Physics - vibrations and waves. Oscillations and waves, laws and formulas Formulas in physics mechanical oscillations of waves

If we look at a wheat field in windy weather, we will see that it is “worried”, that something is moving along it. It’s not clear what, because the stems remain in place. They just bend over, straighten up, bend over again, etc. If we take a cord and fasten one end of it, and set the other in oscillatory motion, we will see that a wave “runs” along the cord. If we throw a stone into the water, then circles will go around the place where the stone fell. These circles are also waves.

The sources of waves are vibrations. Plant stems sway, deformed by the wind, water particles sway, and the end of the cord oscillates. And vibrations that arise in one place are transmitted to other particles. What we call a wave is the propagation of vibrations from point to point, from particle to particle.

A model for the formation of a wave in a cord can be a chain of balls with mass, between which an elastic force acts. Let's imagine that there are small springs between the balls.

Let ball 1 be pulled up and released. The spring connecting it to ball 2 will stretch, and an elastic force will arise, which acts not only on ball 1, but also on ball 2. Consequently, ball 2 will begin to oscillate. This will lead to deformation of the next spring, so it will begin to oscillate and ball 3, etc.
Since all the balls have the same mass and elastic force, all the balls will oscillate - each around its equilibrium position - with the same periods and the same amplitudes. However, all the balls are inert (since they have mass), so the balls will not oscillate at the same time, since it takes time for their speed to change. Therefore, the 2nd point will begin to oscillate later than the 1st, the 3rd later than the 2nd, the 4th later than the 3rd, etc.

If we observe any point on the cord, we will see that each point oscillates with the same period T. Although all points on the cord oscillate with the same frequency, these oscillations are “displaced” relative to each other in time. It is due to this displacement in time that a wave arises. For example, the oscillations of point 2 lag behind the oscillations of point 1 for a quarter period . And the oscillations of point 3 lag behind the oscillations of point 2 by one whole period T. An important conclusion follows: points 2 and 3 move the same way.

The distance between the nearest points of the wave that move the same way is called wavelength and is designated λ .

So, mechanical waves - These are mechanical vibrations that propagate in space over time.

Wave speed

In a time equal to one period T, each point of the medium has completed one oscillation and, therefore, returned to the same position. Consequently, the wave has shifted in space just by one wavelength. Thus, if we denote the speed of wave propagation υ , we find that the wave speed

λ = υ T

Since T = 1/ ν , then we obtain that the wave speed, wave length and wave frequency are related by the relation

υ = λ ν

What do waves carry?

In the above examples it is clear that the substance does not move along the direction of wave propagation, those. waves do not transport matter .
However waves carry energy: after all, a wave is an oscillation that propagates in space, and any oscillations have energy.

Oscillations- these are physical processes that repeat exactly or approximately at regular intervals. If an oscillatory process propagates in space over time, then we speak of wave propagation.

Vibratory movements are often found in nature and technology: trees in the forest, strings of musical instruments, engine pistons, vocal cords, hearts, etc. vibrate. Oscillatory movements occur in life - earthquakes, ebbs and flows, compression and expansion of our Universe.

Oscillations always occur in systems if these systems have stable equilibrium positions. When deviating from the equilibrium position, a “restoring” force arises, which tries to return the system to the equilibrium position. Since bodies are inherently inert, they “overshoot” the equilibrium position and then the deviation occurs in the opposite direction. And then the process begins to repeat itself periodically.

Depending on the physical nature they distinguish mechanical and electromagnetic vibrations. However, oscillations and waves, regardless of their nature, are described quantitatively by the same equations.

Mechanical vibrations - these are the movements of bodies in which, at equal intervals of time, the coordinates of the moving body, its speed and acceleration take on their original values.

Main types of vibrations

1. Available
2. Forced
3. Self-oscillations

Free vibrations

Free vibrations - these are oscillations that occur in a system under the influence of internal forces after the system is removed from its equilibrium position. That is, such oscillations occur only due to the energy reserve imparted to the system.

Conditions for the occurrence of free vibrations:
1. The system is near a stable equilibrium position (for a “returning” force to occur);
2. The friction in the system must be quite low (otherwise the oscillations will quickly die out or not occur at all).

Forced vibrations

Forced vibrations – these are vibrations that arise under the influence of external periodically changing forces.

Difference from free vibrations:
1. The frequency of forced oscillations is always equal to the frequency of the periodic driving force.
2. The amplitude of forced oscillations does not decrease over time, even if there is friction in the system. Since the loss of mechanical energy is replenished due to the work of external forces.

Self-oscillations

Self-oscillations– these are undamped oscillations that can exist in a system without the influence of external periodic forces on it. Such oscillations exist due to the supply of energy from a constant source (which the system has) and are regulated by the system itself.

Self-oscillating systems include: a clock with a pendulum, an electric bell with a breaker, our heart and lungs, etc.

Features of self-oscillations:
1. The frequency of self-oscillations is equal to the frequency of free oscillations of the oscillatory system and does not depend on the energy source (difference from forced oscillations).
2. The amplitude of self-oscillations does not depend on the energy imparted to the system, but is established by the system itself (difference from free oscillations).

Harmonic vibrations

Oscillations in which the displacement depends on time according to the cosine or sine law are called harmonic.

Equation of harmonic vibration

x = X max cosωt

Quantities characterizing oscillatory movements

Amplitude

Oscillation amplitude – the maximum value of a quantity that experiences oscillations according to the harmonic law.

The physical meaning of X max is the maximum value of the displacement of a body from the equilibrium position during harmonic vibrations.

Period and frequency

Harmonic period T– this is the time of one complete oscillation, that is, the period of time through which the movement is completely repeated.

Period unit [ T] = 1s

Oscillation frequency ν is the number of complete oscillations N performed by the body per unit time t.

Frequency unit [ ν ] = 1 Hz = 1/s

Cyclic frequency

Cyclic oscillation frequency ω is the number of complete oscillations completed in 2 π seconds

Cyclic frequency unit [ ω ] = 1 rad/s

Harmonic graph

Example

OSCILLATIONS AND WAVES. Oscillations are processes in which the movements or states of a system are regularly repeated in time. The oscillatory process is most clearly demonstrated by a swinging pendulum, but oscillations are characteristic of almost all natural phenomena. Oscillatory processes are characterized by the following physical quantities.

Oscillation period T– the period of time after which the state of the system takes on the same values: u(t + T) = u(t).

Oscillation frequency n or f– number of oscillations per second, the reciprocal of the period: n = 1/T. It is measured in hertz (Hz), and has units of –1. A pendulum swinging once per second oscillates at a frequency of 1 Hz. Circular or cyclic frequency is often used in calculations w = 2pn.

Oscillation phase j– a value showing how much of the oscillation has passed since the beginning of the process. It is measured in angular units - degrees or radians.

Oscillation amplitude A– the maximum value that the oscillatory system takes, the “span” of oscillation.

Periodic oscillations can have very different shapes, but the most interesting are the so-called harmonic or sinusoidal oscillations. Mathematically they are written in the form

u(t) = A sin j = A sin( w t + j 0),

Where A– amplitude, j– phase, j 0 is its initial value, w– circular frequency, t– function argument, current time. In the case of a strictly harmonic, undamped oscillation, the magnitude A, w And j 0 do not depend on t.

Any periodic oscillation of the most complex form can be represented as a sum of a finite number of harmonic oscillations, and a non-periodic oscillation (for example, a pulse) can be represented as an infinite number of them (Fourier’s theorem).

A system, taken out of balance and left to its own devices, performs free or natural oscillations, the frequency of which is determined by the physical parameters of the system. Natural vibrations can also be represented as a sum of harmonic, so-called normal vibrations, or modes.

Excitation of oscillations can occur in three ways. If a system is subject to a periodic force that varies with frequency f(the pendulum is swung with periodic shocks), the system will oscillate with this – forced – frequency. When the driving force frequency f equal to or a multiple of the natural frequency of the system n, resonance occurs—a sharp increase in the amplitude of oscillations.

If the system parameters (for example, the length of the pendulum suspension) are periodically changed, parametric excitation of oscillations occurs. It is most effective when the frequency of change of the system parameter is equal to twice its natural frequency: f par = 2 n personal

If oscillatory movements occur spontaneously (the system “self-excites”), they speak of the occurrence of self-oscillations that have a complex nature.

During oscillatory processes, the potential energy of the system is periodically converted into kinetic energy. For example, by deflecting the pendulum to the side and, therefore, raising it to a height h, he is given potential energy mgh. It completely transforms into kinetic energy of motion mv 2/2 when the load passes the equilibrium position and its speed is maximum. If there is a loss of energy, the oscillations become damped.

In physics, mechanical and electromagnetic oscillations are considered separately - coupled oscillations of the electric and magnetic fields (light, X-rays, radio). They propagate in space in the form of waves.

A wave is a disturbance (change in the state of the medium) that propagates in space and carries energy without transferring matter. The most common ones are elastic waves, waves on the surface of a liquid, and electromagnetic waves. Elastic waves can only be excited in a medium (gas, liquid, solid), while electromagnetic waves also propagate in a vacuum.

If the disturbance of a wave is directed perpendicular to the direction of its propagation, the wave is called transverse; if it is parallel, it is called longitudinal. Transverse waves include waves traveling along the surface of the water and along a string, as well as electromagnetic waves - the electric and magnetic field strength vectors are perpendicular to the wave speed vector. A typical example of a longitudinal wave is sound.

The equation describing the wave can be derived from the expression for harmonic vibrations. Let periodic motion occur at some point in the medium according to the law A = A 0 sin w t. This movement will be transmitted from layer to layer - an elastic wave will run through the medium. A point at a distance x from the point of excitation, will begin to make oscillatory movements, lagging behind for a while t required for the wave to travel the distance X: t = x/c, Where c– wave speed. Therefore, the law of its motion will be

A x = A 0 sin w(t – x/c),

or, since w= 2p/ T, Where T- period of oscillation,

A x = A 0 sin 2p ( t/T – x/cT).

This is the equation of a sine wave, or monochromatic wave, propagating at a speed With in the direction X. All wave points at a moment in time t have different offsets. But a series of points separated by a distance cT one from the other, at any moment of time are shifted equally (since the arguments of the sines in the equation differ by 2p and, therefore, their values ​​are equal). This distance is the wavelength l = st. It is equal to the path that the wave travels in one oscillation period.

Phases of oscillations of two wave points located at a distance D X one from the other, differ by D j = 2p D X/l, and therefore by 2 p at a distance that is a multiple of the wavelength. A surface at all points of which the wave has the same phases is called a wave front. The propagation of the wave occurs perpendicular to it, so it can be considered as the movement of a wave front in the medium. Wavefront points are formally considered fictitious sources of secondary spherical waves, which when added together give a wave of the original shape (Huygens-Fresnel principle).

The speed of displacement of the elements of the medium changes according to the same law as the displacement itself, but with a phase shift of p/2: Speed ​​reaches maximum when offset drops to zero. That is, the velocity wave is shifted relative to the wave of displacements (deformations of the medium) in time by T/4, and in space by l/4. The velocity wave carries kinetic energy, and the deformation wave carries potential energy. Energy is constantly transferred in the direction of wave propagation + X with speed With.

The speed entered above With corresponds to the propagation of only an infinite sinusoidal (monochromatic) wave. It determines the speed of movement of its phase j and is called phase velocity With f. But in practice, both waves of more complex shapes and waves limited in time (trains), as well as the joint propagation of a large set of waves of different frequencies (for example, white light) are much more common. Like complex oscillations, wave trains and inharmonic waves can be represented as a sum (superposition) of sine waves of different frequencies. When the phase velocities of all these waves are the same, then their entire group (wave packet) moves at the same speed. If the phase velocity of a wave depends on its frequency w, dispersion is observed - waves of different frequencies travel at different speeds. Normal, or negative dispersion, is greater the higher the wave frequency. Due to dispersion, for example, a beam of white light in a prism is decomposed into a spectrum, and in drops of water - into a rainbow. A wave packet, which can be represented as a set of harmonic waves lying in the range w 0±D w, blurred due to dispersion. Its shape - the envelope of the amplitudes of the train components - is distorted, but moves in space at a speed v g, called group velocity. If, during the propagation of a wave packet, the maxima of the waves that comprise it move faster than the envelope, the phase speed of the signal is higher than the group speed: With f > v gr. At the same time, in the tail part of the packet, due to the addition of waves, new maxima appear, which move forward and disappear in its head part. An example of normal dispersion is media that are transparent to light - glass and liquid.

In a number of cases, anomalous (positive) dispersion of the medium is also observed, in which the group velocity exceeds the phase velocity: v gr > With f, and a situation is possible when these velocities are directed in opposite directions. Wave maxima appear at the head of the packet, move backward and disappear in its tail. Anomalous dispersion is observed, for example, during the movement of very small (so-called capillary) waves on water ( v gr = 2With f).

All methods for measuring the time and speed of wave propagation, based on the delay of signals, give the group velocity. It is precisely this that is taken into account in laser, hydro- and radar location, atmospheric sounding, in radio control systems, etc.

When waves propagate in a medium, they are absorbed - an irreversible transfer of wave energy into its other types (in particular, into heat). The absorption mechanism of waves of different natures is different, but absorption in any case leads to a weakening of the wave amplitude according to the exponential law: A 1 /A 0 = e a , where a– the so-called logarithmic damping decrement. For sound waves, as a rule, a ~ w 2: High sounds are absorbed much more than low sounds. Absorption of light - drop in its intensity I- occurs according to Bouguer's law I = I 0 exp(– k l l), where exp( x) = e x, k l – absorption index of vibrations with wavelength l, l– the path traveled by the wave in the medium.

The scattering of sound by obstacles and inhomogeneities in the medium leads to the spreading of the sound beam and, as a consequence, to the attenuation of sound as it propagates. For heterogeneity size L< l/2 wave scattering is absent. Light scattering occurs according to complex laws and depends not only on the size of obstacles, but also on their physical characteristics. Under natural conditions, scattering on atoms and molecules is most pronounced, occurring in proportion to w 4 or, what is the same, l-4 (Rayleigh's law). It is Rayleigh scattering that is responsible for the blue color of the sky and the red color of the Sun at sunset. When the particle size becomes comparable to the wavelength of light ( r ~ l), scattering ceases to depend on wavelength; light scatters more forward than backward. Scattering on large particles ( r >> l) occurs taking into account the laws of optics - reflection and refraction of light.

When adding waves whose phase difference is constant ( cm. COHERENCE) a stable pattern of the intensity of the total oscillations arises - interference. The reflection of a wave from a wall is equivalent to the addition of two waves traveling towards each other with a phase difference p. Their superposition creates a standing wave, in which after each half of the period T/2 there are fixed points (nodes), and between them there are points that oscillate with maximum amplitude A(antinodes).

A wave falling on an obstacle or passing through a hole goes around their edges and enters the shadow area, giving a picture in the form of a system of stripes. This phenomenon is called diffraction; it becomes noticeable when the size of the obstacle (hole diameter) D comparable to the wavelength: D~ l.

In a transverse wave, a polarization phenomenon can be observed, in which a disturbance (displacement in an elastic wave, electric and magnetic field strength vectors in an electromagnetic wave) lies in the same plane (linear polarization) or rotates (circular polarization), while changing intensity (elliptical polarization).

When the wave source moves towards the observer (or, what is the same, the observer towards the source), an increase in frequency is observed f, when removed - a decrease (Doppler effect). This phenomenon can be observed near the railway track when a locomotive with a siren rushes past. The moment he gets close to the observer, there is a noticeable decrease in the tone of the beep. Mathematically, the effect is written as f = f 0 /(1 ± v/c), Where f– observed frequency, f 0 – frequency of the emitted wave, v– relative speed of the source, c– wave speed. The “+” sign corresponds to the approach of the source, the “–” sign to its removal.

Despite the fundamentally different nature of the waves, the laws governing their propagation have much in common. Thus, elastic waves in liquids or gases and electromagnetic waves in a homogeneous space emitted by a small source are described by the same equation, and waves on water, like light and radio waves, experience interference and diffraction.

Sergei Trankovsiy

School No. 283 Moscow

ABSTRACT:

IN PHYSICS

"Vibrations and Waves"

Completed:

Student 9 "b" school No. 283

Grach Evgeniy.

Physics teacher:

Sharysheva

Svetlana

Vladimirovna

Introduction. 3

1. Oscillations. 4

Periodic motion 4

Free swing 4

· Pendulum. Kinematics of its oscillations 4

· Harmonic oscillation. Frequency 5

· Dynamics of harmonic oscillations 6

· Energy conversion during free vibrations 6

· Period 7

8 phase shift

· Forced vibrations 8

Resonance 8

2. Waves. 9

· Transverse waves in cord 9

Longitudinal waves in an air column 10

Sound vibrations 11

· Musical tone. Volume and pitch 11

Acoustic resonance 12

· Waves on the surface of a liquid 13

Wave propagation speed 14

Wave reflection 15

Energy transfer by waves 16

3. Application 17

Acoustic speaker and microphone 17

· Echo sounder 17

· Ultrasound diagnostics 18

4. Examples of problems in physics 18

5. Conclusion 21

6. List of references 22

Introduction

Oscillations are processes that differ in varying degrees of repeatability. This property of repeatability is possessed, for example, by the swinging of a clock pendulum, vibrations of a string or legs of a tuning fork, the voltage between the plates of a capacitor in a radio receiver circuit, etc.

Depending on the physical nature of the repeating process, vibrations are distinguished: mechanical, electromagnetic, electromechanical, etc. This abstract discusses mechanical vibrations.

This branch of physics is key to the question “Why do bridges collapse?” (see page 8)

At the same time, oscillatory processes lie at the very basis of various branches of technology.

For example, all radio technology, and in particular the acoustic speaker, is based on oscillatory processes (see page 17)

About the abstract

The first part of the essay (“Vibrations” pp. 4-9) describes in detail what mechanical vibrations are, what types of mechanical vibrations there are, quantities that characterize vibrations, and also what resonance is.

The second part of the essay (“Waves” pp. 9-16) talks about what waves are, how they arise, what waves are, what sound is, its characteristics, at what speed waves travel, how they are reflected and how energy is transferred by waves .

The third part of the essay (“Application” pp. 17-18) talks about why we need to know all this, and about where mechanical vibrations and waves are used in technology and in everyday life.

The fourth part of the abstract (pp. 18-20) provides several examples of physics problems on this topic.

The abstract ends with a quick summary of everything that has been said (“Conclusion” p. 21) and a list of references (p. 22)

Oscillations.

Periodic motion.

Among all the various mechanical movements occurring around us, repetitive movements are often encountered. Any uniform rotation is a repeating movement: with each revolution, every point of a uniformly rotating body passes through the same positions as during the previous revolution, in the same sequence and at the same speed.

In reality, repetition is not always and not under all conditions exactly the same. In some cases, each new cycle very accurately repeats the previous one, in other cases the difference between successive cycles can be noticeable. Deviations from absolutely exact repetition are very often so small that they can be neglected and the movement can be considered to be repeated quite accurately, i.e. consider it periodic.

Periodic motion is a repeating motion in which each cycle exactly reproduces every other cycle.

The duration of one cycle is called a period. Obviously, the period of uniform rotation is equal to the duration of one revolution.

Free vibrations.

In nature, and especially in technology, oscillatory systems play an extremely important role, i.e. those bodies and devices that are themselves capable of performing periodic movements. “On their own” - this means not being forced to do so by the action of periodic external forces. Such oscillations are therefore called free oscillations, in contrast to forced oscillations occurring under the influence of periodically changing external forces.

All oscillatory systems have a number of common properties:

1. Each oscillatory system has a state of stable equilibrium.

2. If the oscillatory system is removed from a state of stable equilibrium, then a force appears that returns the system to a stable position.

3. Having returned to a stable state, the oscillating body cannot immediately stop.

Pendulum; kinematics of its oscillations.

A pendulum is any body suspended so that its center of gravity is below the point of suspension. A hammer hanging on a nail, scales, a weight on a rope - all these are oscillatory systems, similar to the pendulum of a wall clock.

Any system capable of free oscillations has a stable equilibrium position. For a pendulum, this is the position in which the center of gravity is vertically below the point of suspension. If we remove the pendulum from this position or push it, then it will begin to oscillate, deviating first in one direction, then in the other direction from the equilibrium position. The greatest deviation from the equilibrium position to which the pendulum reaches is called the amplitude of oscillations. The amplitude is determined by the initial deflection or push with which the pendulum was set in motion. This property - the dependence of the amplitude on the conditions at the beginning of the movement - is characteristic not only of free oscillations of a pendulum, but also of free oscillations of many oscillatory systems in general.

Let's attach a hair to the pendulum and move a smoked glass plate under this hair. If you move the plate at a constant speed in a direction perpendicular to the plane of vibration, the hair will draw a wavy line on the plate. In this experiment we have a simple oscilloscope - that’s what instruments for recording vibrations are called. Thus, the wavy line represents an oscillogram of the pendulum's oscillations.

The amplitude of the oscillations is depicted on this oscillogram by segment AB, the period is depicted by segment CD, equal to the distance the plate moves during the period of the pendulum.

Since we move the sooty plate uniformly, any movement of it is proportional to the time during which it occurred. We can therefore say that along the axis x time is delayed on a certain scale. On the other hand, in the direction perpendicular to x a hair marks on the plate the distance of the end of the pendulum from its equilibrium position, i.e. the distance traveled by the end of the pendulum from this position.

As we know, the slope of the line on such a graph represents the speed of movement. The pendulum passes through the equilibrium position at maximum speed. Accordingly, the slope of the wavy line is greatest at those points where it intersects the axis x. On the contrary, at the moments of greatest deviations the speed of the pendulum is zero. Accordingly, the wavy line at those points where it is furthest from the axis x, has a parallel tangent x, i.e. slope is zero

Harmonic oscillation. Frequency.

The oscillation that the projection of this point onto any straight line makes when a point moves uniformly around a circle is called harmonic (or simple) oscillation.

Harmonic oscillation is a special, private type of periodic oscillation. This special type of oscillation is very important, since it is extremely common in a wide variety of oscillatory systems. The oscillation of a load on a spring, a tuning fork, a pendulum, or a clamped metal plate is precisely harmonic in its form. It should be noted that at large amplitudes, the oscillations of these systems have a slightly more complex shape, but the smaller the oscillation amplitude, the closer they are to harmonic.

Oscillations– changes in any physical quantity in which this quantity takes on the same values. Oscillation parameters:

• 1) Amplitude – the value of the greatest deviation from the equilibrium state;
• 2) Period is the time of one complete oscillation, the reciprocal is frequency;
• 3) The law of change of a fluctuating quantity over time;
• 4) Phase – characterizes the state of oscillations at time t.

F x = -r k – restoring force

Harmonic vibrations- oscillations in which the quantity causing the deviation of the system from a stable state changes according to the law of sine or cosine. Harmonic oscillations are a special case of periodic oscillations. Oscillations can be represented graphically, analytically (for example, x(t) = Asin (?t + ?), where? is the initial phase of the oscillation) and in a vector way (the length of the vector is proportional to the amplitude, the vector rotates in the drawing plane with an angular velocity? around the axis, perpendicular to the drawing plane passing through the beginning of the vector, the angle of deviation of the vector from the X axis is the initial phase?). Harmonic vibration equation:

Addition of harmonic vibrations, occurring along the same straight line with the same or similar frequencies. Let's consider two harmonic oscillations occurring with the same frequency: x1(t) = A1sin(?t + ?1); x2(t) = A2sin(?t + ?2).

The vector representing the sum of these oscillations rotates with angular velocity?. The amplitude of the total oscillations is the vector sum of two amplitudes. Its square is equal to A?2 = A12 + A22 + 2A1A2cos(?2 - ?1).

The initial phase is defined as follows:

Those. tangent? is equal to the ratio of the projections of the amplitude of the total oscillation onto the coordinate axes.

If the oscillation frequencies differ by 2?: ?1 = ?0 + ?; ?2 = ?0 - ?, where?<< ?. Положим также?1 = ?2 = 0 и А1 = А2:

X 1 (t)+X 2 (t) = A(Sin(W o +?)t+Sin((W o +?)t) X 1 (t)+X 2 (t) =2ACos?tSinW?.

The quantity 2Аcos?t is the amplitude of the resulting oscillation. It changes slowly over time.

Beats. The result of the sum of such oscillations is called beat. In case A1 ? A2, then the beat amplitude varies from A1 + A2 to A1 – A2.

In both cases (with equal and different amplitudes), the total oscillation is not harmonic, because its amplitude is not constant, but changes slowly over time.

Addition of perpendicular vibrations. Let's consider two oscillations, the directions of which are perpendicular to each other (the oscillation frequencies are equal, the initial phase of the first oscillation is zero):

y= bsin(?t + ?).

From the equation of the first vibration we have: . The second equation can be rearranged as follows

sin?t?cos? +cos?t?sin? = y/b

Let's square both sides of the equation and use the basic trigonometric identity. We get (see below): . The resulting equation is the equation of an ellipse, the axes of which are slightly rotated relative to the coordinate axes. At? = 0 or? = ? the ellipse takes the form of a straight line y = ?bx/a; at? = ?/2 the axes of the ellipse coincide with the coordinate axes.

Lissajous figures . In case?1 ? ?2, the shape of the curve that the radius vector of the total oscillations describes is much more complex; it depends on the ratio ?1/?2. If this ratio is equal to an integer (?2 is a multiple of?1), the addition of oscillations produces figures called Lissajous figures.

Harmonic oscillator – an oscillating system whose potential energy is proportional to the square of the deviation from the equilibrium position.

Pendulum , a rigid body that, under the influence of applied forces, oscillates around a fixed point or axis. In physics, magnetism is usually understood to mean magnetism that oscillates under the influence of gravity; Moreover, its axis should not pass through the center of gravity of the body. The simplest weight consists of a small massive load C suspended on a thread (or light rod) of length l. If we consider the thread to be inextensible and neglect the size of the load compared to the length of the thread, and the mass of the thread compared to the mass of the load, then the load on the thread can be considered as a material point located at a constant distance l from the suspension point O (Fig. 1, a). This kind of M. is called mathematical. If, as is usually the case, the oscillating body cannot be considered as a material point, then the mass is called physical.

Math pendulum . If the magnet, deviated from the equilibrium position C0, is released without an initial speed or imparted to point C a speed directed perpendicular to OC and lying in the plane of the initial deviation, then the magnet will oscillate in one vertical plane along a circular arc (flat, or circular mathematical .). In this case, the position of the magnet is determined by one coordinate, for example, the angle j by which the magnet is tilted from the equilibrium position. In the general case, magnetic vibrations are not harmonic; their period T depends on the amplitude. If the deviations of the magnet are small, it performs oscillations close to harmonic, with a period:

where g is the acceleration of free fall; in this case, the period T does not depend on the amplitude, that is, the oscillations are isochronous.

If the deflected magnet is given an initial speed that does not lie in the plane of the initial deflection, then point C will describe on a sphere of radius l the curves contained between 2 parallels z = z1 and z = z2, a), where the values ​​of z1 and z2 depend on the initial conditions (spherical pendulum). In a particular case, with z1 = z2, b) point C will describe a circle in the horizontal plane (conical pendulum). Among non-circular pendulums, the cycloidal pendulum, whose oscillations are isochronous at any amplitude, is of particular interest.

Physical pendulum . Physical material is usually called a solid body that, under the influence of gravity, oscillates around the horizontal axis of the suspension (Fig. 1, b). The movement of such a magnet is quite similar to the movement of a circular mathematical magnet. At small angles of deflection j, the magnet also performs oscillations close to harmonic, with a period:

where I is the moment of inertia M. relative to the suspension axis, l is the distance from the suspension axis O to the center of gravity C, M is the mass of the material. Consequently, the period of oscillation of a physical material coincides with the period of oscillation of a mathematical material that has a length l0 = I/Ml. This length is called the reduced length of a given physical M.

Spring pendulum- this is a load of mass m, attached to an absolutely elastic spring and performing harmonic oscillations under the action of an elastic force Fupr = - k x, where k is the elasticity coefficient, in the case of a spring it is called. rigidity. Level of movement of the pendulum:, or.

From the above expressions it follows that the spring pendulum performs harmonic oscillations according to the law x = A cos (w0 t +?j), with a cyclic frequency

and period

The formula is valid for elastic vibrations within the limits in which Hooke’s law is satisfied (Fupr = - k x), i.e. when the mass of the spring is small compared to the mass of the body.

The potential energy of a spring pendulum is equal to

U = k x2/2 = m w02 x2/2 .

Forced vibrations. Resonance. Forced oscillations occur under the influence of an external periodic force. The frequency of forced oscillations is set by an external source and does not depend on the parameters of the system itself. The equation of motion of a load on a spring can be obtained by formally introducing into the equation a certain external force F(t) = F0sin?t: . After transformations similar to the derivation of the equation of damped oscillations, we obtain:

Where f0 = F0/m. The solution to this differential equation is the function x(t) = Asin(?t + ?).

Addendum? appears due to the inertia of the system. Let us write f0sin (?t - ?) = f(t) = f0 sin (?t + ?), i.e. the force acts with some advance. Then we can write:

x(t) = A sin ?t.

Let's find A. To do this, we calculate the first and second derivatives of the last equation and substitute them into the differential equation of forced oscillations. After reducing similar ones we get:

Now let’s refresh our memory about the vector recording of oscillations. What do we see? The vector f0 is the sum of the vectors 2??A and A(?02 - ?2), and these vectors are (for some reason) perpendicular. Let's write down the Pythagorean theorem:

4?2?2A2 + A2(?02 - ?2)2 = f02:

From here we express A:

Thus, the amplitude A is a function of the frequency of the external influence. However, what if the oscillating system has weak damping?<< ?, то при близких значениях? и?0 происходит резкое возрастание амплитуды колебаний. Это явление получило название резонанса.