If M.T. It rotates around the circumference, then the force acts on it, then elementary work is performed on some angle:

(22)

If the current force is potential, then

then (24)

Power during rotation

Instant power developing when rotating the body:

Kinetic energy of a rotating body

Kinetic energy material point. Kinetic energy sis material points . Because , We obtain the expression of the kinetic energy of rotation:

With a flat movement (the cylinder rolls along the inclined plane) the total speed is equal to:

where is the speed center of the cylinder.

Full equal to the sum of the kinetic energy of the translational movement of its center of mass and the kinetic energy of the rotational motion of the body relative to the center of the masses, i.e.:

(28)

Conclusion:

And now, considering all the lecture material, summarize, comparable to the magnitude and equation of the rotational and progressive movement of the body:

Protective traffic	Rotary traffic
Weight	M.	Moment of inertia	I.
Way	S.	Angle of rotation
Speed		Angular velocity
Pulse		Moment of impulse
Acceleration		Angular acceleration
Equality external forces	F.	The sum of the moments of external forces	M.
The main equation of speakers		The main equation of speakers
Work	FDS.	Work of rotation
Kinetic energy		Kinetic energy of rotation

Attachment 1:

A man stands in the center of the bench Zhukovsky and along with it rotates inertia. Rotation frequency n. 1 \u003d 0.5 C -1. Moment of inertia j O. human bodies

the axis of rotation is 1.6 kg m 2. In the arms elongated, a man holds a weight m.\u003d 2 kg each. Distance between Garyami l. 1 \u003d L, 6 m. Determine the frequency of rotation n. 2 , benches with man when he lowers his arms and distance l. 2 Between the weights will become equal to 0.4 m. The moment of inertia begins neglected.

Properties of symmetry and conservation laws.

Energy saving.

The laws of conservation considered in the mechanics are based on the properties of space and time.

The preservation of energy is associated with the homogeneity of the time, the preservation of the pulse - with the uniformity of space and, finally, the preservation of the moment of the pulse is due to the isotropy of space.

We begin with the law of conservation of energy. Let the particle system be in constant conditions (this takes place if the system is closed or exposed to a permanent external force field); Communications (if any) are ideal and stationary. In this case the time due to its homogeneity can not be explicitly in the Lagrange function. Really uniformity means the equivalent of all moments of time. Therefore, the replacement of one point of time to another without changing the values \u200b\u200bof coordinates and velocities of particles should not change the mechanical properties of the system. This is certainly right if the replacement of one point of time does not change the conditions in which the system is located, that is, in case of independence from the time of the external field (in particular, this field may be absent).

So for a closed system located in a closed power field ,.

Consider an absolutely solid, rotating around the stationary axis. If mentally break this body on n. Points masses m 1, m 2, ..., m ndistances r 1, R 2, ..., R n From the axis of rotation, then during rotation they will describe the circles and move with different linear speeds v 1, V 2, ..., V N. Since the body is absolutely solid, the angular speed of rotation of the points will be the same:

The kinetic energy of the rotating body is the sum of the kinetic energies of its points, i.e.

Given the relationship between the corner and linear speeds, we get:

Comparison of formula (4.9) with an expression for the kinetic energy of the body moving progressively at speeds v., shows that the moment of inertia is a measure of body inertness in rotational motion.
If the solid is moving progressively at speed v. And at the same time rotates with an angular velocity ω around the axis passing through its center of inertia, its kinetic energy is defined as the sum of the two components:

(4.10)

where v C. - speed center of body mass; J C. - The moment of inertia of the body relative to the axis passing through its center of mass.
Moment of power relative to the fixed axis z. called scalar value M Z.equal to the projection on this axle vector M. The moment of force defined relative to an arbitrary point 0 of this axis. Mother value M Z. does not depend on the choice of point of point 0 on the axis z..
If the axis z. coincides with the direction of the vector M.The moment of force is presented in the form of a vector coinciding with the axis:

M z \u003d [ rf] Z.
We find an expression for working when rotating the body. Let power F. attached to a point in the axis of rotation at a distance r. (Fig. 4.6); α - angle between the direction of force and the radius-vector r.. Since the body is absolutely solid, the work of this force is equal to the work spent on the rotation of the entire body.

When turning the body on an infinitely small angle dφ. application point in goes path ds \u003d rdφ.And the work is equal to the work of the projection of force on the direction of displacement by the amount of displacement:

da \u003d fsinα * rdφ
Considering that FRSINα \u003d M Z can be recorded da \u003d M z dφwhere M Z. - moment of power relative to the axis of rotation. Thus, work during rotation of the body is equal to the moment of the acting force at the angle of rotation.
Work when rotating the body goes to an increase in its kinetic energy:

da \u003d De K
(4.11)

Equation (4.11) is equation of the dynamics of the rotational movement of the solid body relative to the fixed axis.

Work with rotational motion. Moment of power

Consider the work performed during the rotation of the material point around the circumference under the action of the projection of the current force on the movement (the tangential component of force). In accordance with (3.1) and Fig. 4.4, going from the parameters of the translational movement to the parameters of the rotational motion (DS \u003d R DCP)

It introduced the concept of the moment of force relative to the axis of rotation OOI as a work of force F S. on the shoulder strength R:

As can be seen from the ratio (4.8), the moment of force in the rotational movement is an analogue of power in a progressive movementSince both parameters are multiplied by analogs. dCP and ds. give work. Obviously, the moment of force should also be set vector, and relative to the point of its definition, it is given through a vector product and has the appearance

Finally: working with rotational motion is equal to the scalar product of the moment of force on the angular movement:

Kinetic energy with rotational motion. Moment of inertia

Consider an absolutely solid, rotating relative to the fixed axis. Mentally throw this body to infinitely small pieces with infinitely small dimensions and masses Mi, M2, SZ ..., located at a distance R b R 2, R3 ... from the axis. The kinetic energy of the rotating body will find as the amount of the kinetic energies of its small parts

where the moment of the inertia of the solid, relative to this axis Ooj.

From the comparison of the formulas of the kinetic energy of progressive and rotational motion, it can be seen that the moment of inertia in the rotational motion is an analogue of the mass in the translational movement. Formula (4.12) is convenient for calculating the moment of inertia systems consisting of individual material points. To calculate the moment of inertia of solid bodies, using the definition of the integral, can be converted (4.12) to mind

It is easy to see that the moment of inertia depends on the choice of the axis and changes when it is parallel to transfer and turn. We give the values \u200b\u200bof the moments of inertia for some homogeneous bodies.

From (4.12) it can be seen that moment of inertia of the material point Raven

where t. - point of point;

R. - Distance to the axis of rotation.

Easy to calculate the moment of inertia and for hollow thin-walled cylinder (or private cylinder case with low height - thin Ring) Radius r relative to the axis of symmetry. The distance to the axis of rotation of all points for such a body is equally equal to the radius and can be made out of the amount of the amount (4.12):

Solid cylinder (or private cylinder case with low height - disk) R radius for calculating the moment of inertia relative to the symmetry axis requires the calculation of the integral (4.13). In this case, the mass in this case focuses somewhat closer than in the case of a hollow cylinder, and the formula will be similar to (4.15), but there will be a coefficient less than one. We will find this coefficient.

Let a solid cylinder be density r and height h. Throw it on

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

inertia in accordance with (4.15): full moment

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

. Given that the mass of the solid cylinder is associated with

formula density t. = 7IR 2 HP. We have a finally moment of inertia of a solid cylinder:

Similarly Looking for moment of inertia of a thin rod Length L.and masses t, If the axis of rotation is perpendicular to the rod and passes through its middle. We divide such a rod in accordance with Fig. 4.6.

on pieces of thickness dL. The mass of such a piece is equal dm \u003d M DL / L,and the moment of inertia in accordance with the floor

the moment of inertia of the thin rod is obtained by integrating (summation) of the moments of inertia pieces:

For the kinematic description of the process of rotation of the solid, it is necessary to introduce such concepts as an angular movement Δ φ, the angular acceleration ε and the angular velocity ω:

ω \u003d δ φ δ t, (Δ t → 0), ε \u003d δ φ δ t, (Δ t → 0).

Corners are expressed in radians. For a positive direction of rotation, a direction counterclockwise is accepted.

When the solid is rotated relative to the stationary axis, all points of this body are moved with the same angular velocities and accelerations.

Figure 1. The rotation of the disk relative to the axis passing through its center O.

If the angular movement Δ Φ is small, then the linear movement vector module Δ s → some element of the mass Δ m the rotating solid can be expressed by the ratio:

Δ s \u003d R Δ φ,

in which R. - Module radius-vector R →.

Between the modules of angular and linear velocities, you can establish a connection through equality

Linear and angular acceleration modules are also interrelated:

a \u003d a τ \u003d r ε.

Vectors V → and A → \u003d A τ → aimer to tangent to the circle of radius R..

We also need to take into account the emergence of a normal or centripetal acceleration, which always occurs when the bodies in the circumference.

Definition 1.

The acceleration module is expressed by the formula:

a n \u003d v 2 r \u003d ω 2 r.

If splitting the rotating body into small fragments Δ m i, designate the distance to the rotation axis through R I., and linear velocity modules through V i, the record of the formula of the kinesthetic energy of the rotating body will look at:

E k \u003d σ i ν M V i 2 2 \u003d Σ i Δ m (R i Ω) 2 2 \u003d ω 2 2 σ i Δ m i r i 2.

Definition 2.

The physical value of σ i Δ M I R i 2 is called the moment of inertia I of the body relative to the axis of rotation. It depends on the mass distribution of the rotating body relative to the axis of rotation:

I \u003d Σ i Δ m i R i 2.

In the limit at Δ M → 0, this amount goes to the integral. Unit of measurement of the moment of inertia in C and - kilogram - meter in a square (k · m 2). Thus, the kinetic energy of the solid, rotating relative to the fixed axis, can be represented as:

E k \u003d i Ω 2 2.

In contrast to the expression that we used to describe the kinesthetic energy of a translationally moving body M V 2 2, instead of mass M. The formula includes the moment of inertia I.. We also take into account instead of a linear velocity V angular velocity ω.

If the body weight plays the bulk of the body for the dynamics of the translational movement, then the moment of inertia is in the dynamics of the rotational motion. But if the weight is the property of the solid body under consideration, which does not depend on the speed of movement and other factors, the moment of inertia depends on what axis the body rotates. For the same body, the moment of inertia will be determined by various axes of rotation.

In most tasks, it is believed that the axis of rotation of the solid body passes through the center of its mass.

The position X C, Y C of the mass center for a simple case of a system of two particles with M 1 and M 2 masses located in the plane X Y. At points with coordinates x 1, y 1 and x 2, y 2 is determined by expressions:

x c \u003d m 1 x 1 + m 2 x 2 m 1 + m 2, y c \u003d m 1 y 1 + m 2 y 2 m 1 + m 2.

Figure 2. Mass center C system of two particles.

In vector form, this ratio takes the form:

r C → \u003d M 1 R 1 → + M 2 R 2 → M 1 + m 2.

Similarly, for a system from many particles Radius-vector R C → The center of the masses is determined by the expression

r c → \u003d Σ m i R i → Σ m i.

If we are dealing with a solid body consisting of one part, then in the above amount of the amount for R C → must be replaced by integrals.

The center of masses in a homogeneous field of gravity coincides with the center of gravity. This means that if we take the body complex form And suspend it for the center of the masses, then in a uniform field of gravity, this body will be equilibrium. From here, a way to determine the center of masses of the complex body in practice is: it must be sequentially suspended over several points, simultaneously noting the vertical lines on the plumb.

Figure 3. Determination of the position of the center of mass c body complex shape. A 1, A 2, A 3 suspension points.

In the figure, we see the body that is suspended for the mass center. It is in a state of indifferent equilibrium. In a homogeneous field of gravity, gravity is applied to the mass center.

We can imagine any firm movement as the sum of two movements. The first progressive, which is produced at the speed of the center of mass body. The second is rotation relative to the axis, which passes through the center of mass.

Example 1.

Suppose What we have a wheel that rolls along the horizontal surface without slipping. All points of the wheel during movement are moved parallel to one plane. Such a movement we can designate as flat.

Definition 3.

The kinesthetic energy of the rotating solid with a flat movement will be equal to the sum of the kinetic energy of the translational movement and the kinetic energy of rotation relative to the axis, which was carried out through the center of the masses and is perpendicular to the planes in which all points of the body are moving:

E k \u003d m V C 2 2 + I C Ω 2 2,

where M. - full body weight, I C. - the moment of inertia of the body relative to the axis passing through the center of the masses.

Figure 4. Rolting the wheel as the sum of the translational movement at a rate V C → and rotation with an angular velocity ω \u003d V C R with respect to the O axis passing through the center of mass.

The mechanics use the theorem on the movement of the center of mass.

Theorem 1.

Any body or several interacting bodies, which are a single system, possess the center of mass. This center of mass under the influence of external forces is moving in space as a material point in which the entire mass of the system is concentrated.

In the figure, we depicted the movement of a solid, which acts gravity. The center of mass of the body moves along the trajectory, which is close to Parabola, while the trajectory of the remaining points of the body is more complex.

Picture 5. The movement of the solid under the action of gravity.

Consider the case when the solid is moving around some fixed axis. The moment of inertia of this body inertia I. can be expressed after the moment of inertia I C. This body relative to the axis passing through the center of mass body and parallel first.

Figure 6. To the proof of the theorem on the parallel transfer of the rotation axis.

Example 2.

For example, we take a solid, the form of which is arbitrary. Denote the center of the mass of C. We choose the coordinate system of the coordinates with the beginning of the coordinate 0. Compatible Mass Center and Begin Coordinates.

One of the axes passes through the center of mass C. The second axis crosses the arbitrarily selected point of P, which is located at a distance D. from the beginning of the coordinates. We highlight some small element of the mass of this solid body Δ m i.

By definition of the moment of inertia:

I c \u003d σ Δ m i (x i 2 + y i 2), i p \u003d σ m i (x i - a) 2 + y i - b 2

Expression for I P. You can rewrite in the form:

I p \u003d Σ Δ m i (x i 2 + y i 2) + σ Δ m i (a 2 + b 2) - 2 A Σ Δ m i x i - 2 b σ Δ m i y i.

The two recent members of the equation are applied to zero, since the origin of the coordinates in our case coincides with the center of mass body.

So we came to the formula of the Steiner Theorem on the parallel transfer of the axis of rotation.

Theorem 2.

For a body that rotates relative to an arbitrary fixed axis, the moment of inertia, according to the Steiner theorem, is equal to the sum of the moment of the inertia of this body relative to the axis parallel to it passing through the center of mass of the body, and the mass of body mass per square distance between the axes.

I p \u003d i c + m d 2,

where M. - Full body weight.

Figure 7. Model moment inertia.

The figure below shows homogeneous solid bodies of various shapes and the moments of the inertia of these bodies are indicated relative to the axis passing through the center of mass.

Figure 8. Moments of inertia I C some homogeneous solids.

In cases where we are dealing with a solid body, which rotates relatively fixed axis, we can summarize Newton's second law. In the figure below, we were depicted a solid body of an arbitrary shape, rotating relative to some axis passing through the point O. The rotation axis is located perpendicular to the pattern plane.

Δ M i is an arbitrary small element of the mass, which is exposed to external and internal forces. Ensucting all forces is F I →. It can be decomposed into two components: the tangent constituent F I τ → and radial F I R →. Radial component F i R → Creates a centripetal acceleration A N..

Figure 9. Tanner F I τ → and radial F i R → The components of the force F I → active on the element Δ M I of the solid.

Tangency component F I τ → Causes tangential acceleration A I τ → Mass Δ M I.. Newton's second law recorded in a scalar form gives

Δ m i И i τ \u003d f i τ sin θ or δ m i r i ε \u003d f i sin θ

where ε \u003d a i τ r i is an angular acceleration of all points of the solid.

If both parts of the above-written equations are multiplied by R I.then we will get:

Δ m i r i 2 ε \u003d f i r i sin θ \u003d f i l i \u003d m i.

Here L i is the shoulder of power, F I, → M I - the moment of force.

Now you need to record similar ratios for all elements of mass Δ m I. rotating solid body, and then sum up the left and right parts. This gives:

Σ Δ m i R i 2 ε \u003d Σ m i.

The sum of the moments of forces acting on the different points of solid, consists of the sum of all external forces and the sum of all internal forces.

Σ m \u003d σ m i in n e n + σ m i in n y t p.

But the sum of the moments of all internal forces according to the third law of Newton is zero, therefore only the sum of the moments of all foreignses that we denote through the right part M.. So we obtained the basic equation of the dynamics of the rotational movement of the solid.

Definition 4.

Corner acceleration ε and moment of forces M. This equation is algebraic values.

Usually, the positive direction of rotation takes the direction counterclockwise.

The vector form of recording the main equation of the dynamics of the rotational motion is possible, at which the values \u200b\u200bω →, ε →, m → are defined as vectors directed along the axis of rotation.

In a section dedicated to the progressive body movement, we introduced the concept of a body pulse P →. By analogy with progressive movement for rotational motion, we introduce the concept of moment of momentum.

Definition 5.

The moment of the pulse of the rotating body - this is a physical value that is equal to the body of the body inertia I. On the angular velocity ω of its rotation.

To designate the moment of momentum, the Latin letter L is used.

Since ε \u003d δ ω Δ t; Δ T → 0, the rotational motion equation can be represented as:

M \u003d i ε \u003d i δ ω δ t or m δ t \u003d i δ ω \u003d Δ l.

We get:

M \u003d Δ l Δ t; (Δ T → 0).

We received this equation for the case when i \u003d c O n s t. But it will be fair and then when the moment of the body inertia will change during the movement.

If the total moment M. The external forces acting on the body is zero, then the moment of the pulse L \u003d i Ω relative to this axis is preserved: Δ L \u003d 0, if m \u003d 0.

Definition 6.

Hence,

L \u003d L ω \u003d C O n s t.

So we came to the law of preserving the moment of momentum.

Example 3.

As an example, we give the drawing, which shows an inelastic rotational collision of the discs that are planted on the common axis for them.

Figure 10. Incomplete rotational collision of two disks. The law of preservation of the moment of impulse: I 1 ω 1 \u003d (i 1 + i 2) Ω.

We are dealing with a closed system. For any closed system, the moment of preservation of the moment of momentum will be fair. It is performed in the conditions of experiments on mechanics, and in the conditions of space, when the planets move along their orbits around the star.

We can write the equation of rotational movement dynamics for both fixed axis and axis, which moves evenly or with acceleration. The view of the equation will not change in the event that the axis moves accelerated. For this, two conditions should be performed: the axis must pass through the center of body mass, and its direction in space remains unchanged.

Example 4.

Suppose we have a body (ball or cylinder), which rolls on an inclined plane with some friction.

Figure 11. Rolving a symmetric body along the inclined plane.

Axis of rotation O. passes through the center of mass body. Moments of gravity M G → and reaction forces N → relative to the axis O. equal zero. Moment M. Creates only the friction force: m \u003d f t p r.

Rotational motion equation:

I c ε \u003d i c a r \u003d m \u003d f t r r

where ε is the angular acceleration of the rolling body, A. - linear acceleration of its center of mass, I C. - moment of inertia relative to the axis O.passing through the center of mass.

The second law of Newton for the progressive movement of the center of the masses is written in the form:

m a \u003d m g sin α - f t p.

Excluding from these equations F T p, we will finally get:

α \u003d m g sin θ i c r 2 + m.

From this expression it is clear that the body will be sloped faster with the inclined plane, which has a smaller moment of inertia. For example, in a ball I C \u003d 2 5 M R 2, and in a solid homogeneous cylinder I C \u003d 1 2 M R 2. Consequently, the ball will roll faster than the cylinder.

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The friction force is always directed along the surface of contacting the opposite movement. It is always less than the strength of normal pressure.

Here:
F. - gravitational force with which two bodies are attracted to each other (Newton),
m 1. - the mass of the first body (kg),
m 2. - the mass of the second body (kg),
r. - distance between mass centers tel (meter),
γ - gravitational constant 6.67 · 10 -11 (m 3 / (kg · s 2)),

Stroy of gravitational field - vector quantity characterizing the gravitational field at a given point and numerically equal to the ratio of the force acting on the body placed at this point point to the gravitational mass of this body:

12. Studying the mechanics of a solid, we used the concept of an absolutely solid body. But in nature there is no absolutely solid bodies, because All real bodies under the action of forces change their shape and dimensions, i.e. deform.
Deformation called elasticIf after the body has ceased to act on the body, the body restores the initial dimensions and shape on the body. Deformations that persist in the body after the cessation of external forces are called plastic (or residual)

Work and power

Work of force.
The work of constant strength acting on the straightly moving body
where - the movement of the body is the force acting on the body.

In general, the work of a variable force acting on the body moving along the curvilinear trajectory . Work is measured in Joules [J].

Working the moment of forces acting on the body rotating around the stationary axis Where the moment of force is the angle of turning.
In general .
Perfect NAT body work turns into his kinetic energy.
Power- This is a job per unit time (1 s) :. Power is measured in watts [W].

14.Kinetic energy - The energy of the mechanical system, depending on the speeds of its points. Often distinguish the kinetic energy of progressive and rotational tribes.

Consider a system consisting of one particle, and write Newton's second law:

There is a resulting all forces acting on the body. Scalarously multiply the equation for the movement of the particle. Considering that, we get:

If the system is closed, that is, then , and the amount

it remains constant. This value is called kinetic energy Particles. If the system is isolated, the kinetic energy is the integral of the movement.

For absolutely solid body Full kinetic energy can be written in the form of the sum of the kinetic energy of the progressive and rotational movement:

Body mass

Body Mass Center

Moment of inertia body

Corner body velocity.

15.Potential energy - a scalar physical quantity characterizing the ability of a certain body (or material point) to work at the expense of its stay in the field of strength.

16. The stretching or compression of the spring leads to the reserves of its potential energy of elastic deformation. The return of the spring to the position of equilibrium leads to the release of the stored energy of elastic deformation. The magnitude of this energy is:

Potential energy of elastic deformation ..

- work of the strength of elasticity and changing the potential energy of elastic deformation.

17.conservative power (potential forces) - forces whose work does not depend on the form of the trajectory (depends only on the initial and end point of the application of the forces). Hence the definition: Conservative forces - Such forces, whose work on any closed trajectory is 0

Dyssypative forces - Forces, under the action of which, on the mechanical system, its complete mechanical energy decreases (that is, dissipates), moving to other, non-mechanical forms of energy, for example, in heat.

18. Rotation around the stationary axis It is called such a movement of a solid, in which two points remain in all the time of movement remain fixed. Direct, passing through these points is called the axis of rotation. All other points of the body move in planes perpendicular to the axis of rotation, around the circles, the centers of which lie on the axis of rotation.

Moment of inertia - Scalar physical size, inertness measure in rotational motion around the axis, just as the body weight is a measure of its inertness in the translational movement. It is characterized by the mass distribution in the body: the moment of inertia is equal to the amount of the pieces of elementary masses per square of their distances to the base set (points, direct or plane).

Moment of inertia mechanical system relatively fixed axis ("Axial moment of inertia") is called the magnitude J A.equal to the amount of mass of the masses of all n. Material dots of the system on the squares of their distances to the axis:

,

§ m I. - weight i.point,

§ r I. - Distance Ot i.-y point to the axis.

Axial moment of inertia Body J A. It is a measure of the inertness of the body in the rotational motion around the axis is similar to how the body weight is a measure of its inertness in the translational movement.

,