Dynamics rotary motion solid body.
Moment of inertia.
Moment of power. The basic equation of the dynamics of rotational motion.
moment of impulse.
Moment of inertia.
(Consider the experiment with rolling cylinders.)
When considering rotational motion, it is necessary to introduce new physical concepts: moment of inertia, moment of force, moment of impulse.
The moment of inertia is a measure of the inertia of a body during rotation of the body around a fixed axis.
Moment of inertia material point relative to the fixed axis of rotation is equal to the product of its mass by the square of the distance to the considered axis of rotation (Fig. 1):
Depends only on the mass of the material point and its position relative to the axis of rotation and does not depend on the presence of the rotation itself.
Moment of inertia - scalar and additive quantity
The moment of inertia of a body is equal to the sum of the moments of inertia of all its points
.
In the case of a continuous mass distribution, this sum reduces to the integral:
,
where is the mass of a small volume of the body, is the density of the body, is the distance from the element to the axis of rotation.
The moment of inertia is analogous to mass in rotational motion. The greater the moment of inertia of the body, the more difficult it is to change the angular velocity of the rotating body. The moment of inertia is meaningful only for a given position of the axis of rotation.
It is meaningless to speak simply of the “moment of inertia”. It depends:
1) from the position of the axis of rotation;
2) on the distribution of body mass relative to the axis of rotation, i.e. on body shape and size.
Experimental proof of this is the experience with rolling cylinders.
After integrating for some homogeneous bodies, we can obtain the following formulas (the axis of rotation passes through the center of mass of the body):
The moment of inertia of a hoop (we neglect the wall thickness) or a hollow cylinder:
Moment of inertia of a disk or solid cylinder of radius R:
where 
.
Moment of inertia of the ball
Moment of inertia of the rod
E 
If the moment of inertia about the axis passing through the center of mass is known for the body, then the moment of inertia about any axis parallel to the first is found by Steiner theorem: the moment of inertia of the body about an arbitrary axis is equal to the moment of inertia J 0 about an axis parallel to the given one and passing through the center of mass of the body, added to the product of the body's mass by the square of the distance between the axes.
where d distance from the center of mass to the axis of rotation.
The center of mass is an imaginary point, the position of which characterizes the distribution of the mass of a given body. The center of mass of the body moves in the same way as a material point of the same mass would move under the influence of all external forces acting on this body.
The concept of the moment of inertia was introduced into mechanics by the Russian scientist L. Euler in the middle of the 18th century and has since been widely used in solving many problems of rigid body dynamics. The value of the moment of inertia must be known in practice when calculating various rotating units and systems (flywheels, turbines, rotors of electric motors, gyroscopes). The moment of inertia is included in the equations of motion of a body (ship, aircraft, projectile, etc.). It is determined when they want to know the parameters of the rotational motion of the aircraft around the center of mass under the action of an external disturbance (a gust of wind, etc.). For bodies of variable mass (rockets), the mass and moment of inertia change over time.
2 .Moment of power.
The same force can impart different angular accelerations to a rotating body, depending on its direction and point of application. To characterize the rotating action of a force, the concept of a moment of force is introduced.
Distinguish between the moment of force relative to a fixed point and relative to a fixed axis. The moment of force relative to the point O (pole) is a vector quantity equal to the vector product of the radius vector drawn from the point O to the point of application of the force, by the force vector:
Illustrating this definition, Fig. 3 is made on the assumption that the point O and the vector lie in the plane of the drawing, then the vector is also located in this plane, and the vector to it and is directed away from us (as a vector product of 2 vectors; according to the rule of the right gimlet).
The modulus of the moment of force is numerically equal to the product of the force and the arm:
where is the shoulder of the force relative to the point O, is the angle between the directions and, 
.
Shoulder - the shortest distance from the center of rotation to the line of action of the force.
The vector of the moment of force is co-directed with the translational movement of the right gimlet, if its handle is rotated in the direction of the rotating action of the force. The moment of force is an axial (free) vector, it is directed along the axis of rotation, is not associated with a specific line of action, it can be transferred to
space parallel to itself.
The moment of force relative to the fixed axis Z is the projection of the vector onto this axis (passing through the point O).
E 
If several forces act on the body, then the resulting moment of forces about the fixed axis Z is equal to the algebraic sum of the moments about this axis of all forces acting on the body.
If the force applied to the body does not lie in the plane of rotation, it can be decomposed into 2 components: lying in the plane of rotation and to it F n . As can be seen from Figure 4, F n does not create rotation, but only leads to deformation of the body; rotation of the body is due only to the component F .
A rotating body can be represented as a set of material points.
AT 
we choose some point arbitrarily with mass m i, on which the force acts, imparting acceleration to the point (Fig. 5). Since only the tangential component creates rotation, it is directed perpendicular to the rotation axis to simplify the output.
In this case
According to Newton's second law: . Multiply both sides of the equation by r i ;
,
where is the moment of force acting on a material point,
Moment of inertia of a material point.
Hence, .
For the whole body: ,
those. the angular acceleration of a body is directly proportional to the moment of external forces acting on it and inversely proportional to its moment of inertia. The equation
(1) represents the equation of dynamics of rotational motion of a rigid body relative to a fixed axis, or Newton's second law for rotational motion.
3 . moment of impulse.
When comparing the laws of rotational and translational motion, an analogy is seen.
The analogue of momentum is the angular momentum. The concept of angular momentum can also be introduced relative to a fixed point and relative to a fixed axis, but in most cases it can be defined as follows. If a material point rotates around a fixed axis, then its angular momentum relative to this axis is equal in absolute value to
where m i- mass of a material point,
i - its linear speed
r i- distance to the axis of rotation.
Because for rotary motion
where is the moment of inertia of the material point about this axis.
The angular momentum of a rigid body relative to a fixed axis is equal to the sum of the angular momentum of all its points relative to this axis:
G 
de is the moment of inertia of the body.
Thus, the angular momentum of a rigid body relative to a fixed axis of rotation is equal to the product of its moment of inertia relative to this axis by the angular velocity and is co-directed with the angular velocity vector.
Let us differentiate equation (2) with respect to time:
Equation (3) is another form of the basic equation of the dynamics of the rotational motion of a rigid body relative to a fixed axis: the derivative of the moment
momentum of a rigid body about a fixed axis of rotation is equal to the moment of external forces about the same axis
This equation is one of the most important equations of rocket dynamics. In the process of rocket movement, the position of its center of mass is continuously changing, as a result of which various moments of forces arise: drag, aerodynamic force, forces created by the elevator. The equation of rotational motion of the rocket under the action of all moments of forces applied to it, together with the equations of motion of the center of mass of the rocket and the equations of kinematics with known initial conditions, make it possible to determine the position of the rocket in space at any time.
A rigid body rotating around some axes passing through the center of mass, if it is freed from external influences, maintains rotation indefinitely. (This conclusion is similar to Newton's first law for translational motion).
The occurrence of rotation of a rigid body is always caused by the action of external forces applied to individual points of the body. In this case, the appearance of deformations and the appearance of internal forces are inevitable, which in the case of a solid body ensure the practical preservation of its shape. When the action of external forces ceases, the rotation is preserved: internal forces can neither cause nor destroy the rotation of a rigid body.
The result of the action of an external force on a body with a fixed axis of rotation is an accelerated rotational motion of the body. (This conclusion is similar to Newton's second law for translational motion).
The basic law of the dynamics of rotational motion: in an inertial frame of reference, the angular acceleration acquired by a body rotating about a fixed axis is proportional to the total moment of all external forces acting on the body, and inversely proportional to the moment of inertia of the body about a given axis:
It is possible to give a simpler formulation the basic law of the dynamics of rotational motion(also called Newton's second law for rotational motion): the torque is equal to the product of the moment of inertia and the angular acceleration:
angular momentum(angular momentum, angular momentum) of a body is called the product of its moment of inertia times the angular velocity:
The angular momentum is a vector quantity. Its direction is the same as the direction of the vector angular velocity.
The change in angular momentum is defined as follows:
. (I.112)
A change in the angular momentum (with a constant moment of inertia of the body) can occur only as a result of a change in the angular velocity and is always due to the action of the moment of force.
According to the formula, as well as formulas (I.110) and (I.112), the change in angular momentum can be represented as:
. (I.113)
The product in formula (I.113) is called impulse moment of force or driving moment. It is equal to the change in angular momentum.
Formula (I.113) is valid provided that the moment of force does not change with time. If the moment of force depends on time, i.e. , then
. (I.114)
Formula (I.114) shows that: the change in angular momentum is equal to the time integral of the moment of force. In addition, if this formula is presented in the form: , then the definition will follow from it moment of force: the instantaneous moment of force is the first derivative of the moment of momentum with respect to time,
Basic concepts.
Moment of power relative to the axis of rotation is the vector product of the radius vector by 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.
Numerical value given 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 a =0 or , moment of power M=0, i.e. force passing through the axis of rotation or coinciding with it does not cause rotation.
The largest torque moment is created if the force acts at an angle = /2 (M 0) or =3 /2 (M 0).
Using the concept of the shoulder of force (shoulder of force d is a perpendicular dropped 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)
Moment of force rule(equilibrium condition for a body with 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 of the 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 of a material point relative to the axis of rotation is a value equal to the product of the mass of the point and the square of its distance from the axis of rotation:
J =m r 2 (1.18)
The moment of inertia of the body about the 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 about 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 torso is determined empirically and calculated approximately according to the formulas for a cylinder, a round rod or a 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 in the sagittal plane slightly ahead of the second sacral vertebra), depending on the position of the person, has the following values: at attention - 1.2 kg m 2; with the “arabesque” pose - 8 kgm 2; in a horizontal position - 17 kg m 2.
Work in rotary motion occurs when a body rotates under the action 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 the 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)
Angular moment (moment of momentum) 
–
a quantity numerically equal to the product of the momentum of the body and the radius of rotation.
L=p r=m V r (1.24).
After the appropriate transformations, you can write the formula for determining the angular momentum in the form:
(1.25).
angular momentum 
is a vector whose direction is determined by the right screw rule. The unit of angular momentum in SI is kgm 2 /s
Basic laws of rotational motion dynamics.
The basic equation for the dynamics of rotational motion:
The angular acceleration of a rotating body 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 for translational motion. It can be seen from the equation that under the action of external forces, the angular acceleration is the greater, the smaller the moment of inertia of the body.
Newton's second law for the dynamics of rotational motion can be written in a different form:
(1.27),
those. the first derivative of the angular momentum of the body with respect to time is equal to the total moment of all external forces acting on this body.
The law of conservation of momentum of the body:
If the total moment of all external forces acting on the body is zero, i.e.
M =0 , then dL/dt=0 (1.28).
Therefore 
or 
(1.29).
This statement is the essence of the law of conservation of the angular momentum of the 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 skater who performs a rotation around a vertical axis. By pressing his hands, the skater reduces the moment of inertia and increases the angular velocity. To slow down the rotation, on the contrary, he spreads his arms wide; as a result, the moment of inertia increases and the angular velocity of rotation decreases.
In conclusion, we give a comparative table of the main quantities and laws that characterize the dynamics of translational and rotational motions.
Table 1.4.
|
translational movement |
rotational movement |
||
|
Physical quantity |
Formula |
Physical quantity |
Formula |
|
Moment of inertia |
J=m r 2 |
||
|
Moment of power |
M=F
r if |
||
|
Body momentum (momentum) |
p=m V |
momentum of the body |
L=m V r; L=J |
|
Kinetic energy |
|
Kinetic energy |
|
|
mechanical work |
mechanical work |
dA=Md |
|
|
The basic equation of the dynamics of translational motion |
|
The basic equation of the dynamics of rotational motion |
|
|
Law of conservation of body momentum |
|
The law of conservation of momentum of the body |
if
|
,
or
if
or
J
= const,
Lecture plan
Moment of inertia.
Moment of power. The basic equation of the dynamics of rotational motion.
moment of impulse. Law of conservation of angular momentum.
Work and kinetic energy during rotational motion.
Moment of inertia.
When considering rotational motion, it is necessary to introduce new physical concepts: moment of inertia, moment of force, moment of impulse.
The moment of inertia is a measure of the inertia of the body during the rotational movement of the body.
Moment of inertia of a material point relative to a fixed axis of rotation is equal to the product of its mass by the square of the distance to the considered axis of rotation (Fig. 1):
depends only on the mass of the material point and its position relative to the axis of rotation and does not depend on the presence of rotation itself.
The moment of inertia is a scalar and additive quantity, therefore the moment of inertia of a body is equal to the sum of the moments of inertia of all its points:
.
In the case of a continuous mass distribution, this sum reduces to the integral:
,
where is the mass of a small body volume 
,
- density of the body 
- distance from element 
to the axis of rotation.
The moment of inertia is analogous to mass in rotational motion. The greater the moment of inertia of the body, the more difficult it is to change the angular velocity of the rotating body. The moment of inertia is meaningful only for a given position of the axis of rotation. It is meaningless to speak simply of the “moment of inertia”. It depends:
1) from the position of the axis of rotation;
2) on the distribution of body mass relative to the axis of rotation, i.e. on body shape and size.
Experimental proof of this is the experience with rolling cylinders.
By integrating for some homogeneous bodies, we can obtain the following formulas (the axis of rotation passes through the center of mass of the body).
The moment of inertia of a hoop (we neglect the wall thickness) or a hollow cylinder:
Moment of inertia of a disk or solid cylinder of radius R:
.
Moment of inertia of the ball
Moment of inertia of the rod
E 
If the moment of inertia about the axis passing through the center of mass is known for the body, then the moment of inertia about any axis parallel to the first is found by Steiner theorem: the moment of inertia of the body about an arbitrary axis is equal to the moment of inertia J 0 about an axis parallel to the given one and passing through the center of mass of the body, added to the product of the body's mass by the square of the distance between the axes.
where d distance from center of mass O to the axis of rotation (Fig. 2).
Center of mass- an imaginary point, the position of which characterizes the distribution of the mass of a given body. The center of mass of the body moves in the same way as a material point of the same mass would move under the influence of all external forces acting on this body.
The concept of the moment of inertia was introduced into mechanics by the Russian scientist L. Euler in the middle of the 18th century, and has since been widely used in solving many problems of rigid body dynamics. The value of the moment of inertia must be known in practice when calculating various rotating units and systems (flywheels, turbines, rotors of electric motors, gyroscopes). The moment of inertia is included in the equations of motion of a body (ship, aircraft, projectile, etc.). It is determined when they want to know the parameters of the rotational motion of the aircraft around the center of mass under the action of an external disturbance (a gust of wind, etc.).
Let some body, under the action of a 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 the point O (lying on the axis) to the direction of the force, is called shoulder of strength. The product of the force on the shoulder determines the modulus of the moment of force relative to the point O:
M = Fp=Frsinα.
Moment of power is a vector determined by the vector product of the radius-vector of the force application point and the force vector:
(3.1) The unit of the moment of force is the newton meter (N m).
The direction of M can be found using the right screw rule.
angular momentum particle is called the vector product of the radius vector of the particle and its momentum:
or in scalar form L = gPsinα
This quantity is vector and coincides in direction with the vectors ω.
§ 3.2 Moment of inertia. Steiner's theorem
A measure of the inertia of bodies in translational motion is the mass. The inertia of bodies during rotational motion depends not only on the 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 about the axis of rotation.
The moment of inertia of a material point relative to the axis of rotation is the product of the mass of this point and the square of its distance from the axis:
I i =m i r i 2 (3.2)
Moment of inertia of the body about the axis of rotation call the sum of the moments of inertia of the material points that make up this body:
(3.3)
In the general case, if the body is solid and is 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 and how the mass of the body is distributed throughout the volume.
The moment of inertia of bodies that have the correct geometric shape and a uniform distribution of mass over volume is most simply determined.
Moment of inertia of a homogeneous rod relative to the axis passing through the center of inertia and perpendicular to the rod
(3.6)
Moment of inertia of a homogeneous cylinder about an axis perpendicular to its base and passing through the center of inertia,
(3.7)
Moment of inertia of a thin-walled cylinder or a hoop about 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)
Consider an example . Let us determine the moment of inertia of the disk about 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 above 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 about an arbitrary axis, one should use Steiner's theorem : the moment of inertia of the body about an arbitrary axis of rotation is equal to the sum of the moment of inertia of the body about an axis parallel to the given one and passing through the center of mass of the body, and the product of the body's mass by the square of the distance between the axes:
(3.11)
The unit of the moment of inertia is a kilogram-meter squared (kg m 2).
So, the moment of inertia of a homogeneous rod about the axis passing through its end, according to Steiner's theorem, is equal to
(3.12)