School encyclopedia. Physics lesson "Body impulse

In this lesson, everyone will be able to study the topic “Impulse. Law of conservation of momentum." First, we will define the concept of momentum. Then we will determine what the law of conservation of momentum is - one of the main laws, the observance of which is necessary for a rocket to move and fly. Let's consider how it is written for two bodies and what letters and expressions are used in the recording. We will also discuss its application in practice.

Topic: Laws of interaction and motion of bodies

Lesson 24. Impulse. Law of conservation of momentum

Eryutkin Evgeniy Sergeevich

The lesson is devoted to the topic “Momentum and the “law of conservation of momentum”. To launch satellites, you need to build rockets. In order for rockets to move and fly, we must strictly observe the laws by which these bodies will move. The most important law in this sense is the law of conservation of momentum. To go directly to the law of conservation of momentum, let's first define what it is pulse.

is called the product of a body's mass and its speed: . Momentum is a vector quantity; it is always directed in the direction in which the speed is directed. The word “impulse” itself is Latin and is translated into Russian as “push”, “move”. Impulse is denoted by a small letter and the unit of impulse is .

The first person to use the concept of momentum was. He tried to use impulse as a quantity replacing force. The reason for this approach is obvious: measuring force is quite difficult, but measuring mass and speed is quite simple. This is why it is often said that momentum is the amount of movement. And since measuring impulse is an alternative to measuring force, it means that these two quantities need to be connected.

Rice. 1. Rene Descartes

These quantities - impulse and force - are interconnected by the concept. The impulse of a force is written as the product of a force and the time during which that force is applied: impulse of force. There is no special designation for force impulse.

Let's look at the relationship between momentum and force impulse. Let us consider such a quantity as the change in the momentum of a body, . It is the change in the momentum of the body that is equal to the impulse of the force. So we can write: .

Now let's move on to the next important question - law of conservation of momentum. This law is valid for a closed isolated system.

Definition: a closed isolated system is one in which bodies interact only with each other and do not interact with external bodies.

For a closed system, the law of conservation of momentum is valid: in a closed system, the momentum of all bodies remains constant.

Let us turn to how the law of conservation of momentum is written for a system of two bodies: .

We can write the same formula as follows: .

Rice. 2. The total momentum of a system of two balls is conserved after their collision

Please note: this law makes it possible, avoiding consideration of the action of forces, to determine the speed and direction of motion of bodies. This law makes it possible to talk about such an important phenomenon as jet motion.

Derivation of Newton's second law

Using the law of conservation of momentum and the relationship between the momentum of a force and the momentum of a body, Newton's second and third laws can be obtained. The impulse of force is equal to the change in the momentum of the body: . Then we take the mass out of brackets, leaving . Let's move time from the left side of the equation to the right and write the equation as follows: .

Recall that acceleration is defined as the ratio of the change in speed to the time during which the change occurred. If we now substitute the acceleration symbol instead of the expression, we get the expression: - Newton’s second law.

Derivation of Newton's third law

Let's write down the law of conservation of momentum: . Let's move all the quantities associated with m 1 to the left side of the equation, and with m 2 - to the right side: .

Let's take the mass out of brackets: . The interaction of bodies did not occur instantly, but over a certain period. And this period of time for the first and second bodies in a closed system was the same value: .

Dividing the right and left sides by time t, we get the ratio of the change in speed to time - this will be the acceleration of the first and second bodies, respectively. Based on this, we rewrite the equation as follows: . This is Newton’s third law, well known to us: . Two bodies interact with each other with forces equal in magnitude and opposite in direction.

List of additional literature:

Are you familiar with the quantity of motion? // Quantum. - 1991. - No. 6. — P. 40-41. Kikoin I.K., Kikoin A.K. Physics: Textbook. for 9th grade. avg. schools. - M.: Education, 1990. - P. 110-118 Kikoin A.K. Momentum and kinetic energy // Quantum. - 1985. - No. 5. - P. 28-29. Physics: Mechanics. 10th grade: Textbook. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakisheva. - M.: Bustard, 2002. - P. 284-307.

When bodies interact, the impulse of one body can be partially or completely transferred to another body. If a system of bodies is not acted upon by external forces from other bodies, then such a system is called closed.

In a closed system, the vector sum of the impulses of all bodies included in the system remains constant for any interactions of the bodies of this system with each other.

This fundamental law of nature is called law of conservation of momentum . It is a consequence of Newton's second and third laws.

Let us consider any two interacting bodies that are part of a closed system. We denote the interaction forces between these bodies by and According to Newton’s third law

If these bodies interact over time t, then the impulses of the interaction forces are equal in magnitude and directed in opposite directions:

Let us apply Newton's second law to these bodies:

Where and are the impulses of the bodies at the initial moment of time, and are the impulses of the bodies at the end of the interaction. From these relations it follows that as a result of the interaction of two bodies, their total momentum has not changed:

Law of conservation of momentum:

Now considering all possible pair interactions of bodies included in a closed system, we can conclude that the internal forces of a closed system cannot change its total momentum, i.e., the vector sum of the momentum of all bodies included in this system.

Rice. 1.17.1 illustrates the law of conservation of momentum using the example off-central impact two balls of different masses, one of which was at rest before the collision.

Shown in Fig. 1.17.1 the momentum vectors of the balls before and after the collision can be projected onto the coordinate axes OX And OY. The law of conservation of momentum is also true for projections of vectors onto each axis. In particular, from the momentum diagram (Fig. 1.17.1) it follows that the projections of the vectors and momentum of both balls after the collision onto the axis OY must be identical in magnitude and have different signs so that their sum equals zero.

Law of conservation of momentum in many cases it allows one to find the velocities of interacting bodies even when the values ​​of the acting forces are unknown. An example would be jet propulsion .

When firing a gun, a recoil- the projectile moves forward, and the gun rolls back. The projectile and the gun are two interacting bodies. The speed that a gun acquires during recoil depends only on the speed of the projectile and the mass ratio (Fig. 1.17.2). If the velocities of the gun and the projectile are denoted by and and their masses by M And m, then, based on the law of conservation of momentum, we can write in projections onto the axis OX

Based on the principle of giving jet propulsion. IN rocket When fuel burns, gases heated to a high temperature are ejected from the nozzle at high speed relative to the rocket. Let us denote the mass of emitted gases by m, and the mass of the rocket after the exhaust of gases through M. Then for the closed system “rocket + gases”, based on the law of conservation of momentum (by analogy with the problem of firing a gun), we can write:

Where V- the speed of the rocket after the exhaust of gases. In this case, it is assumed that the initial speed of the rocket was zero.

The resulting formula for the rocket speed is valid only under the condition that the entire mass of burnt fuel is ejected from the rocket simultaneously. In fact, the outflow occurs gradually throughout the entire period of accelerated motion of the rocket. Each subsequent portion of gas is ejected from the rocket, which has already acquired a certain speed.

To obtain an accurate formula, the process of gas outflow from a rocket nozzle needs to be considered in more detail. Let the rocket at a moment in time t has mass M and moves at speed (Fig. 1.17.3 (1)). Over a short period of time Δ t a certain portion of gas will be ejected from the rocket with a relative speed of the rocket at the moment t + Δ t will have a speed and its mass will be equal M + Δ M, where Δ M < 0 (рис. 1.17.3 (2)). Масса выброшенных газов будет, очевидно, равна -ΔM> 0. Velocity of gases in the inertial frame OX will be equal to Apply the law of conservation of momentum. At a moment in time t + Δ t the momentum of the rocket is equal to , and the momentum of the emitted gases is equal to . At a moment in time t the momentum of the entire system was equal to Assuming the “rocket + gases” system is closed, we can write:

The value can be neglected, since |Δ M| << M. Dividing both sides of the last relation by Δ t and passing to the limit at Δ t→0, we get:

Magnitude is the fuel consumption per unit time. The quantity is called thrust force The reactive thrust force acts on the rocket from the side of the outflowing gases; it is directed in the direction opposite to the relative speed. Ratio
expresses Newton's second law for a body of variable mass. If gases are ejected from the rocket nozzle strictly backward (Fig. 1.17.3), then in scalar form this relationship takes the form:

Where u- relative velocity module. Using the mathematical operation of integration, from this relation we can obtain formulaTsiolkovskyfor the final speed υ of the rocket:

where is the ratio of the initial and final masses of the rocket.

It follows from it that the final speed of the rocket can exceed the relative speed of the outflow of gases. Consequently, the rocket can be accelerated to the high speeds required for space flights. But this can only be achieved by consuming a significant mass of fuel, constituting a large proportion of the initial mass of the rocket. For example, to achieve the first escape velocity υ = υ 1 = 7.9·10 3 m/s at u= 3·10 3 m/s (gas flow velocities during fuel combustion are on the order of 2-4 km/s) starting mass single stage rocket should be approximately 14 times the final mass. To achieve final speed υ = 4 u the ratio should be 50.

Jet motion is based on the law of conservation of momentum and this is indisputable. Only many problems are solved in different ways. I suggest the following. The simplest jet engine: a chamber in which constant pressure is maintained by burning fuel; in the lower bottom of the chamber there is a hole through which gas flows out at a certain speed. According to the law of conservation of momentum, the camera begins to move (truths). Another way. There is a hole in the bottom bottom of the chamber, i.e. The area of ​​the lower bottom is less than the area of ​​the upper bottom by the area of ​​the hole. The product of pressure and area gives force. The force acting on the upper bottom is greater than on the lower one (due to the difference in areas), we obtain an unbalanced force that sets the chamber in motion. F = p (S1-S2) = pSholes, where S1 is the area of ​​the upper bottom, S2 is the area of ​​the bottom bottom, Sholes is the area of ​​the hole. If you solve problems using the traditional method and the one I proposed, the result will be the same. The method I proposed is more complicated, but it explains the dynamics of jet propulsion. Solving problems using the law of conservation of momentum is simpler, but it does not make it clear where the force that sets the camera in motion comes from.

Lesson objectives:

1. educational: formation of the concepts “body impulse”, “force impulse”; the ability to apply them to the analysis of the phenomenon of interaction of bodies in the simplest cases; to ensure that students understand the formulation and derivation of the law of conservation of momentum;
2. developing: to develop the ability to analyze, establish connections between elements of the content of previously studied material on the basics of mechanics, skills of search cognitive activity, and the ability for self-analysis;
3. educational: development of students’ aesthetic taste, arousing a desire to constantly expand their knowledge; maintain interest in the subject.

Equipment: metal balls on strings, demonstration carts, weights.

Learning tools: test cards.

Lesson progress

1. Organizational stage (1 min)

2. Repetition of the studied material. (10 min)

Teacher: You will learn the topic of the lesson by solving a small crossword puzzle, the key word of which will be the topic of our lesson. (We solve from left to right, we write the words vertically one by one).

1. The phenomenon of maintaining a constant speed in the absence of external influences or when they are compensated.
2. The phenomenon of changing the volume or shape of the body.
3. The force generated during deformation tends to return the body to its original position.
4. An English scientist, a contemporary of Newton, established the dependence of the elastic force on deformation.
5. Unit of mass.
6. English scientist who discovered the basic laws of mechanics.
7. Vector physical quantity, numerically equal to the change in speed per unit time.
8. The force with which the Earth attracts all bodies to itself.
9. A force that arises due to the existence of interaction forces between molecules and atoms of contacting bodies.
10. Measure of interaction between bodies.
11. A branch of mechanics that studies the laws of mechanical motion of material bodies under the influence of forces applied to them.

3. Studying new material. (18 min)

Guys, the topic of our lesson “Body impulse. Law of conservation of momentum"

Lesson Objectives: learn the concept of momentum of a body, the concept of a closed system, study the law of conservation of momentum, learn to solve problems on the law of conservation.

Today in the lesson we will not only perform experiments, but also prove them mathematically.

Knowing the basic laws of mechanics, primarily Newton’s three laws, it would seem that one can solve any problem about the motion of bodies. Guys, I’ll show you some experiments, and you think, is it possible in these cases to solve problems using only Newton’s laws?

Problem experiment.

Experiment No. 1. Rolling a light-moving cart down an inclined plane. She moves a body that is in her path.

Is it possible to find the force between the cart and the body? (no, since the collision between the cart and the body is short-lived and the force of their interaction is difficult to determine).

Experience No. 2. Rolling a loaded cart. Moves the body further.

Is it possible to find the force of interaction between the cart and the body in this case?

Draw a conclusion: what physical quantities can be used to characterize the movement of a body?

Conclusion: Newton's laws make it possible to solve problems related to finding the acceleration of a moving body if all the forces acting on the body are known, i.e. resultant of all forces. But it is often very difficult to determine the resultant force, as was the case in our cases.

If a toy cart is rolling towards you, you can stop it with your toe, but what if a truck is rolling towards you?

Conclusion: to characterize movement, you need to know the mass of the body and its speed.

Therefore, to solve problems, they use another important physical quantity - body impulse.

The concept of momentum was introduced into physics by the French scientist René Descartes (1596-1650), who called this quantity “quantity of motion”: “I accept that in the universe ... there is a certain quantity of motion, which never increases, does not decrease, and, thus, if one body sets another in motion, it loses as much of its motion as it imparts.”

Let's find the relationship between the force acting on the body, the time of its action, and the change in the speed of the body.

Let the body mass m force begins to act F. Then, from Newton’s second law, the acceleration of this body will be A.

Remember how Newton's 2nd law is read?

Let us write the law in the form

On the other side:

Or We obtained the formula for Newton's second law in impulse form.

Let us denote the product through r:

The product of a body's mass and its speed is called the body's momentum.

Pulse r– vector quantity. It always coincides in direction with the body's velocity vector. Any body that moves has momentum.

Definition: The momentum of a body is a vector physical quantity equal to the product of the mass of the body and its speed and having the direction of speed.

Like any physical quantity, momentum is measured in certain units.

Who wants to derive the unit of measurement for impulse? (The student takes notes at the blackboard.)

(p) = (kg m/s)

Let's return to our equality . In physics, the product of force and time of action is called impulse of power.

Impulse force shows how the momentum of a body changes over a given time.

Descartes established the law of conservation of momentum, but he did not clearly understand that momentum is a vector quantity. The concept of momentum was clarified by the Dutch physicist and mathematician Huygens, who, by studying the impact of balls, proved that when they collide, it is not the arithmetic sum that is preserved, but the vector sum of momentum.

Experiment (two balls are suspended on threads)

The right one is rejected and released. Returning to its previous position and hitting a stationary ball, it stops. In this case, the left ball begins to move and deflects at almost the same angle as the right ball was deflected.

Momentum has an interesting property that only a few physical quantities have. This is a conservation property. But the law of conservation of momentum is satisfied only in a closed system.

A system of bodies is called closed if the bodies interacting with each other do not interact with other bodies.

The momentum of each of the bodies that make up a closed system can change as a result of their interaction with each other.

The vector sum of the impulses of the bodies that make up a closed system does not change over time for any movements and interactions of these bodies.

This is the law of conservation of momentum.

Examples: a gun and a bullet in its barrel, a cannon and a shell, a rocket shell and the fuel in it.

Law of conservation of momentum.

The law of conservation of momentum is derived from Newton's second and third laws.

Let's consider a closed system consisting of two bodies - balls with masses m 1 and m 2, which move along a straight line in the same direction with a speed? 1 and? 2. With a slight approximation, we can assume that the balls represent a closed system.

From experience it is clear that the second ball moves at a higher speed (the vector is depicted by a longer arrow). Therefore, he will catch up with the first ball and they will collide. ( View the experiment with teacher's comments).

Mathematical derivation of the conservation law

And now we will motivate the “commanders”, using the laws of mathematics and physics, to make a mathematical derivation of the law of conservation of momentum.

5) Under what conditions is this law fulfilled?

6) What system is called closed?

7) Why does recoil occur when firing a gun?

5. Problem solving (10 min.)

No. 323 (Rymkevich).

Two inelastic bodies, the masses of which are 2 and 6 kg, move towards each other at speeds of 2 m/s each. At what speed and in what direction will these bodies move after the impact?

The teacher comments on the drawing for the problem.

7. Summing up the lesson; homework (2 min)

Homework: § 41, 42 ex. 8 (1, 2).

Literature:

1. V. Ya. Lykov. Aesthetic education in teaching physics. Book for teachers. -Moscow “ENLIGHTENMENT” 1986.
2. V. A. Volkov. Lesson developments in physics, grade 10. - Moscow “VAKO” 2006.
3. Edited by Professor B.I. Spassky. Reader on physics. -MOSCOW “ENLIGHTENMENT” 1987.
4. I. I. Mokrova. Lesson plans based on the textbook by A.V. Peryshkin “Physics. 9th grade.” - Volgograd 2003.

Body impulse

The momentum of a body is a quantity equal to the product of the mass of the body and its speed.

It should be remembered that we are talking about a body that can be represented as a material point. The momentum of a body ($p$) is also called the momentum. The concept of momentum was introduced into physics by René Descartes (1596–1650). The term “impulse” appeared later (impulsus in Latin means “push”). Momentum is a vector quantity (like speed) and is expressed by the formula:

$p↖(→)=mυ↖(→)$

The direction of the momentum vector always coincides with the direction of the velocity.

The SI unit of impulse is the impulse of a body with a mass of $1$ kg moving at a speed of $1$ m/s; therefore, the unit of impulse is $1$ kg $·$ m/s.

If a constant force acts on a body (material point) during a period of time $∆t$, then the acceleration will also be constant:

$a↖(→)=((υ_2)↖(→)-(υ_1)↖(→))/(∆t)$

where $(υ_1)↖(→)$ and $(υ_2)↖(→)$ are the initial and final velocities of the body. Substituting this value into the expression of Newton's second law, we get:

$(m((υ_2)↖(→)-(υ_1)↖(→)))/(∆t)=F↖(→)$

Opening the brackets and using the expression for the momentum of the body, we have:

$(p_2)↖(→)-(p_1)↖(→)=F↖(→)∆t$

Here $(p_2)↖(→)-(p_1)↖(→)=∆p↖(→)$ is the change in momentum over time $∆t$. Then the previous equation will take the form:

$∆p↖(→)=F↖(→)∆t$

The expression $∆p↖(→)=F↖(→)∆t$ is a mathematical representation of Newton's second law.

The product of a force and the duration of its action is called impulse of force. That's why the change in the momentum of a point is equal to the change in the momentum of the force acting on it.

The expression $∆p↖(→)=F↖(→)∆t$ is called equation of body motion. It should be noted that the same action - a change in the momentum of a point - can be achieved by a small force over a long period of time and by a large force over a short period of time.

Impulse of the system tel. Law of Momentum Change

The impulse (amount of motion) of a mechanical system is a vector equal to the sum of the impulses of all material points of this system:

$(p_(syst))↖(→)=(p_1)↖(→)+(p_2)↖(→)+...$

The laws of change and conservation of momentum are a consequence of Newton's second and third laws.

Let us consider a system consisting of two bodies. The forces ($F_(12)$ and $F_(21)$ in the figure with which the bodies of the system interact with each other are called internal.

Let, in addition to internal forces, external forces $(F_1)↖(→)$ and $(F_2)↖(→)$ act on the system. For each body we can write the equation $∆p↖(→)=F↖(→)∆t$. Adding the left and right sides of these equations, we get:

$(∆p_1)↖(→)+(∆p_2)↖(→)=((F_(12))↖(→)+(F_(21))↖(→)+(F_1)↖(→)+ (F_2)↖(→))∆t$

According to Newton's third law, $(F_(12))↖(→)=-(F_(21))↖(→)$.

Hence,

$(∆p_1)↖(→)+(∆p_2)↖(→)=((F_1)↖(→)+(F_2)↖(→))∆t$

On the left side there is a geometric sum of changes in the impulses of all bodies of the system, equal to the change in the impulse of the system itself - $(∆p_(syst))↖(→)$. Taking this into account, the equality $(∆p_1)↖(→)+(∆p_2) ↖(→)=((F_1)↖(→)+(F_2)↖(→))∆t$ can be written:

$(∆p_(syst))↖(→)=F↖(→)∆t$

where $F↖(→)$ is the sum of all external forces acting on the body. The result obtained means that the momentum of the system can only be changed by external forces, and the change in the momentum of the system is directed in the same way as the total external force. This is the essence of the law of change in momentum of a mechanical system.

Internal forces cannot change the total momentum of the system. They only change the impulses of individual bodies of the system.

Law of conservation of momentum

From the equation $(∆p_(syst))↖(→)=F↖(→)∆t$ the law of conservation of momentum follows. If no external forces act on the system, then the right side of the equation $(∆p_(syst))↖(→)=F↖(→)∆t$ becomes zero, which means the total momentum of the system remains unchanged:

$(∆p_(syst))↖(→)=m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=const$

A system on which no external forces act or the resultant of external forces is zero is called closed.

The law of conservation of momentum states:

The total momentum of a closed system of bodies remains constant for any interaction of the bodies of the system with each other.

The result obtained is valid for a system containing an arbitrary number of bodies. If the sum of external forces is not equal to zero, but the sum of their projections to some direction is equal to zero, then the projection of the system's momentum to this direction does not change. So, for example, a system of bodies on the surface of the Earth cannot be considered closed due to the force of gravity acting on all bodies, however, the sum of the projections of impulses on the horizontal direction can remain unchanged (in the absence of friction), since in this direction the force of gravity does not works.

Jet propulsion

Let us consider examples that confirm the validity of the law of conservation of momentum.

Let's take a children's rubber ball, inflate it and release it. We will see that when the air begins to leave it in one direction, the ball itself will fly in the other. The motion of a ball is an example of jet motion. It is explained by the law of conservation of momentum: the total momentum of the “ball plus air in it” system before the air flows out is zero; it must remain equal to zero during movement; therefore, the ball moves in the direction opposite to the direction of flow of the jet, and at such a speed that its momentum is equal in magnitude to the momentum of the air jet.

Jet motion call the movement of a body that occurs when some part of it is separated from it at any speed. Due to the law of conservation of momentum, the direction of movement of the body is opposite to the direction of movement of the separated part.

Rocket flights are based on the principle of jet propulsion. A modern space rocket is a very complex aircraft. The mass of the rocket consists of the mass of the working fluid (i.e., hot gases formed as a result of fuel combustion and emitted in the form of a jet stream) and the final, or, as they say, “dry” mass of the rocket remaining after the working fluid is ejected from the rocket.

When a jet of gas is ejected from a rocket at high speed, the rocket itself rushes in the opposite direction. According to the law of conservation of momentum, the momentum $m_(p)υ_p$ acquired by the rocket must be equal to the momentum $m_(gas)·υ_(gas)$ of the ejected gases:

$m_(p)υ_p=m_(gas)·υ_(gas)$

It follows that the speed of the rocket

$υ_p=((m_(gas))/(m_p))·υ_(gas)$

From this formula it is clear that the greater the speed of the rocket, the greater the speed of the emitted gases and the ratio of the mass of the working fluid (i.e., the mass of the fuel) to the final (“dry”) mass of the rocket.

The formula $υ_p=((m_(gas))/(m_p))·υ_(gas)$ is approximate. It does not take into account that as the fuel burns, the mass of the flying rocket becomes less and less. The exact formula for rocket speed was obtained in 1897 by K. E. Tsiolkovsky and bears his name.

Work of force

The term “work” was introduced into physics in 1826 by the French scientist J. Poncelet. If in everyday life only human labor is called work, then in physics and, in particular, in mechanics it is generally accepted that work is performed by force. The physical quantity of work is usually denoted by the letter $A$.

Work of force is a measure of the action of a force, depending on its magnitude and direction, as well as on the displacement of the point of application of the force. For a constant force and linear displacement, the work is determined by the equality:

$A=F|∆r↖(→)|cosα$

where $F$ is the force acting on the body, $∆r↖(→)$ is the displacement, $α$ is the angle between the force and the displacement.

The work of force is equal to the product of the moduli of force and displacement and the cosine of the angle between them, i.e., the scalar product of the vectors $F↖(→)$ and $∆r↖(→)$.

Work is a scalar quantity. If $α 0$, and if $90°

When several forces act on a body, the total work (the sum of the work of all forces) is equal to the work of the resulting force.

The unit of work in SI is joule($1$ J). $1$ J is the work done by a force of $1$ N along a path of $1$ m in the direction of action of this force. This unit is named after the English scientist J. Joule (1818-1889): $1$ J = $1$ N $·$ m. Kilojoules and millijoules are also often used: $1$ kJ $= 1,000$ J, $1$ mJ $= $0.001 J.

Work of gravity

Let us consider a body sliding along an inclined plane with an angle of inclination $α$ and a height $H$.

Let us express $∆x$ in terms of $H$ and $α$:

$∆x=(H)/(sinα)$

Considering that the force of gravity $F_т=mg$ makes an angle ($90° - α$) with the direction of movement, using the formula $∆x=(H)/(sin)α$, we obtain an expression for the work of gravity $A_g$:

$A_g=mg cos(90°-α) (H)/(sinα)=mgH$

From this formula it is clear that the work done by gravity depends on the height and does not depend on the angle of inclination of the plane.

It follows that:

1. the work of gravity does not depend on the shape of the trajectory along which the body moves, but only on the initial and final position of the body;
2. when a body moves along a closed trajectory, the work done by gravity is zero, i.e., gravity is a conservative force (forces that have this property are called conservative).

Work of reaction forces, is equal to zero, since the reaction force ($N$) is directed perpendicular to the displacement $∆x$.

Work of friction force

The friction force is directed opposite to the displacement $∆x$ and makes an angle of $180°$ with it, therefore the work of the friction force is negative:

$A_(tr)=F_(tr)∆x·cos180°=-F_(tr)·∆x$

Since $F_(tr)=μN, N=mg cosα, ∆x=l=(H)/(sinα),$ then

$A_(tr)=μmgHctgα$

Work of elastic force

Let an external force $F↖(→)$ act on an unstretched spring of length $l_0$, stretching it by $∆l_0=x_0$. In position $x=x_0F_(control)=kx_0$. After the force $F↖(→)$ ceases to act at point $x_0$, the spring is compressed under the action of force $F_(control)$.

Let us determine the work of the elastic force when the coordinate of the right end of the spring changes from $x_0$ to $x$. Since the elastic force in this area changes linearly, Hooke’s law can use its average value in this area:

$F_(control av.)=(kx_0+kx)/(2)=(k)/(2)(x_0+x)$

Then the work (taking into account the fact that the directions $(F_(control av.))↖(→)$ and $(∆x)↖(→)$ coincide) is equal to:

$A_(control)=(k)/(2)(x_0+x)(x_0-x)=(kx_0^2)/(2)-(kx^2)/(2)$

It can be shown that the form of the last formula does not depend on the angle between $(F_(control av.))↖(→)$ and $(∆x)↖(→)$. The work of elastic forces depends only on the deformations of the spring in the initial and final states.

Thus, the elastic force, like gravity, is a conservative force.

Power power

Power is a physical quantity measured by the ratio of work to the period of time during which it is produced.

In other words, power shows how much work is done per unit of time (in SI - per $1$ s).

Power is determined by the formula:

where $N$ is power, $A$ is work done during time $∆t$.

Substituting into the formula $N=(A)/(∆t)$ instead of the work $A$ its expression $A=F|(∆r)↖(→)|cosα$, we obtain:

$N=(F|(∆r)↖(→)|cosα)/(∆t)=Fυcosα$

Power is equal to the product of the magnitudes of the force and velocity vectors and the cosine of the angle between these vectors.

Power in the SI system is measured in watts (W). One watt ($1$ W) is the power at which $1$ J of work is done for $1$ s: $1$ W $= 1$ J/s.

This unit is named after the English inventor J. Watt (Watt), who built the first steam engine. J. Watt himself (1736-1819) used another unit of power - horsepower (hp), which he introduced so that he could compare the performance of a steam engine and a horse: $1$ hp. $= 735.5$ W.

In technology, larger power units are often used - kilowatt and megawatt: $1$ kW $= 1000$ W, $1$ MW $= 1000000$ W.

Kinetic energy. Law of change of kinetic energy

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

The word “energy” (from the Greek energia - action, activity) is often used in everyday life. For example, people who can do work quickly are called energetic, having great energy.

The energy possessed by a body due to motion is called kinetic energy.

As in the case of the definition of energy in general, we can say about kinetic energy that kinetic energy is the ability of a moving body to do work.

Let us find the kinetic energy of a body of mass $m$ moving with a speed $υ$. Since kinetic energy is energy due to motion, its zero state is the state in which the body is at rest. Having found the work necessary to impart a given speed to a body, we will find its kinetic energy.

To do this, let us calculate the work in the area of ​​displacement $∆r↖(→)$ when the directions of the force vectors $F↖(→)$ and displacement $∆r↖(→)$ coincide. In this case the work is equal

where $∆x=∆r$

For the motion of a point with acceleration $α=const$, the expression for displacement has the form:

$∆x=υ_1t+(at^2)/(2),$

where $υ_1$ is the initial speed.

Substituting the expression for $∆x$ into the equation $A=F·∆x$ from $∆x=υ_1t+(at^2)/(2)$ and using Newton’s second law $F=ma$, we obtain:

$A=ma(υ_1t+(at^2)/(2))=(mat)/(2)(2υ_1+at)$

Expressing the acceleration through the initial $υ_1$ and final $υ_2$ velocities $a=(υ_2-υ_1)/(t)$ and substituting in $A=ma(υ_1t+(at^2)/(2))=(mat)/ (2)(2υ_1+at)$ we have:

$A=(m(υ_2-υ_1))/(2)·(2υ_1+υ_2-υ_1)$

$A=(mυ_2^2)/(2)-(mυ_1^2)/(2)$

Now equating the initial speed to zero: $υ_1=0$, we obtain an expression for kinetic energy:

$E_K=(mυ)/(2)=(p^2)/(2m)$

Thus, a moving body has kinetic energy. This energy is equal to the work that must be done to increase the speed of the body from zero to the value $υ$.

From $E_K=(mυ)/(2)=(p^2)/(2m)$ it follows that the work done by a force to move a body from one position to another is equal to the change in kinetic energy:

$A=E_(K_2)-E_(K_1)=∆E_K$

The equality $A=E_(K_2)-E_(K_1)=∆E_K$ expresses theorem on the change in kinetic energy.

Change in body kinetic energy(material point) for a certain period of time is equal to the work done during this time by the force acting on the body.

Potential energy

Potential energy is the energy determined by the relative position of interacting bodies or parts of the same body.

Since energy is defined as the ability of a body to do work, potential energy is naturally defined as the work done by a force, depending only on the relative position of the bodies. This is the work of gravity $A=mgh_1-mgh_2=mgH$ and the work of elasticity:

$A=(kx_0^2)/(2)-(kx^2)/(2)$

Potential energy of the body interacting with the Earth, they call a quantity equal to the product of the mass $m$ of this body by the acceleration of free fall $g$ and the height $h$ of the body above the Earth's surface:

The potential energy of an elastically deformed body is a value equal to half the product of the elasticity (stiffness) coefficient $k$ of the body and the squared deformation $∆l$:

$E_p=(1)/(2)k∆l^2$

The work of conservative forces (gravity and elasticity), taking into account $E_p=mgh$ and $E_p=(1)/(2)k∆l^2$, is expressed as follows:

$A=E_(p_1)-E_(p_2)=-(E_(p_2)-E_(p_1))=-∆E_p$

This formula allows us to give a general definition of potential energy.

The potential energy of a system is a quantity that depends on the position of the bodies, the change in which during the transition of the system from the initial state to the final state is equal to the work of the internal conservative forces of the system, taken with the opposite sign.

The minus sign on the right side of the equation $A=E_(p_1)-E_(p_2)=-(E_(p_2)-E_(p_1))=-∆E_p$ means that when work is performed by internal forces (for example, a fall bodies on the ground under the influence of gravity in the “rock-Earth” system), the energy of the system decreases. Work and changes in potential energy in a system always have opposite signs.

Since work determines only the change in potential energy, then only the change in energy has physical meaning in mechanics. Therefore, the choice of the zero energy level is arbitrary and determined solely by considerations of convenience, for example, the ease of writing the corresponding equations.

The law of change and conservation of mechanical energy

Total mechanical energy of the system the sum of its kinetic and potential energies is called:

It is determined by the position of bodies (potential energy) and their speed (kinetic energy).

According to the kinetic energy theorem,

$E_k-E_(k_1)=A_p+A_(pr),$

where $A_p$ is the work of potential forces, $A_(pr)$ is the work of non-potential forces.

In turn, the work of potential forces is equal to the difference in the potential energy of the body in the initial $E_(p_1)$ and final $E_p$ states. Taking this into account, we obtain an expression for law of change of mechanical energy:

$(E_k+E_p)-(E_(k_1)+E_(p_1))=A_(pr)$

where the left side of the equality is the change in total mechanical energy, and the right side is the work of non-potential forces.

So, law of change of mechanical energy reads:

The change in the mechanical energy of the system is equal to the work of all non-potential forces.

A mechanical system in which only potential forces act is called conservative.

In a conservative system $A_(pr) = 0$. It follows law of conservation of mechanical energy:

In a closed conservative system, the total mechanical energy is conserved (does not change with time):

$E_k+E_p=E_(k_1)+E_(p_1)$

The law of conservation of mechanical energy is derived from Newton's laws of mechanics, which are applicable to a system of material points (or macroparticles).

However, the law of conservation of mechanical energy is also valid for a system of microparticles, where Newton’s laws themselves no longer apply.

The law of conservation of mechanical energy is a consequence of the uniformity of time.

Uniformity of time is that, under the same initial conditions, the occurrence of physical processes does not depend on at what point in time these conditions are created.

The law of conservation of total mechanical energy means that when the kinetic energy in a conservative system changes, its potential energy must also change, so that their sum remains constant. This means the possibility of converting one type of energy into another.

In accordance with the various forms of motion of matter, various types of energy are considered: mechanical, internal (equal to the sum of the kinetic energy of the chaotic movement of molecules relative to the center of mass of the body and the potential energy of interaction of molecules with each other), electromagnetic, chemical (which consists of the kinetic energy of the movement of electrons and electrical the energy of their interaction with each other and with atomic nuclei), nuclear, etc. From the above it is clear that the division of energy into different types is quite arbitrary.

Natural phenomena are usually accompanied by the transformation of one type of energy into another. For example, friction of parts of various mechanisms leads to the conversion of mechanical energy into heat, i.e. internal energy. In heat engines, on the contrary, internal energy is converted into mechanical energy; in galvanic cells, chemical energy is converted into electrical energy, etc.

Currently, the concept of energy is one of the basic concepts of physics. This concept is inextricably linked with the idea of ​​​​the transformation of one form of movement into another.

This is how the concept of energy is formulated in modern physics:

Energy is a general quantitative measure of movement and interaction of all types of matter. Energy does not appear from nothing and does not disappear, it can only move from one form to another. The concept of energy links together all natural phenomena.

Simple mechanisms. Mechanism efficiency

Simple mechanisms are devices that change the magnitude or direction of forces applied to a body.

They are used to move or lift large loads with little effort. These include the lever and its varieties - blocks (movable and fixed), gates, inclined plane and its varieties - wedge, screw, etc.

Lever. Leverage rule

A lever is a rigid body capable of rotating around a fixed support.

The rule of leverage says:

A lever is in equilibrium if the forces applied to it are inversely proportional to their arms:

$(F_2)/(F_1)=(l_1)/(l_2)$

From the formula $(F_2)/(F_1)=(l_1)/(l_2)$, applying the property of proportion to it (the product of the extreme terms of a proportion is equal to the product of its middle terms), we can obtain the following formula:

But $F_1l_1=M_1$ is the moment of force tending to turn the lever clockwise, and $F_2l_2=M_2$ is the moment of force trying to turn the lever counterclockwise. Thus, $M_1=M_2$, which is what needed to be proven.

The lever began to be used by people in ancient times. With its help, it was possible to lift heavy stone slabs during the construction of pyramids in Ancient Egypt. Without leverage this would not be possible. After all, for example, for the construction of the Cheops pyramid, which has a height of $147$ m, more than two million stone blocks were used, the smallest of which weighed $2.5$ tons!

Nowadays, levers are widely used both in production (for example, cranes) and in everyday life (scissors, wire cutters, scales).

Fixed block

The action of a fixed block is similar to the action of a lever with equal arms: $l_1=l_2=r$. The applied force $F_1$ is equal to the load $F_2$, and the equilibrium condition is:

Fixed block used when you need to change the direction of a force without changing its magnitude.

Movable block

The moving block acts similarly to a lever whose arms are: $l_2=(l_1)/(2)=r$. In this case, the equilibrium condition has the form:

where $F_1$ is the applied force, $F_2$ is the load. The use of a moving block gives a double gain in strength.

Pulley hoist (block system)

An ordinary chain hoist consists of $n$ moving and $n$ fixed blocks. Using it gives a gain in strength of $2n$ times:

$F_1=(F_2)/(2n)$

Power chain hoist consists of n movable and one fixed block. The use of a power pulley gives a gain in strength of $2^n$ times:

$F_1=(F_2)/(2^n)$

Screw

A screw is an inclined plane wound around an axis.

The equilibrium condition for the forces acting on the propeller has the form:

$F_1=(F_2h)/(2πr)=F_2tgα, F_1=(F_2h)/(2πR)$

where $F_1$ is the external force applied to the propeller and acting at a distance $R$ from its axis; $F_2$ is the force acting in the direction of the propeller axis; $h$ — propeller pitch; $r$ is the average thread radius; $α$ is the angle of inclination of the thread. $R$ is the length of the lever (wrench) rotating the screw with a force of $F_1$.

Efficiency

Coefficient of efficiency (efficiency) is the ratio of useful work to all work expended.

Efficiency is often expressed as a percentage and is denoted by the Greek letter $η$ (“this”):

$η=(A_п)/(A_3)·100%$

where $A_n$ is useful work, $A_3$ is all expended work.

Useful work always constitutes only a part of the total work that a person expends using one or another mechanism.

Part of the work done is spent on overcoming frictional forces. Since $A_3 > A_n$, the efficiency is always less than $1$ (or $< 100%$).

Since each of the works in this equality can be expressed as a product of the corresponding force and the distance traveled, it can be rewritten as follows: $F_1s_1≈F_2s_2$.

It follows that, winning with the help of a mechanism in force, we lose the same number of times along the way, and vice versa. This law is called the golden rule of mechanics.

The golden rule of mechanics is an approximate law, since it does not take into account the work of overcoming friction and gravity of the parts of the devices used. Nevertheless, it can be very useful in analyzing the operation of any simple mechanism.

So, for example, thanks to this rule, we can immediately say that the worker shown in the figure, with a double gain in the force of lifting the load by $10$ cm, will have to lower the opposite end of the lever by $20$ cm.

Collision of bodies. Elastic and inelastic impacts

The laws of conservation of momentum and mechanical energy are used to solve the problem of the motion of bodies after a collision: from the known impulses and energies before the collision, the values ​​of these quantities are determined after the collision. Let us consider the cases of elastic and inelastic impacts.

An impact is called absolutely inelastic, after which the bodies form a single body moving at a certain speed. The problem of the speed of the latter is solved using the law of conservation of momentum of a system of bodies with masses $m_1$ and $m_2$ (if we are talking about two bodies) before and after the impact:

$m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=(m_1+m_2)υ↖(→)$

It is obvious that the kinetic energy of bodies during an inelastic impact is not conserved (for example, for $(υ_1)↖(→)=-(υ_2)↖(→)$ and $m_1=m_2$ it becomes equal to zero after the impact).

An impact in which not only the sum of impulses is conserved, but also the sum of the kinetic energies of the impacting bodies is called absolutely elastic.

For an absolutely elastic impact, the following equations are valid:

$m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=m_1(υ"_1)↖(→)+m_2(υ"_2)↖(→);$

$(m_(1)υ_1^2)/(2)+(m_(2)υ_2^2)/(2)=(m_1(υ"_1)^2)/(2)+(m_2(υ"_2 )^2)/(2)$

where $m_1, m_2$ are the masses of the balls, $υ_1, υ_2$ are the velocities of the balls before the impact, $υ"_1, υ"_2$ are the velocities of the balls after the impact.

His movements, i.e. size .

Pulse is a vector quantity coinciding in direction with the velocity vector.

SI unit of impulse: kg m/s .

The momentum of a system of bodies is equal to the vector sum of the momentum of all bodies included in the system:

Law of conservation of momentum

If the system of interacting bodies is additionally acted upon by external forces, for example, then in this case the relation is valid, which is sometimes called the law of momentum change:

For a closed system (in the absence of external forces), the law of conservation of momentum is valid:

The law of conservation of momentum can explain the phenomenon of recoil when shooting from a rifle or during artillery fire. Also, the law of conservation of momentum underlies the operating principle of all jet engines.

When solving physical problems, the law of conservation of momentum is used when knowledge of all the details of the movement is not required, but the result of the interaction of bodies is important. Such problems, for example, are problems about the impact or collision of bodies. The law of conservation of momentum is used when considering the motion of bodies of variable mass such as launch vehicles. Most of the mass of such a rocket is fuel. During the active phase of the flight, this fuel burns out, and the mass of the rocket in this part of the trajectory quickly decreases. Also, the law of conservation of momentum is necessary in cases where the concept is not applicable. It is difficult to imagine a situation where a stationary body acquires a certain speed instantly. In normal practice, bodies always accelerate and gain speed gradually. However, when electrons and other subatomic particles move, their state changes abruptly without remaining in intermediate states. In such cases, the classical concept of “acceleration” cannot be applied.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3