Similarly to the moment of force, the moment of impulse (moment of momentum) of a material point is determined

Similarly to the moment of force, the moment of impulse (moment of momentum) of a material point is determined. The angular momentum relative to the point O is equal to

The angular momentum about the z-axis is the component Lz along this axis of angular momentum L relative to the point O lying on the axis (Fig. 97):

where R is the component of the radius vector r perpendicular to the z axis, and p τ is the component of the vector p perpendicular to the plane passing through the z axis and the point m.

Let us find out what determines the change in the angular momentum with time. To do this, we differentiate (37.1) with respect to time t, using the product differentiation rule:

(3 7.5 )

The first term is equal to zero, since it is a vector product of vectors of the same direction. Indeed, the vector is equal to the velocity vector v and, therefore, coincides in direction with the vector p=mv. According to Newton's second law, the vector is equal to the force f acting on the body [see. (22.3)]. Therefore, expression (37.5) can be written as follows:

(3 7.6 )

where M is the moment of forces applied to the material point, taken relative to the same point O, relative to which the angular momentum L is taken.

From relation (37.6) it follows that if the resulting moment of forces acting on a material point relative to any point O is equal to zero, then the angular momentum of the material point, taken relative to the same point O, will remain constant.

Taking the components along the z axis from the vectors included in the formula (37.6), we obtain the expression:

(3 7.7 )

Formula (37.6) is similar to formula (22.3). From a comparison of these formulas, it follows that, just as the time derivative of momentum is equal to the force acting on a material point, the time derivative of the moment of momentum is equal to the moment of force.

Let's look at a few examples.

Example 1. Let the material point m move along the dotted line in Fig.96. Since the motion is rectilinear, the momentum of the material point changes only in absolute value, and

where f is the modulus of the force [in this case, f has the same direction as p (see Fig. 96), so that].

The arm t remains unchanged. Consequently,

which is consistent with formula (37.6) (in this case, L changes only in absolute value, and it increases, therefore ).

Example 2. A material point of mass m moves along a circle of radius R (Fig. 98).

The angular momentum of a material point relative to the center of the circle O is equal in absolute value:

L=mυR

(3 7.8 )

The vector L is perpendicular to the plane of the circle, and the direction of motion of the point and the vector L form a right-handed system.

Since the arm equal to R remains constant, the angular momentum can only be changed by changing the velocity modulus. With a uniform motion of a material point along a circle, the angular momentum remains constant both in magnitude and in direction. It is easy to see that in this case the moment of the force acting on the material point is equal to zero.

Example 3. Consider the motion of a material point in the central field of forces (see § 26). In accordance with (37.6), the angular momentum of a material point, taken relative to the center of forces, must remain constant in magnitude and direction (the moment of the central force relative to the center is zero). The radius vector r drawn from the center of forces to the point m and the vector L are perpendicular to each other. Therefore, the vector r remains all the time in the same plane, perpendicular to the direction L. Consequently, the movement of a material point in the central field of forces will occur along a curve lying in a plane passing through the center of forces.

Depending on the sign of the central forces (i.e., whether they are forces of attraction or repulsion), as well as on the initial conditions, the trajectory is a hyperbola, a parabola, or an ellipse (in particular, a circle). For example, the Earth moves in an elliptical orbit, in one of the focuses of which the Sun is placed.

Law of conservation of angular momentum. Consider a system of N material points. Just as it was done in §23, we divide the forces acting on points into internal and external. The resulting moment of internal forces acting on i-th material point, we denote by the symbol , the resulting moment of external forces acting on the same point, by the symbol M i . Then equation (37.6) for i-th material points will look like:

(i=1, 2,…, N)

This expression is a set of N equations that differ from each other by the values ​​of the index i . Adding these equations, we get:

is called the angular momentum of the system of material points.

The sum of the moments of internal forces [the first of the sums on the right side of formula (37.9)], as shown at the end of §36, is equal to zero. Therefore, denoting the total moment of external forces by the symbol M, we can write that

(3 7.11 )

[the symbols L and M in this formula have a different meaning than the same symbols in formula (37.6)].

For a closed system of material points, M=0, as a result of which the total angular momentum L does not depend on time. Thus, we have come to the law of conservation of angular momentum: the angular momentum of a closed system of material points remains constant.

Note that the angular momentum remains constant for a system subjected to external influences, provided that the total moment of external forces acting on the bodies of the system is equal to zero.

Taking from the vectors on the left and right sides of equation (37.11), their components along the z axis, we arrive at the relation:

(3 7.12 )

It may happen that the resulting moment of external forces relative to the point O is different from zero (M≠0), but the component M z of the vector M in some direction z is equal to zero. Then, according to (37.12), the component L z of the angular momentum of the system along the z axis will be preserved.

According to formula (2.1 1)

where is the projection onto the z-axis of the vector , and L z is the projection onto the z-axis of the vector L . Multiply both sides of the equality by the ort e z z axis and, taking into account that e z does not depend on t, we introduce it on the right side under the derivative sign. As a result, we get:

But the product of e z times the projection of the vector on the z-axis gives the z-component of that vector (see footnote on page 132). Consequently,

where is the component along the axis z vector .

There is the product of its mass and speed:

An analogue of momentum in rotational motion is the moment of momentum, which is the product of the moment of inertia of a material point and its angular velocity:

L \u003d Iω, kg m 2 s -1

The angular momentum is a vector quantity, in direction it coincides with the direction of the angular velocity vector.

Law of conservation of angular momentum

The angular momentum is conserved if the sum of all moments of external forces is equal to zero.

A clear use of the moment of momentum can be seen during the performance of the skaters, when they start the rotation with their arms wide apart, gradually closing their arms, they increase the speed of their rotation. Thus, they decrease their moment of inertia and increase their angular velocity of rotation. Thus, knowing the initial angular velocity of rotation ω 0 and its moment of inertia with outstretched I 0 and closed arms I 1, using the law of conservation of angular momentum, you can find the final angular velocity ω 1:

I 0 ω 0 = I 1 ω 1 ω 1 = (I 0 ω 0)/I 1

Applying the law of conservation of momentum, one can quite simply calculate the parameters of the orbital motion of planets and spacecraft.

On the page "The law of universal gravitation" we calculated the linear velocity of the Moon in an orbit with a radius of 392,500 km (average value). But, as you know, the Moon moves in an elliptical orbit, which at perigee is 356,400 km, and at apogee - 406,700 km. Using the knowledge gained, we calculate the speed of the Moon at perigee and apogee.

Initial data:

• r cf = 392500 km;
• v cf = 3600 km/h;
• r p \u003d 356400 km;
• v p -?;
• r a =406700 km;
• v a -?

According to the law of conservation of momentum, we have the following equalities:

I cf ω cf = I p ω p I cf ω cf = I a ω a

Since the diameter of the Moon (3476 km) is small compared to the distance to the Earth, we will consider the Moon as a material point, which will greatly simplify the calculations without significantly affecting their accuracy.

The moments of inertia for a material point will be equal to:

I cf \u003d mr cf 2 I p \u003d mr p 2 I a \u003d mr a 2

Angular speeds:

ω cf = v cf /r cf ω p = v p /r p ω a = v a /r a

We will carry out the corresponding substitutions in the formula for the law of conservation of momentum:

(mr cf 2) (v cf / r cf) = (mr p 2) (v p / r c) (mr cf 2) (v cf / r cf) = (mr a 2) (v a / r a)

After performing simple algebraic transformations, we get:

V p \u003d v cf (r cf / r p) v a \u003d v cf (r cf / r a)

We substitute numerical values:

V p \u003d 3600 392500 / 356400 \u003d 3964 km / h v a \u003d 3600 392500 / 406700 \u003d 3474 km / h

When solving problems on the movement of bodies in space, formulas for the conservation of kinetic energy and momentum are often used. It turns out that similar expressions exist for rotating bodies. In this article, the law of conservation of angular momentum is considered in detail (corresponding formulas are also given) and an example of solving the problem is given.

Rotation process and angular momentum

Before proceeding to the consideration of the formula for the law of conservation of angular momentum, it is necessary to get acquainted with this physical concept. The easiest way to enter it is to use the figure below.

The figure shows that at the end of the vector r¯, directed from the axis of rotation and perpendicular to it, there is some material point of mass m. This point moves along a circle of the named radius with a linear velocity v¯. It is known from physics that the product of mass and linear velocity is called momentum (p¯). Now it's time to enter a new value:

L¯ = r¯*m*v¯ = r¯*p¯.

Here the vector quantity L¯ represents the angular momentum. To switch to the scalar notation, it is necessary to know the moduli of the corresponding values ​​of r¯ and p¯, as well as the angle θ between them. The scalar formula for L is:

L = r*m*v*sin(θ) = r*p*sin(θ).

In the figure above, the angle θ is a right angle, so you can simply write:

L = r*m*v = r*p.

From the written expressions it follows that the unit for L will be kg * m 2 / s.

Direction of the angular momentum vector

Since the value in question is a vector (the result of a cross product), it will have a certain direction. It follows from the properties of the product of two vectors that their result will give a third vector perpendicular to the plane formed by the first two. At the same time, it will be directed in such a way that if you look from its end, the body will rotate counterclockwise.

The result of applying this rule is shown in the figure in the previous paragraph. It shows that L¯ is directed upwards, because if you look at the body from above, its movement will occur against the clock. When solving problems, it is important to take into account the direction during the transition to the scalar notation. Thus, the considered angular momentum is considered positive. If the body rotated clockwise, then in the scalar formula, L should have been preceded by a minus sign (-L).

Linear momentum analogy

Considering the topic of angular momentum and the law of its conservation, one mathematical trick can be done - transform the expression for L¯ by multiplying and dividing it by r 2. Then you get:

L¯ = r*m*v¯*r 2 /r 2 = m*r 2 *v¯/r.

In this expression, the ratio of velocity to radius of rotation is called angular velocity. It is usually denoted by the letter of the Greek alphabet ω. This value shows how many degrees (radians) the body will rotate along the orbit of its rotation per unit time. In turn, the product of mass times the square of radius is also physical quantity with its own name. Designate it I and call the moment of inertia.

As a result, the formula for the angular momentum is converted into the following form of writing:

L¯ = I *ω¯, where ω¯= v¯/r and I=m*r 2 .

The expression demonstrates that the direction of angular momentum L¯ and angular velocity ω¯ coincide for a system consisting of a rotating material point. The quantity I is of particular interest. It is discussed in more detail below.

moment of inertia of the body

The introduced value I characterizes the "resistance" of the body to any change in the speed of its rotation. That is, it plays exactly the same role as the inertia of the body during the linear movement of the object. In fact, I for circular motion from a physical point of view means the same as mass in linear motion.

As it was shown, for a material point with a mass m, rotating around an axis at a distance r from it, it is easy to calculate the moment of inertia (I = m * r 2), but for any other bodies this calculation will be somewhat complicated, since it involves the use of an integral.

For a body of arbitrary shape, I can be determined using the following expression:

I = ∫ m (r 2 *dm) = ∫ V (r 2 *ρ*dV), where ρ is the density of the material.

The expressions above mean that in order to calculate the moment of inertia, the whole body should be divided into infinitesimal volumes dV, multiplied by the square of the distance to the axis of rotation and by the density and summed up.

For bodies of different shapes, this problem is solved. Below is the data for some of them.

Material point: I = m*r 2 .

Disc or cylinder: I \u003d 1/2 * m * r 2.

A rod of length l, fixed in the center: I \u003d 1/12 * m * l 2.

Ball: I \u003d 2/5 * m * r 2.

The moment of inertia depends on the distributed mass of the body relative to the axis of rotation: the farther away from the axis there is a large part of the mass, the greater will be I for the system.

Change of angular momentum in time

Considering the angular momentum and the law of conservation of the angular momentum in physics, one can solve a simple problem: to determine how and due to what it will change in time. To do this, take the derivative with respect to dt:

dL¯/dt = d(r¯*m*v¯)/dt = m*v¯*dr¯/dt+r*m*dv¯/dt.

The first term here is equal to zero, since dr¯/dt = v¯ and the product of vectors v¯*v¯ = 0 (sin(0) = 0). The second term can be rewritten as follows:

dL¯/dt =r*m*a¯, where acceleration a = dv¯/dt, whence:

dL¯/dt =r*F¯=M¯.

The quantity M¯, according to the definition, is called the moment of force. It characterizes the action of the force F¯ on a material point of mass m located at a distance r from the axis of rotation.

What does the resulting expression show? It demonstrates that a change in the angular momentum L¯ is possible only due to the action of the moment of force M¯. This formula is the law of conservation of angular momentum of a point: if M¯=0, then dL¯/dt = 0 and L¯ is a constant value.

What moments of forces can change L¯ of the system?

There are two types of moments of force M¯: external and internal. The former are associated with the force impact on the elements of the system by any external forces, while the latter arise due to the interaction of the parts of the system.

According to Newton's third law, for every action force there is an opposite reaction force. This means that the total of any interactions within the system is always equal to zero, that is, it cannot affect changes in the angular momentum.

The value of L¯ can change only due to external moments of forces.

The formula for the law of conservation of angular momentum

The formula for writing the expression for the conservation of the value L¯ in case the sum of external moments of forces is equal to zero has the following form:

I 1 *ω 1 = I 2 *ω 2 .

Any changes in the moment of inertia of the system are proportionally reflected in the change in the angular velocity in such a way that the product I * ω does not change its value.

The form of this expression is similar to the law of conservation of linear momentum (I plays the role of mass, and ω plays the role of velocity). If we develop the analogy further, then, in addition to this expression, we can write another one that will reflect the conservation of the kinetic energy of rotation:

E = I *(ω) 2 /2 = const.

The application of the law of conservation of angular momentum finds itself in a number of processes and phenomena, which are briefly described below.

Examples of using the law of conservation of the quantity L¯

The following examples of the law of conservation of angular momentum have importance for the respective fields of activity.

• Any sport where it is necessary to make jumps with rotation. For example, a ballerina or figure skater begins a spinning pirouette by spreading her arms wide and moving her leg away from the center of gravity of her body. Then he presses the leg closer to the supporting one and the arms closer to the body, thereby reducing the moment of inertia (most of the points of the body are located close to the axis of rotation). According to the law of conservation of the value L, its angular velocity of rotation ω should increase.

• To change the direction of orientation relative to the Earth of any artificial satellite. This is done as follows: the satellite has a special heavy "flywheel", it is driven by an electric motor. The total angular momentum must be conserved, so the satellite itself begins to rotate in the opposite direction. When it takes the desired orientation in space, the flywheel is stopped, and the satellite also stops rotating.
• The evolution of the stars. As the star burns its hydrogen fuel, the forces of gravity begin to dominate the pressure of its plasma. This fact leads to a decrease in the radius of the star to a small size and, as a consequence, to a strong increase in the angular rotation rate. For example, it has been established that neutron stars with a diameter of several kilometers rotate at gigantic speeds, making one revolution in a fraction of a millisecond.

Solution of the problem on the conservation law L¯

Scientists have established that in a few billion years the Sun, having exhausted its energy reserves, will turn into a "white dwarf". It is necessary to calculate with what speed it will rotate around the axis.

First you need to write out the values ​​of the required quantities, which can be taken from the literature. So, now this star has a radius of 696,000 km and makes one revolution around its axis in 25.4 Earth days (value for the equatorial region). When it comes to the end of its evolutionary path, it will shrink to a size of 7000 km (about the radius of the Earth).

Assuming that the Sun is a perfect ball, we can use the formula of the law of conservation of angular momentum to solve this problem. You need to convert days to seconds and kilometers to meters, it turns out:

L = I*ω = 2/5*m*r 1 2 *ω 1 = 2/5*m*r 2 2 *ω 2 .

Where does it come from:

ω 2 \u003d (r 1 / r 2) 2 * ω 1 \u003d (696000000 / 7000000) 2 * 2 * 3.1416 / (25.4 * 24 * 3600) \u003d 0.0283 rad / s.

Here the formula for the angular velocity was used (ω = 2*pi/T, where T is the period of rotation in seconds). When performing the calculations, it was also assumed that the mass of the Sun remains constant (this is not true, since it will decrease. Nevertheless, the obtained value of ω 2 is the lower limit, that is, in reality, the dwarf Sun will rotate even faster).

Since a full turn is 2*pi radians, then you get:

T 2 \u003d 2 * pi / ω 2 \u003d 222 s.

That is, at the end of its life cycle, this star will make one rotation around its axis faster than in 222 seconds.

(kinetic moment, angular momentum, orbital momentum, angular momentum) characterizes the amount rotary motion. A quantity that depends on how much mass is rotating, how it is distributed about the axis of rotation, and how fast the rotation occurs.

Angular moment of a material point with respect to the point O is determined by the vector product
, where is the radius vector drawn from the point O, is the momentum of the material point.
The angular momentum of a material point relative to a fixed axis is equal to the projection onto this axis of the angular momentum vector, defined relative to an arbitrary point O of this axis. The value of the angular momentum does not depend on the position of the point O on the axis z.

Momentum of a rigid body relative to the axis is the sum of the angular momentum of the individual particles that make up the body relative to the axis. Considering that , we get
.

If the sum of the moments of forces acting on a body rotating around a fixed axis is equal to zero, then the angular momentum is conserved ( ) :
.

The time derivative of the angular momentum of a rigid body is equal to the sum of the moments of all forces acting on the body:
.

Law of conservation of angular momentum : angular momentum of a closed system of bodies about any fixed point does not change over time.
This is one of the fundamental laws of nature.

Similarly, for a closed system of bodies rotating around the axis z:

From here or .

If the moment of external forces relative to the fixed axis of rotation is identically equal to zero, then the angular momentum relative to this axis does not change during the motion.
The angular momentum is also constant for non-closed systems if the resulting moment of external forces applied to the system is zero.

Law of conservation of angular momentum follows from the basic equation of the dynamics of the rotational motion of a body fixed at a fixed point (Equation 4.8), and is as follows:

If the resulting moment of external forces relative to a fixed point is identically equal to zero, then the angular momentum of the body relative to this point does not change over time.

Indeed, if M= 0, then dL / dt= 0 , whence

(4.14)

In other words, the angular momentum of a closed system does not change over time.
From the basic law of the dynamics of a body rotating around a fixed axis z(Equation 4.13), follows law of conservation of angular momentum of a body about an axis:

If the moment of external forces relative to the fixed axis of rotation of the body is identically equal to zero, then the angular momentum of the body relative to this axis does not change in the process of motion, i.e. if Mz= 0, then dLz / dt= 0, whence

The law of conservation of angular momentum is a fundamental law of nature. The validity of this law is determined by the property of the symmetry of space - its isotropy, i.e. with the invariance of physical laws with respect to the choice of the direction of the coordinate axes of the reference system.

A change in the momentum of a material point is caused by the action of a force on it.

Multiplying equation (1.7) vectorially on the left by the radius vector , we obtain

Where is the vector calledangular momentum of a material point , and the vector - A moment of strength. The change in the angular momentum of a material point is caused by the moment of the force acting on it.

Several bodies, each of which can be considered as a material point, make upThe system of material points. For each material point, one can write the equation Newton's second law

(1.13)

In equation (1.13), the indices give the number of the material point. The forces acting on a material point are divided into external and internal. External forces are forces acting from bodies that are not included in the system of material points. Internal forces are forces acting on a material point from other bodies that make up a system of material points. Here is the force acting on the material point, the index of which is from the side of the material point with the number .

Several important laws follow from equations (1.13). If we sum them over all material points of the system, then we get

(1.14) ,

Value (1.15)

It is called the Impulse of the system of material points. The impulse of the system of material points is equal to the sum of the impulses of individual material points. In equation (1.14), the double sum for internal forces vanishes. For each pair of material points, it includes forces that, according to Newton's third law are equal and oppositely directed. For each pair, the vector sum of these forces vanishes. Therefore, the sum for all forces is also equal to zero.

As a result, we get:

(1.16)

Equation (1.16) expresses the law of change in the momentum of a system of material points. The change in the momentum of the system of material points is caused only by external forces. If external forces do not act on the system, then the momentum of the system of material points is preserved. A system of material points, which is not affected by external forces, is called an isolated, or closed, system of material points.

Similarly, for each material point, the equations (1.8) of the moment of impulses are written

(1.17)

When summing equations (1.17) over all material points of the system of material points, the sum of the moments of internal forces vanishes and it turns out The law of change of the moment of impulse of the system of material points:

(1.18)

Where the designations are introduced:- the moment of impulse of the system of material points, - the moment of external forces. The change in the angular momentum of the system of material points is caused by external forces acting on the system. For a closed system of material points, the angular momentum is conserved

.

A vector equal to the vector product of the radius vector and the force,
is called the moment of force.

Angular moment of a material point with respect to the point O is determined by the vector product
, where is the radius vector drawn from the point O, is the momentum of the material point.
The angular momentum of a material point relative to a fixed axis is equal to the projection onto this axis of the angular momentum vector, defined relative to an arbitrary point O of this axis. The value of the angular momentum does not depend on the position of the point O on the axis z.

Momentum of a rigid body relative to the axis is the sum of the angular momentum of the individual particles that make up the body relative to the axis. Considering that , we get
.

If the sum of the moments of forces acting on a body rotating around a fixed axis is equal to zero, then the angular momentum is conserved ( law of conservation of angular momentum):
.

The time derivative of the angular momentum of a rigid body is equal to the sum of the moments of all forces acting on the body:
.

The vector product of the radius-vector of a material point and its momentum: called angular momentum, this point relative to the point O (Fig. 5.4)

The vector is sometimes also called the angular momentum of a material point. It is directed along the axis of rotation perpendicular to the plane drawn through the vectors and and forms a right triple of vectors with them (when observed from the top of the vector, it can be seen that the rotation along the shortest distance from k occurs counterclockwise).

The vector sum of the angular momentum of all material points of the system is called the angular momentum (momentum) of the system relative to the point O:

The vectors and are mutually perpendicular and lie in the plane perpendicular to the axis of rotation of the body. That's why . Taking into account the relationship between linear and angular quantities

and is directed along the axis of rotation of the body in the same direction as the vector .

In this way.

The angular momentum of the body about the axis of rotation

(5.9)

Therefore, the angular momentum of the body about the axis of rotation is equal to the product of the moment of inertia of the body about the same axis and the angular velocity of rotation of the body about this axis.

Question #16

Three basic laws of motion of bodies:

1st law. Every body maintains its state of rest or uniform and

rectilinear motion until and in so far as the applied forces cause it to

change this state. This law is called the law of inertia. If m is mass

body, and v - its speed, then the law of inertia can be mathematically represented in

in the following form:

If v = 0, then the body is at rest; if v = const, then the body moves

uniform and straight. The product mv is called the momentum of the body.

A change in the momentum of a body can only occur as a result of its

interactions with other bodies, i.e. under the influence of force.

2nd law. The change in momentum is proportional to the applied driving force

force and occurs in the direction of the straight line along which this force acts.

The second law is mathematically written as follows: F = ma

i.e. the product of body mass m and its acceleration is equal to operating force F.

Equation (2.14) is called the basic law of the dynamics of a material point.

3rd law. An action always produces an equal and opposite reaction.

In other words, the effects of two bodies on each other are always equal and directed in

opposite sides.

If some body with mass m1 interacts with another body with mass m2,

then the first body changes the momentum of the second body m2v2 , no and itself

undergoes the same change in its momentum m1v1 from it, but

only backwards, i.e.

Newton's I law

There are such systems of reference, which are called inertial, with respect to which the bodies keep their speed unchanged, if they are not affected by other bodies or the action of other forces is compensated.

Newton's II law

The acceleration of a body is directly proportional to the resultant of the forces applied to the body and inversely proportional to its mass:

Newton's third law

The forces with which two bodies act on each other are equal in magnitude and opposite in direction.

Question #17

momentum change theorem - the change in the momentum of a system over a certain period of time is equal to the sum of the impulses acting on the system of external forces over the same period of time.

Center of Mass Motion Theorem

the system consists of n points, with corresponding masses .

We write for each point the basic law of dynamics

This system of differential equations of motion of the system, since for any point k of the system

Projecting equations (16.1.1) onto the coordinate axes, we obtain 3n equations, which are difficult to integrate in the general case,

Therefore, general dynamics theorems are usually used for which equations (16.1.1) are initial.

Theorem on the change in the kinetic energy of the system: in differential form: dT = , , are elementary works of external and internal forces acting on a point, in final form:

T 2 - T 1 \u003d. For an immutable system and T 2 - T 1 = , i.e. the change in the kinetic energy of a rigid body at some displacement is equal to the sum of the work of external forces acting on the body at this displacement. If the sum of the work of the reactions of the bonds on any possible displacement of the system is equal to zero, then such bonds are called ideal. Efficiency factor (efficiency):< 1, А пол.сопр. – работа полезных сил сопротивления (сил, для которых предназначена машина), А затр = А пол.сопр. + А вр.сопр. – затраченная работа, А вр.сопр. -– работа вредных сил сопротивления (силы трения, сопротивления воздуха и т.п.).

h \u003d N mash / N dv, N mash - the useful power of the machine, N dv - the power of the engine that sets it in motion.

Question #18

Galilean transformations are a limiting (special) case of Lorentz transformations for speeds that are small compared to the speed of light in vacuum and in a limited volume of space. For velocities up to the order of the planetary velocities in solar system(and even larger ones), Galileo's transformations are approximately correct with very high accuracy.

If ISO (inertial reference system) S moving relative to ISO S" with a constant velocity along the axis, and the origins coincide at the initial time in both systems, then the Galilean transformations have the form:

or, using vector notation,

(the last formula remains true for any direction of the coordinate axes).

§ As you can see, these are just formulas for the shift of the origin, linearly dependent on time (assumed to be the same for all reference systems).

From these transformations follow the relationship between the velocities of the point and its accelerations in both frames of reference:

§ Galilean transformations are a limiting (special) case of Lorentz transformations for low speeds (much less than the speed of light).

world broadcast

More than a hundred years ago, the hypothesis of an absolutely immobile space appeared - the world ether. The ether was defined as a kind of homogeneous medium that completely fills the entire matter and vacuum. For this he was called "world ether". What this substance is and what its properties are is a mystery, but it was known that light moves in the ether in exactly the same way as sound in the air. That is, in the form of a wave. Light was considered as a vibration of the world ether. It was also declared that matter moves through the ether without disturbing it, in the same way that a fine mesh with large cells moves inside water. Thus, matter and ether were strictly demarcated.

michelson experience

michelson experience, an experiment first set up by A. Michelson in 1881 to measure the influence of the Earth's motion on the speed of light. Negative result M. about. was one of the main experimental facts that formed the basis of the theory of relativity.

In physics of the late 19th century, it was assumed that light propagates in some universal world medium - the ether. At the same time, a number of phenomena (the aberration of light, the Fizeau experiment) led to the conclusion that the ether is immobile or is partially carried away by bodies during their movement. According to the fixed ether hypothesis, one can observe the "ethereal wind" when the Earth moves through the ether, and the speed of light relative to the Earth should depend on the direction of the light beam relative to the direction of its movement in the ether.

M. o. was carried out using a Michelson interferometer with equal arms; one arm was directed along the motion of the Earth, the other - perpendicular to it. When the entire device is rotated by 90°, the difference in the path of the rays should change sign, as a result of which the interference pattern should shift. The calculation shows that such a shift, expressed in fractions of the width of the interference fringe, is equal to D = ( 2l/ l)(v 2 / c 2), where l is the arm length of the interferometer, l is the wavelength of the applied light (yellow line Na), With - the speed of light in the air, v is the Earth's orbital speed. Since the value v/c for the Earth's orbital motion is about 10 -4 , then the expected displacement is very small and in the first M. o. was only 0.04. Nevertheless, already on the basis of this experience, Michelson came to the conclusion that the fixed ether hypothesis was incorrect.

In the future, M. o. was repeated many times. In the experiments of Michelson and E. W. Morley (1885-87), the interferometer was mounted on a massive plate floating in mercury (for smooth rotation). The optical path length with the help of multiple reflections from mirrors was increased to 11 m. In this case, the expected shift D " 0.4. Measurements confirmed the negative result of M. o. In 1958, the absence of a fixed ether was once again demonstrated at Columbia University (USA). Beams of radiation from two identical quantum generators of microwaves (masers) were directed in opposite directions - along Earth's motion and against motion - and compared their frequencies.With great accuracy (~ 10 -9%) it was found that the frequencies remain the same, while the "ethereal wind" would lead to the appearance of a difference in these frequencies by a value of almost 500 times greater than the measurement accuracy.

In classical physics, a negative result of M. o. could not be understood and agreed with other phenomena of the electrodynamics of moving media. In the theory of relativity, the constancy of the speed of light for all inertial frames of reference is taken as a postulate, confirmed by a large set of experiments.

Postulates of the theory of relativity

1) All laws of nature are the same in inertial frames of reference

2) The speed of light in vacuum is the same in all inertial frames of reference

Lorentz transformation, in the special theory of relativity - the transformation of the coordinates and time of an event during the transition from one inertial frame of reference to another. Obtained in 1904 by Kh. A. Lorentz as transformations with respect to which the equations of classical microscopic electrodynamics (the Lorentz-Maxwell equations) retain their form. In 1905, A. Einstein derived them based on two postulates that formed the basis of the special theory of relativity: the equality of all inertial frames of reference and the independence of the speed of light propagation in vacuum from the movement of the light source.

Let us consider a special case of two inertial frames of reference å and å’ with the x and x’ axes lying on the same straight line and, respectively, the other axes (y and y’, z and z’) being parallel. If the system å’ moves relative to å with a constant velocity u in the direction of the x axis, then the L.p. in the transition from å to å’ have the form:

,

where With is the speed of light in vacuum (dashed coordinates refer to the å’ system, unprimed coordinates to å).

L. p. lead to a number of important consequences, including the dependence of the linear dimensions of bodies and time intervals on the chosen frame of reference, the law of addition of velocities in the theory of relativity, and others. At velocities that are small compared to the speed of light (u<<c), L. p. go over to the Galilean transformations (see Galilean principle of relativity) , fair in classical Newtonian mechanics

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