Simple machines (lever, block, inclined plane, wedge). Construction devices

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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- the simplest mechanical device, which is solid(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 showing 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 adjust 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 amount of force applied. 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.

Inclined plane- 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 90o 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 under 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 strength given by the wedge is equal to the ratio its length to the thickness at the blunt end - the wedging action of the wedge gives a gain in strength 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.

This article talks about how to solve problems about moving along an inclined plane. Reviewed detailed solution problems on the motion of connected bodies on an inclined plane from the Unified State Examination in Physics.

Solving the problem of motion on an inclined plane

Before moving directly to solving the problem, as a tutor in mathematics and physics, I recommend carefully analyzing its condition. You need to start by depicting the forces that act on connected bodies:

Here and are the thread tension forces acting on the left and right bodies, respectively, are the support reaction force acting on the left body, and are the gravity forces acting on the left and right bodies, respectively. Everything is clear about the direction of these forces. The tension force is directed along the thread, the gravity force is vertically downward, and the support reaction force is perpendicular to the inclined plane.

But the direction of the friction force will have to be dealt with separately. Therefore, in the figure it is shown as a dotted line and signed with a question mark. It is intuitively clear that if the right load “outweighs” the left one, then the friction force will be directed opposite to the vector. On the contrary, if the left load “outweighs” the right one, then the friction force will be co-directed with the vector.

The right weight is pulled down by force N. Here we took the acceleration of gravity m/s 2. The left load is also pulled down by gravity, but not all of it, but only a “part” of it, since the load lies on an inclined plane. This “part” is equal to the projection of gravity onto the inclined plane, that is, the leg in right triangle shown in the figure, that is, equal to N.

That is, the right load still “outweighs”. Consequently, the friction force is directed as shown in the figure (we drew it from the center of mass of the body, which is possible in the case when the body can be modeled by a material point):

The second important question that needs to be addressed is whether this coupled system will move at all? What if it turns out that the friction force between the left load and the inclined plane will be so great that it will not allow it to move?

This situation will be possible in the case when the maximum friction force, the modulus of which is determined by the formula (here - the coefficient of friction between the load and the inclined plane - the support reaction force acting on the load from the inclined plane), turns out to be greater than the force that is trying to bring the system into motion. That is, that very “outweighing” force that is equal to N.

The modulus of the support reaction force is equal to the length of the leg in the triangle according to Newton’s 3rd law (with the same magnitude of force the load presses on the inclined plane, with the same magnitude of force the inclined plane acts on the load). That is, the support reaction force is equal to N. Then the maximum value of the friction force is N, which is less than the value of the “overweighing force”.

Consequently, the system will move, and move with acceleration. Let us depict in the figure these accelerations and coordinate axes, which we will need later when solving the problem:

Now, after a thorough analysis of the problem conditions, we are ready to begin solving it.

Let's write down Newton's 2nd law for the left body:

And in the projection onto the axes of the coordinate system we get:

Here, projections are taken with a minus, the vectors of which are directed opposite the direction of the corresponding coordinate axis. Projections whose vectors are aligned with the corresponding coordinate axis are taken with a plus.

Once again we will explain in detail how to find projections and . To do this, consider the right triangle shown in the figure. In this triangle And . It is also known that in this right triangle . Then and.

The acceleration vector lies entirely on the axis, and therefore . As we already mentioned above, by definition, the modulus of the friction force is equal to the product of the friction coefficient and the modulus of the support reaction force. Hence, . Then the original system of equations takes the form:

Let us now write down Newton’s 2nd law for the right body:

In projection onto the axis we get.

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

An inclined plane is used to move heavy objects more high 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 have no meaning of great importance, and it is important to lift the load with the least effort, the inclined plane turns out to be indispensable.

These pictures can help explain how the simple INCLINED PLANE mechanism works.
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, weighing tens and hundreds of tons! All this could be done using, among other simple mechanisms, an inclined plane.

The main lifting device of the Egyptians was an inclined plane - a ramp. The frame of the ramp, that is, its sides and partitions. As the pyramid grew, the ramp was built on. Stones were dragged along these ramps on sleds. 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 lift height of 46 meters, was about half a kilometer long.

A body on an inclined plane is held by a force whose magnitude is as many times less than the weight of this body as the length of the inclined plane is greater than its height."
This condition for the equilibrium of forces on an inclined plane was formulated by the Dutch scientist Simon Stevin (1548-1620).

Drawing on title page books by S. Stevin, with which he confirms his formulation.

The inclined plane at the Krasnoyarsk hydroelectric power station was used very cleverly. Here, instead of locks, there is a ship-carrying chamber moving along an inclined overpass. To move it, a traction force of 4000 kN is required.

Why do mountain roads wind in gentle serpentines?

A wedge is a type of simple mechanism called an inclined plane. The wedge consists of two inclined planes, the bases of which are in contact. It is used to obtain a gain in strength, that is, with the help of a smaller force to counteract a larger force.

When chopping wood, to make the work easier, insert a metal wedge into the crack of the log and hit it with the butt of an ax.

The ideal gain in force given by a wedge is equal to the ratio of its length to its thickness at the blunt end. Due to the high friction, its efficiency is so low that the ideal gain does not matter much

Another type of inclined plane is a screw.
A screw is an inclined plane wound around an axis. The thread of a screw is an inclined plane that is repeatedly wrapped around a cylinder.

Due to the high friction, its efficiency is so low that the ideal gain does not matter much. Depending on the direction of rise of the inclined plane, the screw thread can be left-handed or right-handed.
Examples of simple devices with screw threads are a jack, a bolt with a nut, a micrometer, a vice.

An inclined plane is a flat surface located at a particular angle to the horizontal. It allows you to lift a load with less force than if the load were lifted vertically. On an inclined plane, the load rises along this plane. At the same time, it covers a greater distance than if it rose vertically.

Note 1

Moreover, no matter how many times the gain in strength occurs, the distance that the load will cover will be greater.

Figure 1. Inclined plane

If the height to which the load must be raised is equal to $h$, and at the same time the force $F_h$ would be expended, and the length of the inclined plane is $l$, and at the same time the force $F_l$ is expended, then $l$ is so related to $h $, how $F_h$ relates to $F_l$: $l/h = F_h/F_l$... However, $F_h$ is the weight of the load ($P$). Therefore, it is usually written like this: $l/h = P/F$, where $F$ is the force lifting the load.

The magnitude of the force $F$ that must be applied to a load weighing $P$ in order for the body to be in equilibrium on an inclined plane is equal to $F_1 = P_h/l = Рsin(\mathbf \alpha )$, if the force $P$ is applied parallel to the inclined plane plane (Fig. 2, a), and $F_2$ = $Р_h/l = Рtg(\mathbf \alpha )$, if the force $Р$ is applied parallel to the base of the inclined plane (Fig. 2, b).

Figure 2. Movement of a load along an inclined plane

a) the force is parallel to the plane b) the force is parallel to the base

An inclined plane gives an advantage in strength; with its help, it is easier to lift a load to a height. The smaller the angle $\alpha $, the greater the gain in strength. If the angle $\alpha $ is less than the angle of friction, then the load will not move spontaneously, and force is needed to pull it down.

If we take into account the friction forces between the load and the inclined plane, then for $F_1$ and $F_2$ the following values ​​are obtained: $F_1=Рsin($$(\mathbf \alpha )$$\pm$$(\mathbf \varphi )$) /cos$(\mathbf \varphi )$; $F_2=Рtg($$(\mathbf \alpha )$$\pm$$(\mathbf \varphi )$)

The plus sign refers to upward movement, the minus sign to lowering the load. Inclined plane efficiency $(\mathbf \eta )$1=sin$(\mathbf \alpha )$cos$(\mathbf \alpha )$/sin($(\mathbf \alpha )$+$(\mathbf \varphi )$), if the force $P$ is directed parallel to the plane, and $(\mathbf \eta )$2=tg$(\mathbf \alpha )$/tg($(\mathbf \alpha )$+$(\mathbf \varphi )$), if the force $P$ is directed parallel to the base of the inclined plane.

The inclined plane obeys the “golden rule of mechanics.” The smaller the angle between the surface and the inclined plane (i.e., the flatter it is, not steeply rising), the less force must be applied to lift the load, but the greater the distance will need to be overcome.

In the absence of friction forces, the gain in force is $K = P/F = 1/sin$$\alpha = l/h$. In real conditions, due to the action of friction, the efficiency of the inclined plane is less than 1, the gain in force is less than the ratio $l/h$.

Example 1

A load weighing 40 kg is lifted along an inclined plane to a height of 10 m while applying a force of 200 N (Fig. 3). What is the length of the inclined plane? Ignore friction.

$(\mathbf \eta )$ = 1

When a body moves along an inclined plane, the ratio of the applied force to the weight of the body is equal to the ratio of the length of the inclined plane to its height: $\frac(F)(P)=\frac(l)(h)=\frac(1)((sin (\ mathbf \alpha )\ ))$. Therefore, $l=\frac(Fh)(mg)=\ \frac(200\cdot 10)(40\cdot 9.8)=5.1\ m$.

Answer: The length of the inclined plane is 5.1 m

Example 2

Two bodies with masses $m_1$ = 10 g and $m_2$ = 15 g are connected by a thread thrown over a stationary block installed on an inclined plane (Fig. 4). The plane makes an angle $\alpha $ = 30$()^\circ$ with the horizon. Find the acceleration with which these bodies will move.

$(\mathbf \alpha )$ = 30 degrees

$g$ = 9.8 $m/s_2$

Let's direct the OX axis along the inclined plane, and the OY axis perpendicular to it, and project the vectors $\(\overrightarrow(P))_1\ and\(\overrightarrow(P))_2$ onto these axes. As can be seen from the figure, the resultant of the forces applied to each of the bodies is equal to the difference in the projections of the vectors $\(\overrightarrow(P))_1\ and\(\overrightarrow(P))_2$ onto the OX axis:

\[\left|\overrightarrow(R)\right|=\left|P_(2x)-P_(1x)\right|=\left|m_2g(sin \alpha \ )-m_1g(sin \alpha \ )\right |=g(sin \alpha \left|m_2-m_1\right|\ )\] \[\left|\overrightarrow(R)\right|=9.8\cdot (sin 30()^\circ \ )\cdot \ left|0.015-0.01\right|=0.0245\ H\]\

Answer: Acceleration of bodies $a_1=2.45\frac(m)(s^2);\ \ \ \ \ \ a_2=1.63\ m/s^2$

In addition to the lever and 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 about half a kilometer long.