How is mechanical work defined? Mechanical work is not what you think

1. From the 7th grade physics course, you know that if a force acts on a body and it moves in the direction of the force, then the force does mechanical work A, equal to the product of the modulus of force and the modulus of displacement:

A=fs.

SI unit of work - joule (1 J).

[A] = [F][s] = 1 H 1 m = 1 N m = 1 J.

The unit of work is the work done by the force. 1 N on a way 1m.

It follows from the formula that mechanical work is not performed if the force is zero (the body is at rest or moves uniformly and rectilinearly) or the displacement is zero.

Suppose that the force vector acting on the body makes some angle a with the displacement vector (Fig. 65). Since the body does not move in the vertical direction, the force projection Fy per axle Y does not do work, but the projection of force Fx per axle X does work equal to A = F x s x.

Because the Fx = F cos a, and s x= s, then

A = fs cos a.

In this way,

the work of a constant force is equal to the product of the modules of the vectors of force and displacement and the cosine of the angle between these vectors.

2. Let's analyze the resulting work formula.

If angle a = 0°, then cos 0° = 1 and A = fs. The work done is positive and its value is maximum if the direction of the force coincides with the direction of displacement.

If angle a = 90°, then cos 90° = 0 and A= 0. The force does not do work if it is perpendicular to the direction of movement of the body. Thus, the work of gravity is zero when a body moves along a horizontal plane. Zero is equal to the work of the force imparting centripetal acceleration to the body during its uniform motion in a circle, since this force at any point of the trajectory is perpendicular to the direction of motion of the body.

If angle a = 180°, then cos 180° = –1 and A = –fs. This case occurs when force and displacement are directed in opposite directions. Accordingly, the work done is negative and its value is maximum. Negative work is done, for example, by the force of sliding friction, since it is directed in the direction opposite to the direction of movement of the body.

If the angle a between the force and displacement vectors is acute, then the work is positive; if angle a is obtuse, then the work is negative.

3. We get the formula for calculating the work of gravity. Let the body mass m falls freely to the ground from a point A at the height h relative to the surface of the Earth, and after a while it turns out to be at a point B(Fig. 66, a). The work done by gravity is equal to

A = fs = mgh.

In this case, the direction of motion of the body coincides with the direction of the force acting on it, so the work of gravity in free fall is positive.

If a body moves vertically upward from a point B exactly A(Fig. 66, b), then its movement is directed in the direction opposite to gravity, and the work of gravity is negative:

A= –mgh

4. The work done by a force can be calculated using a force versus displacement graph.

Suppose a body moves under the influence of a constant force of gravity. Plot of the modulus of gravity F cord from the body movement module s is a straight line parallel to the x-axis (Fig. 67). Find the area of ​​the selected rectangle. It is equal to the product of its two sides: S = F heavy h = mgh. On the other hand, the same value is equal to the work of gravity A = mgh.

Thus, the work is numerically equal to the area of ​​the rectangle bounded by the graph, the coordinate axes and the perpendicular raised to the x-axis at the point h.

Consider now the case when the force acting on the body is directly proportional to the displacement. Such a force, as is known, is the force of elasticity. Its modulus is F extr = k D l, where D l- lengthening of the body.

Suppose a spring, the left end of which is fixed, was compressed (Fig. 68, a). At the same time, its right end shifted to D l 1. An elastic force has arisen in the spring F control 1, directed to the right.

If we now leave the spring to itself, then its right end will move to the right (Fig. 68, b), the elongation of the spring will be equal to D l 2, and the elastic force F exercise 2.

Calculate the work of the elastic force when moving the end of the spring from the point with coordinate D l 1 to the point with coordinate D l 2. For this we use the dependency graph F control (D l) (Fig. 69). The work of the elastic force is numerically equal to the area of ​​the trapezoid ABCD. The area of ​​a trapezoid is equal to the product of half the sum of the bases and the height, i.e. S = AD. in a trapeze ABCD grounds AB = F ex 2 = k D l 2 , CD= F ex 1 = k D l 1 and the height AD= D l 1-D l 2. Substitute these quantities into the formula for the area of ​​a trapezoid:

S= (D l 1-D l 2) =– .

Thus, we have obtained that the work of the elastic force is equal to:

A =– .

5 * . Let us assume that a body of mass m moving from point A exactly B(Fig. 70), moving first without friction along an inclined plane from the point A exactly C, and then without friction along the horizontal plane from the point C exactly B. The work of gravity on the site CB is zero because the force of gravity is perpendicular to the displacement. When moving on an inclined plane, the work done by gravity is:

A AC = F heavy l sin a. Because l sin a = h, then A AC = Ft heavy h = mgh.

The work of gravity when a body moves along a trajectory ACB is equal to A ACB = A AC + A CB = mgh + 0.

In this way, A ACB = mgh.

The result obtained shows that the work of gravity does not depend on the shape of the trajectory. It depends only on the initial and final positions of the body.

Let us now assume that the body moves along a closed trajectory ABCA(see fig. 70). When moving a body from a point A exactly B along the trajectory ACB the work done by gravity is A ACB = mgh. When moving a body from a point B exactly A gravity does negative work, which is equal to A BA = –mgh. Then the work of gravity on a closed trajectory A = A ACB + A BA = 0.

The work of the elastic force on a closed trajectory is also equal to zero. Indeed, suppose that a spring that was not deformed at the beginning was stretched and its length increased by D l. The elastic force does work A 1 = . When returning to a state of equilibrium, the elastic force does work A 2 = . The total work of the elastic force during the stretching of the spring and its return to the undeformed state is zero.

6. The work of the force of gravity and the force of elasticity on a closed trajectory is equal to zero.

Forces whose work on any closed trajectory is equal to zero (or does not depend on the shape of the trajectory) are called conservative.

Forces whose work depends on the shape of the trajectory are called non-conservative.

Friction force is non-conservative. For example, a body moves from a point 1 exactly 2 straight ahead first 12 (Fig. 71), and then along a broken line 132 . On each section of the trajectory, the friction force is the same. In the first case, the work of the friction force

A 12 = –F tr l 1 ,

and in the second -

A 132 = A 13 + A 32, A 132 = –F tr l 2 – F tr l 3 .

From here A 12A 132.

7. From the 7th grade physics course, you know that an important characteristic of devices that do work is power.

Power is a physical quantity equal to the ratio of work to the period of time for which it is done:

N = .

Power characterizes the speed of doing work.

Unit of power in SI - watt (1 W).

[N] === 1 W.

The unit of power is the power at which the work 1 J committed for 1 s .

Questions for self-examination

1. What is called work? What is the unit of work?

2. When does a force do negative work? positive work?

3. What is the formula for calculating the work of gravity? elastic force?

5. What forces are called conservative; non-conservative?

6 * . Prove that the work done by the force of gravity and the force of elasticity does not depend on the shape of the trajectory.

7. What is called power? What is the unit of power?

Task 18

1. A boy weighing 20 kg is pulled evenly on a sled, applying a force of 20 N. The rope, by which the sled is pulled, makes an angle of 30 ° with the horizon. What is the work of the elastic force arising in the rope if the sled moved 100 m?

2. An athlete weighing 65 kg jumps into the water from a tower located at a height of 3 m above the surface of the water. What work is done by the force of gravity acting on the athlete as he moves to the surface of the water?

3. Under the action of an elastic force, the length of a deformed spring with a stiffness of 200 N / m decreased by 4 cm. What is the work of the elastic force?

4 * . Prove that the work of a variable force is numerically equal to the area of ​​the figure bounded by the force-coordinate graph and the coordinate axes.

5. What is the traction force of a car engine if, at a constant speed of 108 km/h, it develops a power of 55 kW?

Let the body, on which the force acts, pass, moving along a certain trajectory, the path s. In this case, the force either changes the speed of the body, imparting acceleration to it, or compensates for the action of another force (or forces) that opposes the movement. The action on the path s is characterized by a quantity called work.

Mechanical work is a scalar quantity equal to the product of the projection of the force on the direction of movement Fs and the path s traversed by the point of application of the force (Fig. 22):

A = Fs*s.(56)

Expression (56) is valid if the value of the projection of the force Fs on the direction of movement (i.e., on the direction of speed) remains unchanged all the time. In particular, this occurs when the body moves in a straight line and a force of constant magnitude forms a constant angle α with the direction of motion. Since Fs = F * cos(α), expression (47) can be given the following form:

A = F*s*cos(α).

If is a displacement vector, then the work is calculated as the scalar product of two vectors and :

. (57)

Work is an algebraic quantity. If the force and direction of movement form an acute angle (cos(α) > 0), the work is positive. If the angle α is obtuse (cos(α)< 0), работа отрицательна. При α = π/2 работа равна нулю. Последнее обстоятельство особенно отчетливо показывает, что понятие работы в механике существенно отличается от обыденного представления о работе. В обыденном понимании всякое усилие, в частности и мускульное напряжение, всегда сопровождается совершением работы. Например, для того чтобы держать тяжелый груз, стоя неподвижно, а тем более для того, чтобы перенести этот груз по горизонтальному пути, носильщик затрачивает много усилий, т. е. «совершает работу». Однако это – «физиологическая» работа. Механическая работа в этих случаях равна нулю.

Work when moving under the influence of force

If the magnitude of the projection of the force on the direction of movement does not remain constant during movement, then the work is expressed as an integral:

. (58)

An integral of this kind in mathematics is called a curvilinear integral along the trajectory S. The argument here is a vector variable , which can vary both in absolute value and in direction. Under the integral sign is the scalar product of the force vector and the elementary displacement vector.

A unit of work is the work done by a force equal to one and acting in the direction of movement, on a path equal to one. In SI The unit of work is the joule (J), which is equal to the work done by a force of 1 newton in a path of 1 meter:

1J = 1N * 1m.

In the CGS, the unit of work is the erg, which is equal to the work done by a force of 1 dyne in a path of 1 centimeter. 1J = 10 7 erg.

Sometimes a non-systemic unit kilogrammeter (kg * m) is used. This is the work done by a force of 1 kg on a path of 1 meter. 1kg*m = 9.81 J.

What does it mean?

In physics, "mechanical work" is the work of some force (gravity, elasticity, friction, etc.) on the body, as a result of which the body moves.

Often the word "mechanical" is simply not spelled.
Sometimes you can find the expression "the body has done work", which basically means "the force acting on the body has done the work."

I think - I'm working.

I go - I also work.

Where is the mechanical work here?

If a body moves under the action of a force, then mechanical work is done.

The body is said to do work.
More precisely, it will be like this: the work is done by the force acting on the body.

Work characterizes the result of the action of a force.

The forces acting on a person do mechanical work on him, and as a result of the action of these forces, the person moves.

Work is a physical quantity equal to the product of the force acting on the body and the path taken by the body under the action of the force in the direction of this force.

A - mechanical work,
F - strength,
S - the distance traveled.

Work is done, if 2 conditions are met simultaneously: a force acts on the body and it
moves in the direction of the force.

Work is not done(i.e. equal to 0) if:
1. The force acts, but the body does not move.

For example: we act with force on a stone, but we cannot move it.

2. The body moves, and the force is equal to zero, or all forces are compensated (ie, the resultant of these forces is equal to 0).
For example: when moving by inertia, no work is done.
3. The direction of the force and the direction of motion of the body are mutually perpendicular.

For example: when a train moves horizontally, gravity does no work.

Work can be positive or negative.

1. If the direction of the force and the direction of motion of the body are the same, positive work is done.

For example: gravity, acting on a drop of water falling down, does positive work.

2. If the direction of the force and the movement of the body are opposite, negative work is done.

For example: the force of gravity acting on a rising balloon does negative work.

If several forces act on a body, then the total work of all forces is equal to the work of the resulting force.

Units of work

In honor of the English scientist D. Joule, the unit of work was named 1 Joule.

In the international system of units (SI):
[A] = J = N m
1J = 1N 1m

Mechanical work is equal to 1 J if, under the influence of a force of 1 N, the body moves 1 m in the direction of this force.

When flying from the thumb of a person to the index
a mosquito does work - 0,000,000,000,000,000,000,000,000,001 J.

The human heart performs approximately 1 J of work in one contraction, which corresponds to the work done when lifting a load of 10 kg to a height of 1 cm.

TO WORK, FRIENDS!

Mechanical work. Units of work.

In everyday life, under the concept of "work" we understand everything.

In physics, the concept Work somewhat different. This is a certain physical quantity, which means that it can be measured. In physics, the study is primarily mechanical work .

Consider examples of mechanical work.

The train moves under the action of the traction force of the electric locomotive, while doing mechanical work. When a gun is fired, the pressure force of the powder gases does work - it moves the bullet along the barrel, while the speed of the bullet increases.

From these examples, it can be seen that mechanical work is done when the body moves under the action of a force. Mechanical work is also performed in the case when the force acting on the body (for example, the friction force) reduces the speed of its movement.

Wanting to move the cabinet, we press on it with force, but if it does not move at the same time, then we do not perform mechanical work. One can imagine the case when the body moves without the participation of forces (by inertia), in this case, mechanical work is also not performed.

So, mechanical work is done only when a force acts on the body and it moves .

It is easy to understand that the greater the force acting on the body and the longer the path that the body passes under the action of this force, the greater the work done.

Mechanical work is directly proportional to the applied force and directly proportional to the distance traveled. .

Therefore, we agreed to measure mechanical work by the product of force and the path traveled in this direction of this force:

work = force × path

where BUT- Work, F- strength and s- distance traveled.

A unit of work is the work done by a force of 1 N on a path of 1 m.

Unit of work - joule (J ) is named after the English scientist Joule. In this way,

1 J = 1N m.

Also used kilojoules (kJ) .

1 kJ = 1000 J.

Formula A = Fs applicable when the power F is constant and coincides with the direction of motion of the body.

If the direction of the force coincides with the direction of motion of the body, then this force does positive work.

If the motion of the body occurs in the direction opposite to the direction of the applied force, for example, the sliding friction force, then this force does negative work.

If the direction of the force acting on the body is perpendicular to the direction of motion, then this force does no work, the work is zero:

In the future, speaking of mechanical work, we will briefly call it in one word - work.

Example. Calculate the work done when lifting a granite slab with a volume of 0.5 m3 to a height of 20 m. The density of granite is 2500 kg / m 3.

Given:

ρ \u003d 2500 kg / m 3

Solution:

where F is the force that must be applied to evenly lift the plate up. This force is equal in modulus to the force of the strand Fstrand acting on the plate, i.e. F = Fstrand. And the force of gravity can be determined by the mass of the plate: Ftyazh = gm. We calculate the mass of the slab, knowing its volume and density of granite: m = ρV; s = h, i.e. the path is equal to the height of the ascent.

So, m = 2500 kg/m3 0.5 m3 = 1250 kg.

F = 9.8 N/kg 1250 kg ≈ 12250 N.

A = 12,250 N 20 m = 245,000 J = 245 kJ.

Answer: A = 245 kJ.

Levers.Power.Energy

Different engines take different times to do the same work. For example, a crane at a construction site lifts hundreds of bricks to the top floor of a building in a few minutes. If a worker were to move these bricks, it would take him several hours to do this. Another example. A horse can plow a hectare of land in 10-12 hours, while a tractor with a multi-share plow ( ploughshare- part of the plow that cuts the layer of earth from below and transfers it to the dump; multi-share - a lot of shares), this work will be done for 40-50 minutes.

It is clear that a crane does the same work faster than a worker, and a tractor faster than a horse. The speed of work is characterized by a special value called power.

Power is equal to the ratio of work to the time for which it was completed.

To calculate the power, it is necessary to divide the work by the time during which this work is done. power = work / time.

where N- power, A- Work, t- time of work done.

Power is a constant value, when the same work is done for every second, in other cases the ratio A/t determines the average power:

N cf = A/t . The unit of power was taken as the power at which work in J is done in 1 s.

This unit is called the watt ( Tue) in honor of another English scientist Watt.

1 watt = 1 joule/ 1 second, or 1 W = 1 J/s.

Watt (joule per second) - W (1 J / s).

Larger units of power are widely used in engineering - kilowatt (kW), megawatt (MW) .

1 MW = 1,000,000 W

1 kW = 1000 W

1 mW = 0.001 W

1 W = 0.000001 MW

1 W = 0.001 kW

1 W = 1000 mW

Example. Find the power of the flow of water flowing through the dam, if the height of the water fall is 25 m, and its flow rate is 120 m3 per minute.

Given:

ρ = 1000 kg/m3

Solution:

Mass of falling water: m = ρV,

m = 1000 kg/m3 120 m3 = 120,000 kg (12 104 kg).

The force of gravity acting on water:

F = 9.8 m/s2 120,000 kg ≈ 1,200,000 N (12 105 N)

Work done per minute:

A - 1,200,000 N 25 m = 30,000,000 J (3 107 J).

Flow power: N = A/t,

N = 30,000,000 J / 60 s = 500,000 W = 0.5 MW.

Answer: N = 0.5 MW.

Various engines have capacities ranging from hundredths and tenths of a kilowatt (motor of an electric razor, sewing machine) to hundreds of thousands of kilowatts (water and steam turbines).

Table 5

Power of some engines, kW.

Each engine has a plate (engine passport), which contains some data about the engine, including its power.

Human power under normal working conditions is on average 70-80 watts. Making jumps, running up the stairs, a person can develop power up to 730 watts, and in some cases even more.

From the formula N = A/t it follows that

To calculate the work, you need to multiply the power by the time during which this work was done.

Example. The room fan motor has a power of 35 watts. How much work does he do in 10 minutes?

Let's write down the condition of the problem and solve it.

Given:

Solution:

A = 35 W * 600 s = 21,000 W * s = 21,000 J = 21 kJ.

Answer A= 21 kJ.

simple mechanisms.

Since time immemorial, man has been using various devices to perform mechanical work.

Everyone knows that a heavy object (stone, cabinet, machine), which cannot be moved by hand, can be moved with a fairly long stick - a lever.

At the moment, it is believed that with the help of levers three thousand years ago, during the construction of the pyramids in ancient Egypt, heavy stone slabs were moved and raised to a great height.

In many cases, instead of lifting a heavy load to a certain height, it can be rolled or pulled to the same height on an inclined plane or lifted with blocks.

Devices used to transform power are called mechanisms .

Simple mechanisms include: levers and its varieties - block, gate; inclined plane and its varieties - wedge, screw. In most cases, simple mechanisms are used in order to obtain a gain in strength, i.e., to increase the force acting on the body by several times.

Simple mechanisms are found both in household and in all complex factory and factory machines that cut, twist and stamp large sheets of steel or draw the finest threads from which fabrics are then made. The same mechanisms can be found in modern complex automata, printing and counting machines.

Lever arm. The balance of forces on the lever.

Consider the simplest and most common mechanism - the lever.

The lever is a rigid body that can rotate around a fixed support.

The figures show how a worker uses a crowbar to lift a load as a lever. In the first case, a worker with a force F presses the end of the crowbar B, in the second - raises the end B.

The worker needs to overcome the weight of the load P- force directed vertically downwards. For this, he rotates the crowbar around an axis passing through the only motionless breaking point - its fulcrum O. Strength F, with which the worker acts on the lever, less force P, so the worker gets gain in strength. With the help of a lever, you can lift such a heavy load that you cannot lift it on your own.

The figure shows a lever whose axis of rotation is O(fulcrum) is located between the points of application of forces BUT and AT. The other figure shows a diagram of this lever. Both forces F 1 and F 2 acting on the lever are directed in the same direction.

The shortest distance between the fulcrum and the straight line along which the force acts on the lever is called the arm of the force.

To find the shoulder of the force, it is necessary to lower the perpendicular from the fulcrum to the line of action of the force.

The length of this perpendicular will be the shoulder of this force. The figure shows that OA- shoulder strength F 1; OV- shoulder strength F 2. The forces acting on the lever can rotate it around the axis in two directions: clockwise or counterclockwise. Yes, power F 1 rotates the lever clockwise, and the force F 2 rotates it counterclockwise.

The condition under which the lever is in equilibrium under the action of forces applied to it can be established experimentally. At the same time, it must be remembered that the result of the action of a force depends not only on its numerical value (modulus), but also on the point at which it is applied to the body, or how it is directed.

Various weights are suspended from the lever (see Fig.) on both sides of the fulcrum so that each time the lever remains in balance. The forces acting on the lever are equal to the weights of these loads. For each case, the modules of forces and their shoulders are measured. From the experience shown in Figure 154, it can be seen that the force 2 H balances power 4 H. In this case, as can be seen from the figure, the shoulder of lesser force is 2 times larger than the shoulder of greater force.

On the basis of such experiments, the condition (rule) of the balance of the lever was established.

The lever is in equilibrium when the forces acting on it are inversely proportional to the shoulders of these forces.

This rule can be written as a formula:

F 1/F 2 = l 2/ l 1 ,

where F 1and F 2 - forces acting on the lever, l 1and l 2 , - the shoulders of these forces (see Fig.).

The rule for the balance of the lever was established by Archimedes around 287-212. BC e. (But didn't the last paragraph say that the levers were used by the Egyptians? Or is the word "established" important here?)

It follows from this rule that a smaller force can be balanced with a leverage of a larger force. Let one arm of the lever be 3 times larger than the other (see Fig.). Then, applying a force of, for example, 400 N at point B, it is possible to lift a stone weighing 1200 N. In order to lift an even heavier load, it is necessary to increase the length of the lever arm on which the worker acts.

Example. Using a lever, a worker lifts a slab weighing 240 kg (see Fig. 149). What force does he apply to the larger arm of the lever, which is 2.4 m, if the smaller arm is 0.6 m?

Let's write down the condition of the problem, and solve it.

Given:

Solution:

According to the lever equilibrium rule, F1/F2 = l2/l1, whence F1 = F2 l2/l1, where F2 = P is the weight of the stone. Stone weight asd = gm, F = 9.8 N 240 kg ≈ 2400 N

Then, F1 = 2400 N 0.6 / 2.4 = 600 N.

Answer: F1 = 600 N.

In our example, the worker overcomes a force of 2400 N by applying a force of 600 N to the lever. But at the same time, the arm on which the worker acts is 4 times longer than that on which the weight of the stone acts ( l 1 : l 2 = 2.4 m: 0.6 m = 4).

By applying the rule of leverage, a smaller force can balance a larger force. In this case, the shoulder of the smaller force must be longer than the shoulder of the greater force.

Moment of power.

You already know the lever balance rule:

F 1 / F 2 = l 2 / l 1 ,

Using the property of proportion (the product of its extreme terms is equal to the product of its middle terms), we write it in this form:

F 1l 1 = F 2 l 2 .

On the left side of the equation is the product of the force F 1 on her shoulder l 1, and on the right - the product of the force F 2 on her shoulder l 2 .

The product of the modulus of the force rotating the body and its arm is called moment of force; it is denoted by the letter M. So,

A lever is in equilibrium under the action of two forces if the moment of force rotating it clockwise is equal to the moment of force rotating it counterclockwise.

This rule is called moment rule , can be written as a formula:

M1 = M2

Indeed, in the experiment we have considered (§ 56), the acting forces were equal to 2 N and 4 N, their shoulders respectively amounted to 4 and 2 lever pressures, i.e., the moments of these forces are the same when the lever is in equilibrium.

The moment of force, like any physical quantity, can be measured. A moment of force of 1 N is taken as a unit of moment of force, the shoulder of which is exactly 1 m.

This unit is called newton meter (N m).

The moment of force characterizes the action of the force, and shows that it depends simultaneously on the modulus of the force and on its shoulder. Indeed, we already know, for example, that the effect of a force on a door depends both on the modulus of the force and on where the force is applied. The door is easier to turn, the farther from the axis of rotation the force acting on it is applied. It is better to unscrew the nut with a long wrench than with a short one. The easier it is to lift a bucket from the well, the longer the handle of the gate, etc.

Levers in technology, everyday life and nature.

The lever rule (or the rule of moments) underlies the action of various kinds of tools and devices used in technology and everyday life where gain in strength or on the road is required.

We have a gain in strength when working with scissors. Scissors - it's a lever(rice), the axis of rotation of which occurs through a screw connecting both halves of the scissors. acting force F 1 is the muscular strength of the hand of the person squeezing the scissors. Opposing force F 2 - the resistance force of such a material that is cut with scissors. Depending on the purpose of the scissors, their device is different. Office scissors, designed for cutting paper, have long blades and almost the same length handles. It does not require much force to cut the paper, and it is more convenient to cut in a straight line with a long blade. Shears for cutting sheet metal (Fig.) have handles much longer than the blades, since the resistance force of the metal is large and to balance it, the arm of the acting force must be significantly increased. Even more difference between the length of the handles and the distance of the cutting part and the axis of rotation in wire cutters(Fig.), Designed for wire cutting.

Levers of various types are available on many machines. A sewing machine handle, bicycle pedals or hand brakes, car and tractor pedals, piano keys are all examples of the levers used in these machines and tools.

Examples of the use of levers are the handles of vices and workbenches, the lever of a drilling machine, etc.

The action of lever balances is also based on the principle of the lever (Fig.). The training scale shown in figure 48 (p. 42) acts as equal-arm lever . AT decimal scales the arm to which the cup with weights is suspended is 10 times longer than the arm carrying the load. This greatly simplifies the weighing of large loads. When weighing a load on a decimal scale, multiply the weight of the weights by 10.

The device of scales for weighing freight wagons of cars is also based on the rule of the lever.

Levers are also found in different parts of the body of animals and humans. These are, for example, arms, legs, jaws. Many levers can be found in the body of insects (having read a book about insects and the structure of their body), birds, in the structure of plants.

Application of the law of balance of the lever to the block.

Block It is a wheel with a chute, fixed in a holder. A rope, cable or chain is passed along the gutter of the block.

Fixed block such a block is called, the axis of which is fixed, and when lifting loads it does not rise and does not fall (Fig.

A fixed block can be considered as an equal-arm lever, in which the arms of forces are equal to the radius of the wheel (Fig.): OA = OB = r. Such a block does not give a gain in strength. ( F 1 = F 2), but allows you to change the direction of the force. Movable block is a block. the axis of which rises and falls along with the load (Fig.). The figure shows the corresponding lever: O- fulcrum of the lever, OA- shoulder strength R and OV- shoulder strength F. Since the shoulder OV 2 times the shoulder OA, then the force F 2 times less power R:

F = P/2 .

In this way, movable block gives a gain in strength of 2 times .

This can also be proved using the concept of moment of force. When the block is in equilibrium, the moments of forces F and R are equal to each other. But the shoulder of strength F 2 times more leverage R, which means that the force itself F 2 times less power R.

Usually, in practice, a combination of a fixed block with a movable one is used (Fig.). The fixed block is used for convenience only. It does not give a gain in strength, but changes the direction of the force. For example, it allows you to lift a load while standing on the ground. It comes in handy for many people or workers. However, it gives a power gain of 2 times more than usual!

Equality of work when using simple mechanisms. The "golden rule" of mechanics.

The simple mechanisms we have considered are used in the performance of work in those cases when it is necessary to balance another force by the action of one force.

Naturally, the question arises: giving a gain in strength or path, do not simple mechanisms give a gain in work? The answer to this question can be obtained from experience.

Having balanced on the lever two forces of different modulus F 1 and F 2 (fig.), set the lever in motion. It turns out that for the same time, the point of application of a smaller force F 2 goes a long way s 2, and the point of application of greater force F 1 - smaller path s 1. Having measured these paths and force modules, we find that the paths traversed by the points of application of forces on the lever are inversely proportional to the forces:

s 1 / s 2 = F 2 / F 1.

Thus, acting on the long arm of the lever, we win in strength, but at the same time we lose the same amount on the way.

Product of force F on the way s there is work. Our experiments show that the work done by the forces applied to the lever are equal to each other:

F 1 s 1 = F 2 s 2, i.e. BUT 1 = BUT 2.

So, when using the leverage, the win in the work will not work.

By using the lever, we can win either in strength or in distance. Acting by force on the short arm of the lever, we gain in distance, but lose in strength by the same amount.

There is a legend that Archimedes, delighted with the discovery of the rule of the lever, exclaimed: "Give me a fulcrum, and I will turn the Earth!".

Of course, Archimedes could not cope with such a task, even if he were given a fulcrum (which would have to be outside the Earth) and a lever of the required length.

To raise the earth by only 1 cm, the long arm of the lever would have to describe an arc of enormous length. It would take millions of years to move the long end of the lever along this path, for example, at a speed of 1 m/s!

Does not give a gain in work and a fixed block, which is easy to verify by experience (see Fig.). Paths traversed by points of application of forces F and F, are the same, the forces are the same, and therefore the work is the same.

It is possible to measure and compare with each other the work done with the help of a movable block. In order to lift the load to height h with the help of a movable block, it is necessary to move the end of the rope to which the dynamometer is attached, as experience shows (Fig.), to a height of 2h.

In this way, getting a gain in strength by 2 times, they lose 2 times on the way, therefore, the movable block does not give a gain in work.

Centuries of practice has shown that none of the mechanisms gives a gain in work. Various mechanisms are used in order to win in strength or on the way, depending on the working conditions.

Already ancient scientists knew the rule applicable to all mechanisms: how many times we win in strength, how many times we lose in distance. This rule has been called the "golden rule" of mechanics.

The efficiency of the mechanism.

Considering the device and action of the lever, we did not take into account friction, as well as the weight of the lever. under these ideal conditions, the work done by the applied force (we will call this work complete), is equal to useful lifting loads or overcoming any resistance.

In practice, the total work done by the mechanism is always somewhat greater than the useful work.

Part of the work is done against the friction force in the mechanism and by moving its individual parts. So, using a movable block, you have to additionally perform work on lifting the block itself, the rope and determining the friction force in the axis of the block.

Whatever mechanism we choose, the useful work accomplished with its help is always only a part of the total work. So, denoting the useful work by the letter Ap, the full (spent) work by the letter Az, we can write:

Up< Аз или Ап / Аз < 1.

The ratio of useful work to total work is called the efficiency of the mechanism.

Efficiency is abbreviated as efficiency.

Efficiency = Ap / Az.

Efficiency is usually expressed as a percentage and denoted by the Greek letter η, it is read as "this":

η \u003d Ap / Az 100%.

Example: A 100 kg mass is suspended from the short arm of the lever. To lift it, a force of 250 N was applied to the long arm. The load was lifted to a height h1 = 0.08 m, while the point of application of the driving force fell to a height h2 = 0.4 m. Find the efficiency of the lever.

Let's write down the condition of the problem and solve it.

Given :

Solution :

η \u003d Ap / Az 100%.

Full (spent) work Az = Fh2.

Useful work Аn = Рh1

P \u003d 9.8 100 kg ≈ 1000 N.

Ap \u003d 1000 N 0.08 \u003d 80 J.

Az \u003d 250 N 0.4 m \u003d 100 J.

η = 80 J/100 J 100% = 80%.

Answer : η = 80%.

But the "golden rule" is fulfilled in this case too. Part of the useful work - 20% of it - is spent on overcoming friction in the axis of the lever and air resistance, as well as on the movement of the lever itself.

The efficiency of any mechanism is always less than 100%. By designing mechanisms, people tend to increase their efficiency. To do this, friction in the axes of the mechanisms and their weight are reduced.

Energy.

In factories and factories, machines and machines are driven by electric motors, which consume electrical energy (hence the name).

A compressed spring (rice), straightening out, does work, lifts a load to a height, or makes a cart move. An immovable load raised above the ground does not do work, but if this load falls, it can do work (for example, it can drive a pile into the ground). Every moving body has the ability to do work. So, a steel ball A (rice) rolled down from an inclined plane, hitting a wooden block B, moves it a certain distance. In doing so, work is being done. If a body or several interacting bodies (a system of bodies) can do work, it is said that they have energy. Energy - a physical quantity showing what work a body (or several bodies) can do. Energy is expressed in the SI system in the same units as work, i.e. in joules. The more work a body can do, the more energy it has. When work is done, the energy of bodies changes. The work done is equal to the change in energy. Potential and kinetic energy. Potential (from lat. potency - possibility) energy is called energy, which is determined by the mutual position of interacting bodies and parts of the same body. Potential energy, for example, has a body raised relative to the surface of the Earth, because the energy depends on the relative position of it and the Earth. and their mutual attraction. If we consider the potential energy of a body lying on the Earth to be equal to zero, then the potential energy of a body raised to a certain height will be determined by the work done by gravity when the body falls to the Earth. Denote the potential energy of the body E n because E = A, and the work, as we know, is equal to the product of the force and the path, then A = Fh, where F- gravity. Hence, the potential energy En is equal to: E = Fh, or E = gmh, where g- acceleration of gravity, m- body mass, h- the height to which the body is raised. The water in the rivers held by dams has a huge potential energy. Falling down, the water does work, setting in motion the powerful turbines of power plants. The potential energy of a copra hammer (Fig.) is used in construction to perform the work of driving piles. By opening a door with a spring, work is done to stretch (or compress) the spring. Due to the acquired energy, the spring, contracting (or straightening), does the work, closing the door. The energy of compressed and untwisted springs is used, for example, in wrist watches, various clockwork toys, etc. Any elastic deformed body possesses potential energy. The potential energy of compressed gas is used in the operation of heat engines, in jackhammers, which are widely used in the mining industry, in the construction of roads, excavation of solid soil, etc. The energy possessed by a body as a result of its movement is called kinetic (from the Greek. cinema - movement) energy. The kinetic energy of a body is denoted by the letter E to. Moving water, driving the turbines of hydroelectric power plants, expends its kinetic energy and does work. Moving air also has kinetic energy - the wind. What does kinetic energy depend on? Let's turn to experience (see Fig.). If you roll ball A from different heights, you will notice that the higher the ball rolls from, the greater its speed and the farther it advances the bar, i.e., it does more work. This means that the kinetic energy of a body depends on its speed. Due to the speed, a flying bullet has a large kinetic energy. The kinetic energy of a body also depends on its mass. Let's do our experiment again, but we will roll another ball - a larger mass - from an inclined plane. Block B will move further, i.e. more work will be done. This means that the kinetic energy of the second ball is greater than the first. The greater the mass of the body and the speed with which it moves, the greater its kinetic energy. In order to determine the kinetic energy of a body, the formula is applied: Ek \u003d mv ^ 2 / 2, where m- body mass, v is the speed of the body. The kinetic energy of bodies is used in technology. The water retained by the dam has, as already mentioned, a large potential energy. When falling from a dam, water moves and has the same large kinetic energy. It drives a turbine connected to an electric current generator. Due to the kinetic energy of water, electrical energy is generated. The energy of moving water is of great importance in the national economy. This energy is used by powerful hydroelectric power plants. The energy of falling water is an environmentally friendly source of energy, unlike fuel energy. All bodies in nature, relative to the conditional zero value, have either potential or kinetic energy, and sometimes both. For example, a flying plane has both kinetic and potential energy relative to the Earth. We got acquainted with two types of mechanical energy. Other types of energy (electrical, internal, etc.) will be considered in other sections of the physics course. The transformation of one type of mechanical energy into another. The phenomenon of the transformation of one type of mechanical energy into another is very convenient to observe on the device shown in the figure. Winding the thread around the axis, raise the disk of the device. The disk raised up has some potential energy. If you let it go, it will spin and fall. As it falls, the potential energy of the disk decreases, but at the same time its kinetic energy increases. At the end of the fall, the disk has such a reserve of kinetic energy that it can again rise almost to its previous height. (Part of the energy is expended working against the force of friction, so the disk does not reach its original height.) Having risen up, the disk falls again, and then rises again. In this experiment, when the disk moves down, its potential energy is converted into kinetic energy, and when moving up, kinetic energy is converted into potential. The transformation of energy from one type to another also occurs when two elastic bodies hit, for example, a rubber ball on the floor or a steel ball on a steel plate. If you lift a steel ball (rice) over a steel plate and release it from your hands, it will fall. As the ball falls, its potential energy decreases, and its kinetic energy increases, as the speed of the ball increases. When the ball hits the plate, both the ball and the plate will be compressed. The kinetic energy that the ball possessed will turn into the potential energy of the compressed plate and the compressed ball. Then, due to the action of elastic forces, the plate and the ball will take their original shape. The ball will bounce off the plate, and their potential energy will again turn into the kinetic energy of the ball: the ball will bounce upward with a speed almost equal to the speed that it had at the moment of impact on the plate. As the ball rises, the speed of the ball, and hence its kinetic energy, decreases, and the potential energy increases. bouncing off the plate, the ball rises to almost the same height from which it began to fall. At the top of the ascent, all its kinetic energy will again turn into potential energy. Natural phenomena are usually accompanied by the transformation of one type of energy into another. Energy can also be transferred from one body to another. So, for example, when shooting from a bow, the potential energy of a stretched bowstring is converted into the kinetic energy of a flying arrow.

If a force acts on a body, then this force does work to move this body. Before giving a definition of work in the curvilinear motion of a material point, consider special cases:

In this case, mechanical work A is equal to:

A= F s cos=
,

or A=Fcos× s = F S × s ,

whereF S – projection strength to move. In this case F s = const, and the geometric meaning of the work A is the area of ​​the rectangle constructed in coordinates F S , , s.

Let's build a graph of the projection of force on the direction of movement F S as a function of displacement s. We represent the total displacement as the sum of n small displacements
. For small i -th displacement
work is

or the area of ​​the shaded trapezoid in the figure.

Full mechanical work to move from a point 1 exactly 2 will be equal to:

.

The value under the integral will represent the elementary work on an infinitesimal displacement
:

- basic work.

We break the trajectory of the motion of a material point into infinitesimal displacements and the work of the force by moving a material point from a point 1 exactly 2 defined as a curvilinear integral:

–work with curvilinear motion.

Example 1: The work of gravity
during curvilinear motion of a material point.

.

Further as a constant value can be taken out of the integral sign, and the integral according to the figure will represent a complete displacement . .

If we denote the height of the point 1 from the surface of the earth through , and the height of the point 2 through , then

We see that in this case the work is determined by the position of the material point at the initial and final moments of time and does not depend on the shape of the trajectory or path. The work done by gravity in a closed path is zero:
.

Forces whose work on a closed path is zero is calledconservative .

Example 2 : The work of the friction force.

This is an example of a non-conservative force. To show this, it is enough to consider the elementary work of the friction force:

,

those. the work of the friction force is always negative and cannot be equal to zero on a closed path. The work done per unit of time is called power. If in time
work is done
, then the power is

–mechanical power.

Taking
as

,

we get the expression for power:

.

The SI unit of work is the joule:
= 1 J = 1 N 1 m, and the unit of power is watt: 1 W = 1 J / s.

mechanical energy.

Energy is a general quantitative measure of the movement of the interaction of all types of matter. Energy does not disappear and does not arise from nothing: it can only pass from one form to another. The concept of energy binds together all phenomena in nature. In accordance with the various forms of motion of matter, different types of energy are considered - mechanical, internal, electromagnetic, nuclear, etc.

The concepts of energy and work are closely related to each other. It is known that work is done at the expense of the energy reserve and, conversely, by doing work, it is possible to increase the energy reserve in any device. In other words, work is a quantitative measure of the change in energy:

.

Energy as well as work in SI is measured in joules: [ E]=1 J.

Mechanical energy is of two types - kinetic and potential.

Kinetic energy (or the energy of motion) is determined by the masses and velocities of the considered bodies. Consider a material point moving under the action of a force . The work of this force increases the kinetic energy of a material point
. Let us calculate in this case a small increment (differential) of the kinetic energy:

When calculating
using Newton's second law
, as well as
- modulus of velocity of a material point. Then
can be represented as:

-

- kinetic energy of a moving material point.

Multiplying and dividing this expression by
, and given that
, we get

-

- relationship between momentum and kinetic energy of a moving material point.

Potential energy ( or the energy of the position of bodies) is determined by the action of conservative forces on the body and depends only on the position of the body .

We have seen that the work of gravity
with curvilinear motion of a material point
can be represented as the difference between the values ​​of the function
taken at the point 1 and at the point 2 :

.

It turns out that whenever the forces are conservative, the work of these forces on the way 1
2 can be represented as:

.

Function , which depends only on the position of the body - is called potential energy.

Then for elementary work we get

–work is equal to the loss of potential energy.

Otherwise, we can say that the work is done due to the potential energy reserve.

the value , equal to the sum of the kinetic and potential energies of the particle, is called the total mechanical energy of the body:

–total mechanical energy of the body.

In conclusion, we note that using Newton's second law
, kinetic energy differential
can be represented as:

.

Potential energy differential
, as mentioned above, is equal to:

.

Thus, if the power is a conservative force and there are no other external forces, then , i.e. in this case, the total mechanical energy of the body is conserved.