And the time of movement, you can find the distance traveled:
Substituting the expression into this formula V avg = V/2, we will find the path taken at equals accelerated movement from rest:
If we substitute into formula (4.1) the expression V avg = V 0 /2, then we get the path traveled during braking:
The last two formulas include speeds V 0 and V. Substituting the expression V=at into formula (4.2), and the expression V 0 =at - into formula (4.3), we get
The resulting formula is valid both for uniformly accelerated motion from a state of rest, and for motion with decreasing speed when the body stops at the end of the path. In both of these cases, the distance traveled is proportional to the square of the time of movement (and not just time, as was the case with uniform movement). The first to establish this pattern was G. Galileo.
Table 2 gives the basic formulas describing uniformly accelerated linear motion. 
Galileo did not have a chance to see his book, which outlined the theory of uniformly accelerated motion (along with many of his other discoveries). When was it published? The 74-year-old scientist was already blind. Galileo took the loss of his vision very hard. “You can imagine,” he wrote, “how I grieve when I realize that this sky, this world and the Universe, which by my observations and clear evidence have been expanded a hundred and a thousand times compared to what people thought they were sciences in all the past centuries have now become so diminished and diminished for me.”
Five years earlier, Galileo was tried by the Inquisition. His views on the structure of the world (and he adhered to the Copernican system, in which the central place was occupied by the Sun, not the Earth) had not been liked by church ministers for a long time. Back in 1614, the Dominican priest Caccini declared Galileo a heretic and mathematics an invention of the devil. And in 1616, the Inquisition officially declared that “the doctrine attributed to Copernicus that the Earth moves around the Sun, while the Sun stands at the center of the Universe, not moving from East to West, is disgusting Holy Scripture, and therefore it can neither be defended nor accepted as truth." Copernicus' book outlining his system of the world was banned, and Galileo was warned that if "he does not calm down, he will be imprisoned."
But Galileo “did not calm down.” “There is no greater hatred in the world,” the scientist wrote, “than ignorance for knowledge.” And in 1632 it comes out famous book"Dialogue about two major systems world - Ptolemaic and Copernican", in which he gave numerous arguments in favor of the Copernican system. However, only 500 copies of this work were sold, since after a few months, by order of the Pope
Rimsky, the publisher of the book, received an order to suspend the sale of this work.
In the autumn of the same year, Galileo received an order from the Inquisition to appear in Rome, and after some time the sick 69-year-old scientist was taken to the capital on a stretcher. Here, in the prison of the Inquisition, Galileo was forced to renounce his views on the structure of the world, and on June 22, 1633 in a Roman monastery Minerva Galileo reads and signs the previously prepared text of renunciation
“I, Galileo Galilei, son of the late Vincenzo Galilei of Florence, 70 years of age, brought in person to the court and kneeling before Your Eminences, the most reverend gentlemen cardinals, general inquisitors against heresy throughout Christendom, having before me the sacred Gospel and offering hands on him, I swear that I have always believed, I believe now, and with God’s help I will continue to believe in everything that the Holy Catholic and Apostolic Roman Church recognizes, defines and preaches.”
According to the court decision, Galileo's book was banned, and he himself was sentenced to imprisonment for indefinite term However, the Pope pardoned Galileo and replaced his imprisonment with exile. Galileo moved to Arcetri and here, while under house arrest, wrote the book “Conversations and mathematical proofs concerning two new branches of science related to Mechanics and Local Movement.” In 1636, the manuscript of the book was transported to Holland, where it was published in 1638. With this book, Galileo summed up his many years physical research In the same year, Galileo became completely blind. Talking about the misfortune that befell the great scientist, Viviani (a student of Galileo) wrote: “He suffered severe discharge from his eyes, so that after a few months he was completely left without eyes - yes, I say, without his eyes, which behind short time saw in this world more than all human eyes in all the past centuries were able to see and observe"
The Florentine inquisitor who visited Galileo in his letter to Rome said that he found him in a very serious condition. Based on this letter, the Pope allowed Galileo to return to native home in Florence. Here he was immediately given an order: “Under pain of life imprisonment in a true prison and excommunication, do not go into the city and do not talk to anyone, no matter who it is, about the damned opinion about the double movement of the Earth.”
Galileo did not stay at home for long. After a few months he was again ordered to come to Arcetri. He had about four years to live. On January 8, 1642, at four o'clock in the morning, Galileo died.
1. How does uniformly accelerated motion differ from uniform motion? 2. How does the path formula for uniformly accelerated motion differ from the path formula for uniform motion? 3. What do you know about the life and work of G. Galileo? In which year he was born?
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Materials from physics 8th grade, assignments and answers from physics by grade, notes for preparing for physics lessons, plans for lesson notes on physics 8th grade
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework controversial issues rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for a year guidelines discussion programs Integrated LessonsMechanical movement
Mechanical movement is the process of changing the position of a body in space over time relative to another body, which we consider stationary.
A body conventionally accepted as motionless is a body of reference.
Reference body is a body relative to which the position of another body is determined.
Reference system is a body of reference, a coordinate system rigidly connected to it, and a device for measuring the time of movement.
Trajectory of movement
Body trajectory is a continuous line that is described by a moving body (considered as a material point) in relation to the chosen reference system.
Distance traveled
Distance traveled -scalar quantity, equal to length arc of the trajectory traversed by a body over some time.
Moving
By moving the body called a directed straight line segment connecting the initial position of a body with its subsequent position, vector quantity.
Average and instantaneous speed of movement. Direction and module of speed.
Speed - physical quantity, which characterizes the speed of coordinate change.
Average driving speed- it is a physical quantity equal to the ratio vector of movement of a point to the time interval during which this movement occurred. Vector direction average speed coincides with the direction of the displacement vector ∆S
Instantaneous speed is a physical quantity equal to the limit which he strives for average speed with an infinite decrease in time interval ∆t. Vector instantaneous speed directed tangentially to the trajectory. Module equal to the first derivative of the path with respect to time.
Formula for path with uniformly accelerated motion.
Uniformly accelerated motion-
This is a movement in which the acceleration is constant in magnitude and direction.
Acceleration of movement
Acceleration of movement - a vector physical quantity that determines the rate of change in the speed of a body, that is, the first derivative of speed with respect to time.
Tangential and normal accelerations.
Tangential (tangential) acceleration is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the motion trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.
Direction vector tangential acceleration a lies on the same axis with the tangent circle, which is the trajectory of the body. 
Normal acceleration- this is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body.
Vector
perpendicular linear speed movement, directed along the radius of curvature of the trajectory.
Speed formula for uniformly accelerated motion
Newton's first law (or law of inertia)
There are such reference systems relative to which isolated translationally moving bodies retain their speed unchanged in magnitude and direction.
Inertial system countdown is a reference system relative to which a material point free from external influences, either at rest or moving rectilinearly and uniformly (i.e. at a constant speed).
In nature there are four type of interaction
1. Gravitational (gravitational force) is the interaction between bodies that have mass.
2. Electromagnetic - true for bodies with an electric charge, responsible for mechanical forces such as friction and elasticity.
3. Strong - short-range interaction, that is, it acts at a distance of the order of the size of the nucleus.
4. Weak. Such interaction is responsible for some types of interaction among elementary particles, for some types of β-decay and for other processes occurring inside the atom, atomic nucleus.
Weight - is quantitative characteristics inert properties of the body. It shows how the body reacts to external influences.
Force - is a quantitative measure of the action of one body on another.
Newton's second law.
The force acting on the body is equal to the product of the body mass and the acceleration imparted by this force: F=ma
Measured in
Physical quantity, equal to the product body mass to the speed of its movement is called body impulse (or amount of movement). The momentum of a body is a vector quantity. The SI unit of impulse is kilogram-meter per second (kg m/s).
Expression of Newton's second law through a change in the momentum of a body
Uniform movement – this is movement at a constant speed, that is, when the speed does not change (v = const) and acceleration or deceleration does not occur (a = 0).
Straight-line movement – this is movement in a straight line, that is, a trajectory rectilinear movement- this is a straight line.
Uniformly accelerated motion - movement in which the acceleration is constant in magnitude and direction.
Newton's third law. Examples.
Shoulder of power.
Shoulder of power is the length of the perpendicular from some fictitious point O to the force. We will choose the fictitious center, point O, arbitrarily, and determine the moments of each force relative to this point. It is impossible to choose one point O to determine the moments of some forces, and to choose it in another place to find the moments of other forces!
We select point O in an arbitrary place and do not change its location anymore. Then the gravity arm is the length of the perpendicular (segment d) in the figure
Moment of inertia of bodies.
Moment of inertia J(kgm 2) – parameter similar to physical meaning mass at forward movement. It characterizes the measure of inertia of bodies rotating about a fixed axis of rotation. The moment of inertia of a material point with mass m is equal to the mass multiplied by the square of the distance from the point to the axis of rotation: .
The moment of inertia of a body is the sum of the moments of inertia material points composing this body. It can be expressed in terms of body weight and size
Steiner's theorem.
Moment of inertia J body relative to an arbitrary fixed axis equal to the sum moment of inertia of this body Jc relative to an axis parallel to it, passing through the center of mass of the body, and the product of the body mass m per square of distance d between axes:
Jc- known moment of inertia about an axis passing through the center of mass of the body,
J- the desired moment of inertia relative to parallel axis,
m- body mass,
d- distance between the indicated axes.
Law of conservation of angular momentum. Examples.
If the sum of the moments of forces acting on a body rotating around a fixed axis is equal to zero, then the angular momentum is conserved (law of conservation of angular momentum):
.
The law of conservation of angular momentum is very clear in experiments with a balanced gyroscope - a rapidly rotating body with three degrees of freedom (Fig. 6.9).
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It is the law of conservation of angular momentum that is used by ice dancers to change the speed of rotation. Or more famous example– Zhukovsky bench (Fig. 6.11).
Work of force.
Work of force -measure of force during transformation mechanical movement into another form of movement.
Examples of formulas for the work of forces.
work of gravity; work of gravity on an inclined surface
elastic force work
Work of friction force
Mechanical energy of the body.
Mechanical energy is a physical quantity that is a function of the state of the system and characterizes the system’s ability to do work.
Oscillation characteristics
Phase determines the state of the system, namely coordinate, speed, acceleration, energy, etc.
Cyclic frequency
characterizes the rate of change in the phase of oscillations.
Initial state oscillatory system characterizes initial phase
Oscillation amplitude A- this is the largest displacement from the equilibrium position
Period T- this is the period of time during which the point performs one complete oscillation.
Oscillation frequency is the number of complete oscillations per unit time t.
Frequency, cyclic frequency and period of oscillation are related as
Physical pendulum.
Physical pendulum - a rigid body capable of oscillating about an axis that does not coincide with the center of mass.
Electric charge.
Electric charge is a physical quantity that characterizes the property of particles or bodies to enter into electromagnetic force interactions.
Electric charge is usually represented by the letters q or Q.
The totality of all known experimental facts allows us to make the following conclusions:
· There are two kinds electric charges, conventionally called positive and negative.
· Charges can be transferred (for example, by direct contact) from one body to another. Unlike body mass, electric charge is not an inherent characteristic given body. The same body different conditions may have a different charge.
· Like charges repel, unlike charges attract. This also reveals a fundamental difference electromagnetic forces from gravitational ones. Gravitational forces are always forces of attraction.
Coulomb's law.
The modulus of the force of interaction between two stationary point electric charges in a vacuum is directly proportional to the product of the magnitudes of these charges and inversely proportional to the square of the distance between them.
G is the distance between them, k is the proportionality coefficient, depending on the choice of system of units, in SI
The value showing how many times the force of interaction of charges in a vacuum is greater than in a medium is called the dielectric constant of the medium E. For a medium with dielectric constant e, Coulomb’s law is written in the following way:
In SI, the coefficient k is usually written as follows:
Electric constant, numerically equal
Using electric constant law The pendant looks like:
Electrostatic field.
Electrostatic field - a field created by electric charges that are stationary in space and unchanging in time (in the absence of electric currents). The electric field is special kind matter, associated with electric charges and transmitting the effects of charges on each other.
Main characteristics electrostatic field:
· tension
potential
Examples of formulas for field strength of charged bodies.
1. The intensity of the electrostatic field created by a uniformly charged spherical surface.
Let a spherical surface of radius R (Fig. 13.7) carry a uniformly distributed charge q, i.e. the surface charge density at any point on the sphere will be the same.
Let us enclose our spherical surface in a symmetrical surface S with radius r>R. The flux of the tension vector through the surface S will be equal to
By Gauss's theorem
Hence
Comparing this relationship with the formula for field strength point charge, we can come to the conclusion that the field strength outside the charged sphere is the same as if the entire charge of the sphere was concentrated at its center.
For points located on the surface of a charged sphere of radius R, by analogy with the above equation, we can write
Let us draw through point B, located inside a charged spherical surface, a sphere S of radius r 2. Electrostatic field of the ball. Let us have a ball of radius R, uniformly charged with volume density. At any point A lying outside the ball at a distance r from its center (r>R), its field is similar to the field of a point charge located at the center of the ball. Then out of the ball and on its surface (r=R) At point B, lying inside the ball at a distance r from its center (r>R), the field is determined only by the charge enclosed inside the sphere with radius r. The flux of the tension vector through this sphere is equal to on the other hand, in accordance with Gauss's theorem From comparison latest expressions should Where - the dielectric constant inside the ball. 3. Field strength of a uniformly charged infinite rectilinear thread (or cylinder). Let us assume that a hollow cylindrical surface of radius R is charged with a constant linear density. Let's carry out coaxial cylindrical surface radius The flow of the tension vector through this surface By Gauss's theorem From the last two expressions we determine the field strength created by a uniformly charged thread: Let the plane have infinite extent and the charge per unit area equal to σ. From the laws of symmetry it follows that the field is directed everywhere perpendicular to the plane, and if there are no other external charges, then the fields on both sides of the plane must be the same. Let us limit part of the charged plane to an imaginary cylindrical box, so that the box is cut in half and its constituents are perpendicular, and the two bases, each having an area S, are parallel to the charged plane (Figure 1.10). Total vector flow; tensions equal to the vector, multiplied by the area S of the first base, plus the flux vector through the opposite base. Flow of tension through lateral surface cylinder equal to zero, because lines of tension do not intersect them. Hence But then the field strength of an infinite uniformly charged plane will be equal to This expression does not include coordinates, therefore the electrostatic field will be uniform, and its intensity at any point in the field will be the same. 5. Field strength created by two infinite parallel planes, charged differently with the same densities. Thus, Outside the plate, the vectors from each of them are directed towards opposite sides and are mutually destroyed. Therefore, the field strength in the space surrounding the plates will be zero E=0. Electricity. Electricity - directed (ordered) movement of charged particles Outside forces. Outside forces- forces of a non-electrical nature that cause the movement of electrical charges inside a direct current source. All forces other than Coulomb forces are considered external.
E.m.f. Voltage. Electromotive force(EMF) - a physical quantity characterizing the work of third-party (non-potential) forces in direct or alternating current sources. In a closed conducting EMF circuit equal to the work of these forces to move a unit positive charge along the contour. EMF can be expressed through tension electric field outside forces Voltage (U)
equal to the ratio of the work of the electric field to move the charge SI unit of voltage: Current strength. Current strength (I)-
scalar quantity equal to the ratio of the charge q passed through cross section conductor, to the period of time tduring which the current flowed. The current strength shows how much charge passes through the cross section of the conductor per unit time. Current density. Current density j -
a vector whose modulus is equal to the ratio of the current flowing through a certain area, perpendicular to the direction of the current, to the magnitude of this area. The SI unit of current density is ampere per square meter(A/m2). Ohm's law. Current is directly proportional to voltage and inversely proportional to resistance. Joule-Lenz law. When passing electric current along a conductor, the amount of heat generated in the conductor is directly proportional to the square of the current, the resistance of the conductor and the time during which the electric current flowed through the conductor.
Magnetic interaction- this is the interaction of ordering of moving electric charges. A magnetic field. A magnetic field- this is a special type of matter through which interaction occurs between moving electrically charged particles. Lorentz force and Ampere force. Lorentz force– force acting from outside magnetic field on a positive charge moving at speed (here – the speed of the ordered movement of positive charge carriers). Lorentz force modulus: Ampere power is the force with which a magnetic field acts on a current-carrying conductor. The ampere force module is equal to the product of the current strength in the conductor by the magnitude of the magnetic induction vector, the length of the conductor and the sine of the angle between the magnetic induction vector and the direction of the current in the conductor. The Ampere force is maximum if the magnetic induction vector is perpendicular to the conductor. If the magnetic induction vector is parallel to the conductor, then the magnetic field has no effect on the current-carrying conductor, i.e. Ampere's force is zero. The direction of Ampere's force is determined by the left-hand rule. Biot-Savart-Laplace law. Biot-Savart-Laplace's Law- The magnetic field of any current can be calculated as the vector sum of the fields created by individual sections of currents. Formulation Let D.C. flows along a contour γ located in a vacuum - the point at which the field is sought, then the magnetic field induction at this point is expressed by the integral (in the SI system) The direction is perpendicular to and, that is, perpendicular to the plane in which they lie, and coincides with the tangent to the line of magnetic induction. This direction can be found by the rule for finding magnetic induction lines (the right-hand screw rule): the direction of rotation of the screw head gives the direction if the translational movement of the gimlet corresponds to the direction of the current in the element. The magnitude of the vector is determined by the expression (in SI system) Vector potential given by the integral (in SI system) Loop inductance. SI units of inductance: Magnetic field energy. Electromagnetic induction - the phenomenon of the occurrence of electric current in a closed circuit when changing magnetic flux, passing through it. Lenz's rule. Lenz's rule Occurring in a closed loop induced current its magnetic field counteracts the change in magnetic flux that causes it. Maxwell's first equation Maxwell's second equation: Electromagnetic waves, electromagnetic radiation- disturbance propagating in space (change of state) electromagnetic field. 3.1. Wave
- These are vibrations propagating in space over time. Longitudinal waves arise when particles of the medium oscillate, oriented along the vector of propagation of the disturbance. Transverse waves propagate in perpendicular to the vector impact direction. In short: if in a medium the deformation caused by a disturbance manifests itself in the form of shear, stretching and compression, then we're talking about about a solid body for which both longitudinal and transverse waves. If the appearance of a shift is impossible, then the environment can be any. Each wave travels at a certain speed. Under wave speed
understand the speed of propagation of the disturbance. Since the speed of a wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. Thus, to find the wavelength, you need to multiply the speed of the wave by the period of oscillation in it: Wavelength
- the distance between two points closest to each other in space, in which the vibrations occur in the same phase. The wavelength corresponds to the spatial period of the wave, that is, the distance that a point with a constant phase “travels” in a time interval equal to the period of oscillation, therefore Wave number(also called spatial frequency) is the ratio 2 π
radian to wavelength: spatial analogue circular frequency. Definition: wave number k is the rate of growth of the wave phase φ
by spatial coordinate. 3.2. Plane wave
- a wave whose front has the shape of a plane. The front of a plane wave is unlimited in size, the phase velocity vector is perpendicular to the front. A plane wave is a particular solution to the wave equation and a convenient model: such a wave does not exist in nature, since the front of a plane wave begins at and ends at , which, obviously, cannot exist. The equation of any wave is a solution differential equation, called wave. The wave equation for the function is written as: · - Laplace operator; · - the required function; · - radius of the vector of the desired point; · - wave speed; · - time. wave surface
- locus points experiencing a perturbation of the generalized coordinate in the same phase. Special case wave surface - wave front. A) Plane wave
is a wave whose wave surfaces are a collection parallel friend friend planes. B) Spherical wave
is a wave whose wave surfaces are a collection of concentric spheres. Ray- line, normal and wave surface. The direction of wave propagation refers to the direction of the rays. If the wave propagation medium is homogeneous and isotropic, the rays are straight (and if the wave is plane, they are parallel straight lines). The concept of ray in physics is usually used only in geometric optics and acoustics, since when effects that are not studied in these directions occur, the meaning of the concept of a ray is lost. 3.3. Energy characteristics waves
The medium in which the wave propagates has mechanical energy, consisting of energies oscillatory motion all its particles. The energy of one particle with mass m 0 is found by the formula: E 0 = m 0 Α 2ω 2 /2. A unit volume of the medium contains n = p/m 0 particles (ρ
- density of the medium). Therefore, a unit volume of the medium has energy w р = nЕ 0 = ρ
Α 2ω 2 /2. Bulk Density energy(W р) - energy of vibrational motion of particles of the medium contained in a unit of its volume: Energy flow(F) - a value equal to the energy transferred by a wave through a given surface per unit time: Wave intensity or energy flux density(I) - a value equal to the energy flow transferred by a wave through a unit area perpendicular to the direction of wave propagation: 3.4. Electromagnetic wave
Electromagnetic wave- the process of propagation of an electromagnetic field in space. Occurrence condition electromagnetic waves. Changes in the magnetic field occur when the current strength in the conductor changes, and the current strength in the conductor changes when the speed of movement of electric charges in it changes, i.e. when charges move with acceleration. Consequently, electromagnetic waves should arise from the accelerated movement of electric charges. When the charge speed is zero, there is only an electric field. At constant speed charge creates an electromagnetic field. With the accelerated movement of a charge, an electromagnetic wave is emitted, which propagates in space at a finite speed. Electromagnetic waves propagate in matter with terminal speed. Here ε and μ are the dielectric and magnetic permeabilities of the substance, ε 0 and μ 0 are the electric and magnetic constants: ε 0 = 8.85419·10 –12 F/m, μ 0 = 1.25664·10 –6 H/m. Speed of electromagnetic waves in vacuum (ε = μ = 1): Main characteristics Electromagnetic radiation is generally considered to be frequency, wavelength and polarization. The wavelength depends on the speed of propagation of radiation. The group speed of propagation of electromagnetic radiation in a vacuum is equal to the speed of light; in other media this speed is less. Electromagnetic radiation is usually divided into frequency ranges (see table). There are no sharp transitions between the ranges; they sometimes overlap, and the boundaries between them are arbitrary. Since the speed of radiation propagation is constant, the frequency of its oscillations is strictly related to the wavelength in vacuum. Wave interference. Coherent waves. Conditions for wave coherence. Optical path length (OPL) of light. Relationship between the difference o.d.p. waves with a difference in the phases of the oscillations caused by the waves. Amplitude resulting oscillation when two waves interfere. Conditions for maxima and minima of amplitude during interference of two waves. Interference fringes and interference pattern on a flat screen when illuminated by two narrow long parallel slits: a) red light, b) white light.
Thus, on the other hand, according to Gauss’s theorem
As can be seen from Figure 13.13, the field strength between two infinite parallel planes having surface densities charges and , are equal to the sum of the field strengths created by the plates, i.e.
to the amount of charge moved in a section of the circuit.
Inductance
- physical a value numerically equal to the self-inductive emf that occurs in the circuit when the current changes by 1 Ampere in 1 second.
Inductance can also be calculated using the formula:
where Ф is the magnetic flux through the circuit, I is the current strength in the circuit.
A magnetic field has energy. Just as a charged capacitor has a reserve electrical energy, in the coil through the turns of which current flows, there is a reserve of magnetic energy.
2. Any displaced magnetic field generates a vortex electric field (the basic law of electromagnetic induction).
Mechanical waves can only spread in some medium (substance): in a gas, in a liquid, in a solid. The source of waves are oscillating bodies that create environmental deformation in the surrounding space. A necessary condition for the appearance of elastic waves is the appearance at the moment of disturbance of the medium of forces preventing it, in particular, elasticity. They tend to bring neighboring particles closer together when they move apart, and push them away from each other when they approach each other. Elastic forces, acting on particles remote from the source of disturbance, begin to unbalance them. Longitudinal waves characteristic only of gaseous and liquid media, but transverse– also to solids: the reason for this is that the particles that make up these media can move freely, since they are not rigidly fixed, unlike solids. Accordingly, transverse vibrations are fundamentally impossible.
