Moment of power F, acting on the body relative to the axis of rotation
,
Where 
-
force projection F on a plane perpendicular to the axis of rotation; l - shoulder strength F(shortest distance from the axis of rotation to the line of action of the force).
Moment of inertia about the axis of rotation:
a) material point
J= mr 2 ,
Where T - point mass; r - its distance from the axis of rotation;
b) discrete solid body
Where 
- weight i-th body element; r i is the distance of this element from the axis of rotation; P
-
number of body elements;
c) a solid solid
If the body is homogeneous, i.e. its density 
is the same throughout the entire volume, then
dm=
dV And 
Where V- body volume.
Moments of inertia of some bodies of regular geometric shape:
|
Axis about which the moment of inertia is determined |
Moment of Inertia Formula |
|
|
A homogeneous thin rod of mass T and length l Thin ring, hoop, radius pipe R and mass T, flywheel radius R and mass T, distributed along the rim Round homogeneous disk (cylinder) with radius R and mass T A homogeneous ball of mass T and radius R |
Passes through the center of gravity of the rod perpendicular to the rod Passes through the end of the rod perpendicular to the rod Passes through the center perpendicular to the plane of the base Passes through the center of the disk perpendicular to the plane of the base Passes through the center of the ball |
1/12ml 2 |
Steiner's theorem. Moment of inertia of a body about an arbitrary axis
J= J 0 + ma 2 ,
Where J 0 - the moment of inertia of this body relative to an axis passing through the center of gravity of the body parallel to a given axis; A - distance between axles; m- body mass.
Momentum of momentum of a rotating body relative to the axis
L=
J
.
Law of conservation of angular momentum
Where L i - angular momentum of the i-th body included in the system. Law of conservation of angular momentum for two interacting bodies
Where 
- moments of inertia and angular velocities of bodies before interaction: 
- the same values after interaction.
The law of conservation of angular momentum for one body whose moment of inertia changes,
Where 
- initial and final moments of inertia; 
- initial and final angular velocities of the body.
The basic equation for the dynamics of the rotational motion of a rigid body relative to fixed axis
M d t=d(J 
), Where M- moment of force acting on a body over time dt;
J
-
moment of inertia of the body;
- angular velocity; J
-
moment of impulse.
If the moment of force and moment of inertia are constant, then this equation is written as
M
t=J
.
In the case of a constant moment of inertia, the basic equation for the dynamics of rotational motion takes the form
M=J
, Where 
- angular acceleration.
Work of a constant moment of force M, acting on a rotating body
where is the angle of rotation of the body.
Instantaneous power developed during body rotation
N=
M
.
Kinetic energy of a rotating body
T=1/2
J
.
The kinetic energy of a body rolling along a plane without sliding is
T== 1 / 2
mv 2
+l/2 J
,
Where l /
2
mv 2
-
kinetic energy of translational motion of a body; v
-
speed of the body's center of inertia; l/2 J
, is the kinetic energy of the rotational motion of a body around an axis passing through the center of inertia.
The work done during rotation of a body and the change in its kinetic energy are related by the relation
Let a certain body, under the influence of force F applied at point A, come into rotation around the axis OO" (Fig. 1.14).
The force acts in a plane perpendicular to the axis. The perpendicular p dropped from point O (lying on the axis) to the direction of the force is called shoulder of strength. The product of the force by the arm determines the modulus of the moment of force relative to point O:
M = Fp=Frsinα.
Moment of power is a vector determined by the vector product of the radius vector of the point of application of the force and the force vector:
(3.1) The unit of moment of force is newton meter (N m).
The direction of M can be found using the right screw rule.
moment of impulse particle is the vector product of the radius vector of the particle and its momentum:
or in scalar form L = rPsinα
This quantity is vector and coincides in direction with the vectors ω.
§ 3.2 Moment of inertia. Steiner's theorem
The measure of inertia of bodies during translational motion is mass. The inertia of bodies during rotational motion depends not only on mass, but also on its distribution in space relative to the axis of rotation. The measure of inertia during rotational motion is a quantity calledmoment of inertia of the body relative to the axis of rotation.
Moment of inertia of a material point relative to the axis of rotation, the product of the mass of this point and the square of its distance from the axis is called:
I i =m i r i 2 (3.2)
Moment of inertia of the body relative to the axis of rotation is called the sum of moments of inertia material points, of which this body consists:
(3.3)
In the general case, if the body is solid and represents a collection of points with small masses dm, the moment of inertia is determined by integration:
(3.4)
If the body is homogeneous and its density 
, then the moment of inertia of the body
(3.5)
The moment of inertia of a body depends on which axis it rotates about and how the mass of the body is distributed throughout the volume.
The moment of inertia of bodies that have a regular geometric shape and a uniform distribution of mass over the volume is most easily determined.
Moment of inertia of a homogeneous rod relative to an axis passing through the center of inertia and perpendicular to the rod
(3.6)
Moment of inertia of a homogeneous cylinder relative to an axis perpendicular to its base and passing through the center of inertia,
(3.7)
Moment of inertia of a thin-walled cylinder or hoop relative to an axis perpendicular to the plane of its base and passing through its center,
(3.8)
Moment of inertia ball relative to diameter
(3.9)
Let's look at an example . Let us determine the moment of inertia of the disk relative to the axis passing through the center of inertia and perpendicular to the plane of rotation. Disk mass - m, radius - R.
The area of the ring (Fig. 3.2) enclosed between
r and r + dr, is equal to dS = 2πr·dr. Disk area S = πR 2.
Hence, 
. Then
or 
According to
The given formulas for the moments of inertia of bodies are given under the condition that the axis of rotation passes through the center of inertia. To determine the moments of inertia of a body relative to an arbitrary axis, you should use Steiner's theorem : the moment of inertia of a body relative to an arbitrary axis of rotation is equal to the sum of the moment of inertia of the body relative to an axis parallel to the given one and passing through the center of mass of the body, and the product of the body mass by the square of the distance between the axes:
(3.11)
The unit of moment of inertia is kilogram meter squared (kg m 2).
Thus, the moment of inertia of a homogeneous rod relative to the axis passing through its end, according to Steiner’s theorem, is equal to
(3.12)
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 this point fields, 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- energy 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 bodies.
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 its storage potential energy 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 works 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.
,
§ - mass of a small element of body volume,
Basic concepts.
Moment of power relative to the axis of rotation - this is the vector product of the radius vector and the force.
(1.14)
The moment of force is a vector , the direction of which is determined by the rule of the gimlet (right screw) depending on the direction of the force acting on the body. The moment of force is directed along the axis of rotation and does not have a specific point of application.
The numerical value of this vector is determined by the formula:
M=r F sin (1.15),
where - the angle between the radius vector and the direction of the force.
If =0 or , moment of power M=0, i.e. a force passing through the axis of rotation or coinciding with it does not cause rotation.
The greatest modulus torque is created if the force acts at an angle = /2 (M 0) or =3 /2 (M 0).
Using the concept of leverage d- this is a perpendicular lowered from the center of rotation to the line of action of the force), the formula for the moment of force takes the form:
, Where 
(1.16)
Rule of moments of forces(condition of equilibrium of a body having a fixed axis of rotation):
In order for a body with a fixed axis of rotation to be in equilibrium, it is necessary that the algebraic sum of the moments of forces acting on this body be equal to zero.
M i =0 (1.17)
The SI unit for moment of force is [Nm]
During rotational motion, the inertia of a body depends not only on its mass, but also on its distribution in space relative to the axis of rotation.
Inertia during rotation is characterized by the moment of inertia of the body relative to the axis of rotation J.
Moment of inertia material point relative to the axis of rotation is a value equal to the product of the mass of the point by the square of its distance from the axis of rotation:
J =m r 2 (1.18)
The moment of inertia of a body relative to an axis is the sum of the moments of inertia of the material points that make up the body:
J= m r 2 (1.19)
The moment of inertia of a body depends on its mass and shape, as well as on the choice of the axis of rotation. To determine the moment of inertia of a body relative to a certain axis, the Steiner-Huygens theorem is used:
J=J 0 +m d 2 (1.20),
Where J 0 – moment of inertia about a parallel axis passing through the center of mass of the body, d – distance between two parallel axes . The moment of inertia in SI is measured in [kgm 2 ]
The moment of inertia during the rotational movement of the human body is determined experimentally and calculated approximately using the formulas for a cylinder, round rod or ball.
The moment of inertia of a person relative to the vertical axis of rotation, which passes through the center of mass (the center of mass of the human body is located in the sagittal plane slightly in front of the second sacral vertebra), depending on the position of the person, has the following values: when standing at attention - 1.2 kg m 2; with the “arabesque” pose – 8 kgm 2; in horizontal position – 17 kg m 2.
Work in rotational motion occurs when a body rotates under the influence of external forces.
The elementary work of force in rotational motion is equal to the product of the moment of force and the elementary angle of rotation of the body:
dA =M d (1.21)
If several forces act on a body, then the elementary work of the resultant of all applied forces is determined by the formula:
dA=M d (1.22),
Where M– the total moment of all external forces acting on the body.
Kinetic energy of a rotating bodyW To depends on the moment of inertia of the body and the angular velocity of its rotation:
(1.23)
Angle of impulse (angular momentum) 
–
magnitude, numerically equal to the product momentum of the body per radius of rotation.
L=p r=m V r (1.24).
After appropriate transformations, you can write the formula for determining angular momentum in the form:
(1.25).
Momentum 
– a vector whose direction is determined by the right screw rule. The SI unit of angular momentum iskgm 2 /s
Basic laws of the dynamics of rotational motion.
The basic equation for the dynamics of rotational motion:
The angular acceleration of a body undergoing rotational motion is directly proportional to the total moment of all external forces and inversely proportional to the moment of inertia of the body.
(1.26).
This equation plays the same role in describing rotational motion as Newton's second law does for translational motion. From the equation it is clear that under the action of external forces, the greater the angular acceleration, the smaller the moment of inertia of the body.
Newton's second law for the dynamics of rotational motion can be written in another form:
(1.27),
those. the first derivative of the angular momentum of a body with respect to time is equal to the total moment of all external forces acting on a given body.
Law of conservation of angular momentum of a body:
If the total moment of all external forces acting on the body is equal to zero, i.e.
M =0 , Then dL/dt=0 (1.28).
Therefore 
or 
(1.29).
This statement constitutes the essence of the law of conservation of angular momentum of a body, which is formulated as follows:
The angular momentum of a body remains constant if the total moment of external forces acting on a rotating body is zero.
This law is valid not only for an absolutely rigid body. An example is a figure skater who performs a rotation around a vertical axis. By pressing his hands, the skater reduces the moment of inertia and increases the angular speed. To slow down the rotation, he, on the contrary, spreads his arms wide; As a result, the moment of inertia increases and the angular speed of rotation decreases.
In conclusion, we present a comparative table of the main quantities and laws characterizing the dynamics of translational and rotational movements.
Table 1.4.
|
Forward movement |
Rotational movement |
||
|
Physical quantity |
Formula |
Physical quantity |
Formula |
|
Moment of inertia |
J=m r 2 |
||
|
Moment of power |
M=F
r, if |
||
|
Body impulse (amount of movement) |
p=m V |
Momentum of a body |
L=m V r; L=J |
|
Kinetic energy |
|
Kinetic energy |
|
|
Mechanical work |
Mechanical work |
dA=Md |
|
|
Basic equation of translational motion dynamics |
|
Basic equation for the dynamics of rotational motion |
|
|
Law of conservation of body momentum |
|
Law of conservation of angular momentum of a body |
If
|
,
or
If
or
J
=const,
