acts on the base of the prism, two components appear, perpendicular to the working surfaces. The ideal gain in force given by a wedge is equal to the ratio of its length to the thickness at the blunt end - the wedging action of the wedge gives a gain in force at a small angle and a large length of the wedge. The actual gain of the wedge depends greatly on the frictional force, which changes as the wedge moves.

; where IMA is the ideal gain, W is the width, L is the length. The wedge principle is used in such tools and tools as an axe, chisel, knife, nail, needle, and stake.

I didn’t find anything about construction equipment.

The body that slides down an inclined plane. In this case, the following forces act on it:

Gravity mg directed vertically downward;

Support reaction force N, directed perpendicular to the plane;

The sliding friction force Ftr is directed opposite to the speed (up along the inclined plane when the body slides).

Let us introduce an inclined coordinate system, the OX axis of which is directed downward along the plane. This is convenient, because in this case you will have to decompose only one vector into components - the gravity vector mg, and the vectors of the friction force Ftr and the support reaction force N are already directed along the axes. With this expansion, the x-component of the gravity force is equal to mg sin(α) and corresponds to the “pulling force” responsible for the accelerated downward movement, and the y-component - mg cos(α) = N balances the support reaction force, since the body moves along the OY axis absent.

The sliding friction force Ftr = µN is proportional to the support reaction force. This allows us to obtain the following expression for the friction force: Ftr = µmg cos(α). This force is opposite to the "pulling" component of gravity. Therefore, for a body sliding down, we obtain expressions for the total resultant force and acceleration:

Fx = mg(sin(α) – µ cos(α));

ax = g(sin(α) – µ cos(α)).

acceleration:

speed is

v=ax*t=t*g(sin(α) – µ cos(α))

after t=0.2 s

speed is

v=0.2*9.8(sin(45)-0.4*cos(45))=0.83 m/s

The force with which a body is attracted to the Earth under the influence of the Earth's gravitational field is called gravity. According to the law of universal gravitation, on the surface of the Earth (or near this surface), a body of mass m is acted upon by the force of gravity

Ft=GMm/R2 (2.28)

where M is the mass of the Earth; R is the radius of the Earth.

If only the force of gravity acts on a body, and all other forces are mutually balanced, the body undergoes free fall. According to Newton’s second law and formula (2.28), the gravitational acceleration module g is found by the formula

g=Ft/m=GM/R2. (2.29)

From formula (2.29) it follows that the acceleration of free fall does not depend on the mass m of the falling body, i.e. for all bodies in a given place on the Earth it is the same. From formula (2.29) it follows that Ft = mg. In vector form

In § 5 it was noted that since the Earth is not a sphere, but an ellipsoid of revolution, its polar radius is less than the equatorial one. From formula (2.28) it is clear that for this reason the force of gravity and the acceleration of gravity caused by it at the pole is greater than at the equator.

The force of gravity acts on all bodies located in the gravitational field of the Earth, but not all bodies fall to the Earth. This is explained by the fact that the movement of many bodies is impeded by other bodies, for example supports, suspension threads, etc. Bodies that limit the movement of other bodies are called connections. Under the influence of gravity, the bonds are deformed and the reaction force of the deformed connection, according to Newton’s third law, balances the force of gravity.

In § 5 it was also noted that the acceleration of free fall is affected by the rotation of the Earth. This influence is explained as follows. The reference systems associated with the Earth's surface (except for the two associated with the Earth's poles) are not, strictly speaking, inertial reference systems - the Earth rotates around its axis, and together with it such reference systems move in circles with centripetal acceleration. This non-inertiality of reference systems is manifested, in particular, in the fact that the value of the acceleration of gravity turns out to be different in different places on the Earth and depends on the geographic latitude of the place where the reference system associated with the Earth is located, relative to which the acceleration of gravity is determined.

Measurements carried out at different latitudes showed that the numerical values ​​of the acceleration due to gravity differ little from each other. Therefore, with not very accurate calculations, we can neglect the non-inertiality of the reference systems associated with the Earth’s surface, as well as the difference in the shape of the Earth from spherical, and assume that the acceleration of gravity anywhere on the Earth is the same and equal to 9.8 m/s2.

From the law of universal gravitation it follows that the force of gravity and the acceleration of gravity caused by it decrease with increasing distance from the Earth. At a height h from the Earth's surface, the gravitational acceleration modulus is determined by the formula

It has been established that at an altitude of 300 km above the Earth's surface, the acceleration of gravity is 1 m/s2 less than at the Earth's surface.

Consequently, near the Earth (up to heights of several kilometers) the force of gravity practically does not change, and therefore the free fall of bodies near the Earth is a uniformly accelerated motion.

Body weight. Weightlessness and overload

The force in which, due to attraction to the Earth, a body acts on its support or suspension is called the weight of the body. Unlike gravity, which is a gravitational force applied to a body, weight is an elastic force applied to a support or suspension (i.e., a link).

Observations show that the weight of a body P, determined on a spring scale, is equal to the force of gravity Ft acting on the body only if the scales with the body relative to the Earth are at rest or moving uniformly and rectilinearly; In this case

If a body moves at an accelerated rate, then its weight depends on the value of this acceleration and on its direction relative to the direction of the acceleration of gravity.

When a body is suspended on a spring scale, two forces act on it: the force of gravity Ft=mg and the elastic force Fyp of the spring. If in this case the body moves vertically up or down relative to the direction of acceleration of gravity, then the vector sum of the forces Ft and Fup gives a resultant, causing acceleration of the body, i.e.

Fт + Fуп=ma.

According to the above definition of the concept of “weight”, we can write that P = -Fyп. taking into account the fact that Ft=mg, it follows that mg-ma=-Fyп. Therefore, P=m(g-a).

The forces Fт and Fуп are directed along one vertical straight line. Therefore, if the acceleration of body a is directed downward (i.e., it coincides in direction with the acceleration of free fall g), then in modulus

If the acceleration of the body is directed upward (i.e., opposite to the direction of the acceleration of free fall), then

P = m = m(g+a).

Consequently, the weight of a body whose acceleration coincides in direction with the acceleration of free fall is less than the weight of a body at rest, and the weight of a body whose acceleration is opposite to the direction of the acceleration of free fall is greater than the weight of a body at rest. The increase in body weight caused by its accelerated movement is called overload.

In free fall a=g. it follows that in this case P = 0, i.e. there is no weight. Therefore, if bodies move only under the influence of gravity (i.e., fall freely), they are in a state of weightlessness. A characteristic feature of this state is the absence of deformations and internal stresses in freely falling bodies, which are caused by gravity in bodies at rest. The reason for the weightlessness of bodies is that the force of gravity imparts equal accelerations to a freely falling body and its support (or suspension).

In addition to the lever and the block, simple mechanisms also include an inclined plane and its variations: a wedge and a screw.

INCLINED PLANE

Inclined plane used to move heavy objects to a higher level without directly lifting them.
Such devices include ramps, escalators, conventional stairs and conveyors.
If you need to lift a load to a height, it is always easier to use a gentle lift than a steep one. Moreover, the steeper the slope, the easier it is to complete this work. When time and distance are not of great importance, but lifting the load is important with the least effort, the inclined plane turns out to be irreplaceable.

These pictures can help explain how a simple mechanism works. INCLINED PLANE.
Classical calculations of the action of an inclined plane and other simple mechanisms belong to the outstanding ancient mechanic Archimedes of Syracuse.

When building temples, the Egyptians transported, lifted and installed colossal obelisks and statues that weighed tens and hundreds of tons! All this could be done using, among other simple mechanisms inclined plane.
The main lifting device of the Egyptians was inclined plane - ramp. The frame of the ramp, that is, its sides and partitions, which crossed the ramp at a short distance from each other, was built of brick; the voids were filled with reeds and branches. As the pyramid grows the ramp was being built. Along these ramps, stones were dragged on sleds in the same way as on the ground, helping themselves with levers.

The ramp angle was very slight - 5 or 6 degrees.

Columns of the ancient Egyptian temple in Thebes. Each of these huge columns was pulled by slaves along a ramp-an inclined plane. When the column crawled into the hole, sand was raked out through the hole, and then the brick wall was dismantled and the embankment was removed..

Thus, for example, the inclined road to the Khafre pyramid, with a rise height of 46 meters, had

I will show you with specific examples. Example 1: a body moves uniformly under the influence of traction force (Figure 1).

Students must first learn the algorithm for constructing a drawing. We draw an inclined plane, in the middle of it there is a body in the form of a rectangle, through the middle of the body we draw an axis parallel to the inclined plane. The direction of the axis is not significant, but in the case of uniformly accelerated motion, it is better to show it in the direction of the vector so that in algebraic form in the equation of motion there is a plus sign on the right side in front of it. Next we build strength. We draw the force of gravity vertically downwards of an arbitrary length (I require the drawings to be large so that everyone can understand everything). Then, from the point of application of gravity, a perpendicular to the axis, along which the support reaction force will go. Parallel to this perpendicular, draw a dotted line from the end of the vector until it intersects with the axis. From this point - a dotted line parallel to the intersection with the perpendicular - we obtain a vector of the correct length. Thus, we constructed a parallelogram on the vectors and , automatically indicating the correct magnitude of the support reaction force and constructing, according to all the rules of vector geometry, the resultant of these forces, which I call the rolling resultant (diagonal coinciding with the axis). At this point, using the method from the textbook, in a separate figure I show the reaction force of a support of arbitrary length: first shorter than necessary, and then longer than necessary. I show the resultant force of gravity and the support reaction force: in the first case, it is directed downward at an angle to the inclined plane (Figure 2), in the second case, upward at an angle to the inclined plane (Figure 3).

We draw a very important conclusion: the relationship between the force of gravity and the reaction force of the support must be such that the body, under their action (or under the action of the rolling resultant), in the absence of other forces, moves downwards along inclined plane. Next I ask: what other forces act on the body? The guys answer: traction force and friction force. I ask the following question: which strength will we show first, and which later? I am seeking a correct and reasonable answer: first in this case it is necessary to show the traction force, and then the friction force, the module of which will be equal to the sum of the modules of the traction force and the rolling resultant: , because According to the conditions of the problem, the body moves uniformly, therefore, the resultant of all forces acting on the body must be equal to zero according to Newton’s first law. To control, I ask a provocative question: how much force is acting on the body? The guys must answer - four (not five!): gravity, ground reaction force, traction force and friction force. Now we write the equation of motion in vector form according to Newton’s first law:

We replace the sum of vectors with the rolling resultant:

We obtain an equation in which all vectors are parallel to the axis. Now let's write this equation through projections of vectors onto the axis:

You can skip this entry in the future. Let us replace in the equation the projections of vectors by their modules, taking into account the directions:

Example 2: a body, under the influence of traction, moves onto an inclined plane with acceleration (Figure 4).

In this example, students must say that after constructing the force of gravity, the support reaction force and the rolling resultant, the next one must show the friction force, the last one is the traction force vector, which must be greater than the sum of the vectors, because the resultant of all forces must be directed in the same direction as the acceleration vector according to Newton's second law. The equation of motion of a body must be written according to Newton’s second law:

If there is an opportunity to consider other cases in class, then we do not neglect this opportunity. If not, then I give this task home. Some may consider all the remaining cases, others may consider the right to choose students. In the next lesson, we check, correct errors and move on to solving specific problems, having previously expressed from vector triangles and:

It is advisable to analyze equality (2) for various angles. At we have: as when moving horizontally under the influence of a horizontal traction force. As the angle increases, its cosine decreases, therefore, the support reaction force decreases and the force of gravity becomes less and less. At angle it is equal to zero, i.e. the body does not act on the support and the support, accordingly, “does not react.”

I foresee a question from opponents: how to apply this technique to cases where the traction force is horizontal or directed at an angle to an inclined plane? I will answer with specific examples.

a) The body is pulled with acceleration onto an inclined plane, applying a traction force horizontally (Figure 5).

We decompose the horizontal traction force into two components: along the axis - and perpendicular to the axis - (the reverse operation of constructing the resultant of perpendicular forces). We write down the equation of motion:

We replace the rolling resultant, and instead write:

From vector triangles we express: And : .

Under the influence of horizontal force, the body not only rises up the inclined plane, but is also additionally pressed against it. Therefore, an additional pressure force arises equal to the vector modulus and, according to Newton’s third law, an additional support reaction force: . Then the friction force will be: .

The equation of motion will take the form:

Now we have completely deciphered the equation of motion. Now it remains to express the desired value from it. Try to solve this problem the traditional way and you will get the same equation, only the solution will be more cumbersome.

b) The body is pulled evenly from the inclined plane, applying a traction force horizontally (Figure 6).

In this case, the traction force, in addition to pulling the body down along the inclined plane, also tears it away from the inclined plane. So the final equation is:

c) The body is dragged evenly onto the inclined plane, applying a traction force at an angle to the inclined plane (Figure 7).

I propose to consider specific problems in order to further convincingly advertise my methodological approach to solving such problems. But first, I draw attention to the solution algorithm (I think all physics teachers draw the attention of students to it, and my entire story was subordinated to this algorithm):

1) after carefully reading the problem, find out how the body moves;
2) make a drawing with the correct image of the forces, based on the conditions of the problem;
3) write down the equation of motion in vector form according to Newton’s first or second law;
4) write this equation through the projections of force vectors onto the x-axis (this step can be omitted later, when the ability to solve problems in dynamics is brought to automaticity);
5) express the projections of vectors through their modules, taking into account directions and write the equation in algebraic form;
6) express the force modules using formulas (if necessary);
7) express the desired value.

Task 1. How long does it take for a body of mass to slide down an inclined plane with height and angle of inclination if it moves uniformly along an inclined plane with an angle of inclination?

What would it be like to solve this problem in the usual way!

Task 2. What is easier: to hold the body on an inclined plane or to move it evenly upward along it?

Here, when explaining, one cannot do without the rolling resultant, in my opinion.

As can be seen from the figures, in the first case, the friction force helps to hold the body (directed in the same direction as the holding force), in the second case, it, together with the rolling resultant, is directed against the movement. In the first case, in the second case.