IN Everyday life In order to characterize a person who commits spontaneous actions, the epithet “impulsive” is sometimes used. At the same time, some people don’t even remember, and a significant part don’t even know what physical quantity this word is associated with. What is hidden under the concept of “body impulse” and what properties does it have? Great scientists such as Rene Descartes and Isaac Newton sought answers to these questions.
Like any science, physics operates with clearly formulated concepts. At the moment, the following definition is accepted for a quantity called the momentum of a body: it is a vector quantity, which is a measure (quantity) of the mechanical movement of a body.
Let us assume that the question is considered within the framework of classical mechanics, i.e. it is believed that the body moves with normal, and not with relativistic speed, which means it is at least an order of magnitude less than the speed of light in vacuum. Then the modulus of the body’s momentum is calculated using formula 1 (see photo below).
Thus, by definition, this quantity is equal to the product of the body’s mass and its speed, with which its vector is codirected.
As the SI unit of impulse ( International system units) is taken to be 1 kg/m/s.
Where does the term "impulse" come from?
Several centuries before the concept of the amount of mechanical motion of a body appeared in physics, it was believed that the cause of any movement in space was a special force - impetus.
In the 14th century, Jean Buridan made adjustments to this concept. He suggested that a flying pebble has an impetus directly proportional to its speed, which would be unchanged if there were no air resistance. At the same time, according to this philosopher, bodies with greater weight had the ability to “accommodate” more of this driving force.
Further development of the concept, later called impulse, was given by Rene Descartes, who designated it with the words “quantity of motion.” However, he did not take into account that speed has a direction. That is why the theory he put forward in some cases contradicted experience and did not find recognition.
The English scientist John Wallis was the first to guess that momentum should also have a direction. This happened in 1668. However, it took him another couple of years for him to formulate the well-known law of conservation of momentum. The theoretical proof of this fact, established empirically, was given by Isaac Newton, who used the third and second laws of classical mechanics, discovered by him, and named after him.
Momentum of a system of material points
Let us first consider the case where we are talking about speeds much lower than the speed of light. Then, according to the laws of classical mechanics, the total momentum of a system of material points represents a vector quantity. It is equal to the sum of the products of their masses at speed (see formula 2 in the picture above).
In this case, the momentum of one material point is taken to be a vector quantity (formula 3), which is co-directed with the speed of the particle.
If we are talking about a body of finite size, then first it is mentally divided into small parts. Thus, the system of material points is again considered, but its momentum is calculated not by ordinary summation, but by integration (see formula 4).
As we can see, there is no time dependence, therefore the momentum of the system, which is not affected by external forces (or their influence is mutually compensated), remains unchanged in time.
Proof of the conservation law
Let us continue to consider a body of finite size as a system of material points. For each of them, Newton's Second Law is formulated according to formula 5.
Let us pay attention to the fact that the system is closed. Then, summing over all points and applying Newton’s Third Law, we obtain expression 6.
Thus, the momentum of a closed-loop system is a constant value.
The conservation law is also valid in cases where the total sum of forces that act on the system from the outside is equal to zero. This leads to one important particular statement. It states that the momentum of a body is a constant value if there is no external influence or the influence of several forces is compensated. For example, in the absence of friction, after being hit with a stick, the puck should retain its momentum. This situation will be observed even despite the fact that this body is acted upon by the force of gravity and the reactions of the support (ice), since they, although equal in magnitude, are directed in opposite directions, i.e., they compensate each other.
Properties
The momentum of a body or a material point is an additive quantity. What does it mean? It’s simple: the momentum of a mechanical system of material points consists of the impulses of all material points included in the system.
The second property of this quantity is that it remains unchanged during interactions that change only the mechanical characteristics of the system.
In addition, the impulse is invariant with respect to any rotation of the reference frame.
Relativistic case
Let us assume that we are talking about non-interacting material points with velocities of the order of 10 to the 8th power or slightly less in the SI system. The three-dimensional momentum is calculated using formula 7, where c is understood as the speed of light in a vacuum.
In the case when it is closed, the law of conservation of momentum is true. At the same time, three-dimensional momentum is not a relativistically invariant quantity, since it depends on the reference frame. There is also a four-dimensional option. For one material point it is determined by formula 8.
Momentum and Energy
These quantities, as well as mass, are closely related to each other. In practical problems, relations (9) and (10) are usually used.
Definition via de Broglie waves
In 1924, a hypothesis was put forward that not only photons, but also any other particles (protons, electrons, atoms) have wave-particle duality. Its author was the French scientist Louis de Broglie. If we translate this hypothesis into the language of mathematics, then we can say that with any particle that has energy and momentum, a wave is associated with a frequency and length expressed by formulas 11 and 12, respectively (h is Planck’s constant).
From the last relationship we find that the impulse modulus and the wavelength, denoted by the letter “lambda,” are inversely proportional to each other (13).
If a particle with a relatively low energy is considered, which moves at a speed incommensurable with the speed of light, then the modulus of momentum is calculated in the same way as in classical mechanics (see formula 1). Therefore, the wavelength is calculated according to expression 14. In other words, it is inversely proportional to the product of the mass and speed of the particle, i.e., its momentum.
Now you know that the impulse of a body is a measure of mechanical movement, and you are familiar with its properties. Among them, the Law of Conservation is especially important in practical terms. Even people far from physics observe it in everyday life. For example, everyone knows that firearms and artillery pieces produce recoil when fired. The law of conservation of momentum is clearly demonstrated by the game of billiards. With its help, you can predict the direction of the balls' flight after impact.
The law has found application in calculations necessary to study the consequences of possible explosions, in the field of creating jet vehicles, in the design of firearms and in many other areas of life.
A 22-caliber bullet has a mass of only 2 g. If you throw such a bullet to someone, he can easily catch it even without gloves. If you try to catch such a bullet flying out of the muzzle at a speed of 300 m/s, then even gloves will not help.
If a toy cart is rolling towards you, you can stop it with your toe. If a truck is rolling towards you, you should move your feet out of its path.
Let's consider a problem that demonstrates the connection between a force impulse and a change in the momentum of a body.
Example. The mass of the ball is 400 g, the speed that the ball acquired after 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 impulse of force and the change in momentum of the body for the ball.
Change in body momentum
Example. Estimate the average force from the floor acting on the ball during impact.
1) During a strike, two forces act on the ball: ground reaction force, gravity.
The reaction force changes during the impact time, so it is possible to find the average reaction force of the floor.
BODY IMPULSE
The momentum of a body is a physical vector quantity equal to the product of the mass of the body and its speed.
Pulse 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 bodies of this system: ∑p=p 1 +p 2 +... . Law of conservation of momentum: in a closed system of bodies, during any processes, 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.)
Question 2. Thermodynamic and statistical definition of entropy. 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 absolute temperature:
This formula is only applicable 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 infinitesimal 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 that makes up 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 up. 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 restrictions on the direction of heat transfer processes between bodies.
The second law of thermodynamics states that spontaneous transfer of heat from a less heated body to a more heated body is impossible.
Ticket 6.
§ 2.5. Theorem on the motion of the center of mass
Relationship (16) is very similar to the equation of motion of a material point. Let's try to bring it to even more simple view 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)
Let's multiply and divide (24) by the mass of the entire system and substitute it into equation (16):
. (25)
The expression in parentheses 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) will take the form
(27)
If (26) is substituted into (25), we obtain the theorem on the motion of the center of mass:
those. the center of mass of the system moves, like 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 movement 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 and no matter how complex these particles move, it is always possible to 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.
Product of system mass and speed v The center of mass 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 zero, then the center of mass moves uniformly and rectilinearly or is at rest.
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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 the 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.
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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 stationary center of mass.
Are there any advantages to the law of conservation of momentum over Newton's laws? What is the power of this law?
Its main advantage is that it is integral in nature, i.e. connects the characteristics of a system (its momentum) in two states separated by a finite period of time. This allows you 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 occurring during this process.
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 a precisely given speed v at a given moment in time, but it is possible to count the number of molecules whose speeds have a value lying between some speeds v 1 and v 2 . Based on the theory of probability, Maxwell established a pattern by which it is possible to determine the number of gas molecules whose velocities at a given temperature lie within a certain velocity range. According to Maxwell's distribution, the probable number of molecules per unit volume; the velocity components of which lie in the interval from to, from 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 that the number of molecules whose absolute velocities lie in the interval from v to v + dv has the form
The Maxwell distribution reaches a maximum at speed, i.e. such a speed to which the speeds of most molecules are close. The area of the shaded strip with the base dV will show what part of the total number of molecules has velocities that lie in this interval. The specific form of the Maxwell distribution function depends on the type of gas (molecule mass) and temperature. The pressure and volume of the gas do not affect the velocity distribution of molecules.
The Maxwell distribution curve will allow you to find the arithmetic average speed
Thus,
With increasing temperature, the most probable speed increases, therefore the maximum of the distribution of molecules by speed shifts towards higher speeds, and its absolute value decreases. Consequently, when a 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 equal to:
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). 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 (normalization condition):
In the case when the movement of particles obeys classical mechanics, the energy E i can be considered to consist of the kinetic energy E ikin of the particle (molecule or atom), its internal energy E iin (for example, the excitation energy of electrons) and potential energy E i , sweat in an external field, depending on the position of the particle in space:
E i = E i, kin + E i, int + 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,pot. In accordance with (2), formula (1) can be represented as a product of three exponentials, each of which gives the distribution of particles according to one type of energy.
In a constant gravitational field creating 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 over all possible values of kinetic and internal energies particles, a barometric formula is obtained, expressing the law of decreasing atmospheric density with height.
In astrophysics, especially in the theory of stellar spectra, the Boltzmann distribution is often used to determine the relative electron population of different atomic energy levels. If we designate two energy states of the atom by indices 1 and 2, then the distribution follows:
n 2 /n 1 = (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 kT value characterizing the energy thermal movement 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 within 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 distributions (for particles with half-integer spin). Both of these distributions become a distribution when the average number of quantum states available to 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 filling of quantum states is small. The condition for the applicability of the Boltzmann distribution can be written as the inequality:
where N is the number of particles, V is the volume of the system. This inequality is satisfied at high temperatures and a small number of particles per unit. volume (N/V). It follows from this that the greater the mass of particles, the wider the range of changes in T and N/V the Boltzmann distribution is valid.
ticket 7.
The work done by all applied forces is equal to the work done by 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 forces applied to the body. This connection is most easily established by considering the movement 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 rectilinear uniformly accelerated motion. By directing the coordinate axis along the straight line of motion, we can consider F, s, υ and a as algebraic quantities (positive or negative depending on the direction of the corresponding vector). Then the work of force can be written as A = Fs. With uniformly accelerated motion, the displacement s expressed by the formula
This expression shows that the work done by a 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 a body’s mass and the square of its speed is called kinetic energy body:
This statement is called kinetic energy theorem . The theorem on kinetic energy is also valid in the general case, when a body moves under the influence 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 with a speed equal to the work that must be done by a force applied to a body at rest in order to impart this speed to it:
In physics, along with kinetic energy or energy of motion, the concept plays an important role potential energy or energy of interaction between bodies.
Potential energy is determined by the relative position of bodies (for example, the position of the body relative to the surface of the Earth). The concept of potential energy can be introduced only for forces whose work does not depend on the trajectory of movement and is determined only by the initial and final positions of the body. Such forces are called conservative .
The work done by conservative forces on a closed trajectory is zero. This statement is illustrated by Fig. 1.19.2.
Gravity and elasticity have the property of conservatism. For these forces we can introduce the concept of potential energy.
If a body moves near the surface of the Earth, then it is acted upon 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 part of the path, the work of gravity can be written in projections of the displacement vector onto the axis OY, directed vertically upward:
This work is equal to the change in some physical quantity mgh, taken with the opposite sign. This physical quantity is called potential energy bodies in a gravity field
Potential energy E p depends on the choice of the zero level, i.e. on the choice of the origin of the axis OY. What has a physical meaning is not the potential energy itself, but its change Δ E p = Eр2 – E p1 when moving a body from one position to another. This change is independent of the choice of zero level.
If we consider the movement of bodies in the gravitational field of the Earth at significant 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 universal gravity ). For the forces of universal gravitation, it is convenient to count potential energy from a point at infinity, i.e., to assume the potential energy of a body at an infinitely distant point equal to zero. Formula expressing the potential energy of a body of mass m on distance r from the center of the Earth, has the form ( see §1.24):
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Where M– mass of the Earth, G– gravitational constant.
The concept of potential energy can also be introduced for the elastic force. This force also has the property of being conservative. When stretching (or compressing) a spring, we can do this in various ways.
You can simply lengthen the spring by an amount x, or first lengthen it by 2 x, and then reduce the elongation to the 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 done by 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 determined in this way is not unambiguous characteristic 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 different work. Therefore, the same amount of heat transferred to a body could cause different changes in its internal energy and, consequently, temperature.
This ambiguity in determining heat capacity is typical only for gaseous substances. When liquids and solids are heated, their volume practically does not change, and the work of expansion turns out to be zero. Therefore, the entire amount of heat received by the body goes to change its internal energy. Unlike liquids and solids, 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. Usually two values of the heat capacity of gases are considered: C V – molar heat capacity in an isochoric process (V = const) and C p – molar heat capacity in an isobaric process (p = const).
In the process at a constant volume, the gas does not do any work: A = 0. From the first law of thermodynamics for 1 mole of gas it follows
where ΔV is the change in 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 relation is included in the formula for the adiabatic process (see §3.9).
Between two isotherms with temperatures T 1 and T 2 in 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 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 exchange will turn out to be different for different transition paths. It follows that gas has an infinite number of heat capacities. C p and C V are only partial (and very important for the theory of gases) values of heat capacities.
Ticket 8.
1 Of course, the position of one, even a “special” point does not completely describe the movement of the entire system of bodies under consideration, but it is still better to know the position of at least one point than to know nothing. Nevertheless, let us 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 mass m attached with a weightless rigid rod length r to the fixed axis OO / (Fig. 106).
A material point can move around an axis, remaining at a constant distance from it, therefore, its trajectory will be a circle with a center 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, therefore it is convenient to use the rotation angle as the only coordinate, rather than two Cartesian coordinates; secondly, the system under consideration is acted upon by reaction forces in the axis of rotation, and directly on the material point by the tension force of the rod. Finding these forces is a separate problem, the solution of which is unnecessary to describe rotation. Therefore, it makes sense to obtain, based on Newton’s laws, a special equation that directly describes rotational motion. Let at some moment of time a certain force act on a material point F, lying in a plane perpendicular to the axis of rotation (Fig. 107).
In the kinematic description of curvilinear motion, it is convenient to decompose the total acceleration vector a into two components - normal A n, directed towards 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 tension force of the rod. Let us write the equation of the second law in 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 rotation angle φ directly determined by angular velocity
ω = Δφ/Δt,
the change of which, in turn, is described by the angular acceleration
ε = Δω/Δt.
Angular acceleration is related to the tangential component of acceleration by the relation
A τ = rε.
If we substitute this expression into equation (1), we obtain an equation suitable for determining angular acceleration. It is convenient to introduce a new physical quantity that determines the interaction of bodies when they rotate. To do this, 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 multiplying 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− 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− product of the force modulus and the distance from the line of action of the force to the axis of rotation (force arm) M = Fd− is called the moment of force. The action of force can lead to rotation either clockwise or counterclockwise. In accordance with the chosen positive direction of rotation, the sign of the moment of force should be determined. Note that the moment of force is determined by that 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. When the axis is fixed, this component is compensated by the reaction force in the axis, and therefore does not affect the rotation of the body. Let's write down another useful expression for the moment of force. May the force F applied to a point A, whose Cartesian coordinates are equal X, at(Fig. 109).
Let's break down the power F into two components F X , F at, parallel to the corresponding coordinate axes. The moment of force F relative to the axis passing through the origin of coordinates is obviously equal to the sum of the moments of the components F X , F at, that is
M = xF at − уF X .
In the same way that we introduced the concept of the angular velocity vector, we can also define the concept of the torque vector. The modulus of this vector corresponds to the definition given above, and 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 force moment vector 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 a 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 quantity is called moment of inertia material point relative to 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 fundamental equation of dynamics rotational movement. So, the moment of force in rotational motion plays the same role as the force in translational motion - it is it that determines the change in angular velocity. It turns out (and this is confirmed by our everyday experience), 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 a body with respect to rotation (in simple terms, it shows whether it is easy to spin the body): the farther a material point is from the axis of rotation, 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, in the same way 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 externally coincides with (3), in which I− moment of inertia of the entire body, equal to the sum of the moments of its constituent material points, M− the sum of the 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 size of the body, but also on the position and orientation of the axis of rotation. Formally, the calculation procedure comes down 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 simple shape, such amounts have long been calculated, so it is often enough to remember (or find in a reference book) the corresponding formula for the required moment of inertia. As an example: the moment of inertia of a circular homogeneous cylinder, mass 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 limit ourselves to considering rotation around a fixed axis, because describing the arbitrary rotational motion of a body is a complex mathematical problem that goes far beyond the scope of a high school mathematics course. This description does not require knowledge of other physical laws other than those considered by us.
2 Internal energy body (denoted as E or U) - 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. Therefore, internal energy consists of kinetic energy chaotic movement molecules, potential interaction energy between them and 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 movement of the molecules of the body and the potential energy of their interaction.
Internal energy is a unique function of the state of the system. This means that whenever a system finds itself in a given state, its internal energy takes on the value inherent in this state, regardless of the previous history of the system. 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 took place.
The internal energy of a body cannot be measured directly. You can only determine the change in internal energy:
For quasi-static processes the following relation holds:
1. General information The amount of heat required to warm a unit quantity of gas by 1° is called heat capacity and is designated by the letter With. In technical calculations, heat capacity is measured in kilojoules. When using the old system of units, 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 in kJ/(kg-deg); volumetric heat capacity With V kJ/(m 3 hail). When determining volumetric heat capacity, it is necessary to indicate to what values of temperature and pressure it relates. It is customary to determine the volumetric heat capacity under normal physical conditions. The heat capacity of gases that obey the ideal gas laws depends only on temperature. A distinction is made between the average and true heat capacity of gases. True heat capacity is the ratio of the infinitesimal amount of heat supplied Dd when the temperature increases by an infinitesimal amount At: Average heat capacity determines the average amount of heat supplied when heating a unit amount of gas 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 heat capacity of the gas will be different. If the gas is heated in a vessel of constant volume (V=" = const), then heat is spent 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 produces 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 consumed by 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 values of the molar heat capacities at constant volume can be taken for mono-, di- and polyatomic gases, respectively, equal 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. Big number Such problems involve, for example, finding the acceleration of a moving body if all the forces acting on this body are known. And then other quantities are determined from the acceleration ( instantaneous speed, movement, 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 body's momentum vector always coincides with the direction of the motion velocity vector.
The SI unit of impulse is the impulse of a body weighing 1 kg moving at a speed of 1 m/s. This means that the SI unit of momentum of a body is 1 kg m/s.
When making calculations, use the equation for projections of vectors: р x = mv x.
Depending on the direction of the velocity vector relative to the selected X-axis, the projection of the momentum vector can be either positive or negative.
The word “impulse” (impulsus) translated from Latin means “push”. Some books use the term "momentum" instead of the term "impulse".
This quantity was introduced into science at approximately the same period of time when Newton discovered the laws that were later named after him (i.e. late XVII V.).
When bodies interact, their impulses can change. This can be verified by simple experience.
Two balls of equal mass are suspended on thread loops from a wooden ruler mounted 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 his previous position, he hits ball 1 and stops. In this case, ball 1 begins to move 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, the momentum of ball 1 increased by the same amount.
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 satisfied if the bodies of the system are acted upon by external forces whose vector sum is equal to zero. Let's show this by using Newton's second and third laws to derive the law of conservation of momentum. For simplicity, let us consider a system consisting of only two bodies - balls of masses m 1 and m 2, which move rectilinearly towards each other with speeds 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. This means that the action of these forces can be ignored. The forces of resistance to movement in this case are small, so we will not take their influence into account either. Thus, we can assume that the balls interact only with each other.
From Figure 45 it can be seen that after some time the balls will collide. During a collision lasting for a very short period of time t, interaction forces F 1 and F 2 will arise, applied respectively to the first and second ball. As a result of the action of forces, the speed of the balls will change. Let us denote the velocities of the balls after the collision by the letters v 1 and v 2 .
In accordance with Newton's third law, the interaction forces between the balls are equal in magnitude and directed in opposite directions:
According to Newton's second law, each of these forces can be replaced by the product of the mass and acceleration received by each of the balls during interaction:
m 1 a 1 = -m 2 a 2 .
Accelerations, as you know, are determined from the equalities:
Replacing the acceleration forces in the equation with the corresponding expressions, we obtain:
As a result of reducing both sides of the equality by t, we obtain:
m1(v" 1 - v 1) = -m 2 (v" 2 - v 2).
Let's group the terms of this equation as follows:
m 1 v 1 " + m 2 v 2 " = m 1 v 1 = m 2 v 2 . (1)
Considering that mv = p, we write equation (1) in this form:
P" 1 + P" 2 = P 1 + P 2.(2)
The left sides of equations (1) and (2) represent the total momentum of the balls after their interaction, and the right sides represent the total impulse before interaction.
This means that, despite the fact that the momentum of each of the balls changed during the interaction, the vector sum of their momentum after the interaction remained the same as before the interaction.
Equations (1) and (2) are a mathematical representation of the law of conservation of momentum.
Since this course considers only the interactions of bodies moving along one straight line, to write the law of conservation of momentum in scalar form, one equation is sufficient, which includes projections of vector quantities onto the X axis:
m 1 v" 1x + m 2 v" 2x = m 1 v 1x + m 2 v 2x.
Questions
- What is the impulse of a body?
- What can be said about the directions of the momentum vectors and the speed of a moving body?
- Tell us about the course of the experiment depicted in Figure 44. What does it indicate?
- What does it mean to say that several bodies form a closed system?
- Formulate the law of conservation of momentum.
- For a closed system consisting of two bodies, write 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
- Two wind-up toy cars, each weighing 0.2 kg, move in a straight line towards each other. The speed of each car relative to the ground is 0.1 m/s. Are the impulse vectors of the machines equal? impulse vector modules? Determine the projection of the momentum of each of the cars on the X axis, parallel to their trajectory.
- How much will the impulse of a car weighing 1 ton change (in absolute value) when its speed changes from 54 to 72 km/h?
- A man sits in a boat resting on the surface of a lake. At some point he gets up and walks from the stern to the bow. What will happen to the boat? Explain the phenomenon based on the law of conservation of momentum.
- A railway car weighing 35 tons approaches a stationary car weighing 28 tons standing on the same track and automatically couples with it. After coupling, the cars move straight at a speed of 0.5 m/s. What was the speed of the 35-ton car before the coupling?
Momentum is one of the most fundamental characteristics of a physical system. The momentum of a closed system is conserved during any processes occurring in it.
Let's start getting acquainted with this quantity with the simplest case. The momentum of a material point of mass moving with speed is the product
Law of momentum change. From this definition, using Newton's second law, we can find the law of change in the momentum of a particle as a result of the action of some force on it. By changing the speed of a particle, the force also changes its momentum: . In the case of a constant acting force, therefore
The rate of change of momentum of a material point is equal to the resultant of all forces acting on it. With a constant force, the time interval in (2) can be taken by anyone. Therefore, for the change in momentum of a particle during this interval, it is true
In the case of a force that changes over time, the entire period of time should be divided into small intervals during each of which the force can be considered constant. The change in particle momentum over a separate period is calculated using formula (3):
The total change in momentum over the entire time period under consideration is equal to the vector sum of changes in momentum over all intervals
If we use the concept of derivative, then instead of (2), obviously, the law of change in particle momentum is written as
Impulse of force. The change in momentum over a finite period of time from 0 to is expressed by the integral
The quantity on the right side of (3) or (5) is called the impulse of force. Thus, the change in the momentum Dr of a material point over a period of time is equal to the impulse of the force acting on it during this period of time.
Equalities (2) and (4) are essentially another formulation of Newton's second law. It was in this form that this law was formulated by Newton himself.
The physical meaning of the concept of impulse is closely related to the intuitive idea that each of us has, or one drawn from everyday experience, about whether it is easy to stop a moving body. What matters here is not the speed or mass of the body being stopped, but both together, i.e., precisely its momentum.
System impulse. The concept of momentum becomes especially meaningful when it is applied to a system of interacting material points. The total momentum P of a system of particles is the vector sum of the momenta of individual particles at the same moment in time:
Here the summation is performed over all particles included in the system, so that the number of terms is equal to the number of particles in the system.
Internal and external forces. It is easy to come to the law of conservation of momentum of a system of interacting particles directly from Newton’s second and third laws. We will divide the forces acting on each of the particles included in the system into two groups: internal and external. Internal force is the force with which a particle acts on the External force is the force with which all bodies that are not part of the system under consideration act on the particle.
The law of change in particle momentum in accordance with (2) or (4) has the form
Let us add equation (7) term by term for all particles of the system. Then on the left side, as follows from (6), we obtain the rate of change
total momentum of the system Since the internal forces of interaction between particles satisfy Newton’s third law:
then when adding equations (7) on the right side, where internal forces occur only in pairs, their sum will go to zero. As a result we get
The rate of change of total momentum is equal to the sum of the external forces acting on all particles.
Let us pay attention to the fact that equality (9) has the same form as the law of change in the momentum of one material point, and the right side includes only external forces. In a closed system, where there are no external forces, the total momentum P of the system does not change regardless of what internal forces act between the particles.
The total momentum does not change even in the case when the external forces acting on the system are equal to zero in total. It may turn out that the sum of external forces is zero only along a certain direction. Although the physical system in this case is not closed, the component of the total momentum along this direction, as follows from formula (9), remains unchanged.
Equation (9) characterizes the system of material points as a whole, but refers to a certain point in time. From it it is easy to obtain the law of change in the momentum of the system over a finite period of time. If the acting external forces are constant during this interval, then from (9) it follows
If external forces change with time, then on the right side of (10) there will be a sum of integrals over time from each of the external forces:
Thus, the change in the total momentum of a system of interacting particles over a certain period of time is equal to the vector sum of the impulses of external forces over this period.
Comparison with the dynamic approach. Let's compare approaches to the solution mechanical tasks based on the equations of dynamics and based on the law of conservation of momentum using the following simple example.
A railway car of mass taken from a hump, moving at a constant speed, collides with a stationary car of mass and is coupled with it. At what speed do the coupled cars move?
We know nothing about the forces with which the cars interact during a collision, except for the fact that, based on Newton's third law, they are equal in magnitude and opposite in direction at each moment. With a dynamic approach, it is necessary to specify some kind of model for the interaction of cars. The simplest possible assumption is that the interaction forces are constant throughout the entire time the coupling occurs. In this case, using Newton’s second law for the speeds of each of the cars, after the start of the coupling, we can write
Obviously, the coupling process ends when the speeds of the cars become the same. Assuming that this happens after time x, we have
From here we can express the impulse of force
Substituting this value into any of formulas (11), for example into the second, we find the expression for the final speed of the cars:
Of course, the assumption made about the constancy of the force of interaction between the cars during the process of their coupling is very artificial. The use of more realistic models leads to more cumbersome calculations. However, in reality, the result for the final speed of the cars does not depend on the interaction pattern (of course, provided that at the end of the process the cars are coupled and moving at the same speed). The easiest way to verify this is to use the law of conservation of momentum.
Since no external forces in the horizontal direction act on the cars, the total momentum of the system remains unchanged. Before the collision, it is equal to the momentum of the first car. After coupling, the momentum of the cars is equal. Equating these values, we immediately find
which, naturally, coincides with the answer obtained on the basis of the dynamic approach. The use of the law of conservation of momentum made it possible to find the answer to the question posed using less cumbersome mathematical calculations, and this answer is more general, since no specific interaction model was used to obtain it.
Let us illustrate the application of the law of conservation of momentum of a system using the example of a more complex problem, where choosing a model for a dynamic solution is already difficult.
Task
Shell explosion. The projectile explodes at the top point of the trajectory, located at a height above the surface of the earth, into two identical fragments. One of them falls to the ground exactly below the point of explosion after a time. How many times will the horizontal distance from this point at which the second fragment will fly away change, compared to the distance at which an unexploded shell would fall?
Solution: First of all, let's write an expression for the distance over which an unexploded shell would fly. Since the speed of the projectile at the top point (we denote it by is directed horizontally), then the distance is equal to the product of the time of falling from a height without an initial speed, equal to which an unexploded projectile would fly away. Since the speed of the projectile at the top point (we denote it by is directed horizontally, then the distance is equal to the product of the time of falling from a height without an initial speed, equal to the body considered as a system of material points:
The bursting of a projectile into fragments occurs almost instantly, i.e., the internal forces tearing it apart act within a very short period of time. It is obvious that the change in the velocity of the fragments under the influence of gravity over such a short period of time can be neglected in comparison with the change in their speed under the influence of these internal forces. Therefore, although the system under consideration, strictly speaking, is not closed, we can assume that its total momentum when the projectile ruptures remains unchanged.
From the law of conservation of momentum one can immediately identify some features of the movement of fragments. Momentum is a vector quantity. Before the explosion, it lay in the plane of the projectile trajectory. Since, as stated in the condition, the speed of one of the fragments is vertical, i.e. its momentum remained in the same plane, then the momentum of the second fragment also lies in this plane. This means that the trajectory of the second fragment will remain in the same plane.
Further, from the law of conservation of the horizontal component of the total impulse it follows that the horizontal component of the velocity of the second fragment is equal because its mass is equal to half the mass of the projectile, and the horizontal component of the impulse of the first fragment is equal to zero by condition. Therefore, the horizontal flight range of the second fragment is from
the location of the rupture is equal to the product of the time of its flight. How to find this time?
To do this, remember that the vertical components of the impulses (and therefore the velocities) of the fragments must be equal in magnitude and directed in opposite directions. The flight time of the second fragment of interest to us depends, obviously, on whether the vertical component of its speed is directed upward or downward at the moment the projectile explodes (Fig. 108).
Rice. 108. Trajectory of fragments after a shell burst
This is easy to find out by comparing the time given in the condition for the vertical fall of the first fragment with the time free fall from height A. If then starting speed the first fragment is directed downward, and the vertical component of the velocity of the second is directed upward, and vice versa (cases a and in Fig. 108).