The work of rotation of a rigid body. Kinetic energy of a rotating body Calculate the work of the body during rotational motion

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)

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 in 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- a 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 point of view general theory 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 what force acts on 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 is the principle of physical equality of inertial frames of reference in classical mechanics, manifested in the fact that the laws of mechanics are the same in all such systems.

Mathematically, Galileo's principle of relativity expresses the invariance (constancy) of the equations of mechanics with respect to the transformations of the coordinates of moving points (and time) in the transition from one inertial frame to another - Galileo's 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 the Galilean 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 us 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 movement (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.

If m.t. rotates in a circle, then a force acts on it, then when turning through a certain angle, elementary work is performed:

(22)

If the acting force is potential, then

then (24)

Rotating power

Instantaneous power developed during rotation of the body:

Kinetic energy of a rotating body

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

In flat motion (the cylinder rolls down an inclined plane), the total speed is:

where is the speed of the center of mass of the cylinder.

The total is equal to the sum of the kinetic energy of the translational motion of its center of mass and the kinetic energy of the rotational motion of the body relative to the center of mass, i.e.:

(28)

Conclusion:

And now, having considered all the lecture material, let's summarize, compare the quantities and equations of the rotational and translational motion of the body:

Attachment 1:

A person stands in the center of the Zhukovsky bench and rotates along with it by inertia. Rotation frequency n 1 \u003d 0.5 s -1 . Moment of inertia j o human body relative to

relative to the axis of rotation is 1.6 kg m 2. In arms outstretched to the sides, a person holds a kettlebell with a mass m=2 kg each. Distance between weights l 1 \u003d l.6 m. Determine the speed n 2 , benches with a person when he puts his hands down and the distance l 2 between the weights will be equal to 0.4 m. Neglect the moment of inertia of the bench.

Symmetry properties and conservation laws.

Energy saving.

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

The conservation of energy is related to the homogeneity of time, the conservation of momentum is related to the homogeneity of space, and, finally, the conservation of angular momentum is related to the isotropy of space.

We start with the law of conservation of energy. Let the system of particles be in constant conditions (this takes place if the system is closed or subject to a constant external force field); connections (if any) are ideal and stationary. In this case time, due to its homogeneity, cannot enter explicitly into the Lagrange function. Really homogeneity means the equivalence of all moments of time. Therefore, the replacement of one moment of time by another without changing the values ​​of coordinates and particle velocities should not change the mechanical properties of the system. This is of course true if the replacement of one moment of time by another does not change the conditions in which the system is located, that is, if the external field is independent of time (in particular, this field may be absent).

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

Consider an absolutely rigid body rotating around a fixed axis. If you mentally break this body into n mass points m 1 , m 2 , …, m n located at distances r 1 , r 2 , …, r n from the axis of rotation, then during rotation they will describe circles and move with different linear velocities v 1 , v 2 , …, v n. Since the body is absolutely rigid, the angular velocity of rotation of the points will be the same:

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

Taking into account the relationship between the angular and linear velocities, we get:

Comparison of formula (4.9) with the expression for the kinetic energy of a body moving forward with a speed v, shows that moment of inertia is a measure of the inertia of a body in rotational motion.
If a rigid body is moving forward at a speed v and simultaneously rotates with an angular velocity ω around an axis passing through its center of inertia, then its kinetic energy is determined as the sum of two components:

(4.10)

where vc is the speed of the center of mass of the body; Jc- the moment of inertia of the body about the axis passing through its center of mass.
Moment of force relative to the fixed axis z called a scalar Mz, equal to the projection onto this axis of the vector M moment of force defined relative to an arbitrary point 0 of the given axis. Torque value Mz does not depend on the choice of the position of point 0 on the axis z.
If the axis z coincides with the direction of the vector M, then the moment of force is represented as a vector coinciding with the axis:

Mz = [ RF]z
Let's find an expression for work during the rotation of the body. Let the power F applied to point B, located at a distance from the axis of rotation r(Fig. 4.6); α is the angle between the force direction and the radius vector r. 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 infinitesimal angle dφ attachment point B passes the way ds = rdφ, 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:

dA = Fsinα*rdφ
Given that Frsinα = Mz can be written dA = M z dφ, where Mz- the moment of force about the axis of rotation. Thus, the work during rotation of the body is equal to the product of the moment of the acting force and the angle of rotation.
The work during rotation of the body goes to increase its kinetic energy:

dA = dE k
(4.11)

Equation (4.11) is equation of the dynamics of rotational motion of a rigid body relative to a fixed axis.

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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