1. Moment impulse, Mdt, acting on rotational body, is equal to the change in its angular momentum dL:
Mdt = d(J ω) or Mdt = dL
Where: Mdt – impulse of the moment of force (the product of the moment of force M by the time interval dt)
Jdω = d(Jω) – change in the angular momentum of the body,
Jω = L - the angular momentum of a body is the product of the moment of inertia J and the angular velocity ω ω, and d(Jω) is dL.

2. Kinematic characteristics Rotation of a rigid body as a whole is characterized by an angle φ, measured in angular degrees or radians, angular velocity
ω = dφ/dt(measured in rad/s)
and angular acceleration
ε = d²φ/dt² (measured in rad/s²).
With uniform rotation (T revolutions per second), Rotation frequency is the number of body revolutions per unit time:
f = 1/T =ω/2
The rotation period is the time of one full revolution. The rotation period T and its frequency f are related by the relation
T = 1/f

Linear speed of a point located at a distance R from the axis of rotation

Angular velocity of body rotation
ω = f/Dt = 2/T

Dynamic characteristics The properties of a rigid body during its rotation are described by the moment of inertia solid. This feature is included in differential equations, obtained from Hamilton's or Lagrange's equations. The kinetic energy of rotation can be written as:
E=

In this formula, the moment of inertia plays the role of mass, and angular velocity plays the role normal speed. The moment of inertia expresses the geometric distribution of mass in a body and can be found from the formula:

The moment of inertia of a mechanical system relative to a fixed axis a (“axial moment of inertia”) is a physical quantity Ja equal to the sum of the products of the masses of all n material points systems by the squares of their distances to the axis:
= ∑

Where: mi is the mass of the i-th point, ri is the distance from the i-th point to the axis. The axial moment of inertia of a body Ja is a measure of the inertia of a body in rotational motion around axis a, just as the mass of a body is a measure of its inertia in translational motion.

3. The pendulum represents closed system.
If the pendulum is in extreme point, its potential energy is maximum, and its kinetic energy is zero.
As soon as the pendulum begins to move, its potential energy decreases, and its kinetic energy increases.
At the bottom point, kinetic energy is maximum and potential energy is minimum. After this, the reverse process begins. The accumulated kinetic energy moves the pendulum upward and thereby increases the potential energy of the pendulum. Kinetic energy decreases until the pendulum stops again at the other extreme point.
We can say that during the movement of the pendulum a transition occurs potential energy to kinetic and vice versa.

The sum of the kinetic and potential energy of the bodies that make up a closed system and interact with each other by gravitational and elastic forces remains constant.
Or this: The total mechanical energy of a closed system of bodies interacting with gravitational and elastic forces remains unchanged.
(The sum of the kinetic and potential energy of bodies is called total mechanical energy)

To derive this law, let us consider the simplest case of rotational motion of a material point. Let us decompose the force acting on a material point into two components: normal - and tangent - (Fig. 4.3). The normal component of the force will lead to the appearance of normal (centripetal) acceleration: ; , where r = OA - radius of the circle.

A tangential force will cause a tangential acceleration to appear. In accordance with Newton's second law, F t =ma t or F cos a=ma t.

Let's express the tangential acceleration in terms of the angular acceleration: a t =re. Then F cos a=mre. Let's multiply this expression by the radius r: Fr cos a=mr 2 e. Let us introduce the notation r cos a = l , Where l - leverage of force, i.e. length of the perpendicular lowered from the axis of rotation to the line of action of the force. Sincemr 2 =I - moment of inertia of a material point, and product = Fl = M - moment of force, then

Product of moment of force M for the duration of its validity dt is called the moment impulse. Product of moment of inertia I by angular velocity w is called the angular momentum of the body: L=Iw. Then the basic law of the dynamics of rotational motion in the form (4.5) can be formulated as follows: The momentum of the moment of force is equal to the change in the angular momentum of the body. In this formulation, this law is similar to Newton’s second law in the form (2.2).

End of work -

This topic belongs to the section:

Short course in physics

Ministry of Education and Science of Ukraine.. Odessa National Maritime Academy..

If you need additional material on this topic, or you did not find what you were looking for, we recommend using the search in our database of works:

What will we do with the received material:

If this material was useful to you, you can save it to your page on social networks:

Tweet

All topics in this section:

Basic SI units
Currently, the International System of Units - SI - is generally accepted. This system contains seven basic units: meter, kilogram, second, mole, ampere, kelvin, candela and two additional ones -

Mechanics
Mechanics - the science of mechanical movement material bodies and the resulting interactions between them. Under mechanical movement understand changes in mutual gender over time

Normal and tangential acceleration
Rice. 1.4 Movement of a material point along a curved path

Newton's laws
Dynamics is a branch of mechanics that studies the movement of material bodies under the influence of forces applied to them. Mechanics is based on Newton's laws. Newton's first law

Law of conservation of momentum
Let's consider the derivation of the law of conservation of momentum based on Newton's second and third laws.

Relationship between work and change in kinetic energy
Rice. 3.3 Let a body of mass m move along the x axis under

Relationship between work and change in potential energy
Rice. 3.4 We will establish this connection using the example of the work of gravity

Law of conservation of mechanical energy
Let us consider a closed conservative system of bodies. This means that the bodies of the system are not acted upon by external forces, but internal forces are conservative by nature. Full mechanical

Collisions
Let's consider important occasion interactions of solid bodies - collisions. Collision (impact) is the phenomenon of a finite change in the velocities of solid bodies over very short periods of time when they are not

Law of conservation of angular momentum
Let us consider an isolated body, i.e. a body that is not acted upon by an external moment of force. Then Mdt = 0 and from (4.5) it follows d(Iw)=0, i.e. Iw=const. If an isolated system consists

Gyroscope
A gyroscope is a symmetrical solid body that rotates around an axis that coincides with the axis of symmetry of the body, passing through the center of mass, and corresponding to the greatest moment of inertia.

General characteristics of oscillatory processes. Harmonic vibrations
Oscillations are movements or processes that have varying degrees of repeatability over time. In technology, devices using oscillatory processes can perform op.

Oscillations of a spring pendulum
Rice. 6.1 Let us attach a body of mass m to the end of the spring, which can

Energy of harmonic vibration
Let us now consider, using the example of a spring pendulum, the processes of energy change in a harmonic oscillation. It is obvious that the total energy of the spring pendulum is W=Wk+Wp, where the kinetic

Addition of harmonic vibrations of the same direction
The solution to a number of issues, in particular, the addition of several oscillations of the same direction, is greatly facilitated if the oscillations are depicted graphically, in the form of vectors on a plane. The resulting

Damped oscillations
IN real conditions In systems that oscillate, resistance forces are always present. As a result, the system gradually expends its energy to perform work against resistance forces and

Forced vibrations
In real conditions, an oscillating system gradually loses energy to overcome frictional forces, so the oscillations are damped. For the oscillations to be undamped, it is necessary somehow

Elastic (mechanical) waves
The process of propagation of disturbances in a substance or field, accompanied by the transfer of energy, is called a wave. Elastic waves - the process of mechanical propagation in an elastic medium

Wave interference
Interference is the phenomenon of superposition of waves from two coherent sources, as a result of which a redistribution of wave intensity occurs in space, i.e. interference occurs

Standing waves
A special case of interference is the formation of standing waves. Standing waves arise from the interference of two counterpropagating coherent waves with the same amplitude. This situation may cause trouble

Doppler effect in acoustics
Sound waves are elastic waves with frequencies from 16 to 20,000 Hz, perceived by the human hearing organs. Sound waves in liquid and gaseous media are longitudinal. Into hard

Basic equation of the molecular kinetic theory of gases
Let us consider an ideal gas as the simplest physical model. An ideal gas is one for which the following conditions are met: 1) the dimensions of the molecules are so small that

Distribution of molecules by speed
Fig. 16.1 Let us assume that we were able to measure the speeds of all

Barometric formula
Let us consider the behavior of an ideal gas in a gravity field. As you know, as you rise from the surface of the Earth, the pressure of the atmosphere decreases. Let's find the dependence of atmospheric pressure on altitude

Boltzmann distribution
Let us express the gas pressure at heights h and h0 through the corresponding number of molecules per unit volume and u0, assuming that at different heights T = const: P =

The first law of thermodynamics and its application to isoprocesses
The first law of thermodynamics is a generalization of the law of conservation of energy taking into account thermal processes. Its formulation: the amount of heat imparted to the system is spent on doing work

Number of degrees of freedom. Internal energy of an ideal gas
The number of degrees of freedom is the number of independent coordinates that describe the movement of a body in space. A material point has three degrees of freedom, since when it moves in p

Adiabatic process
Adiabatic is a process that occurs without heat exchange with the environment. In an adiabatic process, dQ = 0, therefore the first law of thermodynamics in relation to this process is

Reversible and irreversible processes. Circular processes (cycles). Operating principle of a heat engine
Reversible processes are those that satisfy the following conditions. 1. After passing through these processes and returning the thermodynamic system to its original state in

Ideal Carnot heat engine
Rice. 25.1 In 1827, the French military engineer S. Carnot, re

Second law of thermodynamics
The first law of thermodynamics, which is a generalization of the law of conservation of energy taking into account thermal processes, does not indicate the direction of the flow various processes in nature. Yes, first

A process is impossible, the only result of which would be the transfer of heat from a cold body to a hot one
IN refrigeration machine heat is transferred from a cold body ( freezer) to a more heated one environment. This would seem to contradict the second law of thermodynamics. Really against it

Entropy
Let us now introduce a new parameter of the state of a thermodynamic system - entropy, which fundamentally differs from other state parameters in the direction of its change. Elementary treason

Discreteness of electric charge. Law of conservation of electric charge
The source of the electrostatic field is an electric charge - internal characteristic elementary particle, which determines its ability to enter into electromagnetic interactions.

Electrostatic field energy
Let's first find the energy of a charged flat capacitor. Obviously, this energy is numerically equal to the work that needs to be done to discharge the capacitor.

Main characteristics of current
Electric current is the ordered (directed) movement of charged particles. The current strength is numerically equal to the charge passing through cross section conductor per unit

Ohm's law for a homogeneous section of a chain
A section of the circuit that does not contain an EMF source is called homogeneous. Ohm experimentally established that the current strength in a homogeneous section of the circuit is proportional to the voltage and inversely proportional

Joule-Lenz law
Joule and, independently of him, Lenz experimentally established that the amount of heat released in a conductor with resistance R during time dt is proportional to the square of the current, resistive

Kirchhoff's rules
Rice. 39.1 For calculation of complex circuits direct current using

Contact potential difference
If two dissimilar metal conductors are brought into contact, then electrons are able to move from one conductor to another and back. The equilibrium state of such a system

Seebeck effect
Rice. 41.1 In a closed circuit of two dissimilar metals per g

Peltier effect
The second thermoelectric phenomenon - the Peltier effect - is that when passing electric current through the contact of two dissimilar conductors, release or absorption occurs in it

Derivation of the basic law of the dynamics of rotational motion. To the derivation of the basic equation of the dynamics of rotational motion. Dynamics of rotational motion of a material point. In projection onto the tangential direction, the equation of motion will take the form: Ft = mt.

15. Derivation of the basic law of the dynamics of rotational motion.

Rice. 8.5. To the derivation of the basic equation of the dynamics of rotational motion.

Dynamics of rotational motion of a material point.Consider a particle of mass m rotating around a current O along a circle of radius R , under the action of the resultant force F (see Fig. 8.5). IN inertial system counting is fair 2 Ouch Newton's law. Let's write it in relation to an arbitrary moment in time:

F = m·a.

The normal component of the force is not capable of causing rotation of the body, so we will consider only the action of its tangential component. In projection onto the tangential direction, the equation of motion will take the form:

F t = m·a t .

Since a t = e·R, then

F t = m e R (8.6)

Multiplying the left and right sides of the equation scalarly by R, we get:

F t R= m e R 2 (8.7)
M = Ie. (8.8)

Equation (8.8) represents 2 Ouch Newton's law (equation of dynamics) for the rotational motion of a material point. It can be given a vector character, taking into account that the presence of a torque causes the appearance of a parallel angular acceleration vector directed along the axis of rotation (see Fig. 8.5):

M = I·e. (8.9)

The basic law of the dynamics of a material point during rotational motion can be formulated as follows:

the product of the moment of inertia and angular acceleration is equal to the resulting moment of forces acting on a material point.

As well as other works that may interest you
3120.		Sets and operations on them	133 KB
	Sets and operations on them Write a program in which, for finite ordered sets, implement all the basic operations using a merge-type algorithm. It is permissible to organize sets as a list or as an array...
3121.		Writing a program that implements parallel operation of several processes	121.5 KB
	It is necessary to write a program that implements the parallel operation of several processes. Each process can consist of one or more threads. Any of the threads running as part of these processes can be suspended and restarted at some point...
3122.		Implementation of parallel operation of several processes using a software method	258 KB
	When writing the program, it turned out that the output functions (Write) available in Borland Pascal are not suitable, because in the case when several processes display information on the screen, it can happen
3123.		Payment cards: Business encyclopedia	115.64 MB
	Payment cards: Business encyclopedia The most important socio-political problem that is being solved today banking system Russia, -increasing the accessibility of financial services for citizens of the country. Banking activity related to...
3124.		Analytical calculation of cutting conditions during turning	42 KB
	Calculation of the cutting mode during turning using an analytical method. Purpose of the work: to study the methodology for calculating the cutting mode using an analytical method. Familiarize yourself with and acquire skills in working with reference literature. Task: On a 16K20 screw-cutting lathe...
3125.		Calculation of cutting conditions during milling	43 KB
	Calculation of cutting modes during milling Purpose of work: To study the methodology for assigning cutting modes using tables of standards. Familiarize yourself with and acquire skills in working with regulations. Task: On a horizontal milling machine 6R82G, produced...
3126.		Bar, public and private law enforcement agencies	93 KB
	Advocacy, public and private law enforcement agencies INTRODUCTION The Bar is a voluntary professional association of citizens that carries out defense in the manner prescribed by law during a preliminary investigation, inquiry, in criminal court...
3127.		Enterprise potential: formation and assessment	433 KB
	Theoretical part: Comparative approach to real estate valuation and its methods: analogue companies, transactions, industry coefficients. The concept of price multipliers and their types The comparative approach is effective if there is an active market with...
3128.		Analysis of solvent enterprises and development of financial rehabilitation methods	268.5 KB
	Introduction A financially stable business entity is one that, using its own funds, covers the funds invested in assets (fixed assets, intangible assets, working capital), does not allow unjustified receivables and credit...

Dynamics of rotational motion

Foundations and foundations are calculated according to 2 limit states

	According to bearing capacity:	N– given design load on the base in the most unfavorable combination; - load bearing capacity(ultimate load) foundations for a given load direction N; - coefficient of operating conditions of the base (<1); - коэффициент надежности (>1).
	According to limiting deformations:	- calculated absolute settlement of the foundation; - calculated relative difference in foundation settlement; , - limit values, respectively, of the absolute and relative difference in foundation settlement (SNiP 2.02.01-83*)

Dynamics of rotational motion

Preface

I draw students' attention to the fact that THIS material was not considered ABSOLUTELY at school (except for the concept of moment of force).

1. Law of dynamics of rotational motion

a. Law of dynamics of rotational motion

b. Moment of power

c. Moment of a couple of forces

d. Moment of inertia

2. Moments of inertia of some bodies:

a. Ring (thin-walled cylinder)

b. Thick wall cylinder

c. Solid cylinder

e. Thin rod

3. Steiner's theorem

4. Momentum of the body. Change in angular momentum of a body. Momentum impulse. Law of conservation of angular momentum

5. Rotary work

6. Kinetic energy of rotation

7. Comparison of quantities and laws for translational and rotational motion

1a. Let's consider a rigid body that can rotate around a fixed axis OO (Fig. 3.1). Let's break this solid body into separate elementary masses Δ m i. The resultant of all forces applied to Δ m i, denote by . It is enough to consider the case when the force lies in a plane perpendicular to the axis of rotation: components of forces parallel to the axis cannot influence the rotation of the body, since the axis is fixed. Then the equation of Newton’s second law for the tangential components of force and acceleration will be written as:

. (3.1)

The normal component of the force provides centripetal acceleration and does not affect angular acceleration. From (1.27): ,where is the radius of rotation i-that point. Then

. (3.2)

Let's multiply both sides (3.2) by:

notice, that

where α is the angle between the force vector and the radius vector of the point (Fig. 3.1), is the perpendicular lowered to the line of action of the force from the center of rotation (force arm). Let's introduce the concept of moment of force.

1b. A moment of power relative to the axis is a vector directed along the axis of rotation and related to the direction of the force by the gimlet rule, the module of which is equal to the product of the force by its arm: . Shoulder of power l relative to the axis of rotation - this is the shortest distance from the line of action of the force to the axis of rotation. Dimension of moment of force:

In vector form, the moment of force about a point:

The vector of the moment of force is perpendicular to both the force and the radius vector of the point of its application:

If the force vector is perpendicular to the axis, then the force moment vector is directed along the axis according to the right screw rule, and the magnitude of the force moment relative to this axis (projection onto the axis) is determined by formula (3.4):

The moment of force depends on both the magnitude of the force and the leverage of the force. If the force is parallel to the axis, then .

1c. Couple of forces - these are two forces equal in magnitude and opposite in direction, the lines of action of which do not coincide (Fig. 3.2). The arm of a force pair is the distance between the lines of action of the forces. Let's find the total moment of the pair of forces u () in projection onto the axis passing through point O:

That is, the moment of a pair of forces is equal to the product of the magnitude of the force by the plccho of the pair:

. (3.6)

Let's return to (3.3). Taking into account (3.4) and (3.6):

. (3.7)

1d. Definition: a scalar quantity equal to the product of the mass of a material point by the square of its distance to the axis is called moment of inertia of a material point relative to the axis OO:

Dimension of moment of inertia

The vectors and coincide in direction with the axis of rotation and are related to the direction of rotation according to the gimlet rule, therefore equality (3.9) can be rewritten in vector form:

. (3.10)

Let us sum (3.10) over all elementary masses into which the body is divided:

. (3.11)

Here it is taken into account that the angular acceleration of all points of a rigid body is the same, and it can be taken out of the sum sign. On the left side of the equality is the sum of the moments of all forces (both external and internal) applied to each point of the body. But according to Newton's third law, the forces with which the points of the body interact with each other (internal forces) are equal in magnitude and opposite in direction and lie on the same straight line, so their moments cancel each other out. Thus, on the left side of (3.11) the total moment remains only external forces: .

The sum of the products of elementary masses by the square of their distances from the axis of rotation is called moment of inertia of a rigid body relative to this axis:

. (3.12)

Thus, ; - this is the basic law of the dynamics of rotational motion of a rigid body (analogue of Newton’s second law): the angular acceleration of a body is directly proportional to the total moment of external forces and inversely proportional to the moment of inertia of the body :

. (3.13)

Moment of inertia Isolid body is a measure of the inert properties of a solid body during rotational motion and is similar to the mass of a body in Newton's second law. It significantly depends not only on the body mass, but also on its distribution relative to the axis of rotation (in the direction perpendicular to the axis).

In the case of continuous mass distribution, the sum in (3.12) is reduced to the integral over the entire volume of the body:

2a. The moment of inertia of a thin ring about an axis passing through its center perpendicular to the plane of the ring.

,

since for any element of the ring its distance to the axis is the same and equal to the radius of the ring: .

2b. Thick-walled cylinder (disc) with an inner radius and an outer radius.

Let us calculate the moment of inertia of a homogeneous disk with density ρ , height h, internal radius and external radius (Fig. 3.3) relative to the axis passing through the center of mass perpendicular to the plane of the disk. Let us divide the disk into thin rings of thickness and height so that the inner radius of the ring is equal to , and the outer radius is equal to . The volume of such a ring, where – area of ​​the base of the thin ring. Its mass:

Let us substitute into (3.14) and integrate over r():

Disk mass, then finally:

. (3.17)

2c. Solid cylinder (disc).

In the particular case of a solid disk or cylinder with a radius R let's substitute into (3.17) R 1 =0, R 2 =R and we get:

. (3.18)

Moment of inertia of a ball of radius R and the mass relative to the axis passing through its center (Fig. 3.4) is equal to (without proof):

2e. The moment of inertia of a thin rod of mass and length relative to an axis passing through its end perpendicular to the rod (Fig. 3.5).

Let us divide the rod into infinitesimal sections of length . The mass of such a section. Let's substitute in (3.14) and integrate from 0 to :

If the axis passes through the center of the rod perpendicular to it, you can calculate the moment of inertia of half the rod using (3.20) and then double it:

. (3.21)

3. If the axis of rotation doesn't work through the center of mass of the body (Fig. 3.6), calculations using formula (3.14) can be quite complex. In this case, the calculation of the moment of inertia is simplified by using Steiner's theorem : the moment of inertia of the body relative to an arbitrary axis is equal to the sum of the moment of inertia I c body relative to an axis passing through the center of mass of the body parallel to this axis, and the product of the body mass by the square of the distance between axes:

. (3.22)

Let's see how Steiner's theorem works if we apply it to a rod:

It is easy to verify that an identity is obtained, since in this case the distance between the axes is equal to half the length of the rod.

4. Momentum of the body. Change in angular momentum of a body. Momentum impulse. Law of conservation of angular momentum.

From the law of dynamics of rotational motion and the definition of angular acceleration it follows:

.

If, then. Let us introduce the angular momentum of a rigid body as

Relationship (3.24) is the basic law of rigid body dynamics for rotational motion. It can be rewritten like this:

and then this will be an analogue of Newton's second law for forward movement in pulse form (2.5)

Expression (3.24) can be integrated:

and formulate the law of change in angular momentum: the change in angular momentum of the body is equal to the impulse of the total moment of external forces . The quantity is called the impulse of the moment of force and is similar to the impulse of force in the formulation of Newton's second law for translational motion (2.2); angular momentum is analogous to momentum.

Dimension of angular momentum

The angular momentum of a rigid body relative to its axis of rotation is a vector directed along the axis of rotation according to the gimlet rule.

The angular momentum of a material point relative to point O (Fig. 3.6) is:

where is the radius vector of the material point, is its momentum. The angular momentum vector is directed according to the gimlet rule perpendicular to the plane in which the vectors and lie: in Fig. 3.7 - towards us due to the figure. Magnitude of angular momentum

Let us divide a rigid body rotating about an axis into elementary masses and sum up the angular momentum of each mass over the entire body (the same can be written in the form of an integral; this is not important):

.

Since the angular velocity of all points is the same and is directed along the axis of rotation, we can write it in vector form:

Thus, the equivalence of definitions (3.23) and (3.26) is proven.

If the total moment of external forces is zero, then the angular momentum of the system does not change(see 3.25):

. This is the law of conservation of angular momentum . This is possible when:

a) the system is closed (or );

b) external forces have no tangential components (the force vector passes through the axis/center of rotation);

c) external forces are parallel to the fixed axis of rotation.

Examples of the use/action of the law of conservation of angular momentum:

1. gyroscope;

2. Zhukovsky bench;

3. figure skater on ice.

5. Work in rotational motion.

Let the body rotate through an angle under the action of a force and the angle between the displacement and the force is equal to ; – radius vector of the point of application of the force (Fig. 3.8), then the work of the force is equal.

In an inertial reference frame, 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 relative to a given axis:

A simpler formulation can be given main law of rotational dynamics (it is also called Newton's second law for rotational motion) : torque is equal to the product of the moment of inertia and angular acceleration:

moment of impulse(angular momentum, angular momentum) of a body is called the product of its moment of inertia and angular velocity:

Momentum– vector quantity. Its direction coincides with the direction of the angular velocity vector.

The change in angular momentum is determined as follows:

. (I.112)

A change in angular momentum (with a constant moment of inertia of the body) can occur only as a result of a change in angular velocity and is always due to the action of a 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 momentum impulse or driving force. It is equal to the change in angular momentum.

Formula (I.113) is valid provided that the moment of force does not change over time. If the moment of force depends on time, i.e. , That

. (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: instantaneous torque is the first derivative of angular momentum with respect to time,

Expression (I.115) is another form basic equation (law ) dynamics of rotational motion of a rigid body relative to the fixed axis: the derivative of the angular momentum of a rigid body relative to an axis is equal to the moment of force relative to the same axis.

Question 15

Moment of inertia

The moment of inertia of a system (body) relative to a given axis is a physical quantity equal to the sum of the products of masses n material points of the system by the squares of their distance to the axis under consideration:

J=

The summation is performed over all elementary masses m(i) into which the body is divided

In the case of a continuous mass distribution, this sum reduces to the integral

where integration is carried out over the entire volume of the body. The value of z in this case is a function of the position of the point with coordinates x, y, z.

As an example, let us find the moment of inertia of a homogeneous solid cylinder of height h and radius R relative to its geometric axis. Let us divide the cylinder into separate hollow concentric cylinders of infinitesimal thickness dr with an internal radius r and an external radius r + dr. The moment of inertia of each hollow cylinder d,/ = r^2 dm (since dr≤r we assume that the distance of all points of the cylinder from the axis is equal to r), where dm is the mass of the entire elementary cylinder; its volume is 2 πr hrd r. If p is the density of the material, then dm = 2πhpr^3d r. Then the moment of inertia of a solid cylinder

but since πR^3h is the volume of the cylinder, then its mass m= πR^2hp, and the moment of inertia

Steiner's theorem

The moment of inertia of a body J relative to an arbitrary axis is equal to its moment of inertia relative to a parallel axis passing through the center of mass C of the body, added to the product of the body mass and the square of the distance a between the axes:

J= +ma^2

1. Moment of inertia of a homogeneous straight thin cylindrical rod length and mass relative to an axis passing through its middle and perpendicular to its length:

2. Moment of inertia of a homogeneous solid cylinder(or disk) radius and mass relative to the axis of symmetry perpendicular to its plane and passing through its center:

3. Moment of inertia of the cylinder radius, mass and height relative to an axis perpendicular to its height and passing through its middle:

4. Moment of inertia of the ball(thin-walled sphere) radius and mass relative to its diameter (or axis passing through the center of the sphere):

5. Moment of inertia of the rod length and mass relative to an axis passing through one of its ends and perpendicular to its length:

6. Moment of inertia of a hollow thin-walled cylinder radius and mass, relative to the cylinder axis:

7. Moment of inertia of a cylinder with a hole(wheel, clutch):

,

where and are the radii of the cylinder and the hole in it. The angular momentum is also constant for open systems if the resulting moment of external forces applied to the system is zero.

A gyroscope (example: spinning top) is a symmetrical body rotating around its axis at high speed.

The angular momentum of the gyroscope coincides with its axis of rotation.

Electric charge is a measure of the participation of bodies in electromagnetic interactions.

There are two kinds electric charges, conventionally called positive and negative.

Coulomb's Law:

.

Electric field is a special form of matter through which interaction between charged particles occurs.

Tension electric field– vector physical quantity. The direction of the tension vector coincides at each point in space with the direction of the force acting on the positive test charge.

Power lines Coulomb fields of positive and negative point charges: