BODY PULSE

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

Momentum vector body is directed in the same way as velocity vector this body.

The impulse of a system of bodies is understood as the sum of the impulses of all the bodies of this system: ∑p=p 1 +p 2 +... . The law of conservation of momentum: in a closed system of bodies, in any process, its momentum remains unchanged, i.e. ∑p = const.

(A closed system is a system of bodies that interact only with each other and do not interact with other bodies.)

Question2. Thermodynamic and statistical definition of entropy. The second law of thermodynamics.

Thermodynamic definition of entropy

The concept of entropy was first introduced in 1865 by Rudolf Clausius. He determined entropy change thermodynamic system at reversible process as the ratio of the change in the total amount of heat to the value of the absolute temperature:

This formula is applicable only for an isothermal process (occurring at a constant temperature). Its generalization to the case of an arbitrary quasi-static process looks like this:

where is the increment (differential) of entropy, and is an infinitely small increment in the amount of heat.

It is necessary to pay attention to the fact that the thermodynamic definition under consideration is applicable only to quasi-static processes (consisting of continuously successive equilibrium states).

Statistical definition of entropy: Boltzmann's principle

In 1877, Ludwig Boltzmann found that the entropy of a system can refer to the number of possible "microstates" (microscopic states) consistent with their thermodynamic properties. Consider, for example, an ideal gas in a vessel. The microstate is defined as the positions and impulses (moments of motion) of each atom constituting the system. Connectivity requires us to consider only those microstates for which: (I) the locations of all parts are located within the vessel, (II) to obtain the total energy of the gas, the kinetic energies of the atoms are summed. Boltzmann postulated that:

where we now know the constant 1.38 10 −23 J/K as the Boltzmann constant, and is the number of microstates that are possible in the existing macroscopic state (statistical weight of the state).

Second law of thermodynamics- a physical principle that imposes a restriction on the direction of the processes of heat transfer between bodies.

The second law of thermodynamics states that spontaneous transfer of heat from a body that is less heated to a body that is more heated is impossible.

Ticket 6.

1. § 2.5. Theorem on the motion of the center of mass

Relation (16) is very similar to the equation of motion of a material point. Let's try to bring it to an even simpler form F=m a. To do this, we transform the left side using the properties of the differentiation operation (y+z) =y +z , (ay) =ay , a=const:

(24)

Multiply and divide (24) by the mass of the entire system and substitute into equation (16):

. (25)

The expression in brackets has the dimension of length and determines the radius vector of some point, which is called center of mass of the system:

. (26)

In projections on the coordinate axes (26) takes the form

(27)

If (26) is substituted into (25), then we obtain a theorem on the motion of the center of mass:

those. the center of mass of the system moves as a material point, in which the entire mass of the system is concentrated, under the action of the sum of external forces applied to the system. The theorem on the motion of the center of mass states that no matter how complex the forces of interaction of the particles of the system with each other and with external bodies are, and no matter how difficult these particles move, you can always find a point (center of mass), the movement of which is described simply. The center of mass is a certain geometric point, the position of which is determined by the distribution of masses in the system and which may not coincide with any of its material particles.

The product of the mass of the system and the velocity v c.m of its center of mass, as follows from its definition (26), is equal to the momentum of the system:

(29)

In particular, if the sum of external forces is equal to zero, then the center of mass moves uniformly and rectilinearly or is at rest.

Example 1 At some point of the trajectory, the projectile breaks into many fragments (Fig. 9). How will their center of mass move?

The center of mass will “fly” along the same parabolic trajectory along which an unexploded projectile would move: its acceleration, in accordance with (28), is determined by the sum of all gravity forces applied to the fragments and their total mass, i.e. the same equation as the motion of a whole projectile. However, as soon as the first fragment hits the Earth, the Earth's reaction force will be added to the external forces of gravity and the movement of the center of mass will be distorted.

Example 2 A "pair" of forces begins to act on a body at rest F and F(Fig. 10). How will the body move?

Since the geometric sum of the external forces is zero, the acceleration of the center of mass is also zero and it will remain at rest. The body will rotate around a fixed center of mass.

Is there any advantage to the law of conservation of momentum over Newton's laws? What is the power of this law?

Its main advantage is that it has an integral character, i.e. relates the characteristics of the system (its momentum) in two states separated by a finite time interval. This allows one to obtain important information immediately about the final state of the system, bypassing the consideration of all its intermediate states and the details of the interactions that occur in this case.

2) The velocities of gas molecules have different values ​​and directions, and due to the huge number of collisions that a molecule experiences every second, its speed is constantly changing. Therefore, it is impossible to determine the number of molecules that have an exactly given speed v at a given moment of time, but it is possible to count the number of molecules whose speeds have values ​​lying between some speeds v 1 and v 2 . Based on the theory of probability, Maxwell established a pattern by which one can determine the number of gas molecules whose velocities at a given temperature are contained in a certain range of velocities. According to the Maxwell distribution, the probable number of molecules per unit volume; whose velocity components lie in the interval from to, from to, and from to, are determined by the Maxwell distribution function

where m is the mass of the molecule, n is the number of molecules per unit volume. It follows from this that the number of molecules whose absolute velocities lie in the interval from v to v + dv has the form

The Maxwell distribution reaches its maximum at the speed , i.e. a speed close to that of most molecules. The area of ​​the shaded strip with the base dV will show what part of the total number of molecules has velocities lying in this interval. The specific form of the Maxwell distribution function depends on the type of gas (the mass of the molecule) and temperature. The pressure and volume of the gas do not affect the distribution of molecules over velocities.

The Maxwell distribution curve will allow you to find the arithmetic mean speed

In this way,

With an increase in temperature, the most probable speed increases, so the maximum of the distribution of molecules in terms of speeds shifts towards higher speeds, and its absolute value decreases. Consequently, when the gas is heated, the proportion of molecules with low velocities decreases, and the proportion of molecules with high velocities increases.

Boltzmann distribution

This is the energy distribution of particles (atoms, molecules) of an ideal gas under conditions of thermodynamic equilibrium. The Boltzmann distribution was discovered in 1868 - 1871. Australian physicist L. Boltzmann. According to the distribution, the number of particles n i with total energy E i is:

n i =A ω i e E i /Kt (1)

where ω i is the statistical weight (the number of possible states of a particle with energy e i). The constant A is found from the condition that the sum of n i over all possible values ​​of i is equal to the given total number of particles N in the system (the normalization condition):

In the case when the movement of particles obeys classical mechanics, the energy E i can be considered as consisting of the kinetic energy E ikin of a particle (molecule or atom), its internal energy E iext (for example, the excitation energy of electrons) and potential energy E i , sweat in the external field depending on the position of the particle in space:

E i = E i, kin + E i, ext + E i, sweat (2)

The velocity distribution of particles is a special case of the Boltzmann distribution. It occurs when the internal excitation energy can be neglected

E i, ext and the influence of external fields E i, sweat. In accordance with (2), formula (1) can be represented as a product of three exponentials, each of which gives the distribution of particles over one type of energy.

In a constant gravitational field that creates an acceleration g, for particles of atmospheric gases near the surface of the Earth (or other planets), the potential energy is proportional to their mass m and height H above the surface, i.e. E i, sweat = mgH. After substituting this value into the Boltzmann distribution and summing it over all possible values ​​of the kinetic and internal energies of the particles, a barometric formula is obtained that expresses the law of decrease in the density of the atmosphere with height.

In astrophysics, especially in the theory of stellar spectra, the Boltzmann distribution is often used to determine the relative electron population of various energy levels of atoms. If we designate two energy states of an atom with indices 1 and 2, then from the distribution it follows:

n 2 / n 1 \u003d (ω 2 / ω 1) e - (E 2 - E 1) / kT (3) (Boltzmann formula).

The energy difference E 2 -E 1 for the two lower energy levels of the hydrogen atom is >10 eV, and the value of kT, which characterizes the energy of the thermal motion of particles for the atmospheres of stars like the Sun, is only 0.3-1 eV. Therefore, hydrogen in such stellar atmospheres is in an unexcited state. Thus, in the atmospheres of stars with an effective temperature Te > 5700 K (the Sun and other stars), the ratio of the numbers of hydrogen atoms in the second and ground states is 4.2 10 -9 .

The Boltzmann distribution was obtained in the framework of classical statistics. In 1924-26. quantum statistics was created. It led to the discovery of the Bose-Einstein (for particles with integer spin) and Fermi-Dirac (for particles with half-integer spin) distributions. Both of these distributions pass into a distribution when the average number of quantum states available for the system significantly exceeds the number of particles in the system, i.e. when there are many quantum states per particle, or, in other words, when the degree of occupation of quantum states is small. The applicability condition for the Boltzmann distribution can be written as an inequality:

where N is the number of particles, V is the volume of the system. This inequality is satisfied at a high temperature and a small number of particles per unit. volume (N/V). It follows from this that the larger the mass of particles, the wider the range of changes in T and N / V, the Boltzmann distribution is valid.

ticket 7.

The work of all applied forces is equal to the work of the resultant force(see fig. 1.19.1).

There is a connection between the change in the speed of a body and the work done by the forces applied to the body. This relationship is easiest to establish by considering the motion of a body along a straight line under the action of a constant force. In this case, the force vectors of displacement, velocity and acceleration are directed along one straight line, and the body performs a rectilinear uniformly accelerated motion. By directing the coordinate axis along the straight line of motion, we can consider F, s, u and a as algebraic quantities (positive or negative depending on the direction of the corresponding vector). Then the work done by the force can be written as A = fs. In uniformly accelerated motion, the displacement s is expressed by the formula

This expression shows that the work done by the force (or the resultant of all forces) is associated with a change in the square of the speed (and not the speed itself).

A physical quantity equal to half the product of the body's mass and the square of its speed is called kinetic energy bodies:

This statement is called kinetic energy theorem . The kinetic energy theorem is also valid in the general case when the body moves under the action of a changing force, the direction of which does not coincide with the direction of movement.

Kinetic energy is the energy of motion. Kinetic energy of a body of mass m moving at a speed is equal to the work that must be done by the force applied to a body at rest to tell it this speed:

In physics, along with the kinetic energy or the energy of motion, the concept plays an important role potential energy or interaction energies of bodies.

Potential energy is determined by the mutual position of the bodies (for example, the position of the body relative to the Earth's surface). The concept of potential energy can be introduced only for forces whose work does not depend on the trajectory of motion and is determined only by the initial and final positions of the body. Such forces are called conservative .

The work of conservative forces on a closed trajectory is zero. This statement is illustrated in Fig. 1.19.2.

The property of conservatism is possessed by the force of gravity and the force of elasticity. For these forces, we can introduce the concept of potential energy.

If a body moves near the surface of the Earth, then it is affected by a force of gravity that is constant in magnitude and direction. The work of this force depends only on the vertical movement of the body. On any section of the path, the work of gravity can be written in projections of the displacement vector onto the axis OY pointing vertically upwards:

This work is equal to a change in some physical quantity mgh taken with the opposite sign. This physical quantity is called potential energy bodies in the field of gravity

Potential energy E p depends on the choice of the zero level, i.e. on the choice of the origin of the axis OY. It is not the potential energy itself that has physical meaning, but its change Δ E p = E p2 - E p1 when moving the body from one position to another. This change does not depend on the choice of the zero level.

If we consider the motion of bodies in the gravitational field of the Earth at considerable distances from it, then when determining the potential energy, it is necessary to take into account the dependence of the gravitational force on the distance to the center of the Earth ( law of gravity). For the forces of universal gravitation, it is convenient to count the potential energy from an infinitely distant point, i.e., to assume that the potential energy of a body at an infinitely distant point is equal to zero. The formula expressing the potential energy of a body with a mass m on distance r from the center of the Earth, has the form ( see §1.24):

where M is the mass of the earth, G is the gravitational constant.

The concept of potential energy can also be introduced for the elastic force. This force also has the property of being conservative. By stretching (or compressing) a spring, we can do this in a variety of ways.

You can simply lengthen the spring by an amount x, or first lengthen it by 2 x, and then reduce the elongation to a value x etc. In all these cases, the elastic force does the same work, which depends only on the elongation of the spring x in the final state if the spring was initially undeformed. This work is equal to the work of the external force A, taken with the opposite sign ( see §1.18):

Potential energy of an elastically deformed body is equal to the work of the elastic force during the transition from a given state to a state with zero deformation.

If in the initial state the spring was already deformed, and its elongation was equal to x 1 , then upon transition to a new state with elongation x 2, the elastic force will do work equal to the change in potential energy, taken with the opposite sign:

In many cases it is convenient to use the molar heat capacity C:

where M is the molar mass of the substance.

The heat capacity thus determined is not unambiguous characterization of a substance. According to the first law of thermodynamics, the change in the internal energy of a body depends not only on the amount of heat received, but also on the work done by the body. Depending on the conditions under which the heat transfer process was carried out, the body could perform various work. Therefore, the same amount of heat transferred to the body could cause different changes in its internal energy and, consequently, temperature.

Such ambiguity in determining the heat capacity is typical only for a gaseous substance. When liquid and solid bodies are heated, their volume practically does not change, and the work of expansion turns out to be equal to zero. Therefore, the entire amount of heat received by the body goes to change its internal energy. Unlike liquids and solids, a gas in the process of heat transfer can greatly change its volume and do work. Therefore, the heat capacity of a gaseous substance depends on the nature of the thermodynamic process. Two values ​​of the heat capacity of gases are usually considered: C V is the molar heat capacity in an isochoric process (V = const) and C p is the molar heat capacity in an isobaric process (p = const).

In the process at a constant volume, the gas does no work: A \u003d 0. From the first law of thermodynamics for 1 mole of gas, it follows

where ΔV is the change in the volume of 1 mole of an ideal gas when its temperature changes by ΔT. This implies:

where R is the universal gas constant. For p = const

Thus, the relationship expressing the relationship between the molar heat capacities C p and C V has the form (Mayer's formula):

The molar heat capacity C p of a gas in a process with constant pressure is always greater than the molar heat capacity C V in a process with constant volume (Fig. 3.10.1).

In particular, this ratio is included in the formula for the adiabatic process (see §3.9).

Between two isotherms with temperatures T 1 and T 2 on the diagram (p, V) different transition paths are possible. Since for all such transitions the change in temperature ΔT = T 2 - T 1 is the same, therefore, the change ΔU of the internal energy is the same. However, the work A performed in this case and the amount of heat Q obtained as a result of heat transfer will be different for different transition paths. It follows that a gas has an infinite number of heat capacities. C p and C V are only particular (and very important for the theory of gases) values ​​of heat capacities.

Ticket 8.

1 Of course, the position of one, even "special", point does not completely describe the movement of the entire system of bodies under consideration, but still it is better to know the position of at least one point than not to know anything. Nevertheless, consider the application of Newton's laws to the description of the rotation of a rigid body around a fixed axes 1 . Let's start with the simplest case: let the material point of the mass m attached with a weightless rigid rod of length r to the fixed axis OO / (Fig. 106).

A material point can move around the axis, remaining at a constant distance from it, therefore, its trajectory will be a circle centered on the axis of rotation. Of course, the motion of a point obeys the equation of Newton's second law

However, the direct application of this equation is not justified: firstly, the point has one degree of freedom, so it is convenient to use the rotation angle as the only coordinate, and not two Cartesian coordinates; secondly, the reaction forces in the axis of rotation act on the system under consideration, and directly on the material point - the tension force of the rod. Finding these forces is a separate problem, the solution of which is redundant for describing rotation. Therefore, it makes sense to obtain, on the basis of Newton's laws, a special equation that directly describes the rotational motion. Let at some point in time a certain force acts on a material point F, lying in a plane perpendicular to the axis of rotation (Fig. 107).

In the kinematic description of curvilinear motion, the total acceleration vector a is conveniently decomposed into two components, the normal a n, directed to the axis of rotation, and tangential a τ directed parallel to the velocity vector. We do not need the value of normal acceleration to determine the law of motion. Of course, this acceleration is also due to acting forces, one of which is the unknown tensile force on the rod. Let us write the equation of the second law in the projection onto the tangential direction:

Note that the reaction force of the rod is not included in this equation, since it is directed along the rod and perpendicular to the selected projection. Changing the angle of rotation φ directly determined by the angular velocity

ω = ∆φ/∆t,

the change of which, in turn, is described by the angular acceleration

ε = ∆ω/∆t.

Angular acceleration is related to the tangential acceleration component by the relation

a τ = rε.

If we substitute this expression into equation (1), we obtain an equation suitable for determining the angular acceleration. It is convenient to introduce a new physical quantity that determines the interaction of bodies during their rotation. To do this, we multiply both sides of equation (1) by r:

mr 2 ε = F τ r. (2)

Consider the expression on its right side F τ r, which has the meaning of the product of the tangential component of the force by the distance from the axis of rotation to the point of application of the force. The same work can be presented in a slightly different form (Fig. 108):

M=F τ r = Frcosα = Fd,

here d is the distance from the axis of rotation to the line of action of the force, which is also called the shoulder of the force. This physical quantity is the product of the modulus of force and the distance from the line of action of the force to the axis of rotation (arm of force) M = Fd− is called the moment of force. The action of a force can result in both clockwise and counterclockwise rotation. In accordance with the chosen positive direction of rotation, the sign of the moment of force should also be determined. Note that the moment of force is determined by the component of the force that is perpendicular to the radius vector of the point of application. The component of the force vector directed along the segment connecting the point of application and the axis of rotation does not lead to untwisting of the body. This component, when the axis is fixed, is compensated by the reaction force in the axis, therefore it does not affect the rotation of the body. Let's write down one more useful expression for the moment of force. Let the power F attached to a point BUT, whose Cartesian coordinates are X, at(Fig. 109).

Let's decompose the force F into two components F X , F at, parallel to the corresponding coordinate axes. The moment of force F about the axis passing through the origin is obviously equal to the sum of the moments of the components F X , F at, that is

M = xF at − yF X .

Similarly, the way we introduced the concept of the vector of angular velocity, we can also define the concept of the vector of the moment of force. The module of this vector corresponds to the definition given above, but it is directed perpendicular to the plane containing the force vector and the segment connecting the point of application of the force with the axis of rotation (Fig. 110).

The vector of the moment of force can also be defined as the vector product of the radius vector of the point of application of the force and the force vector

Note that when the point of application of force is displaced along the line of its action, the moment of force does not change. Let us denote the product of the mass of a material point by the square of the distance to the axis of rotation

mr 2 = I

(this value is called moment of inertia material point about the axis). Using these notations, equation (2) takes on a form that formally coincides with the equation of Newton's second law for translational motion:

Iε = M. (3)

This equation is called the basic equation of rotational motion dynamics. So, the moment of force in rotational motion plays the same role as the force in translational motion - it is he who determines the change in angular velocity. It turns out (and this is confirmed by our everyday experience) that the influence of force on the speed of rotation is determined not only by the magnitude of the force, but also by the point of its application. The moment of inertia determines the inertial properties of the body in relation to rotation (in simple terms, it shows whether it is easy to spin the body): the farther from the axis of rotation is a material point, the more difficult it is to bring it into rotation. Equation (3) can be generalized to the case of rotation of an arbitrary body. When a body rotates around a fixed axis, the angular accelerations of all points of the body are the same. Therefore, just as we did when deriving Newton's equation for the translational motion of a body, we can write equations (3) for all points of a rotating body and then sum them up. As a result, we obtain an equation that outwardly coincides with (3), in which I- the moment of inertia of the whole body, equal to the sum of the moments of its constituent material points, M is the sum of moments of external forces acting on the body. Let us show how the moment of inertia of a body is calculated. It is important to emphasize that the moment of inertia of a body depends not only on the mass, shape and dimensions of the body, but also on the position and orientation of the axis of rotation. Formally, the calculation procedure is reduced to dividing the body into small parts, which can be considered material points (Fig. 111),

and the summation of the moments of inertia of these material points, which are equal to the product of the mass by the square of the distance to the axis of rotation:

For bodies of a simple shape, such sums have long been calculated, so it is often enough to remember (or find in a reference book) the appropriate formula for the desired moment of inertia. As an example: the moment of inertia of a circular homogeneous cylinder, masses m and radius R, for the axis of rotation coinciding with the axis of the cylinder is equal to:

I = (1/2)mR 2 (Fig. 112).

In this case, we restrict ourselves to considering rotation around a fixed axis, because the description of an arbitrary rotational motion of a body is a complex mathematical problem that goes far beyond the scope of a high school mathematics course. Knowledge of other physical laws, except for those considered by us, this description does not require.

2 Internal energy body (referred to as E or U) is the total energy of this body minus the kinetic energy of the body as a whole and the potential energy of the body in the external field of forces. Consequently, the internal energy is made up of the kinetic energy of the chaotic motion of molecules, the potential energy of interaction between them, and the intramolecular energy.

The internal energy of a body is the energy of movement and interaction of the particles that make up the body.

The internal energy of a body is the total kinetic energy of the movement of the molecules of the body and the potential energy of their interaction.

The internal energy is a single-valued function of the state of the system. This means that whenever a system finds itself in a given state, its internal energy assumes the value inherent in this state, regardless of the system's history. Consequently, the change in internal energy during the transition from one state to another will always be equal to the difference in values ​​in these states, regardless of the path along which the transition was made.

The internal energy of a body cannot be measured directly. Only the change in internal energy can be determined:

For quasi-static processes, the following relationship holds:

1. General information The amount of heat required to raise the temperature by 1°C is called heat capacity and is marked with the letter With. In technical calculations, heat capacity is measured in kilojoules. When using the old system of units, the heat capacity is expressed in kilocalories (GOST 8550-61) *. Depending on the units in which the amount of gas is measured, they distinguish: molar heat capacity \xc to kJ/(kmol x X hail); mass heat capacity c kJ/(kg-deg); volumetric heat capacity With in kJ/(m 3 hail). When determining the volumetric heat capacity, it is necessary to indicate to what values ​​​​of temperature and pressure it refers. It is customary to determine the volumetric heat capacity under normal physical conditions. The heat capacity of gases obeying the laws of an ideal gas depends only on temperature. There are average and true heat capacities of gases. The true heat capacity is the ratio of the infinitely small amount of heat supplied Dd with an increase in temperature by an infinitesimal amount At: The average heat capacity determines the average amount of heat supplied when a unit amount of gas is heated by 1 ° in the temperature range from t x before t%: where q- the amount of heat supplied to a unit mass of gas when it is heated from temperature t t up to temperature t%. Depending on the nature of the process in which heat is supplied or removed, the value of the heat capacity of the gas will be different. If the gas is heated in a vessel of constant volume (V\u003d "\u003d const), then heat is consumed only to increase its temperature. If the gas is in a cylinder with a movable piston, then when heat is supplied, the gas pressure remains constant (p == const). At the same time, when heated, the gas expands and performs work against external forces while simultaneously increasing its temperature. In order for the difference between the final and initial temperatures during gas heating in the process R= const would be the same as in the case of heating at V= = const, the amount of heat expended must be greater by an amount equal to the work done by the gas in the process p == const. It follows from this that the heat capacity of a gas at constant pressure With R will be greater than the heat capacity at a constant volume. The second term in the equations characterizes the amount of heat expended on the operation of the gas in the process R= = const when the temperature changes by 1°. When carrying out approximate calculations, it can be assumed that the heat capacity of the working body is constant and does not depend on temperature. In this case, the knowledge of molar heat capacities at constant volume can be taken for one-, two- and polyatomic gases, respectively, equal to 12,6; 20.9 and 29.3 kJ/(kmol-deg) or 3; 5 and 7 kcal/(kmol-deg).

Newton's laws make it possible to solve various practically important problems concerning the interaction and motion of bodies. A large number of such problems are connected, for example, with finding the acceleration of a moving body if all the forces acting on this body are known. And then other quantities are determined by acceleration (instantaneous speed, displacement, etc.).

But it is often very difficult to determine the forces acting on the body. Therefore, to solve many problems, another important physical quantity is used - the momentum of the body.

• The momentum of a body p is a vector physical quantity equal to the product of the mass of the body and its speed

Momentum is a vector quantity. The direction of the momentum vector of the body always coincides with the direction of the velocity vector.

The unit of momentum in SI is the momentum of a body with a mass of 1 kg moving at a speed of 1 m/s. This means that the unit of momentum of a body in SI is 1 kg m/s.

When calculating, they use the equation for projections of vectors: p x \u003d mv x.

Depending on the direction of the velocity vector with respect to the selected X-axis, the projection of the momentum vector can be either positive or negative.

The word "impulse" (impulsus) in Latin means "push". Some books use the term momentum instead of momentum.

This quantity was introduced into science at about the same period of time when Newton discovered the laws that were later named after him (that is, at the end of the 17th century).

When bodies interact, their momenta can change. This can be verified by a simple experiment.

Two balls of the same mass are hung on thread loops to a wooden ruler fixed on a tripod ring, as shown in Figure 44, a.

Rice. 44. Demonstration of the Law of Conservation of Momentum

Ball 2 is deflected from the vertical by an angle a (Fig. 44, b) and released. Returning to the previous position, he hits the ball 1 and stops. In this case, the ball 1 comes into motion and deviates by the same angle a (Fig. 44, c).

In this case, it is obvious that as a result of the interaction of the balls, the momentum of each of them has changed: by how much the momentum of ball 2 decreased, by the same amount the momentum of ball 1 increased.

If two or more bodies interact only with each other (that is, they are not exposed to external forces), then these bodies form a closed system.

The momentum of each of the bodies included in a closed system can change as a result of their interaction with each other. But

• 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.

The law of conservation of momentum is also fulfilled if external forces act on the bodies of the system, the vector sum of which is equal to zero. Let us show this by using Newton's second and third laws to derive the law of conservation of momentum. For simplicity, consider a system consisting of only two bodies - balls with masses m 1 and m 2, which move rectilinearly towards each other with velocities v 1 and v 2 (Fig. 45).

Rice. 45. A system of two bodies - balls moving in a straight line towards each other

The forces of gravity acting on each of the balls are balanced by the elastic forces of the surface on which they roll. Hence, the effect of these forces can be ignored. The forces of resistance to movement in this case are small, so we will not take into account their influence either. Thus, we can assume that the balls interact only with each other.

Figure 45 shows that after some time the balls will collide. During a collision lasting for a very short time t, interaction forces F 1 and F 2 will appear, applied respectively to the first and second balls. As a result of the action of forces, the speeds of the balls will change. Let's designate speeds of balls after collision by letters v 1 and v 2 .

In accordance with Newton's third law, the interaction forces of the balls are equal in absolute value and directed in opposite directions:

According to Newton's second law, each of these forces can be replaced by the product of mass and acceleration received by each of the balls during the interaction:

m 1 a 1 \u003d -m 2 a 2.

Accelerations, as you know, are determined from the equalities:

Replacing the corresponding expressions in the equation for acceleration forces, we obtain:

As a result of reducing both parts of the equality by t, we get:

m1 (v "1 - v 1) \u003d -m 2 (v" 2 - v 2).

We group the terms of this equation as follows:

m 1 v 1 "+ m 2 v 2" = m 1 v 1 = m 2 v 2. (one)

Considering that mv = p, we write equation (1) in the following form:

P "1 + P" 2 \u003d P 1 + P 2. (2)

The left parts of equations (1) and (2) are the total momentum of the balls after their interaction, and the right parts are the total momentum before the interaction.

This means that, despite the fact that the momentum of each of the balls changed during the interaction, the vector sum of their momenta after the interaction remained the same as before the interaction.

Equations (1) and (2) are the mathematical record of the momentum conservation law.

Since this course considers only the interactions of bodies moving along one straight line, then to write the law of conservation of momentum in scalar form, one equation is sufficient, which includes the projections of vector quantities on the X axis:

m 1 v "1x + m 2 v" 2x \u003d m 1 v 1x + m 2 v 2x.

Questions

1. What is called the momentum of the body?
2. What can be said about the directions of the momentum vectors and the speed of a moving body?
3. Tell us about the course of the experiment shown in Figure 44. What does it indicate?
4. What does the statement that several bodies form a closed system mean?
5. Formulate the law of conservation of momentum.
6. For a closed system consisting of two bodies, write down the law of conservation of momentum in the form of an equation that would include the masses and velocities of these bodies. Explain what each symbol in this equation means.

Exercise 20

1. Two toy clockwork machines, each weighing 0.2 kg, move in a straight line towards each other. The speed of each machine relative to the ground is 0.1 m/s. Are the momentum vectors of the machines equal; modules of momentum vectors? Determine the projection of the momentum of each of the machines on the X axis, parallel to their trajectories.
2. By how much will the momentum of a car with a mass of 1 ton change (in absolute value) when its speed changes from 54 to 72 km/h?
3. A man sits in a boat resting on the surface of a lake. At some point, he gets up and goes from stern to bow. What will happen to the boat? Explain the phenomenon based on the law of conservation of momentum.
4. A railway car weighing 35 tons drives up to a stationary car weighing 28 tons standing on the same track and automatically couples with it. After coupling, the cars move in a straight line at a speed of 0.5 m/s. What was the speed of the car weighing 35 tons before coupling?

Having studied Newton's laws, we see that with their help it is possible to solve the main problems of mechanics, if we know all the forces acting on the body. There are situations in which it is difficult or even impossible to determine these quantities. Let's consider several such situations.When two billiard balls or cars collide, we can assert about the acting forces that this is their nature, elastic forces act here. However, we will not be able to accurately establish either their modules or their directions, especially since these forces have an extremely short duration of action.In the movement of rockets and jet aircraft, we can also say little about the forces that set these bodies in motion.In such cases, methods are used that allow one to avoid solving the equations of motion, and immediately use the consequences of these equations. At the same time, new physical quantities are introduced. Consider one of these quantities, called the momentum of the body

An arrow fired from a bow. The longer the contact of the bowstring with the arrow (∆t), the greater the change in the momentum of the arrow (∆), and therefore, the higher its final speed.

Two colliding balls. While the balls are in contact, they act on each other with equal forces, as Newton's third law teaches us. This means that the changes in their momenta must also be equal in absolute value, even if the masses of the balls are not equal.

After analyzing the formulas, two important conclusions can be drawn:

1. The same forces acting for the same period of time cause the same changes in momentum for different bodies, regardless of the mass of the latter.

2. The same change in the momentum of a body can be achieved either by acting with a small force for a long period of time, or by acting for a short time with a large force on the same body.

According to Newton's second law, we can write:

∆t = ∆ = ∆ / ∆t

The ratio of the change in the momentum of the body to the period of time during which this change occurred is equal to the sum of the forces acting on the body.

After analyzing this equation, we see that Newton's second law allows us to expand the class of problems to be solved and include problems in which the mass of bodies changes over time.

If we try to solve problems with a variable mass of bodies using the usual formulation of Newton's second law:

then attempting such a solution would lead to an error.

An example of this is the already mentioned jet aircraft or space rocket, which, when moving, burn fuel, and the products of this burnt material are thrown into the surrounding space. Naturally, the mass of an aircraft or rocket decreases as fuel is consumed.

Despite the fact that Newton's second law in the form "the resultant force is equal to the product of the body's mass and its acceleration" allows solving a fairly wide class of problems, there are cases of body motion that cannot be fully described by this equation. In such cases, it is necessary to apply another formulation of the second law, which relates the change in the momentum of the body to the momentum of the resultant force. In addition, there are a number of problems in which the solution of the equations of motion is mathematically extremely difficult or even impossible. In such cases, it is useful for us to use the concept of momentum.

Using the law of conservation of momentum and the relationship between the momentum of a force and the momentum of a body, we can derive Newton's second and third laws.

Newton's second law is derived from the ratio of the momentum of the force and the momentum of the body.

The impulse of the force is equal to the change in the momentum of the body:

Having made the appropriate transfers, we will get the dependence of force on acceleration, because acceleration is defined as the ratio of the change in speed to the time during which this change occurred:

Substituting the values ​​into our formula, we get the formula for Newton's second law:

To derive Newton's third law, we need the law of conservation of momentum.

Vectors emphasize the vectorial nature of speed, that is, the fact that speed can change in direction. After transformations, we get:

Since the time interval in a closed system was a constant value for both bodies, we can write:

We have obtained Newton's third law: two bodies interact with each other with forces equal in magnitude and opposite in direction. The vectors of these forces are directed towards each other, respectively, the modules of these forces are equal in their value.

Bibliography

1. Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemozina, 2012.
2. Gendenstein L.E., Dick Yu.I. Physics grade 10. - M.: Mnemosyne, 2014.
3. Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.

Homework

1. Define the momentum of a body, the momentum of a force.
2. How is the momentum of a body related to the momentum of a force?
3. What conclusions can be drawn from the formulas for the momentum of the body and the momentum of the force?

1. Internet portal Questions-physics.ru ().
2. Internet portal Frutmrut.ru ().
3. Internet portal Fizmat.by ().

A 22-caliber bullet has a mass of only 2 g. If someone throws such a bullet, he can easily catch it even without gloves. If you try to catch such a bullet that has flown out of the muzzle at a speed of 300 m / s, then even gloves will not help here.

If a toy cart is rolling towards you, you can stop it with your toe. If a truck is rolling towards you, you should keep your feet out of the way.

Let's consider a problem that demonstrates the connection between the momentum of a force and a change in the momentum of a body.

Example. The mass of the ball is 400 g, the speed acquired by the ball after the impact is 30 m/s. The force with which the foot acted on the ball was 1500 N, and the impact time was 8 ms. Find the momentum of the force and the change in the momentum of the body for the ball.

Change in body momentum

Example. Estimate the average force from the side of the floor acting on the ball during impact.

1) During the impact, two forces act on the ball: support reaction force, gravity.

The reaction force changes during the impact time, so it is possible to find the average floor reaction force.

2) Change in momentum body shown in the picture

3) From Newton's second law

The main thing to remember

1) Formulas for body impulse, force impulse;
2) The direction of the momentum vector;
3) Find the change in body momentum

General derivation of Newton's second law

F(t) chart. variable force

The force impulse is numerically equal to the area of ​​the figure under the graph F(t).

If the force is not constant in time, for example, it increases linearly F=kt, then the momentum of this force is equal to the area of ​​the triangle. You can replace this force with such a constant force that will change the momentum of the body by the same amount in the same period of time.

Average resultant force

LAW OF CONSERVATION OF MOMENTUM

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Closed system of bodies

This is a system of bodies that interact only with each other. There are no external forces of interaction.

In the real world, such a system cannot exist, there is no way to remove any external interaction. A closed system of bodies is a physical model, just like a material point is a model. This is a model of a system of bodies that allegedly interact only with each other, external forces are not taken into account, they are neglected.

Law of conservation of momentum

In a closed system of bodies vector the sum of the momenta of the bodies does not change when the bodies interact. If the momentum of one body has increased, then this means that at that moment the momentum of some other body (or several bodies) has decreased by exactly the same amount.

Let's consider such an example. Girl and boy are skating. A closed system of bodies - a girl and a boy (we neglect friction and other external forces). The girl stands still, her momentum is zero, since the speed is zero (see the body momentum formula). After the boy, moving at some speed, collides with the girl, she will also begin to move. Now her body has momentum. The numerical value of the momentum of the girl is exactly the same as the momentum of the boy decreased after the collision.

One body of mass 20kg moves with a speed of , the second body of mass of 4kg moves in the same direction with a speed of . What is the momentum of each body. What is the momentum of the system?

Impulse of the body system is the vector sum of the impulses of all bodies in the system. In our example, this is the sum of two vectors (since two bodies are considered) that are directed in the same direction, therefore

Now let's calculate the momentum of the system of bodies from the previous example if the second body moves in the opposite direction.

Since the bodies move in opposite directions, we get the vector sum of the multidirectional impulses. More on the sum of vectors.

The main thing to remember

1) What is a closed system of bodies;
2) Law of conservation of momentum and its application

Pulse (Quantity of movement) is a vector physical quantity, which is a measure of the mechanical movement of the body. In classical mechanics, the momentum of a body is equal to the product of the mass m this body at its speed v, the direction of the momentum coincides with the direction of the velocity vector:

System momentum particles is the vector sum of the momenta of its individual particles: p=(sums) pi, where pi is the momentum of the i-th particle.

Theorem on the change in the momentum of the system: the total momentum of the system can only be changed by the action of external forces: Fext=dp/dt(1), i.e. the time derivative of the momentum of the system is equal to the vector sum of all external forces acting on the particles of the system. As in the case of a single particle, it follows from expression (1) that the increment of the momentum of the system is equal to the momentum of the resultant of all external forces for the corresponding period of time:

p2-p1= t & 0 F ext dt.

In classical mechanics, complete momentum system of material points is called a vector quantity equal to the sum of the products of the masses of material points at their speed:

accordingly, the quantity is called the momentum of one material point. It is a vector quantity directed in the same direction as the particle's velocity. The unit of momentum in the International System of Units (SI) is kilogram meter per second(kg m/s).

If we are dealing with a body of finite size, which does not consist of discrete material points, to determine its momentum, it is necessary to break the body into small parts, which can be considered as material points and sum over them, as a result we get:

The momentum of a system that is not affected by any external forces (or they are compensated), preserved in time:

The conservation of momentum in this case follows from Newton's second and third laws: having written Newton's second law for each of the material points that make up the system and summing it over all the material points that make up the system, by virtue of Newton's third law we obtain equality (*).

In relativistic mechanics, the three-dimensional momentum of a system of non-interacting material points is the quantity

where m i- weight i-th material point.

For a closed system of non-interacting material points, this value is preserved. However, the three-dimensional momentum is not a relativistically invariant quantity, since it depends on the frame of reference. A more meaningful value will be a four-dimensional momentum, which for one material point is defined as

In practice, the following relationships between the mass, momentum, and energy of a particle are often used:

In principle, for a system of non-interacting material points, their 4-momenta are summed. However, for interacting particles in relativistic mechanics, one should take into account the momenta not only of the particles that make up the system, but also the momentum of the field of interaction between them. Therefore, a much more meaningful quantity in relativistic mechanics is the energy-momentum tensor, which fully satisfies the conservation laws.

Pulse Properties

· Additivity. This property means that the impulse of a mechanical system consisting of material points is equal to the sum of the impulses of all material points included in the system.

· Invariance with respect to the rotation of the frame of reference.

· Preservation. The momentum does not change during interactions that change only the mechanical characteristics of the system. This property is invariant with respect to Galilean transformations. The properties of conservation of kinetic energy, conservation of momentum and Newton's second law are sufficient to derive the mathematical formula for momentum.

Law of conservation of momentum (Law of conservation of momentum)- the vector sum of the impulses of all bodies of the system is a constant value, if the vector sum of the external forces acting on the system is equal to zero.

In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. From Newton's laws, it can be shown that when moving in empty space, momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces.

Like any of the fundamental conservation laws, the law of conservation of momentum is associated, according to Noether's theorem, with one of the fundamental symmetries - the homogeneity of space

The change in momentum of a body is equal to the momentum of the resultant of all forces acting on the body. This is another formulation of Newton's second law.