Basic concepts.
Moment of power relative to the axis of rotation is the vector product of the radius vector by the force.
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× sina(1.15),
where a - the angle between the radius vector and the direction of the force.
If a=0 or p, 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 a=p/2 (M > 0) or a=3p/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.
S M i =0(1.17)
The SI unit for the moment of force is [N×m]
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 i \u003d m i × r i 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=S m i × r i 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 J0– 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 [kg × m 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 in front 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 kg × m 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 i = M i × dj(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× dj(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:
Moment of momentum (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).
The angular momentum is a vector, the direction of which is determined by the rule of the right screw. The SI unit of angular momentum is [kg×m 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.
S M i =0, then dL/dt=0 (1.28).
From this follows 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 |
| Weight | m | Moment of inertia | J=m×r2 |
| Force | F | Moment of power | M=F×r if |
| Body momentum (momentum) | p=m×V | momentum of the body | L=m×V×r; L=J×w |
| Kinetic energy | Kinetic energy | ||
| mechanical work | dA=FdS | mechanical work | dA=Mdj |
| The basic equation of the dynamics of translational motion | The basic equation of the dynamics of rotational motion | , | |
| Law of conservation of body momentum |
or  if
| The law of conservation of momentum of the body | or SJ i w i = const, if |
Centrifugation.
The separation of inhomogeneous systems consisting of particles of different densities can be carried out under the action of gravity and the Archimedes force (buoyancy force). If there is an aqueous suspension of particles of different densities, then the resultant force acts on them
F p \u003d F t - F A \u003d r 1 × V × g - r × V × g, i.e.
F p \u003d (r 1 - r) × V ×g(1.30)
where V is the particle volume, r1 and r are the densities of the substance of the particle and water, respectively. If the densities differ slightly from each other, then the resulting force is small and the separation (deposition) occurs rather slowly. Therefore, forced separation of particles is used due to the rotation of the medium to be separated.
centrifugation called the process of separation (separation) of heterogeneous systems, mixtures or suspensions, consisting of particles of different masses, occurring under the action of the centrifugal force of inertia.
The basis of the centrifuge is a rotor with test-tube seats, located in a closed housing, which is driven by an electric motor. When the centrifuge rotor rotates at a sufficiently high speed, particles of suspension, different in mass, are distributed in layers at different depths under the action of the centrifugal force of inertia, and the heaviest ones settle at the bottom of the test tube.
It can be shown that the force under which separation occurs is determined by the formula:
(1.31)
where w- angular speed of rotation of the centrifuge, r is the distance from the axis of rotation. The effect of centrifugation is the greater, the greater the difference between the densities of the separated particles and liquid, and also significantly depends on the angular velocity of rotation.
Ultracentrifuges operating at a rotor speed of about 10 5 -10 6 revolutions per minute are able to separate particles smaller than 100 nm in size, suspended or dissolved in a liquid. They have found wide application in biomedical research.
Using ultracentrifugation, cells can be separated into organelles and macromolecules. At first, larger parts (nuclei, cytoskeleton) settle (sediment). With a further increase in the centrifugation speed, smaller particles are sequentially deposited - first mitochondria, lysosomes, then microsomes, and finally ribosomes and large macromolecules. During centrifugation, different fractions settle at different rates, forming separate bands in the test tube, which can be isolated and examined. Fractionated cell extracts (cell-free systems) are widely used to study intracellular processes, for example, to study protein biosynthesis and decipher the genetic code.
To sterilize handpieces in dentistry, an oil sterilizer with a centrifuge is used, with which excess oil is removed.
Centrifugation can be used to precipitate particles suspended in urine; separation of formed elements from blood plasma; separation of biopolymers, viruses and subcellular structures; control over the purity of the drug.
Tasks for self-control of knowledge.
Exercise 1 . Questions for self-control.
What is the difference between uniform motion in a circle and uniform rectilinear motion? Under what condition will the body move uniformly in a circle?
Explain the reason why uniform circular motion occurs with acceleration.
Can curvilinear motion occur without acceleration?
Under what condition is the moment of force equal to zero? accepts highest value?
Indicate the limits of applicability of the law of conservation of momentum, angular momentum.
Specify the features of separation under the action of gravity.
Why is it possible to separate proteins with different molecular weights by centrifugation, but the method of fractional distillation is unacceptable?
Task 2 . Tests for self-control.
Insert missing word:
A change in the sign of the angular velocity indicates a change in _ _ _ _ _ rotational motion.
A change in the sign of the angular acceleration indicates a change in _ _ _ rotational motion
The angular velocity is equal to _ _ _ _ _ the derivative of the angle of rotation of the radius vector with respect to time.
Angular acceleration is equal to _ _ _ _ _ _ time derivative of the angle of rotation of the radius vector.
The moment of force is _ _ _ _ _ if the direction of the force acting on the body coincides with the axis of rotation.
Find the correct answer:
The moment of force depends only on the point of application of the force.
The moment of inertia of a body depends only on the mass of the body.
Uniform circular motion occurs without acceleration.
A. Right. B. Wrong.
All of the above quantities are scalar, with the exception of
A. moment of force;
B. mechanical work;
C. potential energy;
D. moment of inertia.
The vector quantities are
A. angular velocity;
B. angular acceleration;
C. moment of force;
D. angular momentum.
Answers: 1 - directions; 2 - character; 3 - the first; 4 - second; 5 - zero; 6 - B; 7 - B; 8 - B; 9 - A; 10 - A, B, C, D.
Task 3. Get the relationship between units of measure :
linear speed cm/min and m/s;
angular acceleration rad/min 2 and rad/s 2;
moment of force kN×cm and N×m;
body momentum g×cm/s and kg×m/s;
moment of inertia g×cm 2 and kg×m 2 .
Task 4. Tasks of medical and biological content.
Task number 1. Why in the flight phase of a jump, an athlete cannot change the trajectory of the center of gravity of the body with any movements? Do the athlete's muscles perform work when the position of body parts in space changes?
Answer: With movements in free flight along a parabola, an athlete can only change the location of the body and its individual parts relative to its center of gravity, which in this case is the center of rotation. The athlete does work to change the kinetic energy of body rotation.
Task number 2. What average power does a person develop when walking if the step duration is 0.5 s? Assume that work is expended on accelerating and decelerating the lower extremities. Angular movement of the legs is about Dj=30 o. The moment of inertia of the lower limb is 1.7kg × m 2. The movement of the legs is considered as equally variable rotational.
Decision:
1) Let's write a brief condition of the problem: Dt= 0.5s; DJ=30 0 =p/ 6; I=1.7kg × m 2
2) Define the work in one step (right and left leg): A= 2×Iw 2 / 2=Iw 2 .
Using the formula for the average angular velocity w av =Dj/Dt, we get: w= 2w cf = 2×Dj/Dt; N=A/Dt= 4×I×(Dj) 2 /(Dt) 3
3) Substitute the numerical values: N=4× 1,7× (3,14) 2 /(0,5 3 × 36)=14.9(W)
Answer: 14.9 W.
Task number 3. What is the role of arm movement in walking?
Answer: The movement of the legs, moving in two parallel planes, located at some distance from each other, creates a moment of force that tends to rotate the human body around a vertical axis. A person swings his arms “towards” the movement of his legs, thereby creating a moment of forces of the opposite sign.
Task number 4. One of the ways to improve drills used in dentistry is to increase the speed of rotation of the drill. The speed of rotation of the boron tip in foot drills is 1500 rpm, in stationary electric drills - 4000 rpm, in turbine drills - already reaches 300,000 rpm. Why are new modifications of drills with a large number of revolutions per unit time being developed?
Answer: Dentin is several thousand times more susceptible to pain than skin: there are 1-2 pain points per 1 mm 2 of skin, and up to 30,000 pain points per 1 mm 2 of incisor dentin. An increase in the number of revolutions, according to physiologists, reduces pain during the treatment of a carious cavity.
W assignment 5 . Fill in the tables:
Table #1. Draw an analogy between the linear and angular characteristics of rotational motion and indicate the relationship between them.
Table number 2.
Task 6. Fill in the indicative action card:
| Main tasks | Directions | Answers |
| Why does the gymnast bend his knees and press them to his chest at the initial stage of the somersault, and straighten his body at the end of the rotation? | Use the concept of angular momentum and the law of conservation of angular momentum to analyze the process. | |
| Explain why standing on tiptoe (or holding a heavy load) is so hard? | Consider the conditions for the balance of forces and their moments. | |
| How will the angular acceleration change with an increase in the moment of inertia of the body? | Analyze the basic equation of rotational motion dynamics. | |
| How does the effect of centrifugation depend on the difference in the densities of the liquid and particles that are separated? | Consider the forces acting during centrifugation and the relationship between them |
Chapter 2. Fundamentals of biomechanics.
Questions.
Levers and joints in the human musculoskeletal system. The concept of degrees of freedom.
Types of muscle contraction. Basic physical quantities that describe muscle contractions.
Principles of motor regulation in humans.
Methods and devices for measuring biomechanical characteristics.
2.1. Levers and joints in the human musculoskeletal system.
The anatomy and physiology of the human motor apparatus have the following features that must be taken into account in biomechanical calculations: body movements are determined not only by muscle forces, but also by external reaction forces, gravity, inertial forces, as well as elastic forces and friction; the structure of the motor apparatus allows only rotational movements. With the help of the analysis of kinematic chains, translational movements can be reduced to rotational movements in the joints; the movements are controlled by a very complex cybernetic mechanism, so that there is a constant change in the accelerations.
The human musculoskeletal system consists of articulated bones of the skeleton, to which muscles are attached at certain points. The bones of the skeleton act as levers that have a fulcrum at the joints and are driven by the traction force that occurs when the muscles contract. Distinguish three types of lever:
1) The lever to which the acting force F and resistance force R attached on opposite sides of the fulcrum. An example of such a lever is the skull viewed in the sagittal plane.
2) A lever whose operating force F and resistance force R applied on one side of the fulcrum, moreover, the force F applied to the end of the lever, and the force R closer to the anchor point. This lever gives a gain in strength and a loss in distance, i.e. is an leverage. An example is the action of the arch of the foot when lifting on the toes, the levers of the maxillofacial region (Fig. 2.1). The movements of the chewing apparatus are very complex. When closing the mouth, lifting the lower jaw from the position of maximum lowering to the position of complete closure of its teeth with the teeth of the upper jaw is carried out by the movement of the muscles that raise the lower jaw. These muscles act on the lower jaw as a second-class lever with a fulcrum at the joint (giving a gain in chewing power).
3) A lever in which the acting force is applied closer to the fulcrum than the resistance force. This lever is speed lever, because gives a loss in strength, but a gain in movement. An example is the bones of the forearm.
Rice. 2.1. The levers of the maxillofacial region and the arch of the foot.
Most of the bones of the skeleton are under the action of several muscles that develop efforts in various directions. Their resultant is found by geometric addition according to the parallelogram rule.
The bones of the musculoskeletal system are connected to each other in joints or joints. The ends of the bones that form the joint are held together with the help of an articular bag tightly covering them, as well as ligaments attached to the bones. To reduce friction, the contact surfaces of the bones are covered with smooth cartilage and there is a thin layer of sticky fluid between them.
The first step in the biomechanical analysis of motor processes is the determination of their kinematics. On the basis of such an analysis, abstract kinematic chains are constructed, the mobility or stability of which can be checked on the basis of geometric considerations. There are closed and open kinematic chains formed by joints and rigid links located between them.
The state of a free material point in three-dimensional space is given by three independent coordinates - x, y, z. Independent variables that characterize the state of a mechanical system are called degrees of freedom. More complex systems may have more degrees of freedom. In general, the number of degrees of freedom determines not only the number of independent variables (which characterizes the state of the mechanical system), but also the number of independent displacements of the system.
Number of degrees freedom is the main mechanical characteristic of the joint, i.e. defines number of axles, around which mutual rotation of articulated bones is possible. It is mainly due geometric shape surfaces of bones that touch at a joint.
The maximum number of degrees of freedom in the joints is 3.
Examples of a uniaxial (flat) articulation in the human body are the humeroulnar, supracalcaneal, and phalangeal joints. They allow only the possibility of flexion and extension with one degree of freedom. So, the ulna, with the help of a semicircular notch, covers a cylindrical protrusion on the humerus, which serves as the axis of the joint. Movement in the joint - flexion and extension in a plane perpendicular to the axis of the joint.
The wrist joint, in which flexion and extension, as well as adduction and abduction, can be attributed to joints with two degrees of freedom.
Joints with three degrees of freedom (spatial articulation) include the hip and scapular-shoulder joints. For example, in the scapular-humeral joint, the spherical head of the humerus enters the spherical cavity of the protrusion of the scapula. Movements in the joint - flexion and extension (in the sagittal plane), adduction and abduction (in the frontal plane) and rotation of the limb around the longitudinal axis.
Closed planar kinematic chains have the number of degrees of freedom f F, which is calculated by the number of links n in the following way:
The situation for kinematic chains in space is more complicated. Here the relation
(2.2)
where fi- number of degrees of freedom constraints i- th link.
In any body, you can choose such axes, the direction of which will be preserved during rotation without any special devices. They have a name free rotation axes
The disk rotates around a vertical axis with angular velocity (experiment)
On the disk, at different distances from the axis of rotation, pendulums are installed (balls of mass m are suspended on threads) .
When the disk rotates, the pendulums deviate from the vertical by some angle a.
INERTIAL REFERENCE SYSTEM(data analysis )__
In a frame of reference associated, for example, with a room, the ball rotates uniformly around a circle with radius R (the distance from the center of the rotating ball to the axis of rotation). Consequently, a force equal to F = m ω 2 R and directed perpendicular to the disk rotation axis acts on it. It is the resultant force of gravity and the force of tension in the thread. For the steady motion of the ball , whence
tg = ω 2 R/g (the more, the larger R and ω).
NONINERTIAL REFERENCE SYSTEM(data analysis )__
In the frame of reference associated with a rotating disk, the ball is at rest, which is possible if the force is balanced by an equal and opposite force , which is nothing more than the force of inertia, since no other forces act on the ball. Force F c, called centrifugal force of inertia , directed horizontally from the axis of rotation of the disk, F c \u003d -m ω 2 R.
The action of centrifugal forces of inertia are, for example, passengers in a moving vehicle on turns, pilots when performing aerobatics. When designing rapidly rotating machine parts (rotors, aircraft propellers, etc.), special measures are taken to balance the centrifugal forces of inertia.
♦ Centrifugal force of inertia ( F q \u003d -m ω 2 R) does not depend on the speed of bodies relative to rotating frames of reference, i.e., it acts on all bodies remote from the axis of rotation at a finite distance, regardless of whether they are at rest in this frame or moving relative to it at some speed.
6.3. FORCES OF INERTIA AFFECTING THE BODY, MOVING IN A ROTATING REFERENCE SYSTEM _
The disk is at rest ( experience)
Ball mass t, directed along the radius of the disk with a speed V" = const, moves along a radial straight line OA.
The disk rotates evenly(co = const) (experience)
A ball of mass m, moving at a speed V "= const (V" ┴ ω), rolls along the curve AB, and its velocity V" relative to the disk changes its direction. This is possible only if the force acting on the ball is perpendicular to the velocity V ".
Analysis of experimental data
In order for the ball to roll along the rotating disk along the radius, a rod rigidly fixed along the radius of the disk is used, on which the ball moves uniformly and rectilinearly without friction with a speed . When the ball is deflected, the rod acts on it with some force .
Relative to the disk (rotating reference frame), the ball moves uniformly and rectilinearly, which can be explained by the fact that the force is balanced by the force of inertia applied to the ball ,
perpendicular speed. This force is called Coriolis force of inertia. Coriolis force 
.
Examples of the manifestation of inertia forces. If a body moves north in the northern hemisphere, then the Coriolis force acting on it directed to the right in relation to the direction of movement, i.e., the body will deviate somewhat to the east. Therefore, in the northern hemisphere, a stronger erosion of the right banks of rivers is observed; the right rails of railway tracks wear out faster than the left rails, etc.
There are three forces acting on the body:
gravity m, support reaction force and friction force tr.
In an inertial frame of reference associated with the Earth, Newton's second law will look like: 
The movement of a body relative to the Earth is a movement in a horizontal plane along a circle with a radius R. The forces acting on it in the vertical direction are compensated. The acceleration vector lies in the horizontal plane, and the acceleration itself is centripetal. Its value is determined by the formula:
Projecting a vector equation onto the X and Y coordinate axes gives two scalar equations: 
The first equation shows that the friction force acts as a centripetal force, the second states that the vertical forces are mutually balanced.
The static friction force obeys the inequality:
so when
Consider the simplest case: a ball of mass t moving uniformly at a speed v 0 along the radius of the rotating disk. To ensure such a movement, we provide the ball with a guide rod along which it could move without friction. The thread attached to the ball will allow it to move in the radial direction at a constant speed. v 0 (Fig. 5.6).
Rice. 5.6
The disk rotates with angular velocity . Let us describe the motion of the ball in a fixed inertial frame of reference S(x,y). In this system, the movement of the ball consists of two movements: uniform rectilinear - along the radius of the disk with a speed v 0 and circular motion with angular velocity .
As a result of the addition of these two movements, the ball will move along a curvilinear trajectory - an unfolding spiral.
At an arbitrary point in time t ball in the distance r from the axis of rotation will have a radial velocity v 0 and tangential - tangential speed associated with disk rotation ( r) (Fig. 5.7).
Rice. 5.7
Let's see how these speeds of the ball change after a short time dt.
First, the whole picture of velocities will rotate through the angle d= dt(Fig. 5.7 b). Secondly, the radial velocity (remaining unchanged in magnitude - V 0) will be incremented:
dV 1 =V 0 d=V 0 dt, (5.5)
associated with the repetition of the velocity vector V 0 per corner d= dt.
The tangential speed will also change. Its change in magnitude is determined by the fact that the ball moves away from the axis of rotation at a distance dr=V 0 dt. So:
dV 2 =( r+dr) – r= dr= V 0 dt. (5.6)
In addition, this speed will change by:
dV 3 = rd = r dt= 2 rdt, (5.7)
due to the rotation of the vector of this velocity by the angle d.
After analyzing all these changes, we come to the conclusion that in the radial direction the change in velocity will be:
dV r =dV 3 = 2 rdt,
and in tangential:
dV = dV 1 +dV 2 = 2 V 0 dt.
Dividing these changes by the time interval dt, we obtain the corresponding acceleration components:
; (5.8)
. (5.9)
It is easy to answer the question: what forces provide these accelerations?
Centripetal acceleration is created by the elastic force of the thread tension ( F c.s. = F ex. = ma c.s. = m 2 r) directed along the radius to the axis of rotation. Tangential acceleration a is supported by the elastic force of the deformed rod ( 
=ma = m 2 V 0). The rod bends during movement and acts on the ball with a force directed in the direction of rotation (Fig. 5.8).
Rice. 5.8
Let us write down the equations of motion of the ball in the inertial frame of reference. These are the equations of Newton's second law for two motions - along the radius:
, (5.10)
and in the perpendicular direction:
. (5.11)
Now let's see how the movement of the same ball appears to an observer rotating with the disk.
This observer sees that the ball in his rotating reference frame is moving uniformly and in a straight line with a speed 
=const along the disk radius. The acceleration of the ball is zero, but at the same time, the elastic force of the thread tension acts on it F c.s. = m 2 r and elastic force of the deformed rod F=m 2 V 0 . Their resultant cannot be equal to zero.
In order to write the equation of motion of this body in a non-inertial frame of reference in the form of equations of Newton's second law, we add two forces of inertia to the actually acting elastic forces (Fig. 5.9):
(5.12)
. (5.13)
Rice. 5.9
Now, both in the radial and tangential directions, the sum of the forces will be equal to zero, which explains the uniform motion of the ball along the radius.
From the first of the forces of inertia 
do we know each other. This is the centrifugal force of inertia.
Second force of inertia 
called the Coriolis force.
These forces can be written in vector form:
.
Summing up the consideration of motions in non-inertial frames of reference, we note the following main points.
The Newtonian equation of motion can also be used in non-inertial frames of reference. But at the same time, the system of really acting forces must be supplemented by inertia forces.
In a non-inertial frame of reference moving rectilinearly and translationally with acceleration 
, the force of inertia is equal to:
. (5.14)
In a non-inertial reference frame rotating with an angular velocity , in the general case, two inertia forces should be introduced:
centrifugal 
, (5.15)
and Coriolis 
. (5.16)
Vladimir.erashov.rf
First, we formulate the combined law of inertia, which applies to all bodies and all types of motion:
The subsequent kinematic state of the body differs from the previous one only if, in the period between states, a new external force or moment of forces began to act on the body, and it differs only by the magnitude of the body's response to this effect.
By this law, we do not open any new pages in the kinematics of bodies, it is obtained on the basis of Newton's laws, but with a complex movement of the body, it helps to simplify the task of describing this movement. We proceed from the fact that in the previous kinematic state, no matter what forces act on the body, it has already responded to the action of these forces and will continue to move according to the acquired laws. For example, in the initial state, the acceleration acts on the body a , the body under the action of this acceleration acquired a speed v , but the acceleration continues until the next state. This means that the body will increase the speed between states by at. If some additional acceleration appears between the states, then it is enough to impose its effect on the previous result obtained, that is, how to use the independence of the action of forces. The main thread of the united law of inertia is that if there is no change in the acting forces between states, then there are no changes in the laws of motion of the body, as in life, the next day is strung on the previous one. If yesterday you didn’t have a penny of money behind your soul, then today you will wake up without a penny of money. If yesterday you went on a long sea voyage on a cruise ship, then today you will wake up on a cruise ship. If you have a clean shirt, then someone has washed it. Neither a speck of dust nor a hair will fall from you by itself, there must be a reason for this (read some kind of force). If before rotation the main axis of inertia of the body was perpendicular to the surface of the Earth and rested relative to this surface, then after the rotation of the body it will rest relative to the Earth, as before (with stable rotation, in the case of unstable rotation, a specific force acts on the body). Changes in the state of the body can occur, but only under the action of a specific force or moment of forces and nothing else.
To make it easier to understand the action of the formulated law, and even try to derive practical benefits from this law, let's consider a specific example - this is our rotating Earth and bodies on its surface.
First, let's clarify, the Earth is affected by Newton's law of gravity so it's round like a ball.
Secondly, centrifugal acceleration from rotation acts on the Earth, under the influence of this acceleration the Earth acquired the shape of a geoid of rotation. To clarify, the property of the Earth-geoid is that at any point on the Earth's surface, any body remains motionless (even if it is able to move freely) due to the fact that the resulting force acting on the body from the forces of attraction and the centrifugal force of inertia is directed perpendicular to the surface and is balanced by the reaction of this surface (property of the geoid). Due to the geoid of rotation, even the ocean on the surface of the Earth came to an equilibrium state and acquired immobility relative to the surface, and therefore the geoid.
Let us return to the body on the surface of the Earth, no one bothers us to assume that a bar in the form of a rectangular parallelepiped lies on the surface of the Earth. The main axis of inertia of this bar passes through the fulcrum on the surface and is perpendicular to the surface. Note that relative to the Earth, the bar lies motionless, and relative to the stars, together with the Earth, it makes one revolution per day.
Let us single out for readers that, relative to the stars, the bar is a rotating body with one revolution per day, the main axis of inertia of this bar is perpendicular to the Earth's surface and is stationary relative to the Earth. Let's spin the bar to high revolutions relative to its main axis of inertia. Will the axis of the bar remain perpendicular and stationary with respect to the Earth? Or, as it is commonly believed, will it acquire motion (rotation) in relation to the Earth and, relative to the stars, change its state from rotation with one revolution per day to a stationary state?
According to the combined law of inertia, after unwinding, the bar must keep the rotation axis (principal axis of inertia) stationary relative to the Earth, and relative to the stars it must still rotate at an angular velocity of one revolution per day. This is motivated by the fact that during the unwinding of the bar, if the bar is bolted relative to the axis of rotation, the same forces will act on the mass cent of the bar as in the previous state (before spinning). Consequently, the subsequent state of the bar (after spinning up) is identical to the previous state (before spinning up) and the bar must retain all the properties of the previous state and not receive any changes, including the axis of rotation of the bar must remain stationary and perpendicular to the Earth's surface.
If someone does not like the combined law of inertia and he does not agree with the conclusions according to the combined law, then the behavior of the bar (rotating body) after spinning to maintain its original state can be explained by the fact that spinning did not add any new forces to the cent of the mass of the bar, and that’s all the parameters of motion of the center of mass of the bar in space remained the same.
In general, under the conditions of the Earth, the following forces act on a body, at least rotating, at least not rotating:
1. Force of attraction of the Earth.
2. The force of inertia.
3. support reaction.
There are no other forces in nature, there is still Coriolis acceleration, but it is a derivative of the forces of inertia (it is not an independent force) and appears only when there is a movement of the body relative to the surface of the Earth. The Coriolis acceleration itself cannot transfer the body from a stationary state relative to the Earth to a mobile state, there is no movement relative to the earth, and there is no Coriolis acceleration.
Bodies that rotate rapidly about the main axis of inertia are called gyroscopes. Gyroscopes have a number of unique properties. Let's take a look at these properties. It is generally accepted that the main property of a gyroscope is that they always keep the position of the axis of rotation fixed relative to the stars.
Our theory introduces a significant refinement of this property of the gyroscope. In inertial coordinate systems, this property of the gyroscope is strictly observed, here we are in solidarity with the accepted theory, but in non-inertial reference systems, in particular those associated with the surface of the rotating Earth, this property does not act otherwise, the axis of the gyroscope, if the rotation is stable, retains its original position and relative to the stars, and relative to the earth. But since in the initial position the axis of the gyroscope rotated relative to the stars, it will continue to rotate relative to the stars at the same speed, and relative to the Earth, as it was motionless, it will remain motionless. The state of the body is inert, the movement of the axis is inert, and not the focus on anything.
The conclusion at the beginning is unusual (inertia of thinking), which requires additional comments. Take a simple wolcho meme to (yule). Let's start the top k. Let's assume that the friction forces at the base of the top's axis are minimal, and it can maintain rotation for a relatively long time. According to our theory, the axis of rotation of the top remains motionless and perpendicular to the surface of the Earth, therefore, nothing prevents the top from stable and long rotation. In life, the top cannot be absolutely isolated from external forces, some external forces, let's call them random, still act on the axis of the top and deviate it from the vertical position. Further, the force of the weight deviates from the fulcrum, a moment of forces arises, to which the wolcho k responds with precession.
If the axis of rotation of the top, as is commonly believed, must remain stationary relative to the stars, then it cannot maintain a long vertical position relative to the surface of the Earth, it will tilt from east to west at a rate of one revolution per day (12 degrees per hour). The axis of rotation of such a top already in five minutes of rotation will deviate from the vertical by about one degree. If earlier, with the vertical position of the axis of rotation, the gravity force acting on the center of mass lay on the axis of rotation and passed through the fulcrum and did not cause any movement of the center of mass, then when the axis of rotation is tilted, an overturning moment should occur. Moreover, the overturning moment circulates not only in direction, but also in magnitude. It is maximum in the lower position of the center of gravity and minimum in the upper position. Thus, this moment should cause not the precession of the top, but its nutation. This contradicts the results of experiments with a top. In a spinning top, the main movement is precession, and nutation appears only at the very end of rotation, when the rotation is already close to disorderly.
There are such units in industry as centrifuges. Due to the very high speed, these units are very sensitive to external forces. If their axis of rotation remained fixed relative to the stars, and tilted relative to the surface of the Earth, then these units would go into spacing and scatter, but they work. Consequently, our version of the interpretation of the behavior of rotating bodies in a non-inertial coordinate system is valid, and not the generally accepted one. Which is accepted on the basis of experiments, and not from theoretical justifications. This means that they did not properly understand the experimental material, they took it for a postulate not what is.
Conclusion
The unified law of inertia is valid in all frames of reference, both inertial and non-inertial ones. On the basis of this law, an erroneous idea was revealed about the existing first law of the gyroscope, according to which the axis of rotation of the gyroscope must always be fixed relative to the stars. It has been established that gyroscopes behave this way only in inertial frames of reference, in non-inertial reference frames it is necessary to use not this rule, but the combined law of inertia.
July 12, 2018