Laws of conservation of momentum and mechanical energy. Start in science 3 laws of conservation of momentum and energy

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Tomsk: TUSUR, 2012.- 136 p.

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TABLE OF CONTENTS
Introduction 6
1 Coordinate method. Vectors 9
1.1 Definitions of primary physical terms 9
1.2 Coordinate system 10
1.3 Speed and acceleration 11
1.4 Coordinate change as an integral of velocity 12
1.5 Generalization to the case of three-dimensional motion 13
1.6 Vectors 14
1.7 Vector Algebra 16
2 Kinematics of material point 19
2.1 Curvilinear speed and acceleration 19
2.2 Cross product 21
2.3 Kinematics of rotary motion 24
2.4 Movement of a body thrown at an angle to the horizontal 26
3 Laws of motion 29
3.1 The concept of force 29
3.2 Newton's second law. Weight 30
3.3 Newton's third law 31
3.4 Inertial frames of reference 33
3.5 Non-inertial frames of reference 34
3.6 Galileo's principle of relativity 35
3.7 Examples of various forces 36
4 Momentum and energy 40
4.1 Center of inertia (center of mass) of an extended body 40
4.2 Determining the position of the center of mass of simple bodies 42
4.3 Body momentum 43
4.4 Mechanical work and kinetic energy 44
4.5 Conservative forces 46
4.6 Potential energy. Gradient 47
4.7 The law of conservation of mechanical energy 49
5 Collision of two particles 51
5.1 Internal energy mechanical system 51
5.2 Classification of double collisions 52
5.3 Absolutely elastic central (frontal) impact 53
5.4 Absolutely inelastic impact 54
5.5 Collision in the C-system 55
5.6 Absolutely elastic non-central impact 55
6 Fluid mechanics 58
6.1 Pascal's law 58
6.2 Hydrostatic pressure. Strength of Archimedes 59
6.3 Stationary flow of an ideal fluid 60
6.4 Examples of using the Bernoulli equation 62
6.5 Viscous friction 64
6.6 The flow of a viscous liquid through a pipe 65
6.7 Turbulent flow. Reynolds number 66
6.8 Resistance forces when bodies move in a viscous fluid 67
7 Elastic properties of solids 69
7.1 Stress and strain 69
7.2 Hooke's law. Young's modulus and Poisson's ratio 71
7.3 Energy of elastic deformation of the medium 72
7.4 All-round compression 72
7.5 Compressive deformation of a fixed bar 73
7.6 Thermal deformation of solids 74
7.7 Shear deformation 75
8 Dynamics of a rigid body 78
8.1 Moment of inertia of a rigid body 78
8.2 Moments of inertia of some simple bodies 79
8.3 Moment of force 81
8.4 Angular torque 82
8.5 Rotational dynamics 83
8.6 Rolling a round body down an inclined plane 84
9 3D rotation of rigid bodies 87
9.1 Tensor of the moment of inertia of a rigid body 87
9.2 Energy and angular momentum of an asymmetric body 89
9.3 Gyroscope 89
9.4 Centrifugal and Coriolis forces 91
10 Strength gravity 94
10.1 Newton's law of gravity 94
10.2 Gravity near extended bodies 96
10.3 Tidal forces 98
10.4 Kepler problem 99
10.5 Parameters of elliptical orbits 101
10.6 Algorithm for calculating the trajectory of a celestial body 103
11 Harmonics 104
11.1 Small vibrations 104
11.2 Energy oscillatory motion 106
11.3 Addition of one-dimensional oscillations. Beats 106
11.4 Addition of mutually perpendicular vibrations 107
11.5 Oscillations of coupled pendulums 108
12 The principle of relativity 112
12.1 The speed of light and Einstein's postulate 112
12.2 Lorentz transformations 114
12.3 Consequences of Lorentz transformations 116
12.3.1 Relativity of simultaneity 116
12.3.2 Relativity of segment lengths 117
12.3.3 Relativity of time intervals between events. . 118
12.4 Speed addition 119
12.5 Light aberration 120
13 Relativistic dynamics 122
13.1 Relativistic momentum 122
13.2 Energy of relativistic particles 123
13.3 Law of conservation of total energy 124
13.4 Inelastic collision of two relativistic particles 126
13.5 Four-dimensional space-time 127
13.6 Dot product of 4-vectors 129
13.7 optical effect Doppler 131
Conclusion 134
Literature 135

This manual contains 13 chapters on the main sections of mechanics, provided by the basic standard of physical education for students of technical specialties of universities.
At the original methodological level, the manual outlines the basics of the coordinate method and the vector conceptual apparatus of mechanics, the basics of kinematics and dynamics of the translational and rotational motion of a rigid body, the laws of conservation of energy and momentum of mechanical systems, the mechanics of fluids and elastic solids, the classical theory of gravity and motion of celestial bodies, basic properties of harmonic oscillations, physical foundations special theory of relativity.
The content of the chapters is a coherent and consistent presentation of the material, in which the most important elements are specially highlighted: definitions of new terms, statements that have the force of theorems, facts or provisions that require special attention from the reader. At the end of each chapter is a list of control questions that the reader should be able to answer during the colloquium or conversation with the teacher.
All vector quantities in formulas and text are indicated in bold, for example, the velocity vector v. The scalar product of vectors is denoted by a dot between the factor vectors - Fv, and the vector product by a cross - g xp. Parentheses in mathematical formulas are used only for standard grouping of mathematical operations and designation of function arguments.
This manual on physics is presented in the most concise, but quite informative language. In general, this manual seems to be useful not only for first-year students, but also for all graduates of technical universities. Physics teachers will also find new approaches in the presentation of some sections.

work and energy. Laws of conservation of energy and momentum

Work and power

Law of conservation of momentum.

Energy. Potential and kinetic energy. Law of energy conservation.

Work and power

When a body moves under the action of a certain force, the action of the force is characterized by a quantity called mechanical work.

mechanical work- a measure of the action of a force, as a result of which the bodies make a movement.

The work of a constant force. If the body moves in a straight line under the action of a constant force that makes a certain angle  with the direction of movement (Fig. 1), the work is equal to the product of this force by the displacement of the point of application of the force and by the cosine of the angle  between the vectors and; or the work is equal to the scalar product of the force vector and the displacement vector:

Variable force work. To find the work of a variable force, the distance traveled is divided into big number small sections so that they can be considered rectilinear, and the force acting at any point of this section is constant.

Elementary work (i.e. work on an elementary section) is , and all work of a variable force along the entire path S is found by integration: .

As an example of the work of a variable force, consider the work done during the deformation (stretching) of a spring that obeys Hooke's law.

If the initial strain x 1 =0, then.

When a spring is compressed, the same work is done.

G graphic image of the work (Fig. 3).

On the graphs, the work is numerically equal to the area of the shaded figures.

To characterize the speed of doing work, the concept of power is introduced.

The power of a constant force is numerically equal to the work done by this force per unit of time.

1 W is the power of a force that does 1 J of work in 1 second.

In the case of variable power (different work is done for small equal time intervals), the concept of instantaneous power is introduced:

where is the velocity of the force application point.

That. power is equal to the scalar product of the force and the speed of the point of its application.

2. Law of conservation of momentum.

A mechanical system is a set of bodies allocated for consideration. The bodies that form a mechanical system can interact both with each other and with bodies that do not belong to this system. In accordance with this, the forces acting on the bodies of the system are divided into internal and external.

internal called the forces with which the bodies of the system interact with each other

External are called forces due to the influence of bodies that do not belong to this system.

Closed(or isolated) is a system of bodies that is not acted upon by external forces.

For closed systems, three physical quantities turn out to be unchanged (conserved): energy, momentum, and angular momentum. In accordance with this, there are three laws of conservation: energy, momentum, angular momentum.

R Let's consider a system consisting of 3 bodies, the impulses of which and on which external forces act (Fig. 4). According to Newton's 3rd law, the internal forces are equal in pairs and oppositely directed:

Internal forces:

We write down the basic equation of dynamics for each of these bodies and add these equations term by term

For N bodies:

The sum of the impulses of the bodies that make up the mechanical system is called the impulse of the system:

Thus, the time derivative of the impulse of a mechanical system is equal to the geometric sum of external forces acting on the system,

For a closed system .

Law of conservation of momentum: momentum of a closed system of material points remains constant.

From this law follows the inevitability of recoil when firing from any weapon. A bullet or projectile at the moment of a shot receives an impulse directed in one direction, and a rifle or a gun receives an impulse directed in the opposite direction. To reduce this effect, special recoil devices are used, in which the kinetic energy of the gun is converted into the potential energy of elastic deformation and into the internal energy of the recoil device.

The law of conservation of momentum underlies the movement of ships (submarines) with the help of paddle wheels and propellers, and water-jet marine engines (the pump sucks in outboard water and throws it behind the stern). In this case, a certain amount of water is thrown back, taking with it a certain momentum, and the ship acquires the same forward momentum. The same law underlies jet propulsion.

Absolutely inelastic impact- a collision of two bodies, as a result of which the bodies are combined, moving on as a whole. With such an impact, the mechanical energy is partially or completely converted into the internal energy of the colliding bodies, i.e. the law of conservation of energy is not fulfilled, only the law of conservation of momentum is fulfilled.

The theory of absolutely elastic and absolutely inelastic impacts is used in theoretical mechanics to calculate stresses and strains caused in bodies by impact forces. When solving many impact problems, they often rely on the results of various bench tests, analyzing and generalizing them. Impact theory is widely used in calculations of explosive processes; It is used in elementary particle physics in calculations of collisions of nuclei, in the capture of particles by nuclei, and in other processes.

A great contribution to the theory of impact was made by the Russian academician Ya.B. Zel'dovich, who, developing the physical foundations of rocket ballistics in the 1930s, solved the difficult problem of hitting a body flying at high speed over the surface of a medium.

3. Energy. Potential and kinetic energy. Law of energy conservation.

All previously introduced values characterized only mechanical motion. However, there are many forms of motion of matter; there is a constant transition from one form of motion to another. You must enter physical quantity characterizing the motion of matter in all forms of its existence, with the help of which it would be possible to quantitatively compare various forms of motion of matter.

Energy- a measure of the movement of matter in all its forms. The main property of all types of energy is interconvertibility. The amount of energy that a body possesses is determined by the maximum work that the body can do, having used up its energy completely. Energy is numerically equal to the maximum work that the body can do, and is measured in the same units as the work. During the transition of energy from one type to another, it is necessary to calculate the energy of the body or system before and after the transition and take their difference. This difference is called work:

Thus, the physical quantity characterizing the ability of a body to perform work is called energy.

The mechanical energy of a body can be due either to the movement of the body at a certain speed, or to the presence of the body in a potential field of forces.

Kinetic energy.

The energy possessed by a body due to its motion is called kinetic. The work done on the body is equal to the increment of its kinetic energy.

Let's find this work for the case when the resultant of all forces applied to the body is equal to .

The work done by the body due to kinetic energy is equal to the loss of this energy.

Potential energy.

If at each point in space other bodies act on the body with a force, the magnitude of which may be different at different points, the body is said to be in a field of forces or a force field.

If the lines of action of all these forces pass through one point - the force center of the field - and the magnitude of the force depends only on the distance to this center, then such forces are called central, and the field of such forces is called central (gravitational, electric field of a point charge).

The field of forces constant in time is called stationary.

A field in which the lines of action of forces are parallel straight lines located at the same distance from each other is homogeneous.

All forces in mechanics are divided into conservative and non-conservative (or dissipative).

Forces whose work does not depend on the shape of the trajectory, but is determined only by the initial and final position of the body in space, are called conservative.

The work of conservative forces along a closed path is zero. All central forces are conservative. The forces of elastic deformation are also conservative forces. If only conservative forces act in the field, the field is called potential (gravitational fields).

Forces whose work depends on the shape of the path are called non-conservative (friction forces).

Potential energy is called the part of the total mechanical energy of the system, which is determined only mutual arrangement bodies that make up the system, and the nature of the forces of interaction between them. Potential energy is the energy possessed by bodies or body parts due to their relative position.

concept potential energy is entered as follows. If the body is in a potential field of forces (for example, in the gravitational field of the Earth), each point of the field can be associated with some function (called potential energy) so that the work BUT 12 , performed over the body by the forces of the field when it moves from an arbitrary position 1 to another arbitrary position 2, was equal to the decrease of this function on the path 12:

where and are the values of the potential energy of the system in positions 1 and 2.

W

The written relation allows one to determine the potential energy value up to some unknown additive constant. However, this circumstance does not matter, because. all ratios include only the difference in potential energies corresponding to two positions of the body. In each specific problem, it is agreed to consider the potential energy of a certain position of the body equal to zero, and take the energy of other positions relative to the zero level. The specific form of the function depends on the nature of the force field and the choice of the zero level. Since the zero level is chosen arbitrarily, it can have negative values. For example, if we take as zero the potential energy of a body located on the surface of the Earth, then in the field of gravity forces near the earth's surface, the potential energy of a body of mass m, raised to a height h above the surface, is (Fig. 5).

where is the displacement of the body under the action of gravity;

The potential energy of the same body lying at the bottom of a well with depth H is equal to

In the considered example, it was about the potential energy of the Earth-body system.

Potential energy can be possessed not only by a system of interacting bodies, but by a single body. In this case, the potential energy depends on the relative position of the body parts.

Let us express the potential energy of an elastically deformed body.

The potential energy of elastic deformation, if we assume that the potential energy of an undeformed body is zero;

where k- coefficient of elasticity, x- deformation of the body.

In the general case, a body can simultaneously possess both kinetic and potential energies. The sum of these energies is called full mechanical energy bodies:

The total mechanical energy of a system is equal to the sum of its kinetic and potential energies. The total energy of the system is equal to the sum of all types of energy that the system possesses.

The law of conservation of energy is the result of a generalization of many experimental data. The idea of this law belongs to Lomonosov, who stated the law of conservation of matter and motion, and the quantitative formulation was given by the German physician Mayer and the naturalist Helmholtz.

Law conservation of mechanical energy: in the field of only conservative forces, the total mechanical energy remains constant in an isolated system of bodies. The presence of dissipative forces (forces of friction) leads to dissipation (scattering) of energy, i.e. converting it into other types of energy and violating the law of conservation of mechanical energy.

The law of conservation and transformation of total energy: the total energy of an isolated system is a constant value.

Energy never disappears and does not appear again, but only changes from one form to another in equivalent quantities. This is the physical essence of the law of conservation and transformation of energy: the indestructibility of matter and its motion.

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Energy and momentum are the most important concepts in physics. It turns out that conservation laws play an important role in nature in general. The search for conserved quantities and the laws from which they can be obtained is the subject of research in many branches of physics. Let us derive these laws in the simplest way from Newton's second law.

Law of conservation of momentum.Pulse, or amount of movementp defined as the product of the mass m material point per speed V: p= mV. Newton's second law, using the definition of momentum, is written as

= dp= F, (1.3.1)

here F is the resultant of the forces applied to the body.

closed system called a system in which the sum of external forces acting on the body is equal to zero:

F= å Fi= 0 . (1.3.2)

Then the change in the momentum of the body in a closed system according to Newton's second law (1.3.1), (1.3.2) is

dp= 0 . (1.3.3)

In this case, the momentum of the particle system remains constant:

p= å pi= const . (1.3.4)

This expression is law of conservation of momentum, which is formulated as follows: when the sum of external forces acting on a body or system of bodies is equal to zero, the momentum of the body or system of bodies is a constant value.

Law of energy conservation. In everyday life, by the concept of "work" we understand any useful work of a person. In physics, it is studied mechanical work, which occurs only when the body moves under the action of a force. Mechanical work ∆A is defined as the scalar product of the force F applied to the body, and body displacement Δ r as a result of this force:

A A= (F, Δ r) = F A r cosα. (1.3.5)

In formula (1.3.5), the sign of work is determined by the sign of cos α.

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

in this case no mechanical work is done either. If a system of bodies can do work, then it has energy.

Energy is one of the most important concepts not only in mechanics, but also in other areas of physics: thermodynamics and molecular physics, electricity, optics, atomic, nuclear and particle physics.

In any system that belongs physical world, energy is conserved in any process. Only the form into which it passes can change. For example, when a bullet hits a brick, part of the kinetic energy (moreover, more) is converted into heat. The reason for this is the presence of a frictional force between the bullet and the brick, in which it moves with great friction. When the turbine rotor rotates, mechanical energy is converted into electrical energy, and at the same time, a current appears in a closed circuit. The energy released during the combustion of chemical fuels, i.e. energy molecular bonds, is converted into thermal energy. The nature of chemical energy is the energy of intermolecular and interatomic bonds, essentially representing molecular or atomic energy.

Energy is a scalar quantity that characterizes the ability of a body to do work:

E2-E1= ∆A. (1.3.6)

When mechanical work is performed, the energy of a body changes from one form to another. The energy of a body can be in the form of kinetic or potential energy.

The energy of mechanical movement

W kin = .

called kinetic energy forward movement of the body. Work and energy in the SI system of units are measured in joules (J).

Energy can be determined not only by the movement of bodies, but also by their mutual arrangement and shape. This energy is called potential.

Potential energy is possessed relative to each other by two loads connected by a spring, or by a body located at a certain height above the Earth. This last example refers to the gravitational potential energy when a body moves from one height above the Earth to another. It is calculated according to the formula

Presnyakova I.A. oneBondarenko M.A. one

Atayan L.A. 1

1 Municipal educational institution"Secondary school No. 51 named after the Hero Soviet Union A. M. Chislov Traktorozavodsky district of Volgograd "

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Introduction

In the world in which we live, everything flows and changes, but a person always hopes to find something unchanged. This unchanging must be the primary source of any movement - it is energy.

Relevance of the problem stems from an increased interest in the exact sciences. Objective possibilities for the formation of cognitive interest - experimental justification, as the main condition for scientific knowledge.

Object of study- energy and momentum.

Subject: laws of conservation of energy and momentum.

Objective:

Investigate the implementation of the laws of conservation of energy and momentum in various mechanical processes;

Develop skills research work learn how to analyze the result.

To achieve this goal, the following tasks:

- conducted an analysis of theoretical material on the topic of the study;

We studied the specifics of the operation of conservation laws;

Considered practical significance these laws.

Hypothesis research lies in the fact that the laws of conservation and the transformation of energy and momentum - the universal laws of nature.

Significance of work is to use the results of research in physics lessons, which determines the possibility of increasing new skills and abilities; the development of the project is expected through the creation of a site where further experimental studies will be disclosed.

Chapter I.

1.1 Types of mechanical energy

Energy is a general measure of various processes and types of interaction. Mechanical energy is a physical quantity that characterizes the ability of a body or system of bodies to do work. The energy of a body or system of bodies is determined by the maximum work that they are capable of doing under given conditions. Mechanical energy includes two types of energy - kinetic and potential. Kinetic energy is the energy of a moving body. To calculate the kinetic energy, we assume that on a body of mass m for a time t an unchanging force F, which causes a change in speed by an amount v-v 0 , and work is done A = fs(1), where s is the path traveled by the body in time t in the direction of the force. According to Newton's second law, we write Ft = m(v - v 0), whence F=m.The path traveled by the body during the time is determined through average speed:s =v Wed t.Since the motion is equally variable, then s = t.We can conclude that the kinetic energy of a body of mass m, moving forward at a speed v, provided that v 0 = 0 is equal to: E k \u003d (3). Under appropriate conditions, a change in potential energy is possible, due to which work is done.

Let's do an experiment: Let's compare the potential energy of the spring with the potential energy of the raised body. Equipment: a tripod, a training dynamometer, a ball weighing 50 g, threads, a measuring ruler, training scales, weights. Let's determine the height of the ball's rise due to the potential energy of the stretched spring, using the law of conservation of mechanical energy. Let's conduct an experiment and compare the results of calculation and experience.

Work order .

1. Let's measure the mass with the help of scales m ball.

2. We fix the dynamometer on a tripod and tie the ball to the hook. Note the initial deformation x 0 springs corresponding to the dynamometer reading F 0 =mg.

3. Hold the ball on the table surface, raise the foot of the tripod with the dynamometer so that the dynamometer shows the force F 0 + F 1 , where F 1 = 1 N, with a dynamometer spring extension equal to x 0 + x 1 .

4. Calculate the height H T, to which the ball must rise under the action of the elastic force of the stretched spring in the field of gravity: H T =

5. Let's release the ball and note the height with a ruler H E to which the ball rises.

6. Repeat the experiment, raising the dynamometer so that its elongation is equal to x 0 + x 2 , x 0 + x 3 , which corresponds to the readings of the dynamometer F 0 + F 2 and F 0 + F 3 , where F 2 = 2 N, F 3 = 3 N.

7. Calculate the height of the ball in these cases and make the appropriate height measurements using a ruler.

8. the results of measurements and calculations are entered in the reporting table.

				H T, m	H E, m

kx 2 /2= mgH (0.0125 J= 0,0125J)

9. For one of the experiments, we estimate the reliability of the verification of the law of conservation of energy = mgH .

1.2. Law of energy conservation

Consider the process of changing the state of a body raised to a height h. However, its potential energy E p= mh. The body began to fall freely ( v 0 = 0). At the start of the fall E p = max, and E k \u003d 0. However, the sum of kinetic and potential energies at all intermediate points of the path remains unchanged if energy is not dissipated by friction, etc. therefore, if there is no conversion of mechanical energy into other forms of energy, then Ep+E k = const. Such a system is classified as conservative. The energy of a closed conservative system remains constant for all processes and transformations occurring in it. Energy can pass from one type to another (mechanical, thermal, electrical, etc.), but its total amount remains constant. This provision is called the law of conservation and transformation of energy. .

Let's do an experiment: Let's compare the changes in the potential energy of a stretched spring with the change in the kinetic energy of the body.

	F at							E k	Δ E k

Equipment : two tripods for frontal work, a training dynamometer, a ball, threads, sheets of white and carbon paper, a measuring ruler, training scales with a tripod, weights. Based on the law of conservation and transformation of energy during the interaction of bodies by elastic forces, the change in the potential energy of a stretched spring should be equal to the change kinetic energy of the body connected with it, taken with the opposite sign: Δ E p= - ∆ E k.For experimental verification of this statement, you can use the installation. We fix the dynamometer in the foot of the tripod. We tie a ball to its hook on a thread 60-80 cm long. On another tripod, at the same height with a dynamometer, we fix a gutter in the foot. Having installed the ball on the edge of the chute, and holding it, we move the second tripod away from the first one by the length of the thread. If you move the ball away from the edge of the chute by x, then as a result of deformation, the spring will acquire a supply of potential energy Δ E p = , where k- spring stiffness. Then we release the ball. Under the action of the elastic force, the ball acquires speed υ . Neglecting the losses caused by the action of the friction force, we can assume that the potential energy of the stretched spring is completely converted into the kinetic energy of the ball:. The speed of a ball can be determined by measuring its flight range s in free fall from a height h. From expressions v= and t= it follows that v= s. Then Δ E k= = . Taking into account equality F at = kx we get: =.

kx2/2 = (mv) 2 /2

0.04 \u003d 0.04. Let's estimate the error limits for measuring the potential energy of an extended spring. Since E p =, then the relative error limit is: = + = +. The absolute error limit is: Δ Ep=E p. Let us estimate the limits of errors in measuring the kinetic energy of the ball. Because E k = , then the relative error limit is equal to: = + ? +? g + ? h.Inaccuracies,? g and? h compared with the error?s can be neglected. In this case, ≈ 2? = 2. The conditions of the experiment on measuring the flight range are such that the deviations of the results of individual measurements from the average are much higher than the boundary of the systematic error (Δs random Δ s syst), so we can assume that ∆s cf ≈ ∆s random. The boundary of the random error of the arithmetic mean with a small number of measurements N is found by the formula: Δs av = ,

where is calculated by the formula:

Thus, = 6. The limit of the absolute error in measuring the kinetic energy of the ball is: Δ E k = E k .

Chapter II.

2.1. Law of conservation of momentum

The momentum of a body (momentum) is the product of the body's mass and its speed. Momentum is a vector quantity. SI unit of momentum: = kg*m/s = N*s. If p is the momentum of the body, m- body mass, v is the speed of the body, then = m(one). A change in the momentum of a body of constant mass can only occur as a result of a change in speed and is always due to the action of a force. If Δp is a change in momentum, m- body weight, Δ v = v 2 -v 1 - speed change, F- constant force accelerating the body, Δ t- the duration of the force, then according to the formulas = m and = . We have = m= m,

Taking into account expression (1), we obtain: Δ = mΔ = Δ t (2).

Based on (6), we can conclude that the changes in the momenta of two interacting bodies are identical in absolute value, but opposite in direction (if the momentum of one of the interacting bodies increases, then the momentum of the other body decreases by the same amount), and based on (7), that the sums the impulses of the bodies before the interaction and after the interaction are equal, i.e. the total momentum of the bodies does not change as a result of the interaction. The law of conservation of momentum is valid for a closed system with any number of bodies: = = constant. The geometric sum of the impulses of a closed system of bodies remains constant for any interactions of the bodies of this system with each other, i.e. momentum of a closed system of bodies is conserved.,

Let's experiment: Let us check the fulfillment of the law of conservation of momentum.

Equipment: tripod for frontal work; the tray is arched; balls with a diameter of 25mm-3pcs; measuring ruler 30 cm long with millimeter divisions; sheets of white and carbon paper; educational scales; weights. Let us check the fulfillment of the law of conservation of momentum in the case of a direct central collision of balls. According to the law of conservation of momentum, for any interactions of bodies, the vector sum

m 1 kg

m 2 kg

l 1. m

v 1 .m/s

p 1. kg*m/s

l ′ 1

l ′ 2

v ′ 1

v ′ 2

p ′ 1

p ′ 2

central

impulses before the interaction is equal to the vector sum of the impulses of the bodies after the interaction. The validity of this law can be verified experimentally by studying the collisions of balls on the installation. To communicate a certain momentum to the ball in the horizontal direction, we use an inclined tray with a horizontal section. The ball, having rolled down from the tray, moves along a parabola until it hits the table surface. Speed projections

the ball and its momentum on the horizontal axis during free fall do not change, since there are no forces acting on the ball in the horizontal direction. Having determined the momentum of one ball, we conduct an experiment with two balls, placing the second ball on the edge of the tray, and launch the first ball in the same way as in the first experiment. After the collision, both balls fly off the tray. According to the law of conservation of momentum, the sum of the momenta of the first and second balls before the collision should be equal to the sum of the momentum and these balls after the collision: + = + (1). balls), and both balls after the collision move along the same straight line and in the same direction in which the first ball moved before the collision, then from the vector form of the law of conservation of momentum we can go to the algebraic form: p 1 +p 2 = p ′ 1 +p ′ 2 , or m 1 v 1 + m 2 v 2 = m 1 v ′ 1 + m 2 v ′ 2 (2). Since the speed v 2 of the second ball before the collision was equal to zero, then expression (2) is simplified: m 1 v 1 = m 1 v ′ 1 + m 2 v ′ 2 (3)

To check the fulfillment of equality (3), we measure the masses m 1 and m 2 balls and calculate the speed v 1 , v ′ 1 and v ′ 2 . During the movement of the ball along the parabola, the projection of the velocity on the horizontal axis will not change; it can be found by range l ball flight in horizontal direction and time t its free fall ( t=):v= = l(4). p1 = p′1 + p′2

0.06 kg*m/s = (0.05+0.01) kg*m/s

0.06 kg*m/s=0.06 kg*m/s

We have verified that the law of conservation of momentum is satisfied in the case of a direct central collision of balls.

Let's experiment: compare the momentum of the elastic force of the spring with the change in the momentum of the projectile. Equipment: double-sided ballistic pistol; technical scales with weight; calipers; level; measuring tape; plumb; spring dynamometer for a load of 4 N; tripod laboratory with the coupling; a plate with a wire loop; two sheets of writing and carbon paper each. It is known that the momentum of the force is equal to the change in the momentum of the body, which is affected by a constant force, i.e. Δ t = m- m. In this work, the elastic force of the spring acts on the projectile, which at the beginning of the experiment is at rest ( v 0 = 0): the shot is fired by projectile 2, and projectile 1 at this time is firmly held by hand on the platform. Therefore, this relation in scalar form can be rewritten as follows: ft=mv, where F- the average elastic force of the spring, equal to, t- time of action of the spring force, m- projectile mass 2, v is the horizontal component of the projectile velocity. The maximum elastic force of the spring and the mass of projectile 2 are measured. Speed v we calculate from the relation v=, where is a constant value, and h- height and s - range of the projectile are taken from experience. The time of action of the force is calculated from two equations: v=at and v 2 = 2ax, i.e. t=, where x- the amount of deformation of the spring. To find the value x measure the length of the protruding part of the spring at the first projectile l, and the second one has the length of the protruding rod and add them up: x = l 1 +l 2 . We measure the flight range s (distance from the plumb line to the average point) and the height of the fall h. Then we determine the mass of the projectile on the scales m 2 and measuring with a caliper l 1 and l 2 , we calculate the value of the deformation of the spring x. After that, at projectile 1, we unscrew the ball and clamp it with a plate with a loop of wire. We connect the shells and hook the dynamometer hook to the loop. Holding the shell with 2 hand, we compress the spring with a dynamometer (while the shells should connect) and determine the elastic force of the spring Knowing the flight range and the height of the fall, we calculate the speed of the projectile

						mv, 10 -2 kg*m/s	ft, 10 -2 kg*m/s

v=, and then the time of action of the force t = . Finally, we calculate the change in momentum of the projectile mv and momentum of force ft. The experiment is repeated three times, changing the force of elasticity of the spring, and all the results of measurements and calculations are entered in the table. The results of the experiment at h= 0.2 m and m= 0.28 kg will be: mv=Ft (3.47*10-2 kg*m/s =3.5*10-2 kg*m/s)

	F max, N					s(from experience)m

The coincidence of the final results within the measurement accuracy confirms the momentum conservation law. mv=Ft(3.47*10 -2 kg * m / s \u003d 3.5 * 10 -2 kg*m/s). Substituting these expressions into formula (1) and expressing the acceleration in terms of the average spring force, i.e. a=, we get the formula for calculating the range of the projectile: s = . Thus, by measuring F max, projectile weight m, drop height h and spring deflection x = l 1 +l 2 , we calculate the range of the projectile and check it experimentally. We perform the experiment at least twice, changing the elasticity of the spring, the mass of the projectile or the height of the fall.

Chapter III.

3.1. Devices according to the laws of conservation of energy and momentum

Newton's pendulum

Newton's cradle (Newton's pendulum) is a mechanical system named after Isaac Newton to demonstrate the transformation of various types of energy into each other: kinetic into potential and vice versa. In the absence of opposing forces (friction), the system could operate forever, but in reality this is unattainable. If the first ball is deflected and released, then its energy and momentum will be transferred unchanged through the three middle balls to the last one, which will acquire the same speed and rise to the same height. According to Newton's calculations, two balls with a diameter of 30 cm, located at a distance of 0.6 cm, will converge under the action of the force of mutual attraction one month after the start of movement (the calculation is made in the absence of external resistance). Newton took the density of the balls equal to the average density of the earth: p 5 * 10^3 kg/m^3 .

At a distance l = 0.6 cm = 0.006 m between the surfaces of balls with a radius R = 15 cm = 0.15 m, the force acts on the balls

F? \u003d GM² / (2R + l)². When the balls come into contact, a force acts on them

F? = GM²/(2R)². F?/F? = (2R)²/(2R+l)² = (2R/(2R+l))² = (0.3/(0.3 + 0.006))² = 0.996 ≈ 1 so the assumption is correct. The mass of the ball is:

M \u003d ρ (4/3) pR³ \u003d 5000 * 4 * 3.14 * 0.15³ / 3 \u003d 70.7 kg. The interaction force is

F = GM²/(2R)² = 6.67.10?¹¹.70.7²/0.3² = 3.70.10?? H. The acceleration due to gravity is: a = F/M = 3.70.10??/70.7 = 5.24.10?? m/s². Path: s = l/2 = 0.6/2 = 0.3 cm = 0.003 m the ball will pass in time t equal to t = √2S/a = √(2*0.003/5.24.10??) = 338 c = 5.6 min. So Newton was wrong: it looks like the balls will converge quite quickly - in 6 minutes.

Maxwell pendulum

Maxwell's pendulum is a disk (1) tightly mounted on a rod (2) on which threads (3) are wound (Fig. 2.1). The disk of the pendulum is directly the disk itself and replaceable rings that are fixed on the disk. When the pendulum is released, the disk begins to move: translational down and rotational around its axis of symmetry. The rotation, continuing by inertia at the lowest point of the movement (when the threads are already unwound), leads again to the winding of the threads on the rod, and, consequently, to the rise of the pendulum. After that, the movement of the pendulum slows down again, the pendulum stops and starts its downward movement again, etc. The acceleration of the translational movement of the center of mass of the pendulum (a) can be obtained from the measured time t and the distance h traveled by the pendulum from the equation. .The mass of the pendulum m is the sum of the masses of its parts (axis m0, disk md and ring mk):

The moment of inertia of the pendulum J is also an additive quantity and is determined by the formula

Where, - respectively, the moments of inertia of the axis, disk and ring of the pendulum.

The moment of inertia of the pendulum axis is, where r- axis radius, m 0 = 0.018 kg - mass of the axle. The moments of inertia of the disk can be found as

Where R d - disk radius, m d \u003d 0.018 kg - mass of the disk. The moment of inertia of the ring is calculated by the formula average ring radius, m k is the mass of the ring, b is the width of the ring. Knowing the linear acceleration a and angular acceleration ε(ε · r), can be found angular velocity its rotation ( ω ):, The total kinetic energy of the pendulum is the sum of the energy of the translational movement of the center of mass and the energy of rotation of the pendulum around the axis:

Conclusion.

Conservation laws form the foundation on which the continuity of physical theories is based. Indeed, considering the evolution of the most important physical concepts in the field of mechanics, electrodynamics, the theory of heat, modern physical theories, we were convinced that these theories invariably contain either the same classical conservation laws (energy, momentum, etc.), or along with with them new laws appear, forming the core around which the interpretation of experimental facts takes place. "The commonality of conservation laws in old and new theories is another form of internal interconnection of the latter." It is difficult to overestimate the role of the momentum conservation law. He is general rule obtained by a person on the basis of long experience. Skillful use of the law makes it relatively easy to solve such practical problems as forging products in a blacksmith shop, driving piles in the construction of buildings.

Application.

Our compatriots I. V. Kurchatov, L. A. Artsimovich investigated one of the first nuclear reactions, proved the validity of the law of conservation of momentum in such reactions. Currently controlled chain nuclear reactions solve the energy problems of mankind.

Literature

1. World Encyclopedia

2. Dick Yu.I., Kabardin O.F. "Physical workshop for classes with in-depth study of physics." Moscow: "Enlightenment", 1993 - p. 93.

3. Kuhling H. Handbook of physics; translated from German 2nd ed. M, Mir, 1985 - p.120.

4. Pokrovsky A.A. "Workshop on physics in high school". Moscow: "Enlightenment", 1973 - p. 45.

5. Pokrovsky A.A. "Workshop on physics in high school". Moscow: 2nd edition, "Enlightenment", 1982 - p. 76.

6. Rogers E. “Physics for the Curious. Volume 2. "Moscow: "Mir", 1969 - p.201.

7. Shubin A.S. "Course of General Physics". Moscow: " graduate School", 1976 - p.224.

mechanical energy.

Dependences of momentum on the speed of motion of two bodies. Which body has the greater mass and by how much? 1) The masses of the bodies are the same 2) The mass of the body 1 is 3.5 times greater 3) The mass of the body 2 is 3.5 times greater 4) According to the graphs, the masses of the bodies cannot be compared

Moving at a speed v, it collides with a resting plasticine ball of mass 2t. After the impact, the balls stick together and move together. What is the speed of their movement? 1) v/3 2) 2v/3 3) v/2 4) Not enough data to answer

They move along a rectilinear railway track with speeds, the dependence of the projections of which on an axis parallel to the tracks on time is shown in the figure. After 20 seconds, an automatic coupling occurred between the cars. At what speed and in which direction will the coupled wagons go? 1) 1.4 m/s, towards initial movement 1. 2) 0.2 m/s, towards initial movement 1. 3) 1.4 m/s, towards initial movement 2. 4) 0.2 m/s, in the direction of the initial movement 2.

The value showing what work can be done by the body Perfect work is equal to the change in the energy of the body

According to the equation x: = 2 + 30 t - 2 t2, written in SI. Body weight 5 kg. What is the kinetic energy of the body 3 seconds after the start of motion? 1) 810 J 2) 1440 J 3) 3240 J 4) 4410 J

deformed body

This is done work 2 J. What work should be done to stretch the spring another 4 cm. 1) 16 J 2) 4 J 3) 8 J 4) 2 J

Determine the kinetic energy Ek that the body has at the top of the trajectory (see figure)? 1) EK=mgH 2) EK=m(V0)2/2 + mgh-mgH 3) EK=mgH-mgh 4) EK=m(V0)2/2 + mgH

same initial speed. The first time the ball's velocity vector was directed vertically downwards, the second time - vertically upwards, the third time - horizontally. Ignore air resistance. The modulus of the ball's speed when approaching the ground will be: 1) more in the first case 2) more in the second case 3) more in the third case 4) the same in all cases

Photo of the setup for studying the sliding of a carriage weighing 40 g along an inclined plane at an angle of 30º. At the moment of the start of the movement, the upper sensor turns on the stopwatch. When the carriage passes the bottom sensor, the stopwatch stops. Estimate the amount of heat released as the carriage slides down the inclined plane between the sensors.

It descends from point 1 to point 3 (Fig.). At which point of the trajectory does its kinetic energy have highest value? 1) At point 1. 2) At point 2. 3) At point 3. 4) At all points, the energy values are the same.

They rise along its opposite slope to a height of 2 m (to point 2 in the figure) and stop. The weight of the sled is 5 kg. Their speed at the bottom of the ravine was 10 m/s. How did the total mechanical energy of the sled change when moving from point 1 to point 2? 1) Hasn't changed. 2) Increased by 100 J. 3) Decreased by 100 J. 4) Decreased by 150 J. 2

Laws conservation as a reflection of symmetry in physics

Laws conservation energy in macroscopic processes

Law conservation momentum

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