Rotary work. Moment of power

Consider the work done during the rotation of a material point around a circle under the action of the projection of the acting force on the displacement (the tangential component of the force). In accordance with (3.1) and Fig. 4.4, passing from the parameters of translational motion to the parameters of rotational motion (dS = Rdcp)

Here, the concept of the moment of force about the axis of rotation OOi is introduced as the product of the force F s on the shoulder of force R:

As can be seen from relation (4.8), moment of force in rotational motion is analogous to force in translational motion, since both parameters when multiplied by analogues dcp and dS give work. Obviously, the moment of force must also be specified vectorially, and with respect to the point O, its definition is given through the vector product and has the form

Finally: work during rotational motion is equal to the scalar product of the moment of force and the angular displacement:

Kinetic energy during rotational motion. Moment of inertia

Consider an absolutely rigid body rotating about a fixed axis. Let's mentally divide this body into infinitely small pieces with infinitely small sizes and masses mi, m2, Shz..., located at a distance R b R 2 , R3 ... from the axis. We find the kinetic energy of a rotating body as the sum of the kinetic energies of its small parts

where Y is the moment of inertia of a rigid body, relative to a given axis OOj.

From a comparison of the formulas for the kinetic energy of translational and rotational motions, it can be seen that moment of inertia in rotational motion is analogous to mass in translational motion. Formula (4.12) is convenient for calculating the moment of inertia of systems consisting of individual material points. To calculate the moment of inertia of solid bodies, using the definition of the integral, we can transform (4.12) to the form

It is easy to see that the moment of inertia depends on the choice of axis and changes with its parallel translation and rotation. We present the values ​​of the moments of inertia for some homogeneous bodies.

From (4.12) it is seen that moment of inertia of a material point equals

where t- point mass;

R- distance to the axis of rotation.

It is easy to calculate the moment of inertia for hollow thin-walled cylinder(or a special case of a cylinder with a small height - thin ring) radius R about the axis of symmetry. The distance to the axis of rotation of all points for such a body is the same, equal to the radius and can be taken out from under the sign of the sum (4.12):

solid cylinder(or a special case of a cylinder with a small height - disk) radius R to calculate the moment of inertia about the axis of symmetry requires the calculation of the integral (4.13). The mass in this case, on average, is concentrated somewhat closer than in the case of a hollow cylinder, and the formula will be similar to (4.15), but a coefficient less than one will appear in it. Let's find this coefficient.

Let a solid cylinder have a density R and height h. Let's break it down into

hollow cylinders (thin cylindrical surfaces) thick dr(Fig. 4.5) shows a projection perpendicular to the axis of symmetry). The volume of such a hollow cylinder of radius G is equal to the surface area multiplied by the thickness: weight: and the moment

inertia according to (4.15): Total moment

of inertia of a solid cylinder is obtained by integrating (summing) the moments of inertia of hollow cylinders:

. Considering that the mass of a solid cylinder is related to

density formula t = 7iR 2 hp we finally have the moment of inertia of a solid cylinder:

Similarly searched moment of inertia of a thin rod length L and the masses t, if the axis of rotation is perpendicular to the rod and passes through its middle. Let us split such a rod in accordance with Fig. 4.6

into thick pieces dl. The mass of such a piece is dm=m dl/L, and the moment of inertia according to Paul

The new moment of inertia of a thin rod is obtained by integrating (summing) the moments of inertia of the pieces:

« 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.

If a body is brought into rotation by a force, then its energy increases by the amount of work expended. As in translational motion, this work depends on the force and the displacement produced. However, the displacement is now angular and the expression for working when moving a material point is not applicable. Because the body is absolutely rigid, then the work of the force, although it is applied at a point, is equal to the work expended on turning the whole body.

When turning through an angle, the point of application of the force travels a path. In this case, the work is equal to the product of the projection of the force on the direction of displacement by the magnitude of the displacement: ; From fig. it can be seen that is the arm of the force, and is the moment of the force.

Then elementary work: . If , then .

The work of rotation goes to increase the kinetic energy of the body

; Substituting , we get: or taking into account the equation of dynamics: , it is clear that , i.e. the same expression.

6. Non-inertial frames of reference

End of work -

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All topics in this section:

mechanical movement
Matter, as is known, exists in two forms: in the form of substance and field. The first type includes atoms and molecules, from which all bodies are built. The second type includes all types of fields: gravity

Space and time
All bodies exist and move in space and time. These concepts are fundamental to all natural sciences. Any body has dimensions, i.e. its spatial extent

Reference system
To unambiguously determine the position of a body at an arbitrary point in time, it is necessary to choose a reference system - a coordinate system equipped with a clock and rigidly connected to an absolutely rigid body, according to

Kinematic equations of motion
When t.M moves, its coordinates and change with time, therefore, to set the law of motion, it is necessary to specify the type of

Movement, elementary movement
Let point M move from A to B along a curved path AB. At the initial moment, its radius vector is equal to

Acceleration. Normal and tangential accelerations
The movement of a point is also characterized by acceleration - the speed of change in speed. If the speed of a point in an arbitrary time

translational movement
The simplest form of mechanical motion of a rigid body is translational motion, in which the straight line connecting any two points of the body moves with the body, remaining parallel | its

Law of inertia
Classical mechanics is based on Newton's three laws, formulated by him in the work "Mathematical Principles of Natural Philosophy", published in 1687. These laws were the result of a genius

Inertial frame of reference
It is known that mechanical motion is relative and its nature depends on the choice of reference frame. Newton's first law is not valid in all frames of reference. For example, bodies lying on a smooth surface

Weight. Newton's second law
The main task of dynamics is to determine the characteristics of the motion of bodies under the action of forces applied to them. It is known from experience that under the influence of force

The basic law of the dynamics of a material point
The equation describes the change in the motion of a body of finite dimensions under the action of a force in the absence of deformation and if it

Newton's third law
Observations and experiments show that the mechanical action of one body on another is always an interaction. If body 2 acts on body 1, then body 1 necessarily counteracts those

Galilean transformations
They allow one to determine the kinematic quantities in the transition from one inertial frame of reference to another. Let's take

Galileo's principle of relativity
The acceleration of any point in all frames of reference moving relative to each other in a straight line and uniformly is the same:

Conserved quantities
Any body or system of bodies is a collection of material points or particles. The state of such a system at some point in time in mechanics is determined by setting the coordinates and velocities in

Center of mass
In any system of particles, you can find a point called the center of mass

Equation of motion of the center of mass
The basic law of dynamics can be written in a different form, knowing the concept of the center of mass of the system:

Conservative forces
If a force acts on a particle placed there at each point in space, it is said that the particle is in a field of forces, for example, in the field of gravity, gravitational, Coulomb and other forces. Field

Central Forces
Any force field is caused by the action of a certain body or system of bodies. The force acting on a particle in this field is about

Potential energy of a particle in a force field
The fact that the work of a conservative force (for a stationary field) depends only on the initial and final positions of the particle in the field allows us to introduce the important physical concept of potentially

Relationship between potential energy and force for a conservative field
The interaction of a particle with surrounding bodies can be described in two ways: using the concept of force or using the concept of potential energy. The first method is more general, because it applies to forces

Kinetic energy of a particle in a force field
Let a particle with mass move in forces

Total mechanical energy of a particle
It is known that the increment in the kinetic energy of a particle when moving in a force field is equal to the elementary work of all forces acting on the particle:

Law of conservation of mechanical energy of a particle
It follows from the expression that in a stationary field of conservative forces, the total mechanical energy of a particle can change

Kinematics
Rotate the body through some angle

The angular momentum of the particle. Moment of power
In addition to energy and momentum, there is another physical quantity with which the conservation law is associated - this is the angular momentum. Particle angular momentum

Moment of momentum and moment of force about the axis
Let us take in the frame of reference we are interested in an arbitrary fixed axis

The law of conservation of momentum of the system
Let us consider a system consisting of two interacting particles, which are also acted upon by external forces and

Thus, the angular momentum of a closed system of particles remains constant, does not change with time
This is true for any point in the inertial frame of reference: . Angular moments of individual parts of the system m

Moment of inertia of a rigid body
Consider a rigid body that can

Rigid Body Rotation Dynamics Equation
The equation of the dynamics of rotation of a rigid body can be obtained by writing the equation of moments for a rigid body rotating around an arbitrary axis

Kinetic energy of a rotating body
Consider an absolutely rigid body rotating around a fixed axis passing through it. Let's break it down into particles with small volumes and masses

Centrifugal force of inertia
Consider a disk that rotates with a ball on a spring, put on a spoke, Fig.5.3. The ball is

Coriolis force
When a body moves relative to a rotating CO, in addition, another force appears - the Coriolis force or the Coriolis force

Small fluctuations
Consider a mechanical system whose position can be determined using a single quantity, say x. In this case, the system is said to have one degree of freedom. The value of x can be

Harmonic vibrations
The equation of Newton's 2nd Law in the absence of friction forces for a quasi-elastic force of the form has the form:

Mathematical pendulum
This is a material point suspended on an inextensible thread with a length that oscillates in a vertical plane.

physical pendulum
This is a rigid body that oscillates around a fixed axis associated with the body. The axis is perpendicular to the drawing and

damped vibrations
In a real oscillatory system, there are resistance forces, the action of which leads to a decrease in the potential energy of the system, and the oscillations will be damped. In the simplest case

Self-oscillations
With damped oscillations, the energy of the system gradually decreases and the oscillations stop. In order to make them undamped, it is necessary to replenish the energy of the system from the outside at a certain moment

Forced vibrations
If the oscillatory system, in addition to the resistance forces, is subjected to the action of an external periodic force that changes according to the harmonic law

Resonance
The curve of the dependence of the amplitude of forced oscillations on leads to the fact that for some specific for a given system

Wave propagation in an elastic medium
If a source of oscillations is placed in any place of an elastic medium (solid, liquid, gaseous), then due to the interaction between particles, the oscillation will propagate in the medium from particle to hour

Equation of plane and spherical waves
The wave equation expresses the dependence of the displacement of an oscillating particle on its coordinates,

wave equation
The wave equation is a solution to a differential equation called the wave equation. To establish it, we find the second partial derivatives with respect to time and coordinates from the equation

Work and power during rotation of a rigid body.

Let's find an expression for work during the rotation of the body. Let the force be applied at a point located at a distance from the axis - the angle between the direction of the force and the radius vector . Since the body is absolutely rigid, the work of this force is equal to the work expended on turning the whole body. When the body rotates through an infinitely small angle, the point of application passes the path and the work is equal to the product of the projection of the force on the direction of displacement by the magnitude of the displacement:

The modulus of the moment of force is equal to:

then we get the following formula for calculating the work:

Thus, the work during rotation of a rigid body is equal to the product of the moment of the acting force and the angle of rotation.

Kinetic energy of a rotating body.

Moment of inertia mat.t. called physical the value is numerically equal to the product of the mass of mat.t. by the square of the distance of this point to the axis of rotation. W ki \u003d m i V 2 i / 2 V i -Wr i Wi \u003d miw 2 r 2 i / 2 \u003d w 2 / 2 * m i r i 2 I i \u003d m i r 2 i moment of inertia of a rigid body is equal to the sum of all mat.t I=S i m i r 2 i the moment of inertia of a rigid body is called. physical value equal to the sum of the products of mat.t. by the squares of the distances from these points to the axis. W i -I i W 2 /2 W k \u003d IW 2 /2

W k \u003d S i W ki moment of inertia during rotational motion yavl. analogue of mass in translational motion. I=mR 2 /2

21. Non-inertial reference systems. Forces of inertia. The principle of equivalence. Equation of motion in non-inertial frames of reference.

Non-inertial frame of reference- an arbitrary reference system that is not inertial. Examples of non-inertial frames of reference: a frame moving in a straight line with constant acceleration, as well as a rotating frame.

When considering the equations of motion of a body in a non-inertial frame of reference, it is necessary to take into account additional inertial forces. Newton's laws are valid only in inertial frames of reference. In order to find the equation of motion in a non-inertial frame of reference, it is necessary to know the laws of transformation of forces and accelerations during the transition from an inertial frame to any non-inertial one.

Classical mechanics postulates the following two principles:

time is absolute, that is, the time intervals between any two events are the same in all arbitrarily moving frames of reference;

space is absolute, that is, the distance between any two material points is the same in all arbitrarily moving frames of reference.

These two principles make it possible to write down the equation of motion of a material point with respect to any non-inertial frame of reference in which Newton's First Law does not hold.

The basic equation of the dynamics of the relative motion of a material point has the form:

where is the mass of the body, is the acceleration of the body relative to the non-inertial frame of reference, is the sum of all external forces acting on the body, is the portable acceleration of the body, is the Coriolis acceleration of the body.

This equation can be written in the familiar form of Newton's Second Law by introducing fictitious inertial forces:

Portable inertia force

Coriolis force

inertia force- fictitious force that can be introduced in a non-inertial frame of reference so that the laws of mechanics in it coincide with the laws of inertial frames.

In mathematical calculations, the introduction of this force occurs by transforming the equation

F 1 +F 2 +…F n = ma to the form

F 1 + F 2 + ... F n –ma = 0 Where F i is the actual force, and –ma is the “force of inertia”.

Among the forces of inertia are the following:

simple force of inertia;

centrifugal force, which explains the tendency of bodies to fly away from the center in rotating frames of reference;

the Coriolis force, which explains the tendency of bodies to deviate from the radius during radial motion in rotating frames of reference;

From the point of view of general relativity, gravitational forces at any point are the forces of inertia at a given point in Einstein's curved space

Centrifugal force- the force of inertia, which is introduced in a rotating (non-inertial) frame of reference (in order to apply Newton's laws, calculated only for inertial FRs) and which is directed from the axis of rotation (hence the name).

The principle of equivalence of forces of gravity and inertia- a heuristic principle used by Albert Einstein in deriving the general theory of relativity. One of the options for his presentation: “The forces of gravitational interaction are proportional to the gravitational mass of the body, while the forces of inertia are proportional to the inertial mass of the body. If the inertial and gravitational masses are equal, then it is impossible to distinguish which force acts on a given body - gravitational or inertial force.

Einstein's formulation

Historically, the principle of relativity was formulated by Einstein as follows:

All phenomena in the gravitational field occur in exactly the same way as in the corresponding field of inertial forces, if the strengths of these fields coincide and the initial conditions for the bodies of the system are the same.

22. Galileo's principle of relativity. Galilean transformations. Classical velocity addition theorem. Invariance of Newton's laws in inertial frames of reference.

Galileo's principle of relativity- this is the principle of physical equality of inertial reference systems in classical mechanics, which manifests itself in the fact that the laws of mechanics are the same in all such systems.

Mathematically, Galileo's principle of relativity expresses the invariance (invariance) of the equations of mechanics with respect to transformations of the coordinates of moving points (and time) when moving from one inertial frame to another - Galilean transformations.
Let there be two inertial frames of reference, one of which, S, we will agree to consider as resting; the second system, S", moves with respect to S with a constant speed u as shown in the figure. Then the Galilean transformations for the coordinates of a material point in the systems S and S" will have the form:
x" = x - ut, y" = y, z" = z, t" = t (1)
(the primed quantities refer to the S frame, the unprimed quantities refer to S). Thus, time in classical mechanics, as well as the distance between any fixed points, is considered the same in all frames of reference.
From Galileo's transformations, one can obtain the relationship between the velocities of a point and its accelerations in both systems:
v" = v - u, (2)
a" = a.
In classical mechanics, the motion of a material point is determined by Newton's second law:
F = ma, (3)
where m is the mass of the point, and F is the resultant of all forces applied to it.
In this case, forces (and masses) are invariants in classical mechanics, i.e., quantities that do not change when moving from one frame of reference to another.
Therefore, under Galilean transformations, equation (3) does not change.
This is the mathematical expression of the Galilean principle of relativity.

GALILEO'S TRANSFORMATIONS.

In kinematics, all frames of reference are equal to each other and motion can be described in any of them. In the study of movements, sometimes it is necessary to move from one reference system (with the coordinate system OXYZ) to another - (О`Х`У`Z`). Let's consider the case when the second frame of reference moves relative to the first uniformly and rectilinearly with the speed V=const.

To facilitate the mathematical description, we assume that the corresponding coordinate axes are parallel to each other, that the velocity is directed along the X axis, and that at the initial time (t=0) the origins of both systems coincide with each other. Using the assumption, which is fair in classical physics, about the same flow of time in both systems, it is possible to write down the relations connecting the coordinates of some point A(x, y, z) and A (x`, y`, z`) in both systems. Such a transition from one reference system to another is called the Galilean transformation):

OXYZ O`X`U`Z`

x = x` + V x t x` = x - V x t

x = v` x + V x v` x = v x - V x

a x = a` x a` x = a x

The acceleration in both systems is the same (V=const). The deep meaning of Galileo's transformations will be clarified in dynamics. Galileo's transformation of velocities reflects the principle of independence of displacements that takes place in classical physics.

Addition of speeds in SRT

The classical law of addition of velocities cannot be valid, because it contradicts the statement about the constancy of the speed of light in vacuum. If the train is moving at a speed v and a light wave propagates in the car in the direction of the train, then its speed relative to the Earth is still c, but not v+c.

Let's consider two reference systems.

In system K 0 the body is moving at a speed v one . As for the system K it moves at a speed v 2. According to the law of addition of speeds in SRT:

If a v<<c and v 1 << c, then the term can be neglected, and then we obtain the classical law of addition of velocities: v 2 = v 1 + v.

At v 1 = c speed v 2 equals c, as required by the second postulate of the theory of relativity:

At v 1 = c and at v = c speed v 2 again equals speed c.

A remarkable property of the law of addition is that at any speed v 1 and v(not more c), resulting speed v 2 does not exceed c. The speed of movement of real bodies is greater than the speed of light, it is impossible.

Addition of speeds

When considering a complex motion (that is, when a point or body moves in one frame of reference, and it moves relative to another), the question arises about the relationship of velocities in 2 frames of reference.

classical mechanics

In classical mechanics, the absolute velocity of a point is equal to the vector sum of its relative and translational velocities:

In plain language: The speed of a body relative to a fixed frame of reference is equal to the vector sum of the speed of this body relative to a moving frame of reference and the speed of the most mobile frame of reference relative to a fixed frame.

When rotating a rigid body with an axis of rotation z, under the influence of a moment of force Mz work is done about the z-axis

The total work done when turning through the angle j is

At a constant moment of forces, the last expression takes the form:

Energy

Energy - measure of a body's ability to do work. Moving bodies have kinetic energy. Since there are two main types of motion - translational and rotational, then the kinetic energy is represented by two formulas - for each type of motion. Potential energy is the energy of interaction. The decrease in the potential energy of the system occurs due to the work of potential forces. Expressions for the potential energy of gravity, gravity and elasticity, as well as for the kinetic energy of translational and rotational motions are given in the diagram. Complete mechanical energy is the sum of kinetic and potential.

momentum and angular momentum

Impulse particles p The product of the mass of a particle and its velocity is called:

angular momentumLrelative to point O is called the vector product of the radius vector r, which determines the position of the particle, and its momentum p:

The modulus of this vector is:

Let a rigid body have a fixed axis of rotation z, along which the pseudovector of the angular velocity is directed w.

Table 6

Kinetic energy, work, impulse and angular momentum for various models of objects and movements

Ideal	Physical quantities
model	Kinetic energy	Pulse	angular momentum	Work
A material point or rigid body moving forward. m- mass, v - speed.			,	. At
A rigid body rotates with an angular velocity w. J- the moment of inertia, v c - the speed of the center of mass.				. At
A rigid body performs a complex plane motion. J ñ - the moment of inertia about the axis passing through the center of mass, v c - the speed of the center of mass. w is the angular velocity.

The angular momentum of a rotating rigid body coincides in direction with the angular velocity and is defined as

The definitions of these quantities (mathematical expressions) for a material point and the corresponding formulas for a rigid body with various forms of motion are given in Table 4.

Law formulations

Kinetic energy theorem

particles is equal to the algebraic sum of the work of all forces acting on the particle.

Increment of kinetic energy body systems is equal to the work done by all the forces acting on all the bodies of the system:

. (1)