Mechanics.

Question #1

Reference system. Inertial reference systems. Galileo-Einstein's principle of relativity.

reference system- this is a set of bodies in relation to which the movement of a given body and the coordinate system associated with it are described.

Inertial Reference System (ISO)- a system in which a freely moving body is at rest or uniform rectilinear motion.

Galileo-Einstein's principle of relativity- All phenomena of nature in any inertial frame of reference occur in the same way and have the same mathematical form. In other words, all ISOs are equal.

Question #2

The equation of motion. Types of motion of a rigid body. The main task of kinematics.

Equations of motion of a material point:

- kinematic equation of motion

Types of motion of a rigid body:

1) Translational motion - any straight line drawn in the body moves parallel to itself.

2) Rotational movement - any point of the body moves in a circle.

φ = φ(t)

The main task of kinematics- this is obtaining the time dependences of the velocity V= V(t) and the coordinates (or radius vector) r = r(t) of a material point from the known time dependence of its acceleration a = a(t) and the known initial conditions V 0 and r 0 .

Question #7

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

In theoretical mechanics generalized momentum is the partial derivative of the Lagrangian of the system with respect to the generalized velocity

If the Lagrangian of the system does not depend on some generalized coordinate, then due to Lagrange equations .

For a free particle, the Lagrange function has the form: , hence:

The independence of the Lagrangian of a closed system from its position in space follows from the property homogeneity of space: for a well-isolated system, its behavior does not depend on where in space we place it. By Noether's theorem this homogeneity implies the conservation of some physical quantity. This quantity is called the impulse (ordinary, not generalized).

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

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

If we are dealing with a body of finite size, to determine its momentum, it is necessary to break the body into small parts, which can be considered material points and sum over them, as a result we get:

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

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

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

,

where m i- weight i-th material point.

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

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

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

Question #8

Moment of inertia- a scalar physical quantity, a measure of the inertia of a body in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion. It is characterized by the distribution of masses in the body: the moment of inertia is equal to the sum of the products of elementary masses and the square of their distances to the base set

Axial moment of inertia

Axial moments of inertia of some bodies.

The moment of inertia of a mechanical system relative to a fixed axis ("axial moment of inertia") is called the value J a equal to the sum of the products of the masses of all n material points of the system into the squares of their distances to the axis:

,

• m i- weight i-th point,
• r i- distance from i-th point to the axis.

Axial moment of inertia body J a is a measure of the inertia of a body in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion.

,

• dm = ρ dV- mass of a small volume element of the body dV,
• ρ - density,
• r- distance from element dV to axis a.

If the body is homogeneous, that is, its density is the same everywhere, then

Formula derivation

dm and moments of inertia DJ i. Then

Thin-walled cylinder (ring, hoop)

Formula derivation

The moment of inertia of a body is equal to the sum of the moments of inertia of its constituent parts. Dividing a thin-walled cylinder into elements with a mass dm and moments of inertia DJ i. Then

Since all elements of a thin-walled cylinder are at the same distance from the axis of rotation, formula (1) is converted to the form

Steiner's theorem

Moment of inertia of a rigid body relative to any axis depends not only on the mass, shape and dimensions of the body, but also on the position of the body with respect to this axis. According to the Steiner theorem (Huygens-Steiner theorem), moment of inertia body J relative to an arbitrary axis is equal to the sum moment of inertia this body Jc relative to the axis passing through the center of mass of the body parallel to the considered axis, and the product of the body mass m per square distance d between axles:

If is the moment of inertia of the body about an axis passing through the center of mass of the body, then the moment of inertia about a parallel axis located at a distance from it is equal to

,

where is the total mass of the body.

For example, the moment of inertia of a rod about an axis passing through its end is:

Rotational energy

Kinetic energy of rotational motion- the energy of the body associated with its rotation.

The main kinematic characteristics of the rotational motion of a body are its angular velocity (ω) and angular acceleration. The main dynamic characteristics of rotational motion are the angular momentum about the rotation axis z:

Kz = Izω

and kinetic energy

where I z is the moment of inertia of the body about the axis of rotation.

A similar example can be found when considering a rotating molecule with principal axes of inertia I 1, I 2 and I 3. The rotational energy of such a molecule is given by the expression

where ω 1, ω 2, and ω 3 are the principal components of the angular velocity.

In the general case, the energy during rotation with angular velocity is found by the formula:

, where I is the inertia tensor.

Question #9

moment of impulse (angular momentum, angular momentum, orbital momentum, angular momentum) characterizes the amount of rotational 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.

It should be noted that rotation here is understood in a broad sense, not only as a regular rotation around an axis. For example, even with a rectilinear motion of a body past an arbitrary imaginary point that does not lie on the line of motion, it also has an angular momentum. Perhaps the greatest role is played by the angular momentum in describing the actual rotational motion. However, it is extremely important for a much wider class of problems (especially if the problem has central or axial symmetry, but not only in these cases).

Law of conservation of momentum(law of conservation of angular momentum) - the vector sum of all angular momenta about any axis for a closed system remains constant in the case of equilibrium of the system. In accordance with this, the angular momentum of a closed system with respect to any non-time derivative of the angular momentum is the moment of force:

Thus, the requirement of system closure can be weakened to the requirement that the main (total) moment of external forces be equal to zero:

where is the moment of one of the forces applied to the system of particles. (But of course, if there are no external forces at all, this requirement is also met).

Mathematically, the law of conservation of angular momentum follows from the isotropy of space, that is, from the invariance of space with respect to rotation through an arbitrary angle. When rotating through an arbitrary infinitesimal angle , the radius vector of the particle with the number will change by , and the velocities - . The Lagrange function of the system will not change during such a rotation, due to the isotropy of space. That's why

« Physics - Grade 10 "

Why does the skater stretch along the axis of rotation to increase the angular velocity of rotation.
Should a helicopter rotate when its propeller rotates?

The questions asked suggest that if external forces do not act on the body or their action is compensated and one part of the body begins to rotate in one direction, then the other part must rotate in the other direction, just as when fuel is ejected from a rocket, the rocket itself moves in the opposite direction.

moment of impulse.

If we consider a rotating disk, it becomes obvious that the total momentum of the disk is zero, since any particle of the body corresponds to a particle moving with an equal speed in absolute value, but in the opposite direction (Fig. 6.9).

But the disk is moving, the angular velocity of rotation of all particles is the same. However, it is clear that the farther the particle is from the axis of rotation, the greater its momentum. Therefore, for rotational motion it is necessary to introduce one more characteristic, similar to an impulse, - the angular momentum.

The angular momentum of a particle moving in a circle is the product of the particle's momentum and the distance from it to the axis of rotation (Fig. 6.10):

The linear and angular velocities are related by v = ωr, then

All points of a rigid matter move relative to a fixed axis of rotation with the same angular velocity. A rigid body can be represented as a collection of material points.

The angular momentum of a rigid body is equal to the product of the moment of inertia and the angular velocity of rotation:

The angular momentum is a vector quantity, according to formula (6.3), the angular momentum is directed in the same way as the angular velocity.

The basic equation of the dynamics of rotational motion in impulsive form.

The angular acceleration of a body is equal to the change in angular velocity divided by the time interval during which this change occurred: Substitute this expression into the basic equation for the dynamics of rotational motion hence I(ω 2 - ω 1) = MΔt, or IΔω = MΔt.

In this way,

∆L = M∆t. (6.4)

The change in the angular momentum is equal to the product of the total moment of forces acting on the body or system and the time of action of these forces.

Law of conservation of angular momentum:

If the total moment of forces acting on a body or system of bodies with a fixed axis of rotation is equal to zero, then the change in the angular momentum is also equal to zero, i.e., the angular momentum of the system remains constant.

∆L=0, L=const.

The change in the momentum of the system is equal to the total momentum of the forces acting on the system.

The spinning skater spreads his arms out to the sides, thereby increasing the moment of inertia in order to decrease the angular velocity of rotation.

The law of conservation of angular momentum can be demonstrated using the following experiment, called the "experiment with the Zhukovsky bench." A person stands on a bench with a vertical axis of rotation passing through its center. The man holds dumbbells in his hands. If the bench is made to rotate, then a person can change the speed of rotation by pressing the dumbbells to his chest or lowering his arms, and then spreading them apart. Spreading his arms, he increases the moment of inertia, and the angular velocity of rotation decreases (Fig. 6.11, a), lowering his hands, he reduces the moment of inertia, and the angular velocity of rotation of the bench increases (Fig. 6.11, b).

A person can also make a bench rotate by walking along its edge. In this case, the bench will rotate in the opposite direction, since the total angular momentum must remain equal to zero.

The principle of operation of devices called gyroscopes is based on the law of conservation of angular momentum. The main property of a gyroscope is the preservation of the direction of the axis of rotation, if external forces do not act on this axis. In the 19th century gyroscopes were used by navigators to navigate the sea.

Kinetic energy of a rotating rigid body.

The kinetic energy of a rotating solid body is equal to the sum of the kinetic energies of its individual particles. Let us divide the body into small elements, each of which can be considered a material point. Then the kinetic energy of the body is equal to the sum of the kinetic energies of the material points of which it consists:

The angular velocity of rotation of all points of the body is the same, therefore,

The value in brackets, as we already know, is the moment of inertia of the rigid body. Finally, the formula for the kinetic energy of a rigid body with a fixed axis of rotation has the form

In the general case of motion of a rigid body, when the axis of rotation is free, its kinetic energy is equal to the sum of the energies of translational and rotational motions. So, the kinetic energy of a wheel, the mass of which is concentrated in the rim, rolling along the road at a constant speed, is equal to

The table compares the formulas of the mechanics of the translational motion of a material point with similar formulas for the rotational motion of a rigid body.

The main dynamic characteristics of rotational motion are the angular momentum about the rotation axis z:

and kinetic energy

In the general case, the energy during rotation with angular velocity is found by the formula:

, where is the inertia tensor .

In thermodynamics

By exactly the same reasoning as in the case of translational motion, equipartition implies that at thermal equilibrium the average rotational energy of each particle of a monatomic gas is: (3/2)k B T. Similarly, the equipartition theorem allows one to calculate the root-mean-square angular velocity of molecules.

see also

Wikimedia Foundation. 2010 .

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Since a rigid body is a special case of a system of material points, the kinetic energy of the body during rotation around a fixed Z axis will be equal to the sum of the kinetic energies of all its material points, that is

All material points of a rigid body rotate in this case along circles with radii and with the same angular velocities. The linear speed of each material point of a rigid body is equal to . The kinetic energy of a rigid body takes the form

The sum on the right side of this expression, in accordance with (4.4), is the moment of inertia of this body about the given axis of rotation. Therefore, the formula for calculating the kinetic energy of a rigid body rotating relative to a fixed axis will take the final form:

. (4.21)

It is taken into account here that

The calculation of the kinetic energy of a rigid body in the case of arbitrary motion becomes much more complicated. Consider a plane motion, when the trajectories of all material points of the body lie in parallel planes. The speed of each material point of a rigid body, according to (1.44), can be represented as

,

where as the instantaneous axis of rotation we choose the axis passing through the center of inertia of the body perpendicular to the plane of the trajectory of some point of the body. In this case, in the last expression is the speed of the center of inertia of the body, - the radii of the circles along which the points of the body rotate with an angular velocity around the axis passing through the center of its inertia. Since with such a movement ^, then the vector equal to lies in the plane of the trajectory of the point.

Based on the above, the kinetic energy of the body during its plane motion is equal to

.

Raising the expression in parentheses to the square and taking out the constant values ​​for all points of the body beyond the sum sign, we obtain

Here it is taken into account that ^.

Consider each term on the right side of the last expression separately. The first term, due to the obvious equality, is equal to

The second term is equal to zero, since the sum determines the radius vector of the center of inertia (3.5), which in this case lies on the axis of rotation. The last term, taking into account (4.4), takes the form . Now, finally, the kinetic energy for an arbitrary, but plane motion of a rigid body can be represented as the sum of two terms:

, (4.23)

where the first term is the kinetic energy of a material point with a mass equal to the mass of the body and moving at a speed that the center of mass of the body has;

the second term is the kinetic energy of a body rotating about an axis (moving at speed) passing through its center of inertia.

Conclusions: So, the kinetic energy of a rigid body during its rotation around a fixed axis can be calculated using one of the relations (4.21), and in the case of a plane motion using (4.23).

Test questions.

4.4. In what cases does (4.23) go over to (4.21)?

4.5. What will the formula for the kinetic energy of a body look like during its plane motion if the instantaneous axis of rotation does not pass through the center of inertia? What is the meaning of the quantities included in the formula?

4.6. Show that the work of internal forces during the rotation of a rigid body is zero.

Tasks

1. Determine how many times the effective mass is greater than the gravitating mass of a train with a mass of 4000 tons, if the mass of the wheels is 15% of the mass of the train. Consider the wheels as disks with a diameter of 1.02 m. How will the answer change if the diameter of the wheels is half that?

2. Determine the acceleration with which a wheel pair of mass 1200 kg rolls down a hill with a slope of 0.08. Consider wheels as disks. Rolling resistance coefficient 0.004. Determine the adhesion force of the wheels to the rails.

3. Determine the acceleration with which a wheel pair with a mass of 1400 kg rolls up a hill with a slope of 0.05. Drag coefficient 0.002. What should be the coefficient of adhesion so that the wheels do not slip. Consider wheels as disks.

4. Determine the acceleration with which a wagon weighing 40 tons rolls down a hill with a slope of 0.020 if it has eight wheels weighing 1200 kg and a diameter of 1.02 m. Determine the force of adhesion of the wheels to the rails. Drag coefficient 0.003.

5. Determine the pressure force of the brake shoes on the tires, if a train weighing 4000 tons slows down with an acceleration of 0.3 m/s 2 . The moment of inertia of one wheelset is 600 kg m 2 , the number of axles is 400, the sliding friction coefficient of the block is 0.18, the rolling resistance coefficient is 0.004.

6. Determine the braking force acting on a four-axle wagon with a mass of 60 tons on the brake pad of a marshalling yard if the speed on a 30 m track decreased from 2 m/s to 1.5 m/s. The moment of inertia of one wheelset is 500 kg m 2 .

7. The speedometer of the locomotive showed an increase in the speed of the train within one minute from 10 m/s to 60 m/s. Probably, there was a slipping of the leading wheelset. Determine the moment of forces acting on the armature of the electric motor. Moment of inertia of wheelset 600 kg m 2 , anchors 120 kg m 2 . Gear ratio gear 4.2. The pressure force on the rails is 200 kN, the sliding friction coefficient of the wheels along the rail is 0.10.

11. KINETIC ENERGY OF THE ROTATOR

MOVEMENTS

We derive the formula for the kinetic energy of rotational motion. Let the body rotate with angular velocity ω about the fixed axis. Any small particle of the body performs translational motion in a circle with a speed , where r i - distance to the axis of rotation, radius of the orbit. Kinetic energy of a particle masses m i is equal to . The total kinetic energy of a system of particles is equal to the sum of their kinetic energies. Let us sum up the formulas for the kinetic energy of the particles of the body and take out the sign of the sum of half the square of the angular velocity, which is the same for all particles, . The sum of the products of the masses of particles and the squares of their distances to the axis of rotation is the moment of inertia of the body about the axis of rotation . So, the kinetic energy of a body rotating about a fixed axis is equal to half the product of the moment of inertia of the body about the axis and the square of the angular velocity of rotation:

Rotating bodies can store mechanical energy. Such bodies are called flywheels. Usually these are bodies of revolution. The use of flywheels in the potter's wheel has been known since antiquity. In internal combustion engines, during the working stroke, the piston imparts mechanical energy to the flywheel, which then performs work on the rotation of the engine shaft for the next three cycles. In stamps and presses, the flywheel is driven by a relatively low-power electric motor, accumulates mechanical energy for almost a full revolution and, in a short moment of impact, gives it to the work of stamping.

There are numerous attempts to use rotating flywheels to drive vehicles: cars, buses. They are called mahomobiles, gyro carriers. Many such experimental machines were created. It would be promising to use flywheels for energy storage during braking of electric trains in order to use the accumulated energy during subsequent acceleration. Flywheel energy storage is known to be used on New York City subway trains.