Topic 3. Elements of solid body mechanics.
Lecture No. 5.
Kinematic relations
Determination of moment of force.
Moment of inertia, moment of momentum of a rigid body.
Kinematic relations.
A solid body can be considered as a system of material points rigidly fastened to each other. The nature of its movement may be different.
Mainly distinguish translational and rotational movements .
At progressive In motion, all points of the body move along parallel trajectories, so to describe the motion of the body as a whole, it is enough to know the law of motion of one point. In particular, the center of mass of a rigid body can serve as such a point
At rotational(more complex!) In motion, all points of the body describe concentric circles, the centers of which lie on the same axis. The velocities of points on any circle are related to the radii of these circles and the angular velocity
rotation: 
. Since a rigid body retains its shape during rotation, the radii of rotation remain constant and the linear acceleration will be equal to:
. (1)
Determination of moment of force.
To describe the dynamics of the rotational motion of a rigid body, it is necessary to introduce the concepts of moments of force.
Definition 1.
moment - strength – , applied to a material point T. A, relative to an arbitrary point T. ABOUT , drawn from the point T. ABOUT to the point T. A:
Note.
The modulus of the vector product, that is, the actual magnitude of the moment, is determined by the product - 
,
and the direction of the moment is given by the definition of the right triple of vectors.
Definition 2.
moment– strength – , applied at point t.A, relative to an arbitrary axis is called the cross product of the radius vector and force component , lying in a plane perpendicular to the axis and passing through the point T. A:
.
Basic equation for the dynamics of rotational motion.
Let there be a rigid body of arbitrary shape that can rotate around an axis OO. Breaking the body into small elements, you can see that they all rotate around an axis OO in planes perpendicular to the axis of rotation with the same angular velocity w.
The movement of each of the individual elements of small mass m i described by Newton's second law.
For i th element we have:
Where f ik (k = 1,2, ...N) represent the internal forces of interaction of all
Items with selected and F i- the resultant of all external forces acting on i- element.
Speed v i each element, generally speaking, can change as desired, but since the body is solid, the displacement of points in the direction of the radii of rotation need not be considered. Therefore, we project equation (1) onto the direction of the tangent to the circle of rotation and multiply both sides of the equation by r i:
On the right side of the resulting equation, products of the type represent the moments of internal forces relative to the axis of rotation, since r i And f it mutually perpendicular. Similarly, the products are the moments of external forces acting on i-element.
Let us sum up in the equation of motion over all the elements into which the body was divided.
The sum of the moments of internal forces can be divided into pairs of terms, which owe their origin to the interaction of two symmetrical elements of the body with each other. Their moments are equal and oppositely directed. Based on this, we can conclude that when adding up all the moments of internal forces, they will be destroyed in pairs. Let us denote the total moment of all external forces S M i, Where M i = [ r i × F i ].
The left side of equation (2), taking into account relation (1) in the previous section, is presented as follows:
=
= 
, (3)
where is the moment of inertia.
Equation (3) is basic equation of rotational motion.
4.Moment of inertia of a rigid body.
Definition 1.
Magnitude is called the moment of inertia of a rigid body about a given axis.
Dynamics of rotational motion of a rigid body. Basic equation for the dynamics of rotational motion. Moment of inertia of a rigid body about an axis. Steiner's theorem. Moment of impulse. Moment of power. Law of conservation and change of angular momentum.
In the last lesson we discussed impulse and energy. Let's consider the magnitude of angular momentum - it characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed relative to the axis of rotation, and at what speed the rotation occurs. Let's consider particle A. r is the radius vector characterizing the position relative to some point O, the chosen reference system. P-pulse in this system. Vector quantity L is the angular momentum of particle A relative to point O: Module of vector L: where α is the angle between r and p, l=r sin α arm of vector p relative to point O.
Let's consider the change in vector L with time: = because dr/dt =v, v is directed in the same way as p, since dp/dt=F is the resultant of all forces. Then: Moment of force: M= Modulus of the moment of force: where l is the arm of the vector F relative to point O Equation of moments: the time derivative of the moment of momentum L of the particle relative to some point O is equal to the moment M of the resultant force F relative to the same point O: If M = 0, then L=const – if the moment of the resultant force is equal to 0 during the period of time of interest, then the momentum of the particle remains constant during this time.
The moment equation allows you to: Find the moment of force M relative to point O at any time t if the time dependence of the angular momentum L(t) of the particle is known relative to the same point; Determine the increment of the angular momentum of a particle relative to point O for any period of time, if the time dependence of the moment of force M(t) acting on this particle (relative to the same point O) is known. We use the equation of moments and write down the elementary increment of the vector L: Then, by integrating the expression, we find the increment of L for a finite period of time t: the right side is the momentum of the moment of force. The increment in the angular momentum of a particle over any period of time is equal to the angular momentum of the force over the same time.
Moment of impulse and moment of force about the axis Let's take the z axis. Let's choose point O. L is the angular momentum of particle A relative to the point, M is the moment of force. The angular momentum and the moment of force relative to the z axis are the projection of the vectors L and M onto this axis. They are denoted by Lz and Mz - they do not depend on the selection point O. The time derivative of the angular momentum of the particle relative to the z axis is equal to the moment of force relative to this axis. In particular: Mz=0 Lz=0. If the moment of force relative to some moving axis z is equal to zero, then the angular momentum of the particle relative to this axis remains constant, while the vector L itself can change.
Law of conservation of angular momentum Let's choose an arbitrary system of particles. The angular momentum of a given system will be the vector sum of the angular momentum of its individual particles: Vectors are defined relative to the same axis. The angular momentum is an additive value: the angular momentum of a system is equal to the sum of the angular impulses of its individual parts, regardless of whether they interact with each other or not. Let's find the change in angular momentum: - the total moment of all internal forces relative to point O.; - the total moment of all external forces relative to point O. The time derivative of the angular momentum of the system is equal to the total moment of all external forces! (using Newton's 3rd law):
The angular momentum of a system can change under the influence only of the total moment of all external forces. Law of conservation of momentum: the angular momentum of a closed system of particles remains constant, that is, it does not change with time. : Valid for angular momentum taken relative to any point in the inertial reference system. There may be changes within the system, but the increase in the angular momentum of one part of the system is equal to the decrease in the angular momentum of its other part. The law of conservation of angular momentum is not a consequence of Newton’s 3rd law, but represents an independent general principle; one of the fundamental laws of nature. The law of conservation of angular momentum is a manifestation of the isotropy of space with respect to rotation.
Dynamics of a rigid body Two main types of motion of a rigid body: Translational: all points of the body receive movement equal in magnitude and direction over the same period of time. Specify the movement of one point Rotational: all points of a rigid body move in circles, the centers of which lie on the same straight line, called the axis of rotation. Set the axis of rotation and angular velocity at each moment of time. Any movement of a rigid body can be represented as the sum of these two movements!
Arbitrary movement of a rigid body from position 1 to position 2 can be represented as the sum of two movements: translational movement from position 1 to position 1’ or 1’’ and rotation around the O’ axis or O’ axis. Elementary movement ds: - “translational” - “rotational” Speed of a point: - the same speed of translational motion for all points of the body - the speed associated with the rotation of the body is different for different points of the body
Let the reference frame be stationary. Then the movement can be considered as a rotational movement with angular velocity w in a reference system moving relative to a stationary system translationally with a speed v 0. Linear speed v' due to the rotation of a rigid body: The speed of a point in complex motion: There are points that, with vector multiplication of vectors r and w give the vector v 0. These points lie on the same straight line and form the instantaneous axis of rotation.
The motion of a rigid body in the general case is determined by two vector equations: The equation of motion of the center of mass: The equation of moments: The laws of acting external forces, the points of their application and the initial conditions, the speed and position of each point of the rigid body at any time. The points of application of external forces can be moved along the direction of action of the forces. Resultant force is a force that is equal to the resultant forces F acting on a rigid body and creates a moment equal to the total moment M of all external forces. The case of a gravity field: the resultant of gravity passes through the center of mass. Force acting on a particle: The total moment of gravity relative to any point:
Conditions for the equilibrium of a rigid body: a body will remain at rest if there are no reasons causing its movement. According to the two basic equations of body motion, this requires two conditions: The resultant external forces is equal to zero: The sum of the moments of all external forces acting on the body relative to any point must be equal to zero: If the system is non-inertial, then in addition to external forces it is necessary to take into account inertial forces (forces , caused by the accelerated motion of the non-inertial reference system relative to the inertial reference system). Three cases of motion of a rigid body: Rotation around a fixed axis Plane motion Rotation around free axes
Rotation around a fixed axis Momentum of momentum of a solid body relative to the axis of rotation OO’: where mi and pi are the mass and distance from the axis of rotation of the i-th particle of the solid body, wz is its angular velocity. Let us introduce the notation: where I is the moment of inertia of a solid body relative to the OO’ axis: The moment of inertia of a body is found as: where dm and dv are the mass and volume of an element of the body located at a distance r from the z axis of interest to us; ρ is the density of the body at a given point.
Moments of inertia of homogeneous solid bodies relative to an axis passing through the center of mass: Steiner's theorem: the moment of inertia I relative to an arbitrary axis z is equal to the moment of inertia Ic relative to the axis Ic parallel to the given one and passing through the center of mass C of the body, plus the product of the mass m of the body by the square of the distance a between axes:
Equation of the dynamics of rotation of a rigid body: where Mz is the total moment of all external forces relative to the axis of rotation. The moment of inertia I determines the inertial properties of a rigid body during rotation: for the same value of the moment of force Mz, a body with a large moment of inertia acquires a smaller angular acceleration βz. Mz also includes moments of inertia forces. Kinetic energy of a rotating rigid body (the axis of rotation is stationary): let the speed of a particle of a rotating rigid body be – Then: where I is the moment of inertia relative to the axis of rotation, w is its angular velocity. The work of external forces during rotation of a rigid body around a fixed axis is determined by the action of the moment Mz of these forces relative to this axis.
Plane motion of a rigid body In plane motion, the center of mass of a rigid body moves in a certain plane, stationary in a given reference frame K, and the vector of its angular velocity w is perpendicular to this plane. The movement is described by two equations: where m is the mass of the body, F is the result of all external forces, Ic and Mcz are the moment of inertia and the total moment of all external forces, both relative to the axis passing through the center of the body. The kinetic energy of a rigid body in plane motion consists of the energy of rotation in the system around an axis passing the center of mass, the energy associated with the movement of the center of mass: where Ic is the moment of inertia relative to the axis of rotation (through the CM), w is the angular velocity of the body, m is its mass , Vc – velocity of the center of mass of the body in the reference system K.
Rotation around free axes The axis of rotation, the direction of which in space remains unchanged without any external forces acting on it, is called the free axis of rotation of the body. The main axes of a body are three mutually perpendicular axes passing through its center of mass, which can serve as free axes. To hold the axis of rotation in a constant direction, it is necessary to apply a moment M of some external forces F to it: If the angle is 90 degrees, then L coincides in direction with w, i.e. M = 0! - the direction of the axis of rotation will remain unchanged without external influence When a body rotates around any main axis, the angular momentum vector L coincides in direction with the angular velocity w: where I is the moment of inertia of the body relative to a given axis.
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Light wave. Interference of light waves.
Light - in physical optics, electromagnetic radiation perceived by the human eye. The region with wavelengths in vacuum of 380-400 nm (750-790 THz) is taken as the short-wave boundary of the spectral range occupied by light, and the region 760-780 nm (385-395 THz) is taken as the long-wave boundary. In the broad sense used outside physical optics, often called light
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1. Kinematics of rotational motion. Relationship between vectors v and ω.
The rotational motion of an absolutely rigid body around a fixed axis is such a movement in which all points of the body move in planes perpendicular to a fixed straight line, called the axis of rotation, and describe circles whose centers lie on this axis. The angular velocity of rotation is a vector that is numerically equal to the first derivative of the angle of rotation of the body with respect to time and directed along the axis of rotation according to the right-hand screw rule:
The unit of angular velocity is radians per second (rad/s).
So the vector ω
determines the direction and speed of rotation. If ω=const, then the rotation is called uniform.
Angular velocity can be related to linear velocity υ
arbitrary point A. Let it take time Δt a point passes along an arc of a circle the length of the path Δs. Then the linear speed of the point will be equal to:
/////////////
With uniform rotation, it can be characterized by the rotation period T– the time during which a point of the body makes one full revolution, i.e. rotates through an angle of 2π:
/////////////////
The number of complete revolutions made by a body during uniform circular motion per unit time is called the rotation frequency:
….....................
Where
To characterize the uneven rotation of a body, the concept of angular acceleration is introduced. Angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect to time:
////////////////////////(1.20)
Let us express the tangential and normal components of the acceleration of a point A of a rotating body through angular velocity and angular acceleration:
////////////////(1.21)
/////////////////(1.22)
In the case of uniform motion of a point along a circle ( ε=const):
////////////////////////////
Where ω0
- initial angular velocity. Translational and rotational motions of a rigid body are only the simplest types of its motion. In general, the motion of a rigid body can be very complex. However, in theoretical mechanics it is proven that any complex motion of a rigid body can be represented as a combination of translational and rotational motions.
The kinematic equations of translational and rotational motions are summarized in table. 1.1 .
Table 1.1
2. Maxwell's equations. 06
The first pair of Maxwell's equations is formed by
The first of these equations connects the values of E with temporary changes in the vector B and is essentially an expression of the law of electromagnetic induction. The second equation reflects the property of vector B that its lines are closed (or go to infinity)
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Job. Power.
Work is a scalar quantity equal to the product of the projection of force on the direction of movement and path s traversed by the force application point A fs cos (1.53)If the force and direction of movement form an acute angle (cosα>0), the work is positive. If angle α is obtuse (cosα<0),работа отрицательна. При α = π/2 работаравна нулю
The scalar product of two vectors is equal to:AB AB cos.The expression for work (1.54) can be written as a scalar product
Where by Δs we mean the vector of elementary displacement, which we previously denoted by Δr. s v t ////////////
Power W is a quantity equal to the work ratio ΔA to a period of time Δt for which it is committed: ////////////////////////
If the work changes over time, then the instantaneous power value is entered ///////////
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Maxwell's equations.
2. Fresnel diffraction from the simplest obstacles.
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In a state of balance
force mg is balanced by elastic force kΔ l0:
mg kl 0 (1.129)
0 f mg k(l x)
f kx(1.130)
Forces of this type are accepted
Call them quasi-elastic
Amplitude of oscillation.
The value in parentheses under the sign
The initial phase of oscillation.
time period T during which the phase
oscillations receive an increment equal to 2π
Cyclic frequency.
0 2 (1.139)
Harmonic energy
Oscillations
Having differentiated (1.135) with respect to time,
Same as average
meaning Ep and equal E/ 2.
The current is inductive.
The magnitude of the induction current is determined
only by the rate of change of Φ, i.e. the value
derivative dΦ/ d t. When changing sign
Current.
The phenomenon of electromagnetic
Induction.
Lenz's philosophy states that the induced current is always
Calling him.
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Zero
Dividing this expression by L and replacing through
(2.188);
Replacing ω0 using formula (2.188), we obtain
Free fading
Oscillations.
The equation of oscillations can be obtained based on the fact that
has the form:
Where ….
Substituting the value (2.188) for ω0 and (2.196) for β,
We find that
Dividing (2.198) by capacity WITH, we get the voltage
on the capacitor:
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The Lorentz force is equal to
So the movement
Circle radius, by
Which rotates
Determined by the formula
(2.184) with replacement v on v = v
Spiral pitch l can be found
multiplying v║ to the determined
Formula (2.185) period
appeals T:
…............
2. Polarization with birefringence. Birefringence is the effect of splitting a light beam into two components in anisotropic media. It was first discovered by Danish scientist Rasmus Bartholin on a crystal of Iceland spar. If a ray of light falls perpendicular to the surface of the crystal, then on this surface it is split into two rays. The first ray continues to propagate straight and is called ordinary ( o- ordinary), the second one deviates to the side and is called extraordinary ( e- extraordinary). The direction of oscillation of the electric field vector of the extraordinary beam lies in the plane of the main section (the plane passing through the beam and the optical axis of the crystal). The optical axis of a crystal is the direction in an optically anisotropic crystal along which a beam of light propagates without experiencing birefringence.
Violation of the law of refraction of light by an extraordinary ray is due to the fact that the speed of propagation of light (and therefore the refractive index) of waves with such polarization as that of an extraordinary ray depends on the direction. For an ordinary wave, the speed of propagation is the same in all directions.
It is possible to select conditions under which ordinary and extraordinary rays propagate along the same trajectory, but at different speeds. Then the effect of polarization change is observed. For example, linearly polarized light incident on a plate can be represented as two components (ordinary and extraordinary waves) moving at different speeds. Due to the difference in the speeds of these two components, there will be some phase difference between them at the exit from the crystal, and depending on this difference, the light at the output will have different polarizations. If the thickness of the plate is such that at the exit from it one ray lags behind the other by a quarter of a wave (quarter of a period), then the polarization will turn into circular (such a plate is called a quarter-wave), if one ray lags behind the other by half a wave, then the light will remain linearly polarized , but the plane of polarization will rotate by a certain angle, the value of which depends on the angle between the plane of polarization of the incident beam and the plane of the main section (such a plate is called half-wave). Qualitatively, the phenomenon can be explained as follows. From Maxwell's equations for a material medium it follows that the phase speed of light in the medium is inversely proportional to the value of the dielectric constant ε of the medium. In some crystals, the dielectric constant - a tensor quantity - depends on the direction of the electric vector, that is, on the polarization state of the wave, therefore the phase velocity of the wave will depend on its polarization. According to the classical theory of light, the occurrence of the effect is due to the fact that the alternating electromagnetic field of light causes the electrons of a substance to oscillate, and these vibrations affect the propagation of light in the medium, and in some substances it is easier to make electrons oscillate in some certain directions. Artificial birefringence. In addition to crystals, birefringence is also observed in visotropic media placed in an electric field (Kerr effect), in a magnetic field (Cotton-Mouton effect, Faraday effect), under the influence of mechanical stress (photoelasticity). Under the influence of these factors, an initially isotropic medium changes its properties and becomes anisotropic. In these cases, the optical axis of the medium coincides with the direction of the electric field, magnetic field, and the direction of application of force. Negative crystals are uniaxial crystals in which the speed of propagation of an ordinary ray of light is less than the speed of propagation of an extraordinary ray. In crystallography, Negative crystals are also called liquid inclusions in crystals that have the same shape as the crystal itself. Positive crystals are uniaxial crystals in which the speed of propagation of an ordinary ray of light is greater than the speed of propagation of an extraordinary ray.
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Dipole radiation.06
Called elementary
Dipole electric
The moment of such a system is equal to
p ql cos tn p m cos t, (2.228)
Where l– double amplitude
Lined along the dipole axis,
p m= ql n
The wave front in the so-called wave zone, i.e.
Addiction
Wave intensity from
angle θ is depicted with
Using a diagram
Dipole directivity
(Fig. 246).
Energy emitted in all directions in
Radiation.
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This point.
Negative
Dipole axis.
Let's find the voltage
Field size on the axis
Dipole, as well as on
Straight, passing-
Cabbage soup through the center
Dipoles and perpendicular
Dicular to him
axis (Fig. 4).
Point position
We will characterize
Keep them at a distance
eat r from the center of the diplomatic
la. Let us remind you that
r >> l.
On the dipole axis, the vectors E+ and E– have opposite
Follows that
….........
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Energy
Physical quantity characterizing
Speed and
secondly, the presence of the body in
Potential field of forces.
The first type of energy is called
Vector v.
Multiplying by m numerator and denominator,
equation (1.65) can be rewritten as:
Kinetic energy
…..........
A T 2T1(1.67)
Potential energy
Bodies forming a system
…...........
Law of energy conservation
E E 2 E 1 A n. k. (1.72)
For a system from N bodies between which
Line of tension.
Tension vector flow
The density of the lines is chosen so that the number
Vector E.
Lines E of a point charge represent
radial straight lines.
Therefore, the total number of lines N equals
If the site dS oriented so that the normal to
forms an angle α with the vector E, then the quantity
Normals to the site
numerically equal
…..........
where the expression for Ф is called the flow of the vector E
In those places where vector E
The volume covered by the surface
ness), En and correspondingly d F
will be negative (Fig. 10)
Gauss's theorem
It can be shown that, as for the spherical
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Changes.
Inertial systems
Countdown
The reference system in which
Non-inertial.
An example of an inertial system
Inertial
Group velocity is a quantity that characterizes the speed of propagation of a “group of waves” - that is, a more or less well-localized quasi-monochromatic wave (waves with a fairly narrow spectrum). The group velocity in many important cases determines the speed of energy and information transfer by a quasi-sinusoidal wave (although this statement in the general case requires serious clarifications and reservations).
The group velocity is determined by the dynamics of the physical system in which the wave propagates (a specific medium, a specific field, etc.). In most cases, the linearity of this system (exactly or approximately) is assumed.
For one-dimensional waves, the group velocity is calculated from the dispersion law:
,
Where - angular frequency, - wave number.
The group velocity of waves in space (for example, three-dimensional or two-dimensional) is determined by the frequency gradient along the wave vector :
Note: the group velocity generally depends on the wave vector (in the one-dimensional case, on the wave number), that is, generally speaking, it is different for different values and for different directions of the wave vector.
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Work of forces
Electrostatic field
….......
…........
…........
we took into account that
….....
From here, for work on path 1–2, we get
Therefore, the forces acting on the charge q" V
stationary charge field q, are
potential.
Where El– projection of vector E onto the direction
elementary movement d l
Circulation along the circuit.
Thus, for electrostatic
Potential.
For different trial values q′ attitude
Wp/qpr will be constant
vedicina φ ─ called field potential
Electric fields
From 225 and 226 we get
Taking (2.23) into account, we obtain
….......
For potential charge energy q′ in field
Separateness
From 226 it follows that
Wednesdays
Homogeneous substance
Examples of turbid media:
– smoke (tiny solid particles in gas)
– fog (drops of liquid in air, gas)
– cell suspension
– emulsion (dispersed system consisting of
Other types of energy
Absorbing substance
….......
…........
….....
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Newton's second law.02
Bodies.
The connection between tensions
Direction r is equal to
You can write
Along the tangent to
surface τ by the amount dτ
The potential won't change, so
that φ/τ = 0. But φ/τ is equal
The cial surface will be
Match the direction
Same point.
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Capacitors
The capacitance of a capacitor is understood as the physical
quantity proportional to charge q and back
Connection of capacitors
With a parallel connection (Fig. 50) on each of
Voltage
Coverings.
Therefore, the voltage at each
capacitors:
Kirchhoff's law.
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Can be given a different look
…..............
Vector quantity
p m v(1.44)
Law of conservation of momentum
The impulse of the system p is called
Forming a system
…....................
The center of gravity of the system.
The speed of the center of inertia is obtained
by differentiating r With By
time:
.................
Considering that mi vi is pi and Σрi gives
the impulse of the system p can be written
p m v c(1.50)
Thus, the momentum of the system is equal to
Each of the internal forces
According to the third law
Newton can be written f ij
= – f ji
Symbol F i designated
Resultant external
force acting on a body i
Equation (1.45)
…......
….........
…..........
Zero, as a result
P is constant
Permanent
p m v c(1.50)
Energy of the charge system.02
Consider a system of two point charges q 1 and q 2,
located at a distance r 12.
Charge transfer work q 1 from infinity to point,
away from q 2 on r 12 is equal to:
Where φ 1 – potential created by charge q 2 in that
the point to which the charge moves q 1
Similarly for the second charge we get:
…........
Equal to the energy of three charges
…...............
….....................
where φ1 is the potential created by the charges q 2 and q 3 in that
point where the charge is located q 1, etc.
By adding charges to the system sequentially
q4, q 5, etc., you can make sure that in
case N charges potential energy
The system is equal
Where φi– potential created at that point,
where is qi, all charges except i th.
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Force
Expression (2.147) coincides with (2.104), if we put
k = 1. Therefore, in SI Ampere’s law has the form
df id lB (2.148)
df iB dl sin (2.149)
Lorentz force
According to (2.148) per current element d l operates in
magnetic field strength
df id lB (2.150)
Replacing id l through S j dl[cm. (2.111)], expression of the law
Ampere can be given the form
df SDLjB jB dV
Where dV– volume of the conductor to which it is attached
force d f.
By dividing d f on dV, we get the “force density”, i.e.
force acting on a unit volume of a conductor:
f units about jB (2.151)
Let's find that
fed. about ne"uB
This force is equal to the sum of the forces applied to the carriers
per unit volume. Such carriers n, investigator-
It is important to note that the law only talks about the total energy emitted. The energy distribution over the emission spectrum is described by Planck's formula, according to which there is a single maximum in the spectrum, the position of which is determined by Wien's law.
Wien's displacement law gives the dependence of the wavelength at which the radiation flux of a black body's energy reaches its maximum on the temperature of the black body. λmax = b/T≈ 0.002898 mK × T−1(K),
Where T is the temperature, and λmax is the wavelength with maximum intensity. Coefficient b, called Wien's constant, in the SI system has a value of 0.002898 m K.
For light frequency (in hertz) Wien's displacement law is:
α ≈ 2.821439… is a constant value (the root of the equation 
),
k - Boltzmann constant,
h - Planck's constant,
T - temperature (in kelvins).
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Newton's third law.
Direction.
f12 f21 (1.42)
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Planck's formula.
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Joule-Lenz law.
Photo effect.
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Compton effect.
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Basic equation for the dynamics of rotational motion.
This is the basic equation for the dynamics of the rotational motion of a body: the angular acceleration of a rotating body is directly proportional to the sum of the moments of all forces acting on it relative to the axis of rotation of the body and inversely proportional to the moment of inertia of the body relative to this axis of rotation. The resulting equation is similar in form to the expression of Newton's second law for the translational motion of a body.
Newton's second law for rotational motion By definition, angular acceleration and then this equation can be rewritten as follows, taking into account (5.9) or
This expression is called the basic equation of the dynamics of rotational motion and is formulated as follows: the change in the angular momentum of a rigid body is equal to the angular momentum of all external forces acting on this body.
Let us remind you that basic workdAstrengthFcalled the scalar product of forceFfor infinitesimal displacementdl:where is the angle between the direction of force and the direction of movement.
Note that the normal component of the force F n(unlike tangential F τ ) and ground reaction force N no work is done, since they are perpendicular to the direction of movement.
Element dl=rd at small angles of rotation d (r – radius vector of the body element). Then the work of this force is written as follows:
. (19)
The expression Fr cos is the moment of force (the product of force F by the arm p=r cos):
(20)
Then the work is equal
. (21)
This work is spent on changing the kinetic energy of rotation:
. (22)
If I=const, then after differentiating the right side we get:
or, since 
, (23)
Where 
- angular acceleration.
Expression (23) is equation of the dynamics of the rotational motion of a rigid body relative to a fixed axis, which is better represented from the point of view of cause-and-effect relationships as:
. (24)
The angular acceleration of a body is determined by the algebraic sum of the moments of external forces relative to the axis of rotation divided by the moment of inertia of the body relative to this axis.
Let's compare the basic quantities and equations that determine the rotation of a body around a fixed axis and its translational motion (see Table 1):
Table 1
|
Forward movement |
Rotational movement |
|
Moment of inertia I |
|
|
Speed |
Angular velocity |
|
Acceleration |
Angular acceleration |
|
Force |
Moment of power |
|
Basic equation of dynamics: |
Basic equation of dynamics: |
|
Job |
Job |
|
Kinetic energy |
Kinetic energy |
or 
The dynamics of the translational motion of a rigid body is completely determined by force and mass as a measure of their inertia. In the rotational motion of a rigid body, the dynamics of motion are determined not by force as such, but by its moment; inertia is determined not by mass, but by its distribution relative to the axis of rotation. The body does not acquire angular acceleration if a force is applied, but its moment will be zero.
Method of doing the work
A schematic diagram of the laboratory setup is shown in Fig. 6. It consists of a disk with mass m d, four rods with masses m 2 attached to it, and four weights with masses m 1, located symmetrically on the rods. A thread is wound around a disk, from which a load of mass m is suspended.
According to Newton’s second law, let’s create an equation for the translational motion of a load m without taking into account friction forces:
|
|
(25)or in scalar form, i.e. in projections onto the direction of movement:
. (26)
, (27)
where T is the tension force of the thread. According to the basic equation of the dynamics of rotational motion (24), the moment of force T, under the influence of which the system of bodies m d, m 1, m 2 performs rotational motion, is equal to the product of the moment of inertia I of this system and its angular acceleration :
or 
, (28)
where R is the arm of this force equal to the radius of the disk.
Let us express the tension force of the thread from (28):
(29)
and equate the right-hand sides of (27) and (29):
. (30)
Linear acceleration is related to the angular acceleration by the following relation a=R, therefore:
. (31)
Where does the acceleration of the load m without taking into account the friction forces in the block equal to:
. (32)
Let us consider the dynamics of the system's motion, taking into account the friction forces that act in the system. They occur between the rod on which the disk is attached and the stationary part of the installation (inside the bearings), as well as between the moving part of the installation and the air. We will take into account all these friction forces using the moment of friction forces.
Taking into account moment of friction forces The rotation dynamics equation is written as follows:
, (33)
where a’ is linear acceleration under the action of friction forces, Mtr is the moment of friction forces.
Subtracting equation (33) from equation (28), we obtain:
,
. (34)
Acceleration without taking into account the friction force (a) can be calculated using formula (32). The acceleration of the weight, taking into account friction forces, can be calculated from the formula for uniformly accelerated motion, measuring the distance traveled S and time t:
. (35)
Knowing the values of accelerations (a and a’), using formula (34) we can determine the moment of friction forces. For calculations, it is necessary to know the magnitude of the moment of inertia of the system of rotating bodies, which will be equal to the sum of the moments of inertia of the disk, rods and loads.
The moment of inertia of the disk according to (14) is equal to:
. (36)
The moment of inertia of each of the rods (Fig. 6) relative to the O axis according to (16) and Steiner’s theorem is equal to:
where a c =l/2+R, R is the distance from the center of mass of the rod to the axis of rotation O; l is the length of the rod; I oc is its moment of inertia relative to the axis passing through the center of mass.
The moments of inertia of the loads are calculated in the same way:
, (38)
where h is the distance from the center of mass of the load to the axis of rotation O; d – load length; I 0 r is the moment of inertia of the load relative to the axis passing through its center of mass. By adding up the moments of inertia of all bodies, we obtain a formula for calculating the moment of inertia of the entire system.
This article describes an important section of physics - “Kinematics and dynamics of rotational motion”.
Basic concepts of kinematics of rotational motion
Rotational motion of a material point around a fixed axis is called such motion, the trajectory of which is a circle located in a plane perpendicular to the axis, and its center lies on the axis of rotation.
Rotational motion of a rigid body is a motion in which all points of the body move along concentric (the centers of which lie on the same axis) circles in accordance with the rule for the rotational motion of a material point.
Let an arbitrary rigid body T rotate around the O axis, which is perpendicular to the plane of the drawing. Let us select point M on this body. When rotated, this point will describe a circle with radius around the O axis r.
After some time, the radius will rotate relative to its original position by an angle Δφ.
The direction of the right screw (clockwise) is taken as the positive direction of rotation. The change in the angle of rotation over time is called the equation of rotational motion of a rigid body:
φ = φ(t).
If φ is measured in radians (1 rad is the angle corresponding to an arc of length equal to its radius), then the length of the circular arc ΔS, which the material point M will pass in time Δt, is equal to:
ΔS = Δφr.
Basic elements of the kinematics of uniform rotational motion
A measure of the movement of a material point over a short period of time dt serves as an elementary rotation vector dφ.
The angular velocity of a material point or body is a physical quantity that is determined by the ratio of the vector of an elementary rotation to the duration of this rotation. The direction of the vector can be determined by the rule of the right screw along the O axis. In scalar form:
ω = dφ/dt.
If ω = dφ/dt = const, then such motion is called uniform rotational motion. With it, the angular velocity is determined by the formula
ω = φ/t.
According to the preliminary formula, the dimension of angular velocity
[ω] = 1 rad/s.
The uniform rotational motion of a body can be described by the period of rotation. The period of rotation T is a physical quantity that determines the time during which a body makes one full revolution around the axis of rotation ([T] = 1 s). If in the formula for angular velocity we take t = T, φ = 2 π (one full revolution of radius r), then
ω = 2π/T,
Therefore, we define the rotation period as follows:
T = 2π/ω.
The number of revolutions that a body makes per unit time is called the rotation frequency ν, which is equal to:
ν = 1/T.
Frequency units: [ν]= 1/s = 1 s -1 = 1 Hz.
Comparing the formulas for angular velocity and rotation frequency, we obtain an expression connecting these quantities:
ω = 2πν.
Basic elements of the kinematics of uneven rotational motion
The uneven rotational motion of a rigid body or material point around a fixed axis is characterized by its angular velocity, which changes with time.
Vector ε , characterizing the rate of change of angular velocity, is called the angular acceleration vector:
ε = dω/dt.
If a body rotates, accelerating, that is dω/dt > 0, the vector has a direction along the axis in the same direction as ω.
If the rotational movement is slow - dω/dt< 0 , then the vectors ε and ω are oppositely directed.
Comment. When uneven rotational motion occurs, the vector ω can change not only in magnitude, but also in direction (when the axis of rotation is rotated).
Relationship between quantities characterizing translational and rotational motion
It is known that the arc length with the angle of rotation of the radius and its value are related by the relation
ΔS = Δφ r.
Then the linear speed of a material point performing rotational motion
υ = ΔS/Δt = Δφr/Δt = ωr.
The normal acceleration of a material point that performs rotational translational motion is defined as follows:
a = υ 2 /r = ω 2 r 2 /r.
So, in scalar form
a = ω 2 r.
Tangential accelerated material point that performs rotational motion
a = ε r.
Momentum of a material point
The vector product of the radius vector of the trajectory of a material point of mass m i and its momentum is called the angular momentum of this point about the axis of rotation. The direction of the vector can be determined using the right screw rule.
Momentum of a material point ( L i) is directed perpendicular to the plane drawn through r i and υ i, and forms a right-hand triple of vectors with them (that is, when moving from the end of the vector r i To υ i the right screw will show the direction of the vector L i).
In scalar form
L = m i υ i r i sin(υ i , r i).
Considering that when moving in a circle, the radius vector and the linear velocity vector for the i-th material point are mutually perpendicular,
sin(υ i , r i) = 1.
So the angular momentum of a material point for rotational motion will take the form
L = m i υ i r i .
The moment of force that acts on the i-th material point
The vector product of the radius vector, which is drawn to the point of application of the force, and this force is called the moment of force acting on the i-th material point relative to the axis of rotation.
In scalar form
M i = r i F i sin(r i , F i).
Considering that r i sinα = l i ,M i = l i F i .
Magnitude l i, equal to the length of the perpendicular lowered from the point of rotation to the direction of action of the force, is called the arm of the force F i.
Dynamics of rotational motion
The equation for the dynamics of rotational motion is written as follows:
M = dL/dt.
The formulation of the law is as follows: the rate of change of angular momentum of a body that rotates around a fixed axis is equal to the resulting moment relative to this axis of all external forces applied to the body.
Moment of impulse and moment of inertia
It is known that for the i-th material point the angular momentum in scalar form is given by the formula
L i = m i υ i r i .
If instead of linear speed we substitute its expression through angular speed:
υ i = ωr i ,
then the expression for the angular momentum will take the form
L i = m i r i 2 ω.
Magnitude I i = m i r i 2 is called the moment of inertia relative to the axis of the i-th material point of an absolutely rigid body passing through its center of mass. Then we write the angular momentum of the material point:
L i = I i ω.
We write the angular momentum of an absolutely rigid body as the sum of the angular momentum of the material points that make up this body:
L = Iω.
Moment of force and moment of inertia
The law of rotational motion states:
M = dL/dt.
It is known that the angular momentum of a body can be represented through the moment of inertia:
L = Iω.
M = Idω/dt.
Considering that the angular acceleration is determined by the expression
ε = dω/dt,
we obtain a formula for the moment of force, represented through the moment of inertia:
M = Iε.
Comment. A moment of force is considered positive if the angular acceleration that causes it is greater than zero, and vice versa.
Steiner's theorem. Law of addition of moments of inertia
If the axis of rotation of a body does not pass through its center of mass, then relative to this axis one can find its moment of inertia using Steiner’s theorem:
I = I 0 + ma 2,
Where I 0- initial moment of inertia of the body; m- body mass; a- distance between axes.
If a system that rotates around a fixed axis consists of n bodies, then the total moment of inertia of this type of system will be equal to the sum of the moments of its components (the law of addition of moments of inertia).