Determine the amount of work. Mechanical work

1. Mechanical work\(A \) - physical quantity, equal to the product the vector of force acting on the body and the vector of its displacement:\(A=\vec(F)\vec(S) \) . Job - scalar quantity, characterized by a numerical value and a unit.

A unit of work is taken to be 1 joule (1 J). This is the work done by a force of 1 N along a path of 1 m.

\[ [\,A\,]=[\,F\,][\,S\,]; [\,A\,]=1Н\cdot1m=1J\]

2. If the force acting on the body makes a certain angle \(\alpha \) with the displacement, then the projection of the force \(F \) onto the X axis is equal to \(F_x \) (Fig. 42).

Since \(F_x=F\cdot\cos\alpha \) , then \(A=FS\cos\alpha \) .

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

3. If force \(F \) = 0 or displacement \(S \) = 0, then mechanical work is zero \(A \) = 0. Work is zero if the force vector is perpendicular to the displacement vector, t .e. \(\cos90^\circ \) = 0. So, the work of the force imparting to the body is equal to zero centripetal acceleration with him uniform motion along a circle, since this force is perpendicular to the direction of movement of the body at any point of the trajectory.

4. The work done by a force can be either positive or negative. The work is positive \(A \) > 0, if the angle is 90° > \(\alpha \) ≥ 0°; if the angle 180° > \(\alpha \) ≥ 90°, then the work is negative \(A \) < 0.

If the angle \(\alpha \) = 0°, then \(\cos\alpha \) = 1, \(A=FS \) . If the angle \(\alpha \) = 180°, then \(\cos\alpha \) = -1, \(A=-FS \) .

5. In a free fall from a height \(h\) a body of mass \(m\) moves from position 1 to position 2 (Fig. 43). In this case, the force of gravity does work equal to:

\[ A=F_тh=mg(h_1-h_2)=mgh \]

When a body moves vertically downward, the force and displacement are directed in one direction, and the force of gravity does positive work.

If a body rises up, then the force of gravity is directed downward, and if it moves upward, then the force of gravity is directed negative work, i.e.

\[ A=-F_тh=-mg(h_1-h_2)=-mgh \]

6. The work can be presented graphically. The figure shows a graph of the dependence of gravity on the height of a body relative to the surface of the Earth (Fig. 44). Graphically, the work done by gravity is equal to the area of the figure (rectangle), limited by schedule, coordinate axes and perpendicular to the abscissa axis
at point \(h\) .

The graph of the elastic force versus the elongation of the spring is a straight line passing through the origin of coordinates (Fig. 45). By analogy with the work of gravity, the work of the elastic force is equal to the area of the triangle bounded by the graph, the coordinate axes and the perpendicular to the abscissa at the point \(x\) .
\(A=Fx/2=kx\cdot x/2 \) .

7. The work done by gravity does not depend on the shape of the trajectory along which the body moves; it depends on the initial and final positions of the body. Let the body first move from point A to point B along the trajectory AB (Fig. 46). The work of gravity in this case

\[ A_(AB)=mgh \]

Let now the body move from point A to point B, first along the inclined plane AC, then along the base of the inclined plane BC. The work done by gravity when moving along the aircraft is zero. The work of gravity when moving along the AC is equal to the product of the projection of gravity onto the inclined plane \(mg\sin\alpha \) and the length of the inclined plane, i.e. \(A_(AC)=mg\sin\alpha\cdot l \). Product \(l\cdot\sin\alpha=h \) . Then \(A_(AC)=mgh \) . The work of gravity when moving a body along two different trajectories does not depend on the shape of the trajectory, but depends on the initial and final positions of the body.

The work of the elastic force also does not depend on the shape of the trajectory.

Suppose that a body moves from point A to point B along the trajectory ACB, and then from point B to point A along the trajectory BA. When moving along the ACB trajectory, gravity does positive work; when moving along the BA trajectory, the work of gravity is negative, equal in magnitude to the work when moving along the ACB trajectory. Therefore, the work done by gravity along a closed path is zero. The same applies to the work of the elastic force.

Forces whose work does not depend on the shape of the trajectory and is equal to zero along a closed trajectory are called conservative. TO conservative forces include gravity and elastic force.

8. Forces whose work depends on the shape of the path are called non-conservative. The friction force is non-conservative. If a body moves from point A to point B (Fig. 47) first along a straight line and then along a broken line ACB, then in the first case the work of the friction force \(A_(AB)=-Fl_(AB) \) , and in the second \(A_(ABC)=A_(AC)+A_(CB) , \(A_(ABC)=-Fl_(AC)-Fl_(CB) \) .

Therefore, work \(A_(AB) \) is not equal to work \(A_(ABC) \) .

9. Power is a physical quantity equal to the ratio of work to the period of time during which it is performed. Power characterizes the speed at which work is done.

Power is denoted by the letter \(N\) .

Power unit: \([N]=[A]/[t] \) . \([N] \) = 1 J/1 s = 1 J/s. This unit is called the watt (W). One watt is the power at which 1 J of work is performed in 1 s.

10. The power developed by the engine is equal to: \(N = A/t \) , \(A=F\cdot S \) , whence \(N=FS/t \) . The ratio of movement to time is the speed of movement: \(S/t = v \) . Where \(N = Fv \) .

From the resulting formula it is clear that when constant force resistance, the speed of movement is directly proportional to the engine power.

Transformation occurs in various machines and mechanisms mechanical energy. Due to energy, work is done during its transformation. At the same time, to commit useful work only part of the energy is consumed. Some of the energy is spent doing work against friction forces. Thus, any machine is characterized by a value indicating what part of the energy transferred to it is used usefully. This quantity is called efficiency factor (efficiency).

The efficiency factor is the quantity equal to the ratio useful work \((A_п) \) to all completed work \((A_с) \) : \(\eta=A_п/A_с \) . Efficiency is expressed as a percentage.

Part 1

1. The work is determined by the formula

1) \(A=Fv \)
2) \(A=N/t\)
3) \(A=mv\)
4) \(A=FS \)

2. The load is evenly lifted vertically upward by a rope tied to it. The work of gravity in this case

1) equal to zero
2) positive
3) negative
4) more work of elastic force

3. The box is pulled by a rope tied to it, making an angle of 60° with the horizontal, applying a force of 30 N. What is the work done by this force if the modulus of displacement is 10 m?

1) 300 J
2) 150 J
3) 3 J
4) 1.5 J

4. Artificial satellite The Earth, whose mass is equal to \(m\) , moves uniformly in a circular orbit with radius \(R\) . The work done by gravity in time is equal to the period circulation is equal to

1) \(mgR \)
2) \(\pi mgR \)
3) \(2\pi mgR \)
4) \(0 \)

5. A car weighing 1.2 tons travels 800 m along a horizontal road. How much work was done by the friction force if the friction coefficient is 0.1?

1) -960 kJ
2) -96 kJ
3) 960 kJ
4) 96 kJ

6. A spring with a stiffness of 200 N/m is stretched by 5 cm. How much work will the elastic force do when the spring returns to equilibrium?

1) 0.25 J
2) 5 J
3) 250 J
4) 500 J

7. Balls of the same mass roll down a slide along three different chutes, as shown in the figure. In which case will the work done by gravity be greatest?

1) 1
2) 2
3) 3
4) the work is the same in all cases

8. Work along a closed path is zero

A. Friction forces
B. Elastic forces

The correct answer is

1) both A and B
2) only A
3) only B
4) neither A nor B

9. The SI unit of power is

1) J
2) W
3) J s
4) Nm

10. What is the useful work if the work done is 1000 J and the engine efficiency is 40%?

1) 40000 J
2) 1000 J
3) 400 J
4) 25 J

11. Establish a correspondence between the work of force (in the left column of the table) and the sign of work (in the right column of the table). In your answer, write down the selected numbers under the corresponding letters.

WORK OF FORCE
A. Work of elastic force when stretching a spring
B. Work of friction force
B. Work of gravity when a body falls

WORK SIGN
1) positive
2) negative
3) equal to zero

12. From the statements below, choose two correct ones and write their numbers in the table.

1) The work of gravity does not depend on the shape of the trajectory.
2) Work is done during any movement of the body.
3) The work done by the sliding friction force is always negative.
4) The work done by the elastic force along a closed contour is not zero.
5) The work of the friction force does not depend on the shape of the trajectory.

Part 2

13. A winch uniformly lifts a load weighing 300 kg to a height of 3 m in 10 s. What is the power of the winch?

Answers

Every body that makes a movement can be characterized by work. In other words, it characterizes the action of forces.

Work is defined as:
The product of the modulus of force and the path traveled by the body, multiplied by the cosine of the angle between the direction of force and movement.

Work is measured in Joules:
1 [J] = = [kg* m2/s2]

For example, body A, under the influence of a force of 5 N, traveled 10 m. Determine the work done by the body.

Since the direction of movement and the action of the force coincide, the angle between the force vector and the displacement vector will be equal to 0°. The formula will be simplified because the cosine of an angle of 0° is equal to 1.

Substituting the initial parameters into the formula, we find:
A= 15 J.

Let's consider another example: a body weighing 2 kg, moving with an acceleration of 6 m/s2, has traveled 10 m. Determine the work done by the body if it moved upward along an inclined plane at an angle of 60°.

To begin with, let's calculate how much force needs to be applied to impart an acceleration of 6 m/s2 to the body.

F = 2 kg * 6 m/s2 = 12 H.
Under the influence of a force of 12N, the body moved 10 m. The work can be calculated using the already known formula:

Where, a is equal to 30°. Substituting the initial data into the formula we get:
A= 103.2 J.

Power

Many machines and mechanisms perform the same work in different periods of time. To compare them, the concept of power is introduced.
Power is a quantity that shows the amount of work performed per unit of time.

Power is measured in Watts, after the Scottish engineer James Watt.
1 [Watt] = 1 [J/s].

For example, a large crane lifted a load weighing 10 tons to a height of 30 m in 1 minute. A small crane lifted 2 tons of bricks to the same height in 1 minute. Compare crane capacities.
Let's define the work performed by cranes. The load rises 30m, while overcoming the force of gravity, so the force expended on lifting the load will be equal to the force of interaction between the Earth and the load (F = m * g). And work is the product of forces by the distance traveled by the loads, that is, by the height.

For a large crane A1 = 10,000 kg * 30 m * 10 m/s2 = 3,000,000 J, and for a small crane A2 = 2,000 kg * 30 m * 10 m/s2 = 600,000 J.
Power can be calculated by dividing work by time. Both cranes lifted the load in 1 minute (60 seconds).

From here:
N1 = 3,000,000 J/60 s = 50,000 W = 50 kW.
N2 = 600,000 J/ 60 s = 10,000 W = 10 kW.
From the above data it is clearly seen that the first crane is 5 times more powerful than the second.

What does this mean?

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

Often the word “mechanical” is simply not written.
Sometimes you can come across the expression “the body has done work,” which in principle means “the force acting on the body has done work.”

I think - I'm working.

I'm going - I'm working too.

Where is the mechanical work here?

If a body moves under the influence of a force, then mechanical work is performed.

They say that the body does work.
Or more precisely, it will be like this: the work is done by the force acting on the body.

Work characterizes the result of a force.

The forces acting on a person perform 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 a body and the path made by the body under the influence of a force in the direction of this force.

A - mechanical work,
F - strength,
S - 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.

No work is done(i.e. equal to 0), if:
1. The force acts, but the body does not move.

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

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

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

Work can be positive and negative

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

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

2. If the direction of force and movement of the body is 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 done by all forces is equal to the work done by the resulting force.

Units of work

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

IN international system 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, a body moves 1 m in the direction of this force.

When flying from thumb man's hands on the index
the mosquito does work - 0.000 000 000 000 000 000 000 000 001 J.

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

GET TO WORK, FRIENDS!

Mechanical work. Units of work.

IN everyday life By the concept of “work” we mean everything.

In physics, the concept Job somewhat different. It is a definite physical quantity, which means it can be measured. In physics it is studied primarily mechanical work .

Let's look at examples of mechanical work.

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

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

Wanting to move the cabinet, we press hard on it, but if it does not move, then we do not perform mechanical work. One can imagine a case when a 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 a body and it moves .

It is not difficult to understand that the greater the force acts on the body and the more longer way which the body passes under the influence of this force, the more work is done.

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

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

work = force × path

Where A- Job, F- strength and s- the distance traveled.

A unit of work is taken to be the work done by a force of 1N over a path of 1 m.

Unit of work - joule (J ) named after the English scientist Joule. Thus,

1 J = 1N m.

Also used kilojoules (kJ) .

1 kJ = 1000 J.

Formula A = Fs applicable when the force F constant and coincides with the direction of movement of the body.

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

If the body moves in the direction opposite direction 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 movement, then this force does no work, the work is zero:

In the future, speaking about 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/m3.

Given:

ρ = 2500 kg/m 3

Solution:

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

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

F = 9.8 N/kg · 1250 kg ≈ 12,250 N.

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

Answer: A =245 kJ.

Levers.Power.Energy

To perform the same work, different engines require different times. For example, a crane at a construction site lifts hundreds of bricks to the top floor of a building in a few minutes. If these bricks were moved by a worker, 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-ploughshare - many ploughshares), this work will be completed in 40-50 minutes.

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

Power is equal to the ratio of work to the time during which it was performed.

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

Where N- power, A- Job, t- time of work performed.

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

N avg = A/t . The unit of power is taken to be the power at which J of work is done in 1 s.

This unit is called the watt ( W) 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 technology - 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 water flow 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 by flow 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 powers 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 indicates some information about the engine, including its power.

Human power at normal conditions work on average is 70-80 W. When jumping or running up stairs, a person can develop power up to 730 W, and in in some cases and even greater.

From the formula N = A/t it follows that

To calculate the work, it is necessary to multiply the power by the time during which this work was performed.

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 conditions of the problem and solve it.

Given:

Solution:

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

Answer A= 21 kJ.

Simple mechanisms.

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

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

On 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 moved and raised heavy stone slabs to great heights.

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

Devices used to convert force 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 to gain strength, that is, to increase the force acting on the body several times.

Simple mechanisms are found both in household and in all complex industrial and industrial 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 automatic machines, printing and counting machines.

Lever. Balance of forces on the lever.

Let's consider the simplest and most common mechanism - the lever.

The lever is solid, which can rotate around a fixed support.

The pictures show how a worker uses a crowbar as a lever to lift a load. In the first case, the worker with 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. To do this, he turns the crowbar around an axis passing through the only motionless the breaking point is the point of its support ABOUT. Strength F with which the worker acts on the lever is less force P, thus the worker receives gain in strength. Using 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 ABOUT(fulcrum) is located between the points of application of forces A And IN. Another picture shows a diagram of this lever. Both forces F 1 and F 2 acting on the lever are directed in one direction.

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

To find the arm of the force, you need to lower the perpendicular from the fulcrum to the line of action of the force.

The length of this perpendicular will be the arm of this force. The figure shows that OA- shoulder strength F 1; OB- shoulder strength F 2. The forces acting on the lever can rotate it around its axis in two directions: clockwise or counterclockwise. Yes, strength 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 influence of forces applied to it can be established experimentally. 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 figure) 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 force modules and their shoulders are measured. From the experience shown in Figure 154, it is clear that force 2 N balances the force 4 N. In this case, as can be seen from the figure, the shoulder of lesser strength is 2 times larger than the shoulder of greater strength.

Based on such experiments, the condition (rule) of lever equilibrium was established.

A lever is in equilibrium when the forces acting on it are inversely proportional to the arms 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 figure).

The rule of lever equilibrium was established by Archimedes around 287 - 212. BC e. (but in the last paragraph it was said that levers were used by the Egyptians? Or here important role plays on the word "installed"?)

From this rule it follows that a smaller force can be used to balance a larger force using a lever. Let one arm of the lever be 3 times larger than the other (see figure). Then, by applying a force of, for example, 400 N at point B, you can lift a stone weighing 1200 N. To lift an even heavier load, you need 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 lever arm of 2.4 m if the smaller arm is 0.6 m?

Let's write down the conditions 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, applying a force of 600 N to the lever. But in this case, the arm on which the worker acts is 4 times longer than the one 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 lesser strength should be longer than the shoulder greater strength.

Moment of power.

You already know the rule of lever equilibrium:

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

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

F 1l 1 = F 2 l 2 .

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

The product of the modulus of the force rotating the body and its shoulder is called moment of force; it is designated by the letter M. This means

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

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

M1 = M2

Indeed, in the experiment we 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. The unit of moment of force is taken to be a moment of force of 1 N, the arm of which is exactly 1 m.

This unit is called newton meter (N m).

The moment of force characterizes the action of a force, and shows that it depends simultaneously on both the modulus of the force and its leverage. Indeed, we already know, for example, that the action of a force on a door depends both on the magnitude of the force and on where the force is applied. The easier it is to turn the door, 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 rule of leverage (or the rule of moments) underlies the action of various kinds of tools and devices used in technology and everyday life where a gain in strength or travel is required.

We have a gain in strength when working with scissors. Scissors - this is a lever(fig), 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 gripping the scissors. Counterforce F 2 is the resistance force of the material being cut with scissors. Depending on the purpose of the scissors, their design varies. Office scissors, designed for cutting paper, have long blades and handles that are almost the same length. No paper cutting required great strength, and with a long blade it is more convenient to cut in a straight line. Shears for cutting sheet metal (fig.) have handles much longer than the blades, since the resistance force of the metal is large and the shoulder is used to balance it acting force has to be increased significantly. More 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 cutting wire.

Levers various types available on many cars. The handle of a sewing machine, the pedals or handbrake of a bicycle, the pedals of a car and tractor, and the keys of a piano are all examples of 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 scales is based on the principle of the lever (Fig.). The training scales shown in Figure 48 (p. 42) act as equal-arm lever . IN decimal scales The shoulder from which the cup with weights is suspended is 10 times longer than the shoulder carrying the load. This makes weighing large loads much easier. When weighing a load on a decimal scale, you should multiply the mass of the weights by 10.

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

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

Application of the law of equilibrium of a lever to a block.

Block It is a wheel with a groove, mounted in a holder. A rope, cable or chain is passed through the block groove.

Fixed block This is called a block whose axis is fixed and does not rise or fall when lifting loads (Fig.).

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

F = P/2 .

Thus, the movable block gives a 2-fold gain in strength .

This can be proven using the concept of moment of force. When the block is in equilibrium, the moments of forces F And R equal to each other. But the shoulder of strength F 2 times the leverage R, and, therefore, the power itself F 2 times less force R.

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

Equality of work when using simple mechanisms. "Golden rule" of mechanics.

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

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

By balancing two different magnitude forces on a lever F 1 and F 2 (fig.), set the lever in motion. It turns out that at the same time the point of application of the smaller force F 2 goes further s 2, and the point of application of the greater force F 1 - shorter 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 gain in strength, but at the same time we lose by the same amount along 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 is equal to each other:

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

So, When using leverage, you won’t be able to win at work.

By using leverage, we can gain either in strength or in distance. By applying force to the short arm of the lever, we gain in distance, but lose by the same amount in strength.

There is a legend that Archimedes, delighted with the discovery of the rule of leverage, exclaimed: “Give me a fulcrum and I will turn the Earth over!”

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

To raise the earth just 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!

A stationary block does not give any gain in work, which is easy to verify experimentally (see figure). Ways, passable points application of forces F And F, are the same, the forces are the same, which means the work is the same.

You can measure and compare the work done with the help of a moving block. In order to lift a load to a height h using 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.

Thus, getting a 2-fold gain in strength, they lose 2-fold on the way, therefore, the movable block does not give a gain in work.

Centuries-old practice has shown that None of the mechanisms gives a gain in performance. They use various mechanisms in order to win in strength or in travel, depending on the working conditions.

Already ancient scientists knew a rule applicable to all mechanisms: no matter how many times we win in strength, the same number of times we lose in distance. This rule has been called the "golden rule" of mechanics.

Efficiency of the mechanism.

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

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

Part of the work is done against the friction force in the mechanism and by moving it individual parts. So, when using a movable block, you have to additionally do work to lift the block itself, the rope and determine the friction force in the axis of the block.

Whatever mechanism we take, the useful work done with its help always constitutes only a part of the total work. This means, denoting useful work by the letter Ap, total (expended) work by the letter Az, we can write:

Up< Аз или Ап / Аз < 1.

The ratio of useful work to full time job called the efficiency of the mechanism.

The efficiency factor is abbreviated as efficiency.

Efficiency = Ap / Az.

Efficiency is usually expressed as a percentage and is denoted Greek letterη, it is read as “this”:

η = Ap / Az · 100%.

Example: A load weighing 100 kg is suspended on the short arm of a lever. To lift it, a force of 250 N is applied to the long arm. The load is raised to a height h1 = 0.08 m, and the point of application driving force dropped to a height h2 = 0.4 m. Find the efficiency of the lever.

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

Given :

Solution :

η = Ap / Az · 100%.

Total (expended) work Az = Fh2.

Useful work Ap = Рh1

P = 9.8 100 kg ≈ 1000 N.

Ap = 1000 N · 0.08 = 80 J.

Az = 250 N · 0.4 m = 100 J.

η = 80 J/100 J 100% = 80%.

Answer : η = 80%.

But " golden rule"is carried out in this case as well. 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%. When designing mechanisms, people strive to increase their efficiency. To achieve this, friction in the axes of the mechanisms and their weight are reduced.

Energy.

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

A compressed spring (Fig.), when straightened, does work, raises a load to a height, or makes a cart move.

A stationary 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. Thus, a steel ball A (fig) rolling down from an inclined plane, hitting a wooden block B, moves it a certain distance. At the same time, work is done.

If a body or several interacting bodies (a system of bodies) can do work, they are said to have energy.

Energy - a physical quantity showing how much 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.

How great job a body can accomplish, the more energy it possesses.

When work is done, the energy of bodies changes. Perfect work equal to the change in energy.

Potential and kinetic energy.

Potential (from lat. potency - possibility) energy is the energy that is determined by the relative position of interacting bodies and parts of the same body.

Potential energy, for example, is possessed by a body raised relative to the surface of the Earth, because energy depends on mutual position him and the Earth. and them mutual attraction. If we consider the potential energy of a body lying on the Earth, 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. Let us denote the potential energy of the body E n, because E = A, and work, as we know, is equal to the product of force and path, then

A = Fh,

Where F- gravity.

This means that the potential energy En is equal to:

E = Fh, or E = gmh,

Where g- acceleration free fall, m- body weight, h- the height to which the body is raised.

Water in rivers held by dams has enormous potential energy. Falling down, the water does work, driving powerful turbines of power plants.

The potential energy of a copra hammer (Fig.) is used in construction to carry out the work of driving piles.

When 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 work, closing the door.

The energy of compressed and untwisted springs is used, for example, in watches, various wind-up toys, etc.

Any elastic deformed body has 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 road construction, excavation of hard soil, etc.

The energy that a body possesses as a result of its movement is called kinetic (from the Greek. kinema - movement) energy.

The kinetic energy of a body is denoted by the letter E To.

Moving water, driving the turbines of hydroelectric power plants, consumes its kinetic energy and does the work. Moving air, the wind, also has kinetic energy.

What does kinetic energy depend on? Let's turn to experience (see figure). If you roll ball A from different heights, you will notice that the greater height The ball rolls down, the greater its speed and the further it moves the block, i.e., it does more work. This means that the kinetic energy of a body depends on its speed.

Due to its speed, a flying bullet has high kinetic energy.

The kinetic energy of a body also depends on its mass. Let's do our experiment again, but we'll roll another ball of greater mass from the inclined plane. Bar 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.

How more mass body and the speed at which it moves, the greater its kinetic energy.

In order to determine the kinetic energy of a body, the formula is used:

Ek = mv^2 /2,

Where m- body weight, v- speed of body movement.

The kinetic energy of bodies is used in technology. The water retained by the dam has, as already mentioned, great potential energy. When falling from a dam, water moves and has the same high kinetic energy. It drives a turbine connected to a generator electric current. Due to the kinetic energy of water, electrical energy is generated.

The energy of moving water has great value V national economy. This energy is used using 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 conventional zero value, have either potential or kinetic energy, and sometimes both together. For example, a flying airplane has both kinetic and potential energy relative to the Earth.

We became acquainted with two types of mechanical energy. Other types of energy (electrical, internal, etc.) will be discussed in other sections of the physics course.

Conversion of one type of mechanical energy into another.

The phenomenon of transformation of one type of mechanical energy into another is very convenient to observe on the device shown in the figure. By winding the thread onto the axis, the device disk is lifted. A disk raised upward has some potential energy. If you let go of it, it will spin and begin to 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 rise again to almost its previous height. (Part of the energy is spent working against the frictional force, 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 downward, its potential energy turns into kinetic energy, and when it moves up, the kinetic energy turns into potential energy.

The transformation of energy from one type to another also occurs when two objects collide. elastic bodies, for example, a rubber ball on the floor or a steel ball on a steel plate.

If you lift a steel ball (rice) above 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 potential energy of the compressed plate and the compressed ball. Then through action elastic forces the plate and the ball will return to 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 up at a speed of almost equal speed, which he possessed at the moment of impact with the slab. As the ball rises upward, the speed of the ball, and therefore its kinetic energy, decreases, while the potential energy increases. Having bounced off the plate, the ball rises to almost the same height from which it began to fall. At the top point of the rise, all its kinetic energy will again turn into potential.

Natural phenomena are usually accompanied by the transformation of one type of energy into another.

Energy can be transferred from one body to another. For example, when archery, the potential energy of a drawn bowstring is converted into the kinetic energy of a flying arrow.

The energy characteristics of motion are introduced on the basis of the concept of mechanical work or work of force.

Definition 1

Work A done constant force F → is a physical quantity equal to the product of the force and displacement modules multiplied by the cosine of the angle α , located between the force vectors F → and the displacement s →.

This definition discussed in Figure 1. 18. 1.

The work formula is written as,

A = F s cos α .

Work is a scalar quantity. This makes it possible to be positive at (0° ≤ α< 90 °) , отрицательной при (90 ° < α ≤ 180 °) . Когда задается прямой угол α , тогда совершаемая сила равняется нулю. Единицы измерения работы по системе СИ - джоули (Д ж) .

A joule is equal to the work done by a force of 1 N to move 1 m in the direction of the force.

Figure 1. 18. 1. Work of force F →: A = F s cos α = F s s

When projecting F s → force F → onto the direction of movement s → the force does not remain constant, and the calculation of work for small movements Δ s i is summed up and produced according to the formula:

A = ∑ ∆ A i = ∑ F s i ∆ s i .

This amount work is calculated from the limit (Δ s i → 0), after which it goes into the integral.

The graphic representation of the work is determined from the area curvilinear figure, located under the graph F s (x) in Figure 1. 18. 2.

Figure 1. 18. 2. Graphic definition of work Δ A i = F s i Δ s i .

An example of a force that depends on the coordinate is the elastic force of a spring, which obeys Hooke's law. To stretch a spring, it is necessary to apply a force F →, the modulus of which is proportional to the elongation of the spring. This can be seen in Figure 1. 18. 3.

Figure 1. 18. 3. Stretched spring. Direction external force F → coincides with the direction of movement s →. F s = k x, where k denotes the spring stiffness.

F → y p = - F →

The dependence of the external force modulus on the x coordinates can be plotted using a straight line.

Figure 1. 18. 4. Dependence of the external force modulus on the coordinate when the spring is stretched.

From the above figure it is possible to find work on external force the right free end of the spring, using the area of the triangle. The formula will take the form

This formula is applicable to express the work done by an external force when compressing a spring. Both cases show that the elastic force F → y p is equal to the work of the external force F → , but with the opposite sign.

Definition 2

If several forces act on a body, then the formula for the total work will look like the sum of all the work done on it. When a body moves translationally, the points of application of forces move equally, that is general work of all forces will be equal to the resultant work of the applied forces.