1. Definition of oscillatory motion

oscillatory motion is a movement that repeats exactly or approximately at regular intervals. The doctrine of oscillatory motion in physics is singled out especially. This is due to the commonality of the laws of oscillatory motion of various nature and methods of its study. Mechanical, acoustic, electromagnetic oscillations and the waves are viewed from a unified point of view. oscillatory motion common to all natural phenomena. Rhythmically repeating processes, for example, the beating of the heart, continuously occur inside any living organism.

Mechanical vibrationsOscillations are any physical process characterized by repeatability in time.

The roughness of the sea, the swing of the pendulum of a clock, the vibrations of the ship's hull, the beating of the human heart, sound, radio waves, light, alternating currents - all these are vibrations.

In the process of fluctuations, the values ​​of physical quantities that determine the state of the system are repeated at equal or unequal intervals. The fluctuations are called periodical, if the values ​​of changing physical quantities are repeated at regular intervals.

The smallest time interval T, after which the value of a changing physical quantity repeats (in magnitude and direction, if this quantity is vector, in magnitude and sign, if it is scalar), is called period fluctuations.

The number of complete oscillations nperformed per unit of time is called frequency fluctuations of this quantity and is denoted by ν. The period and frequency of oscillations are related by the relation:

Any oscillation is due to one or another effect on the oscillating system. Depending on the nature of the impact that causes oscillations, the following types of periodic oscillations are distinguished: free, forced, self-oscillations, parametric.

Free vibrations- these are oscillations that occur in a system left to itself, after removing it from a state of stable equilibrium (for example, oscillations of a load on a spring).

Forced vibrations- these are oscillations due to external periodic influences (for example, electromagnetic oscillations in a TV antenna).

Mechanicalfluctuations

Self-oscillations- free oscillations supported by an external source of energy, the inclusion of which at the right time is carried out by the oscillating system itself (for example, oscillations of the pendulum of a clock).

Parametric vibrations- these are oscillations, during which a periodic change in any parameter of the system occurs (for example, swinging of a swing: crouching in extreme positions and straightening in the middle position, a person on a swing changes the moment of inertia of the swing).

Oscillations that are different in nature show much in common: they obey the same laws, are described by the same equations, and are studied by the same methods. This makes it possible to create a unified theory of oscillations.

The simplest of periodic oscillations

are harmonic vibrations.

Harmonic oscillations are oscillations in the course of which the values ​​of physical quantities change over time according to the law of sine or cosine. Most oscillatory processes are described by this law or can be added as a sum of harmonic oscillations.

Another “dynamic” definition of harmonic vibrations is also possible as a process performed under the action of an elastic or “quasi-elastic”

2. periodic Oscillations are called oscillations in which an exact repetition of the process occurs at regular intervals.

Period periodic oscillation is the minimum time after which the system returns to its original state.

x - an oscillating value (for example, the current strength in the circuit, the state and the repetition of the process begins. The process occurring in one period of oscillation is called "one complete oscillation."

periodic oscillations is called the number of complete oscillations per unit time (1 second) - it may not be an integer.

T - oscillation period Period - the time of one complete oscillation.

To calculate the frequency v, you need to divide 1 second by the time T of one oscillation (in seconds) and you get the number of oscillations in 1 second or the coordinate of the point) t - time

harmonic oscillation

This is a periodic oscillation, in which the coordinate, speed, acceleration, characterizing the movement, change according to the sine or cosine law.

Harmonic waveform

The graph establishes the dependence of the displacement of the body over time. Install a pencil to the spring pendulum, behind the pendulum a paper tape that moves evenly. Or let's force the mathematical pendulum to leave a trace. A graph will appear on the paper.

The graph of a harmonic oscillation is a sine wave (or cosine wave). According to the schedule of oscillations, you can determine all the characteristics of the oscillatory movement.

Harmonic Wave Equation

The harmonic oscillation equation establishes the dependence of the body coordinate on time

The cosine graph has a maximum value at the initial moment, and the sine graph has a zero value at the initial moment. If we begin to investigate the oscillation from the equilibrium position, then the oscillation will repeat the sinusoid. If we begin to consider the oscillation from the position of the maximum deviation, then the oscillation will describe the cosine. Or such an oscillation can be described by the sine formula with an initial phase.

Change in speed and acceleration during harmonic oscillation

Not only the coordinate of the body changes with time according to the law of sine or cosine. But such quantities as force, speed and acceleration also change in a similar way. The force and acceleration are maximum when the oscillating body is in the extreme positions where the displacement is maximum, and are equal to zero when the body passes through the equilibrium position. The speed, on the contrary, in the extreme positions is equal to zero, and when the body passes the equilibrium position, it reaches its maximum value.

If the oscillation is described according to the law of cosine

If the oscillation is described according to the sine law

Maximum speed and acceleration values

After analyzing the equations of dependence v(t) and a(t), one can guess that the maximum values ​​of speed and acceleration are taken when the trigonometric factor is equal to 1 or -1. Determined by the formula

How to get dependencies v(t) and a(t)

1. Movement is called oscillatory if during movement there is a partial or complete repetition of the state of the system in time. If the values ​​of the physical quantities characterizing a given oscillatory motion are repeated at regular intervals, the oscillations are called periodic.

2. What is the oscillation period? What is the oscillation frequency? What is the connection between them?

2. The period is the time during which one complete oscillation takes place. Oscillation frequency - the number of oscillations per unit time. The oscillation frequency is inversely proportional to the oscillation period.

3. The system oscillates at a frequency of 1 Hz. What is the oscillation period?

4. At what points of the trajectory of an oscillating body is the speed equal to zero? Is the acceleration equal to zero?

4. At the points of maximum deviation from the equilibrium position, the speed is zero. The acceleration is zero at the equilibrium points.

5. What quantities characterizing the oscillatory motion change periodically?

5. Speed, acceleration and coordinate in oscillatory motion change periodically.

6. What can be said about the force that must act in an oscillatory system in order for it to perform harmonic oscillations?

6. The force must change over time according to the harmonic law. This force must be proportional to the displacement and directed opposite to the displacement towards the equilibrium position.

oscillatory processes are called in which the parameters characterizing the state of the oscillatory system have a certain repeatability in time. Such processes, for example, can be daily and annual fluctuations in the temperature of the atmosphere and the Earth's surface, oscillations of pendulums, etc.

If the time intervals after which the state of the system repeats are equal to each other, then the oscillations are called periodical, and the time interval between two successive identical states of the system is period of oscillation.

For periodic oscillations, the function that determines the state of the oscillating system is repeated after an oscillation period:

Among periodic oscillations, a special place is occupied by oscillations harmonic, i.e. oscillations in which the characteristics of the motion of the system change according to a harmonic law, for example:

(308)

The greatest attention paid in the theory of oscillations to harmonic processes that are often encountered in practice is explained both by the fact that the analytical apparatus is most well developed for them, and by the fact that any periodic oscillations (and not only periodic ones) can be considered as a certain combination of harmonic components. For these reasons, we will consider mainly harmonic oscillations below. In the analytical expression for harmonic oscillations (308), the value x of the deviation material point from the equilibrium position is called displacement.

Obviously, the maximum deviation of a point from the equilibrium position is a, this value is called oscillation amplitude. Physical quantity equal to:

and which determines the state of the oscillating system at a given moment of time, is called oscillation phase. Phase value at the time of the start from the time count

called initial phase of oscillations. The value w in the expression of the oscillation phase, which determines the speed of the oscillatory process, is called its circular or cyclic oscillation frequency.

The state of motion during periodic oscillations should be repeated at intervals equal to the oscillation period T. In this case, obviously, the oscillation phase should change by 2p (the period of the harmonic function), i.e.:

It follows that the period of oscillation and the cyclic frequency are related by the relation:

The speed of the point, the law of motion of which is determined by (301), also changes according to the harmonic law

(309)

Note that the displacement and velocity of the point do not simultaneously vanish or take on maximum values, i.e. mixing and velocity are out of phase.

Similarly, we obtain that the acceleration of a point is equal to:

It can be seen from the expression for acceleration that it is out of phase with respect to displacement and velocity. Although displacement and acceleration simultaneously pass through zero, at this point in time they have opposite directions, i.e. shifted to p. Plots of displacement, velocity and acceleration versus time for harmonic vibrations are presented on a conditional scale in Fig. 81.

Vibrations are one of the most common processes in nature and technology.

The wings of insects and birds oscillate in flight, high-rise buildings and high voltage wires under the action of the wind, the pendulum of a wound clock and a car on springs during movement, the level of the river during the year and the temperature of the human body during illness.

Sound is fluctuations in the density and pressure of air, radio waves are periodic changes in the strength of electric and magnetic fields, visible light- also electromagnetic oscillations, only with slightly different wavelength and frequency.

Earthquakes - soil vibrations, tides - changes in the level of the seas and oceans, caused by the attraction of the Moon and reaching 18 meters in some areas, pulse beats - periodic contractions of the human heart muscle, etc.

The change of wakefulness and sleep, work and rest, winter and summer... Even our daily going to work and returning home falls under the definition of fluctuations, which are interpreted as processes that repeat exactly or approximately at regular intervals.

Vibrations are mechanical, electromagnetic, chemical, thermodynamic and various others. Despite this diversity, they all have much in common and are therefore described by the same equations.

Free oscillations are called oscillations that occur due to the initial supply of energy given to the oscillating body.

In order for a body to oscillate freely, it must be brought out of equilibrium.

NEED TO KNOW

A special branch of physics - the theory of oscillations - deals with the study of the laws of these phenomena. Shipbuilders and aircraft builders, industry and transport specialists, creators of radio engineering and acoustic equipment need to know them.

The first scientists who studied oscillations were Galileo Galilei (1564...1642) and Christian Huygens (1629...1692). (It is believed that the relationship between the length of the pendulum and the time of each swing was discovered by Gallileo. One day in the church he watched how a huge chandelier was swinging, and noted the time by his pulse. Later, he discovered that the time for which one swing occurs depends on the length of the pendulum - time is halved if the pendulum is shortened by three quarters.).
Huygens invented the first pendulum clock (1657) and in the second edition of his monograph "Pendulum Clock" (1673) investigated a number of problems associated with the movement of the pendulum, in particular, found the center of swing of a physical pendulum.

A great contribution to the study of oscillations was made by many scientists: English - W. Thomson (Lord Kelvin) and J. Rayleigh, Russians - A.S. Popov and P.N. Lebedev and others

The gravity vector is shown in red, the reaction force in blue, the resistance force in yellow, and the resultant force in burgundy. To stop the pendulum, press the "Stop" button in the "Control" window or click the mouse button inside the main program window. To continue the movement, repeat the action.

Further oscillations of the thread pendulum, taken out of equilibrium, occur
under the action of the resulting force, which is the sum of two vectors: gravity
and elastic forces.
The resulting force in this case is called the restoring force.

FOUCAULT PENDULUM IN THE PARIS PANTHEON

What did Jean Foucault prove?

The Foucault pendulum is used to demonstrate the rotation of the Earth around its axis. A heavy ball is suspended on a long cable. It swings back and forth over a round platform with divisions.
After some time, it begins to seem to the audience that the pendulum is already swinging over other divisions. It seems that the pendulum has turned, but it has not. It turned with the Earth the circle itself!

For everyone, the fact of the rotation of the Earth is obvious, if only because the day replaces the night, that is, in 24 hours one complete rotation of the planet around its axis takes place. The rotation of the Earth can be proved by many physical experiments. The most famous of these was the experiment carried out by Jean Bernard Léon Foucault in 1851 at the Paris Pantheon in the presence of Emperor Napoleon. Under the dome of the building, a physicist suspended a metal ball weighing 28 kg on a steel wire 67 m long. Distinctive feature of this pendulum was that it could swing freely in all directions. A fence with a radius of 6 m was made under it, inside which sand was poured, whose surface was touched by the tip of the pendulum. After the pendulum was set in motion, it became obvious that the swing plane rotated clockwise relative to the floor. This followed from the fact that with each subsequent swing, the tip of the pendulum made a mark 3 mm further than the previous one. This deviation explains why the Earth rotates around its axis.

In 1887, the principle of the pendulum was demonstrated both in and in St. Isaac's Cathedral in St. Petersburg. Although today it cannot be seen, since now it is kept in the fund of the museum-monument. This was done in order to restore the original internal architecture of the cathedral.

MAKE A MODEL OF THE FOUCAULT PENDULUM YOURSELF

Turn the stool upside down and put some rail on the ends of its legs (diagonally). And in the middle of it, hang a small load (for example, a nut) or a thread. Make it swing so that the swing plane passes between the legs of the stool. Now slowly rotate the stool around its vertical axis. You will notice that the pendulum is swinging in the other direction. In fact, it is still rocking, and the change was due to the turn of the stool itself, which in this experiment plays the role of the Earth.

TORSIVE PENDULUM

This is Maxwell's pendulum, it allows you to identify a number of interesting patterns of motion solid body. Threads are tied to a disk mounted on an axle. If you twist the thread around the axis, the disk will rise. Now we release the pendulum, and it begins to make a periodic movement: the disk lowers, the thread unwinds. Having reached the bottom point, by inertia the disk continues to rotate, but now it twists the thread and rises up.

Typically, a torsion pendulum is used in mechanical wristwatches. The wheel-balancer under the action of the spring rotates in one direction or the other. His uniform movements ensure the accuracy of the watch.

MAKE A TWISTING PENDULUM YOURSELF

Cut out a small circle with a diameter of 6-8 cm from thick cardboard. Draw an open notebook on one side of the circle, and the number "5" on the other side. On both sides of the circle, make 4 holes with a needle and insert 2 strong threads. Secure them so that they do not pop out with knots. Next, you just need to spin the circle 20 - 30 turns and pull the threads to the sides. As a result of the rotation, you will see the picture "5 in my notebook".
Nicely?

mercury heart

A small drop is a puddle of mercury, the surface of which in its center is touched by an iron wire - a needle, filled with a weak aqueous solution of hydrochloric acid, in which the salt of potassium dichromate is dissolved .. mercury in a solution of hydrochloric acid receives electric charge and the surface tension at the boundary of the contacting surfaces decreases. When the needle comes into contact with the surface of mercury, the charge decreases and, consequently, the surface tension changes. In this case, the drop acquires a more spherical shape. The top of the drop creeps onto the needle, and then, under the action of gravity, jumps off it. Externally, the phenomenon gives the impression of shuddering mercury. This first impulse gives rise to vibrations, the drop swings and the "heart" begins to pulsate. The mercury "heart" is not a perpetual motion machine! Over time, the length of the needle decreases, and it must again be placed in contact with the mercury surface.

is one of the special cases uneven movement. There are many examples of oscillatory motion in life: swing swing, minibus swing on springs, and piston movement in the engine ... These movements are different, but they have common property: Once in a while, the movement is repeated.

This time is called period of oscillation.

Consider one of the simplest examples of oscillatory motion - a spring pendulum. A spring pendulum is a spring connected at one end to a fixed wall, and at the other end to a movable load. For simplicity, we will assume that the load can only move along the axis of the spring. This is a realistic assumption - in real elastic mechanisms, the load usually moves along the guide.

If the pendulum does not oscillate and no forces act on it, then it is in a position of equilibrium. If it is taken away from this position and released, then the pendulum will begin to oscillate - it will overshoot the equilibrium point at maximum speed and freeze at extreme points. The distance from the equilibrium point to the extreme point is called amplitude, period in this situation there will be a minimum time between visits to the same extreme point.

When the pendulum is at its extreme point, an elastic force acts on it, tending to return the pendulum to its equilibrium position. It decreases as it approaches equilibrium, and at the equilibrium point it becomes equal to zero. But the pendulum has already picked up speed and overshoots the point of equilibrium, and the force of elasticity begins to slow it down.

At the extreme points, the pendulum has the maximum potential energy, and at the equilibrium point, the maximum kinetic energy.

AT real life oscillations usually die out, as there is resistance in the medium. In this case, the amplitude decreases from oscillation to oscillation. Such fluctuations are called fading.

If there is no damping, and oscillations occur due to the initial energy reserve, then they are called free vibrations.

The bodies participating in the oscillation, and without which the oscillations would be impossible, are collectively called oscillatory system. In our case, the oscillatory system consists of a weight, a spring and a fixed wall. In general, an oscillatory system can be called any group of bodies capable of free oscillations, that is, those in which, during deviations, forces appear that return the system to equilibrium.