Work done when a body rotates. Kinetic energy of a rotating body. Kinematics of translational motion
Rotating work and power solid.
Let's find an expression for work when the body rotates. 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 solid, the work done by this force is equal to the work expended on turning the entire body. When a body rotates through an infinitesimal angle, the point of application travels a path and the work is equal to the product of the projection of the force on the direction of the 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 work:
Thus, the work done 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 rotating body.
Moment of inertia mat.t. called physical a value 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 =m i V 2 i /2 V i -Wr i Wi=miw 2 r 2 i /2 =w 2 /2*m i r i 2 I i =m i r 2 i moment of inertia of a rigid body equal to the sum of all mat.t I=S i m i r 2 i the moment of inertia of a solid body is called. physical quantity equal to the sum of the products of mathematical t. by the squares of the distances from these points to the axis. W i -I i W 2 /2 W k =IW 2 /2
W k =S i W ki moment of inertia at rotational movement yavl. analogous to mass in translational motion. I=mR 2 /2
21. Non-inertial reference systems. Inertia forces. The principle of equivalence. Equation of motion in non-inertial reference systems.
Non-inertial reference frame- an arbitrary reference system that is not inertial. Examples of non-inertial reference systems: a system moving in a straight line with constant acceleration, as well as a rotating system.
When considering the equations of motion of a body in a non-inertial reference frame, it is necessary to take into account additional inertial forces. Newton's laws are satisfied only in inertial frames of reference. In order to find the equation of motion in a non-inertial reference frame, you need 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 reference frames;
space is absolute, that is, the distance between any two material points is the same in all arbitrarily moving reference frames.
These two principles allow us to write the equation of motion material point relative to any non-inertial frame of reference in which Newton's First Law is not satisfied.
The basic equation for 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 a non-inertial reference frame, 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:
Transferable 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 view
F 1 +F 2 +…F n –ma = 0 Where F i is the actual force, and –ma is the “force of inertia”.
Among the inertial forces the following are distinguished:
simple inertia force;
centrifugal force, which explains the desire of bodies to fly away from the center in rotating reference frames;
the Coriolis force, which explains the tendency of bodies to leave the radius during radial motion in rotating reference frames;
From the point of view of general relativity, gravitational forces at any point- these are the forces of inertia at a given point in Einstein’s curved space
Centrifugal force - inertial force, which is introduced in a rotating (non-inertial) reference frame (in order to apply Newton’s laws, calculated only for inertial reference frames) and which is directed from the axis of rotation (hence the name).
The principle of equivalence of the forces of gravity and inertia- a heuristic principle used by Albert Einstein in deducing the general theory of relativity. One of the options for its 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 inert and gravitational mass are equal, it is impossible to distinguish what 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 a gravitational field occur in exactly the same way as in the corresponding field of inertial forces, if the intensities of these fields coincide and the initial conditions for the bodies of the system are the same.
22.Galileo's principle of relativity. Galileo's transformations. Classical velocity addition theorem. Invariance of Newton's laws in inertial reference systems.
Galileo's principle of relativity is the principle of physical equality of inertial reference systems 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 (immutability) of the equations of mechanics with respect to transformations of the coordinates of moving points (and time) during the transition from one inertial system to another - Galilean transformations.
Let there be two inertial reference systems, one of which, S, we agree to consider at rest; the second system, S", moves relative 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)
(shaded values refer to the S system, unprimed ones - to S). Thus, time in classical mechanics, like the distance between any fixed points, is considered the same in all reference systems.
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.
Moreover, forces (and masses) are invariants in classical mechanics, i.e. quantities that do not change when moving from one reference system to another.
Therefore, under Galilean transformations, equation (3) does not change.
This is the mathematical expression of Galileo's principle of relativity.
GALILEO'S TRANSFORMATIONS.
In kinematics, all reference systems are equal to each other and motion can be described in any of them. When studying movements, sometimes it is necessary to move from one reference system (with the coordinate system OXYZ) to another - (O`X`U`Z`). Let's consider the case when the second frame of reference moves relative to the first uniformly and rectilinearly with speed V=const.
To facilitate the mathematical description, we assume that the corresponding coordinate axes are parallel to each other, that the speed is directed along the X axis, and that at the initial moment of time (t=0) the origins of coordinates of both systems coincided with each other. Using the assumption that is valid in classical physics about the same flow of time in both systems, we can write down relations connecting the coordinates of a certain point A(x,y,z) and A(x`,y`,z`) in both systems. Such a transition from one reference system to another is called Galilean transformation):
ОХУZ О`Х`У`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 found in classical physics.
Addition of speeds in service station
Classical law addition of speeds cannot be fair, because it contradicts the statement about the constancy of the speed of light in a vacuum. If the train is moving at a speed v and in the carriage in the direction of movement of the train it spreads light wave, then its speed relative to the Earth is still c, but not v+c.
Let's consider two reference systems.
In system K 0 body moves with speed v 1 . Regarding the system K it moves at speed v 2. According to the law of adding speeds in the service station: 
If 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 is equal c, as required by the second postulate of the theory of relativity: 
At v 1 = c and at v = c speed v 2 is again equal to speed c. 
A remarkable property of the addition law 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 greater than the speed of light is impossible.
Speed addition
When considering complex motion (that is, when a point or body moves in one reference system, and it moves relative to another), the question arises about the connection between velocities in 2 reference systems.
Classical mechanics
In classical mechanics, the absolute speed of a point is equal to the vector sum of its relative and portable speeds:
In simple terms: The speed of movement of a body relative to a stationary reference frame 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 reference system relative to a stationary frame.
Kinetic energy- the quantity is additive. Therefore, the kinetic energy of a body moving in an arbitrary manner is equal to the sum of the kinetic energies of all P material points into which this body can be mentally divided: If the body rotates around a fixed axis z with an angular velocity of 1 m I 1...(PHYSICS. MECHANICS)
Kinetic energy of a rotating rigid body
The kinetic energy of a body moving in an arbitrary manner is equal to the sum of the kinetic energies of all P material points (particles) into which this body can be mentally divided (Fig. 6.8) If a body rotates around a fixed axis Oz with an angular velocity co, then the linear speed of any /-th particle,...(CLASSICAL AND RELATIVISTIC MECHANICS)
(THEORETICAL MECHANICS.)
Rotation of a body around a fixed axis
Let the rigid body in time sk made an infinitesimal rotation through an angle s/f relative to an axis that is motionless in a given reference system. This angle of rotation с/ср is a measure of the change in the position of a body rotating relative to a fixed axis. By analogy with s/r, we will call s/f angular displacement....(PHYSICS: MECHANICS, ELECTRICITY AND MAGNETISM)
Analogy between translational and rotational motion
This analogy was discussed above and follows from the similarity of the basic equations of translational and rotational motion. Just as acceleration is given by the time derivative of velocity and the second derivative of displacement, so angular acceleration is given by the time derivative of angular velocity and the second derivative of angular displacement....(PHYSICS)
Translational and rotational motion
Translational motion Translational motion is a motion of a rigid body in which any straight line drawn in this body moves while remaining parallel to its original position. The properties of translational motion are determined by the following theorem: during translational motion of a body...(APPLIED MECHANICS)
Here is the angular momentum relative to the axis of rotation, that is, the projection onto the axis of the angular momentum defined relative to some point belonging to the axis (see lecture 2). 
- this is the moment of external forces relative to the axis of rotation, that is, the projection onto the axis of the resulting moment of external forces, determined relative to some point belonging to the axis, and the choice of this point on the axis, as in the case of c, does not matter. Indeed (Fig. 3.4),
 |
| where is the component of the force applied to the rigid body, perpendicular to the axis of rotation, and is the arm of the force relative to the axis. |
Rice. 3.4.
| (3.8) |
Since ( is the moment of inertia of the body relative to the axis of rotation), then instead we can write
The vector is always directed along the axis of rotation, and is the component of the vector of the moment of force along the axis. In the case we obtain, accordingly, the angular momentum relative to the axis is conserved. Moreover, the vector itself L
 |
| , defined relative to any point on the rotation axis, can change. An example of such movement is shown in Fig. 3.5. |
Rice. 3.5. In the case we obtain, accordingly, the angular momentum relative to the axis is conserved. Moreover, the vector itself Rod AB, hinged at point A, rotates by inertia around a vertical axis in such a way that the angle between the axis and the rod remains constant. Momentum vector In the case we obtain, accordingly, the angular momentum relative to the axis is conserved. Moreover, the vector itself, relative to point A, moves along a conical surface with a half-opening angle; however, the projection
on the vertical axis remains constant, since the moment of gravity about this axis is zero.
Kinetic energy of a rotating body and the work of external forces (the axis of rotation is stationary).
| (3.11) |
Speed of the i-th particle of the body
| (3.12) |
where is the distance of the particle to the axis of rotation Kinetic energy because angular velocity
rotation for all points is the same. In accordance with law of change of mechanical energy
system, the elementary work of all external forces is equal to the increment of the kinetic energy of the body:
let us assume that the sharpener disk rotates by inertia at an angular speed and we stop it by pressing an object against the edge of the disk with constant force. In this case, a constant force will act on the disk, directed perpendicular to its axis. The work of this force
where is the moment of inertia of the sharpening disk together with the electric motor armature. Comment.
If the forces are such that they do not produce work.
Free axles. Stability of free rotation.
If a solid body is spun around an arbitrary axis rigidly connected to the body and the axis is released from the bearings, then its direction in space, generally speaking, will change. In order for an arbitrary axis of rotation of a body to maintain its direction unchanged, certain forces must be applied to it. The situations that arise in this case are shown in Fig. 3.6.
 |
| Rice. 3.6. |
A massive homogeneous rod AB is used here as a rotating body, attached to a fairly elastic axis (depicted by double dashed lines). The elasticity of the axle allows you to visualize the dynamic loads it experiences. In all cases, the axis of rotation is vertical, rigidly connected to the rod and secured in bearings; the rod is untwisted around this axis and left to its own devices.
In the case shown in Fig. 3.6a, the axis of rotation is the main one for point B of the rod, but not the central one. The axis bends; from the side of the axis, a force acts on the rod to ensure its rotation (in the NISO associated with the rod, this force balances the centrifugal force of inertia). From the side of the rod, a force acts on the axis that is balanced by the forces from the bearings.
In the case of Fig. 3.6b the axis of rotation passes through the center of mass of the rod and is central for it, but not the main one. The angular momentum relative to the center of mass O is not conserved and describes a conical surface. The axis is deformed (broken) in a complex way; forces act on the rod from the side of the axis and the moment of which provides an increment (In the NISO associated with the rod, the moment of elastic forces compensates for the moment of centrifugal inertial forces acting on one and the other halves of the rod). From the side of the rod, forces act on the axis and are directed opposite to the forces and The moment of forces and is balanced by the moment of forces and arising in the bearings.
And only in the case when the axis of rotation coincides with the main central axis of inertia of the body (Fig. 3.6c), the untwisted and left to itself rod does not have any effect on the bearings. Such axes are called free axes because if the bearings are removed, they will maintain their direction in space unchanged.
Whether this rotation will be stable with respect to small disturbances, which always occur in real conditions, is another matter. Experiments show that rotation around the main central axes with the largest and smallest moments of inertia is stable, and rotation around an axis with an intermediate value of the moment of inertia is unstable. This can be verified by throwing up a body in the form of a parallelepiped, untwisted around one of the three mutually perpendicular main central axes (Fig. 3.7). The axis AA" corresponds to the largest, the axis BB" - the average, and the axis CC" - the smallest moment of inertia of the parallelepiped. If you throw such a body, giving it a rapid rotation around the axis AA" or around the axis CC", you can make sure that this rotation is quite stable. Attempts to force the body to rotate around the BB" axis do not lead to success - the body moves in a complex way, tumbling in flight.
- rigid body - Euler angles
The friction force is always directed along the contact surface in the direction opposite to the movement. It is always less than the force of normal pressure.
Here:
F- gravitational force with which two bodies attract each other (Newton),
m 1- mass of the first body (kg),
m 2- mass of the second body (kg),
r- distance between the centers of mass of bodies (meter),
γ
- gravitational constant 6.67 10 -11 (m 3 /(kg sec 2)),
Gravitational field strength- a vector quantity characterizing the gravitational field at a given point and numerically equal to the ratio of the gravitational force acting on a body placed at a given point in the field to the gravitational mass of this body:
12. While studying rigid body mechanics, we used the concept of an absolutely rigid body. But in nature there are no absolutely solid bodies, because... all real bodies, under the influence of forces, change their shape and size, i.e. deformed.
Deformation called elastic, if, after external forces have ceased to act on the body, the body restores its original size and shape. Deformations that remain in the body after the cessation of external forces are called plastic(or residual)
OPERATION AND POWER
Work of force.
Work done by a constant force acting on a rectilinearly moving body
, where is the displacement of the body, is the force acting on the body. 
In general, the work done by a variable force acting on a body moving along a curved path 
. Work is measured in Joules [J].
The work of a moment of force acting on a body rotating around a fixed axis, where is the moment of force and is the angle of rotation.
In general .
The work done by the body turns into its kinetic energy.
Power- this is work per unit of time (1 s): . Power is measured in Watts [W].
14.Kinetic energy- the energy of a mechanical system, depending on the speed of movement of its points. The kinetic energy of translational and rotational motion is often released.
Let's consider a system consisting of one particle and write Newton's second law:
There is a resultant of all forces acting on a body. Let us scalarly multiply the equation by the displacement of the particle. Considering that , we get:
If the system is closed, that is, then 
, and the value
remains constant. This quantity is called kinetic energy particles. If the system is isolated, then kinetic energy is the integral of motion.
For an absolutely rigid body, the total kinetic energy can be written as the sum of the kinetic energy of translational and rotational motion:
Body mass
Speed of the body's center of mass
Moment of inertia of the body
Angular velocity of the body.
15.Potential energy- a scalar physical quantity that characterizes the ability of a certain body (or material point) to do work due to its presence in the field of action of forces.
16. Stretching or compressing a spring leads to the storage of its potential energy of elastic deformation. The return of the spring to its equilibrium position results in the release of the stored elastic deformation energy. The magnitude of this energy is:
Potential energy of elastic deformation..
- work of the elastic force and change in the potential energy of elastic deformation.
17.conservative forces(potential forces) - forces whose work does not depend on the shape of the trajectory (depends only on the starting and ending points of application of forces). This implies the definition: conservative forces are those forces whose work along any closed trajectory is equal to 0
Dissipative forces- forces, under the action of which on a mechanical system, its total mechanical energy decreases (that is, dissipates), turning into other, non-mechanical forms of energy, for example, into heat.
18. Rotation around a fixed axis This is the motion of a rigid body in which two of its points remain motionless during the entire movement. The straight line 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, along circles whose centers lie on the axis of rotation.
Moment of inertia- a scalar physical quantity, a measure of inertia 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 by the square of their distances to the base set (point, line or plane).
Moment of inertia of a mechanical system relative to a fixed axis (“axial moment of inertia”) is the quantity J a, equal to the sum of the products of the masses of all n material points of the system by 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.
,
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 active force is potential, then
then (24)
Rotating power
Instantaneous power developed during body rotation:
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 plane motion (the cylinder rolls down an inclined plane), the total speed is equal to:
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 and compare the quantities and equations of rotational and translational motion of a body:
| Forward movement | Rotational movement | ||
| Weight | m | Moment of inertia | I |
| Path | S | Angle of rotation | |
| Speed | Angular velocity | ||
| Pulse | Momentum | ||
| Acceleration | Angular acceleration | ||
| Resultant of external forces | F | Sum of moments of external forces | M |
| Basic equation of dynamics | Basic equation of dynamics | ||
| Job | Fds | Rotation work | |
| Kinetic energy | Kinetic energy of rotation |
Annex 1:
A man stands in the center of the Zhukovsky bench and rotates with it by inertia. Rotation frequency n 1 =0.5 s -1 . Moment of inertia j o human body relative
relative to the axis of rotation is 1.6 kg m 2. In arms extended to the sides, a person holds a weight of m=2 kg each. Distance between weights l 1 =l.6 m. Determine the rotation speed n 2 , benches with a person when he lowers his hands and distance l 2 between the weights will become equal to 0.4 m. Neglect the moment of inertia of the bench.
Properties of symmetry and conservation laws.
Energy saving.
The conservation laws considered in mechanics are based on the properties of space and time.
Conservation of energy is associated with the homogeneity of time, conservation of momentum is associated with the homogeneity of space, and, finally, conservation of angular momentum is associated with the isotropy of space.
We start with the law of conservation of energy. Let the system of particles be in constant conditions (this occurs if the system is closed or subject to the influence of a constant external force field); connections (if any) are ideal and stationary. In this case time, due to its homogeneity, cannot be included explicitly in the Lagrange function. Really homogeneity means the equivalence of all points in time. Therefore, replacing one moment of time with 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 replacing one moment of time with 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, .