Quantities characterizing the oscillatory motion. Harmonic vibrations

Amplitude

Amplitude denoted by a capital letter A and measured in meters.

Definition: amplitude called the maximum displacement from the equilibrium position.

Often the amplitude is confused with the range of oscillations. A swing is when a body swings from one extreme point to another. And the amplitude is the displacement, i.e. the distance from the point of balance, from the line of balance to the extreme point at which it hit. In addition to amplitude, there is another characteristic - displacement. This is the current deviation from the equilibrium position.

A - amplitude - [m]

x - displacement - [m]

Definition: period of oscillation is the time interval during which one complete oscillation takes place.

Please note that the value of "period" is denoted by a capital letter T, it is defined as follows: - period [s]. The period is measured in seconds. Here I would like to add one more interesting thing. It consists in the fact that the more we take oscillations, the number of oscillations over a longer time, the more accurately we will determine the period of oscillations.

Frequency

Definition: The number of oscillations per unit of time is called the oscillation frequency.

Frequency - Þ [Hz]

Designated frequency Greek letter, which is read as "nu". We define the frequency, how many oscillations occurred per unit of time. The frequency is measured by the value , or. This unit is called hertz in honor of the German physicist Heinrich Hertz. Look, it is no accident that we placed two quantities - period and frequency - side by side. If you look at these quantities, you will see how they are related to each other: - period [c]. - frequency - Þ [Hz]

The period and frequency are related through the number of oscillations and the time during which this oscillation takes place. For each oscillatory system, the frequency and period are constant values. The relationship between these quantities is quite simple: .

Oscillation phase

In conclusion, consider another characteristic of oscillations - phase. We will talk about what a phase is in more detail in the senior classes. Today we have to consider with what this characteristic can be compared, contrasted and how to determine it for ourselves. It is most convenient to compare the phase of oscillations with the speed of the pendulum.

Our example shows two different pendulums. The first pendulum was deflected to the left by a certain angle, the second was also deflected to the left by a certain angle, the same as the first one. Both pendulums will make exactly the same oscillations. In this case, we can say the following, that the pendulums oscillate with the same phase, since the speeds of the pendulum are the same.

Two similar pendulums, but one is deflected to the left and the other to the right. They also have the same modulus of speed, but the direction is opposite. In this case, the pendulums are said to oscillate in antiphase.

Of course, in addition to oscillations and those characteristics that we talked about, there are other equally important characteristics of oscillatory motion. But we'll talk about them in high school.

Pendulums oscillate in phase

(with identical phases)

Pendulums swing

out of phase

HARMONIC OSCILLATIONS

Oscillations in which changes in physical quantities occur according to the cosine or sine law are called harmonic oscillations.

Graph of harmonic oscillations of the pendulum - shows the dependence of the coordinates of the pendulum on time.

KSU "Suvorovskaya high school»

(grade 9)

Prepared by: Kochutova G.A.

Lesson topic: Oscillatory motion. Basic quantities,

characterizing the oscillatory motion.

Lesson Objectives :

Formed students' ideas about oscillatory motion; to study the properties and main characteristics of periodic (oscillatory) movements. Introduce the main characteristics of the oscillatory motion.

Find out what determines the period of oscillation of a mathematical pendulum.
Develop logical thinking, speech of students, independence in the experiment.

Cultivate interest in the subject.

Lesson type: Learning new material

Teaching method: practical

Equipment: presentation, flipchat, video material

During the classes.

Organizing time.

Learning new material.

1) We divide the class into two groups (colored stickers). I remind you of the rule of working in a group.

Crossword. Make a question according to the given words.

1. The value that characterizes the speed of movement (speed);

2.Speed of change of speed (acceleration);

3.Measure of interaction of bodies (force);

4. A segment connecting the initial position with its subsequent position (moving);

5. Fall in the absence of medium resistance (free);

6. Price division of the thermometer (degree);

7. Changing the position of the body in space (movement);

8. Force directed against the movement (friction);

9. What the clock shows (time).

2) Each group gives examples of "Oscillations of bodies".

1. The conclusion must be made by the guys: the movements are repeated or the oscillatory movement is characterized by periodicity.

Demonstration of bodies that oscillate: a mathematical pendulum and a spring pendulum.

Vibrations are a very common type of movement. This is the swaying of tree branches in the wind, the vibration of strings musical instruments, the movement of the piston in the cylinder of the car engine, the swing of the pendulum in wall clock and even our heartbeats.
Consider the oscillatory movement on the example of two pendulums - mathematical and spring.
a mathematical pendulum is a ball attached to a thin, light thread. If this ball is shifted away from the equilibrium position and released, then it will begin to oscillate, i.e., make repeated movements, periodically passing through the equilibrium position.
A spring pendulum is a weight that can oscillate under the action of the elastic force of a spring.

2. conclusion: What conditions are necessary for the occurrence of oscillatory motion? First, there must be a force returning the body to its original position and the absence of friction, which is directed against the movement.

A - amplitude; T - period; v - frequency.

Oscillation amplitude is the maximum distance that an oscillating body moves away from its equilibrium position. The oscillation amplitude is measured in units of length - meters, centimeters, etc.
Oscillation period is the time it takes to complete one oscillation. The oscillation period is measured in units of time - seconds, minutes, etc.
Oscillation frequency is the number of oscillations in 1 second. The SI unit of frequency is named hertz (Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency is equal to! 1 Hz, this means that one oscillation is made for every second. If, for example, the frequency v \u003d 50 Hz, then this means that 50 oscillations are made in every second.
For the period T and frequency ν of oscillations, the same formulas are valid as for the period and frequency of revolution, which were considered in the study uniform motion around the circumference.
1. To find the period of oscillations, it is necessary to divide the time t, during which several oscillations are made, by the number n of these oscillations:

2. To find the frequency of oscillations, it is necessary to divide the number of oscillations by the time during which they occurred:

When counting the number of oscillations in practice, it should be clearly understood what constitutes one (full) oscillation. If, for example, the pendulum starts moving from position 1, then one oscillation is its movement when it, having passed the equilibrium position 0, and then extreme position 2 returns through equilibrium position 0 again to position 1.
The period and frequency of oscillations are mutually inverse quantities, i.e.

T = 1/v
In the process of oscillations, the position of the body is constantly changing. A graph of the dependence of the coordinate of an oscillating body on time is called an oscillation graph. The time t is plotted along the horizontal axis on this graph, and the x coordinate is plotted along the vertical axis. The module of this coordinate shows at what distance from the equilibrium position is the oscillating body (material point) in this moment time. When the body passes through the equilibrium position, the sign of the coordinate changes to the opposite, thereby indicating that the body is on the other side of the average position.
With sufficiently small friction and over short time intervals, the oscillation graph of each of the pendulums is a sinusoidal curve, or briefly a sinusoid.
According to the schedule of oscillations, you can determine all the characteristics of the oscillatory movement. So, for example, the graph describes oscillations with amplitude A = 5 cm, period T = 4 s and frequency ν = 1 / T = 0.25 Hz.

Fizminutka page 91.

Consolidation.

Answer the questions with average motivation (Aizhan, Zhenya, Masha):

What movement is called oscillatory?

What is body vibration?

What is the oscillation frequency? What is the unit of intent?

What is called the amplitude of oscillations?

What is called the period of oscillation?

What is the unit of measure for the period of oscillation?

What is a pendulum? What kind of pendulum is called mathematical?

Which pendulum is called a spring pendulum?

Which of the movements listed below are rolled by mechanical vibrations a) swing movement; b) the movement of the ball falling to the ground; c) the movement of a sounding guitar string?

With low motivation (Vagin A., Matyash A.): practical task: The shape of the oscillation graph can be judged on the basis of the following experiments.

Let's connect a spring pendulum to a writing device (for example, a brush) and start moving the paper tape evenly in front of the oscillating body. The brush will draw a line on the tape, which will coincide in shape with the oscillation graph.
Solve problems with high motivation (Yanna, Nurzhan, Asker): exercise 21 p. 91

Summarizing. Grading. Homework§24,25

Learning new material

Anchoring

Answered all questions 2 points

Experience 1 point

Problem solved 3 points

Total:

10-12 points score "5"

7-9 points score "4"

4-6 points score "3"

1-3 points score "2"

Group assessment sheet.

Learning new material

1. Concluded what an oscillatory movement is - 1 point

2. Made a conclusion about the condition for the occurrence of oscillatory movements - 2 points

3. They gave a definition, designation and units of measurement of the values of oscillatory motion -3 points

Anchoring

Answered all questions - 2 points

Conducted experience -1 point

Solved problems -3 points

Total:

10-12 points score - "5"

7-9 points score - "4"

4-6 points score - "3"

1-3 points score - "2"

With the help of this video tutorial, you can independently study the topic "Quantities characterizing the oscillatory motion." In this lesson, you will learn how and by what quantities oscillatory movements are characterized. The definition of such quantities as amplitude and displacement, period and frequency of oscillation will be given.

Let us discuss the quantitative characteristics of oscillations. Let's start with the most obvious characteristic - amplitude. Amplitude denoted by a capital letter A and measured in meters.

Definition

Amplitude called the maximum displacement from the equilibrium position.

Often the amplitude is confused with the range of oscillations. A swing is when a body oscillates from one extreme point to another. And the amplitude is the maximum displacement, that is, the distance from the equilibrium point, from the equilibrium line to the extreme point at which it hit. In addition to amplitude, there is another characteristic - displacement. This is the current deviation from the equilibrium position.

BUT – amplitude –

X – offset –

Rice. 1. Amplitude

Let's see how the amplitude and offset differ in an example. The mathematical pendulum is in a state of equilibrium. The line of location of the pendulum at the initial moment of time is the line of equilibrium. If you take the pendulum to the side, this will be its maximum displacement (amplitude). At any other time, the distance will not be an amplitude, but simply a displacement.

Rice. 2. Difference between amplitude and offset

Next Feature, to which we pass, is called oscillation period.

Definition

Oscillation period is the time interval during which one complete oscillation takes place.

Please note that the "period" value is denoted by a capital letter , it is defined as follows: , .

Rice. 3. Period

It is worth adding that the more we take the number of oscillations over a longer time, the more accurately we will determine the period of oscillations.

The next value is frequency.

Definition

The number of oscillations per unit time is called frequency fluctuations.

Rice. 4. Frequency

The frequency is indicated by the Greek letter, which is read as "nu". Frequency is the ratio of the number of oscillations to the time during which these oscillations occurred:.

Frequency units. This unit is called "hertz" in honor of the German physicist Heinrich Hertz. Note that period and frequency are related in terms of the number of oscillations and the time during which this oscillation takes place. For each oscillatory system, the frequency and period are constant values. The relationship between these quantities is quite simple: .

In addition to the concept of "oscillation frequency", the concept of "cyclic oscillation frequency" is often used, that is, the number of oscillations per second. It is denoted by a letter and is measured in radians per second.

Graphs of free undamped oscillations

We already know the solution to the main problem of mechanics for free oscillations - the law of sine or cosine. We also know that graphs are a powerful research tool. physical processes. Let's talk about how the graphs of the sinusoid and cosine wave will look like when applied to harmonic oscillations.

To begin with, let's define the singular points during oscillations. This is necessary in order to correctly choose the scale of construction. Consider a mathematical pendulum. The first question that arises is: which function to use - sine or cosine? If the oscillation starts from the top point - the maximum deviation, the cosine law will be the law of motion. If you start moving from the point of equilibrium, the law of motion will be the law of sine.

If the law of motion is the law of cosine, then after a quarter of the period the pendulum will be in an equilibrium position, after another quarter - in extreme point, after another quarter - again in the equilibrium position, and after another quarter it will return to its original position.

If the pendulum oscillates according to the sine law, then after a quarter of the period it will be at the extreme point, after another quarter - in the equilibrium position. Then again at the extreme point, but on the other side, and after another quarter of the period, it will return to the equilibrium position.

So, the time scale will not be an arbitrary value of 5 s, 10 s, etc., but a fraction of the period. We will build a chart in quarters of the period.

Let's move on to construction. varies either according to the law of sine or according to the law of cosine. The ordinate axis is , the abscissa axis is . The time scale is equal to quarters of the period: The chart will lie in the range from to .

Rice. 5. Dependency graphs

The graph for oscillation according to the sine law goes out of zero and is indicated in dark blue (Fig. 5). The graph for oscillation according to the law of cosine leaves the position of maximum deviation and is indicated blue color on the image. The graphs look absolutely identical, but are shifted in phase relative to each other by a quarter of a period or radians.

Dependence graphs and will have a similar look, because they also change according to the harmonic law.

Features of the oscillations of a mathematical pendulum

Mathematical pendulum is a material point of mass suspended on a long inextensible weightless thread of length .

Pay attention to the formula for the period of oscillation of a mathematical pendulum: , where is the length of the pendulum, is the acceleration free fall.

The longer the pendulum, the longer the period of its oscillations (Fig. 6). The longer the thread, the longer the pendulum swings.

Rice. 6 Dependence of the period of oscillation on the length of the pendulum

The greater the free fall acceleration, the shorter the oscillation period (Fig. 7). The greater the free fall acceleration, the stronger heavenly body attracts the weight and the faster it tends to return to the equilibrium position.

Rice. 7 Dependence of the oscillation period on the free fall acceleration

Please note that the oscillation period does not depend on the mass of the load and the oscillation amplitude (Fig. 8).

Rice. 8. The oscillation period does not depend on the oscillation amplitude

Galileo Galilei was the first to draw attention to this fact. Based on this fact, a pendulum clock mechanism is proposed.

It should be noted that the accuracy of the formula is maximum only for small, relatively small deviations. For example, for the deviation, the error of the formula is . For larger deviations, the accuracy of the formula is not so great.

Consider qualitative problems that describe a mathematical pendulum.

Task.How will the course of pendulum clocks change if they are: 1) transported from Moscow to the North Pole; 2) transport from Moscow to the equator; 3) lift high uphill; 4) take it out of the heated room into the cold.

In order to correctly answer the question of the problem, it is necessary to understand what is meant by the “running of a pendulum clock”. Pendulum clocks are based on a mathematical pendulum. If the oscillation period of the clock is less than we need, the clock will start to rush. If the oscillation period becomes longer than necessary, the clock will lag behind. The task is reduced to answering the question: what will happen to the period of oscillation of a mathematical pendulum as a result of all the actions listed in the task?

Let's consider the first situation. The mathematical pendulum is transferred from Moscow to the North Pole. We recall that the Earth has the shape of a geoid, that is, a ball flattened at the poles (Fig. 9). This means that at the Pole the magnitude of the free fall acceleration is somewhat greater than in Moscow. And since the acceleration of free fall is greater, then the period of oscillation will become somewhat shorter and the pendulum clock will start to rush. Here we neglect the fact that it is colder at the North Pole.

Rice. 9. Acceleration of free fall is greater at the poles of the Earth

Let's consider the second situation. We move the clock from Moscow to the equator, assuming that the temperature does not change. The free fall acceleration at the equator is slightly less than in Moscow. This means that the period of oscillation of the mathematical pendulum will increase and the clock starts to slow down.

In the third case, the clock is raised high uphill, thereby increasing the distance to the center of the Earth (Fig. 10). This means that the free fall acceleration at the top of the mountain is less. The period of oscillation increases the clock will be behind.

Rice. 10 Gravity is greater at the top of the mountain

Let's consider the last case. The clock is taken out warm room to frost. When the temperature drops linear dimensions bodies decrease. This means that the length of the pendulum will be slightly reduced. Since the length has become smaller, the period of oscillation has also decreased. The clock will rush.

We have considered the most typical situations that allow us to understand how the formula for the oscillation period of a mathematical pendulum works.

Figure 11 shows two identical pendulums. The first pendulum was deflected to the left by a certain angle, the second was also deflected to the left by a certain angle, the same as the first one. Both pendulums will make exactly the same oscillations. In this case, we can say that the pendulums oscillate with the same phase, since the speeds of the pendulum have the same direction and equal modules.

Figure 12 shows two similar pendulums, but one is tilted to the left and the other to the right. They also have the same velocities modulo, but the direction is opposite. In this case, the pendulums are said to oscillate in antiphase.

In all other cases, as a rule, mention is made of the phase difference.

Rice. 13 Phase difference

The phase of oscillations at an arbitrary point in time can be calculated by the formula , that is, as the product of the cyclic frequency and the time that has elapsed since the beginning of the oscillations. The phase is measured in radians.

Features of oscillations of a spring pendulum

The formula for the oscillation of a spring pendulum: . Thus, the period of oscillation of a spring pendulum depends on the mass of the load and the stiffness of the spring.

The greater the mass of the load, the greater its inertia. That is, the pendulum will accelerate more slowly, the period of its oscillations will be longer (Fig. 14).

Rice. 14 Dependence of the oscillation period on the mass

The greater the stiffness of the spring, the faster it tends to return to its equilibrium position. The period of the spring pendulum will be less.

Rice. 15 Dependence of the period of oscillation on the stiffness of the spring

Consider the application of the formula on the example of the problem.

Rice. 17 Oscillation period

If we now substitute all the necessary values \u200b\u200bin the formula for calculating the mass, we get:

Answer: weight of the weight is approximately 10 g.

Just as in the case of a mathematical pendulum, for a spring pendulum the oscillation period does not depend on its amplitude. Naturally, this is true only for small deviations from the equilibrium position, when the deformation of the spring is elastic. This fact was the basis for the construction of spring clocks (Fig. 18).

Rice. 18 Spring watch

Conclusion

Bibliography

Kikoin A.K. On the law of oscillatory motion // Kvant. - 1983. - No. 9. - S. 30-31.
Kikoin I.K., Kikoin A.K. Physics: textbook. for 9 cells. avg. school - M.: Enlightenment, 1992. - 191 p.
Chernoutsan A.I. Harmonic vibrations- ordinary and amazing // Kvant. - 1991. - No. 9. - S. 36-38.
Peryshkin A.V., Gutnik E.M. Physics. Grade 9: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300 p.

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Homework

What are mathematical and spring pendulums? What's the difference between them?
What is harmonic oscillation, oscillation period?
A weight of 200 g oscillates on a spring with a stiffness of 200 N/m. Find the total mechanical energy of oscillations and the maximum speed of movement of the load if the amplitude of oscillations is 10 cm (neglect friction).

Questions.

1. What is called the amplitude of the oscillation; period of oscillation; oscillation frequency? What letter stands for and in what units is each of these quantities measured?

The oscillation amplitude is the largest deviation of the oscillating body from the equilibrium position in absolute value. It is denoted by the letter A and in the SI system it is measured in meters (m), but it can also be measured in centimeters, as well as in degrees.
The period of oscillation is the period of time during which the body makes a complete oscillation. It is denoted by the letter T and in the SI system is measured in seconds (s).
The oscillation frequency is the number of oscillations per unit time. It is denoted by the letter ∪ (nu) and in the SI system is measured in Hertz (Hz, 1Hz = 1s -1).

2. What is one complete oscillation?

A complete oscillation is an oscillation in time T (period of oscillation).

3. What mathematical relationship exists between the period and frequency of oscillation?

4. How do they depend: a) frequency; b) the period of free oscillations of the pendulum on the length of its thread?

a) the oscillation frequency of the pendulum ∪ decreases with increasing length of the thread l; b) the period T of the pendulum oscillation increases with the length of the thread l.

5. What is called the natural frequency of an oscillatory system?

The frequency of free oscillations is called the natural frequency of the oscillatory system. For example, if the weight of a thread pendulum is deflected from the equilibrium position and released, then it will oscillate with its own frequency, but if the weight is given a certain, non-zero speed, then it will oscillate with a different frequency.

6. How are the speeds of two pendulums directed relative to each other at any moment of time if these pendulums oscillate in opposite phases? in the same phase?

If the pendulums oscillate in opposite phases, then at any moment of time their velocities will be directed oppositely to each other, and vice versa, if they oscillate in the same phases, then their velocities are co-directed.

Exercises.

1. Figure 58 shows pairs of oscillating pendulums. In what cases do two pendulums oscillate: in the same phases with respect to each other? in opposite phases?

The system b) oscillates in identical phases. In opposite phases a), c), d).

2. The oscillation frequency of a hundred-meter railway bridge is 2 Hz. Determine the period of these oscillations.

3. Period of vertical oscillations rail car equals 0.5 s. Determine the oscillation frequency of the car.

4. Needle sewing machine makes 600 complete oscillations in one minute. What is the oscillation frequency of the needle, expressed in hertz?

5. The amplitude of the oscillations of the load on the spring is 3 cm. What distance from the equilibrium position will the load pass in 1/4 T, 1/2 T, 3/4 T, T?

6. The amplitude of the load oscillations on the spring is 10 cm, the frequency is 0.5 Hz. What is the distance traveled by the load in 2 s?

7. The horizontal spring pendulum, shown in Figure 49, performs free vibrations. Which quantities that characterize this movement (amplitude, frequency, period, speed, force, under the action of which oscillations occur) are constant, and which are variables? (Disregard friction).

Constant values are - amplitude, frequency, period. The variables are speed and strength.

fluctuations called movements or processes that are characterized by a certain repetition in time.

Free (natural) vibrations oscillations are called that occur in the absence of variable external influences on an oscillatory system and arise as a result of any initial deviation of this system from a state of stable equilibrium; vibrations that are made due to the initially communicated energy with the subsequent absence of external influences on the oscillatory system.

compelled oscillations that occur in any system under the influence of a variable external influence are called.

Oscillation period (T) - the smallest period of time after which the oscillating system returns to the same state in which it was at the initial arbitrarily chosen moment.

Oscillation frequency is the number of complete oscillations per unit time. ν=1/T.

Oscillation amplitude is the maximum value of the fluctuating quantity.

Oscillation phase is the value of the fluctuating quantity at an arbitrary moment of time (ω 0 t+φ).

The most important quantities characterizing mechanical vibrations are:

number of vibrations for some period of time t. Denoted by letter N;

coordinate material point or its bias(deviation) - a value that characterizes the position of the oscillating point at time t relative to the equilibrium position and is measured by the distance from the equilibrium position to the position of the point at a given time. Denoted by letter x, measured in meters(m);

amplitude- the maximum displacement of a body or system of bodies from an equilibrium position. Denoted by letter A or x max , measured in meters(m);

period is the time it takes to complete one complete oscillation. Denoted by letter T, measured in seconds(with);

frequency is the number of complete oscillations per unit time. Denoted by the letter ν, measured in hertz(Hz);

cyclic frequency, the number of complete oscillations of the system during 2π seconds. Denoted by the letter ω, measured in radians per second(rad/s);

phase- argument of a periodic function that determines the value of a physical quantity at any time t. Denoted by the letter φ, measured in radians(glad);

initial phase- the argument of the periodic function, which determines the value of the physical quantity at the initial moment of time ( t= 0). Denoted by the letter φ 0, measured in radians(glad).

These quantities are interconnected by the following relationships:

T=tN, ν =1T=Nt,

ω =2π ⋅ν =2πT, φ =ω ⋅t+φ 0.

Harmonic vibrations

Harmonic vibrations- these are oscillations in which the coordinate (displacement) of the body changes with time according to the cosine or sine law and is described by the formulas:

x=A sin( ω ⋅t+φ 0) or x=A cos( ω ⋅t+φ 0).

Coordinate versus time x(t) is called kinematic law of harmonic oscillation(law of motion).

Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).

Let the body perform harmonic oscillations according to the law x=A⋅ cos ω ⋅t(φ 0 = 0). Figure 2, a shows a graph of the dependence of the coordinate x from time t.

Let us find out how the projection of the velocity of an oscillating point changes with time. To do this, we find the time derivative of the law of motion:

υx=x′=( A⋅ cos ω ⋅t)′=− ω ⋅A⋅sin ω ⋅t=ω ⋅A cos( ω ⋅t+π 2),

where ω ⋅A=υx max - amplitude of velocity projection on the axis x.

This formula shows that during harmonic oscillations, the projection of the body velocity on the axis x also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by π/2 (Fig. 2, b).

To find out the dependence of the acceleration a x (t) we find the time derivative of the velocity projection:

ax=υ ′ x=x′′=( A⋅ cos ω ⋅t)′′=(− ω ⋅A⋅sin ω ⋅t)′= =− ω 2⋅A⋅ cos ω ⋅t=ω 2⋅A cos( ω ⋅t+π ), (1)

where ω 2⋅A=ax max - acceleration projection amplitude on the axle x.

For harmonic oscillations, the acceleration projection leads the phase shift by π (Fig. 2, c).

Similarly, you can build dependency graphs x(t), υ x (t) and a x (t), if x=A⋅sin ω ⋅t(φ 0 = 0).

Given that A⋅ cos ω ⋅t=x, from equation (1) for acceleration we can write

ax=−ω 2⋅x,

those. for harmonic oscillations, the acceleration projection is directly proportional to the displacement and opposite in sign to it, the acceleration is directed in the direction opposite to the displacement. This relation can be rewritten as

ax+ω 2⋅x=0.

The last equality is called equation of harmonic oscillations.

A physical system in which harmonic oscillations can exist is called harmonic oscillator, and the equation of harmonic oscillations - harmonic oscillator equation.

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Quantities characterizing the oscillatory motion. Harmonic vibrations

Harmonic vibrations

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