The main quantities characterizing the oscillatory motion. oscillatory motion

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

The next feature we move on to 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 tool for studying 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 the equilibrium position, after another quarter - at the 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 cosine law leaves the position of maximum deviation and is indicated in blue in the figure. 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 of 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 acceleration of free fall, the stronger the celestial 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 of the warm room into the cold. As the temperature decreases, the linear dimensions of the 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 examined the most typical situations that allow us to understand how the formula for the oscillation period of a mathematical pendulum works.

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.

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

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.

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).

Let us compare the oscillations of two identical pendulums shown in Figure 58. The first pendulum oscillates with a large swing, that is, its extreme positions are farther from the equilibrium position than that of the second pendulum.

Rice. 58. Oscillations of pendulums occurring with different amplitudes

The largest (modulo) deviation of an oscillating body from the equilibrium position is called the oscillation amplitude

We will consider oscillations that occur with small amplitudes (Fig. 59), at which the length of the arc AB can be considered equal to the segment AB and even the half-chord CB. Therefore, the amplitude of oscillations of a thread pendulum can be understood as an arc or any of these segments. So, the amplitude of oscillations of the first pendulum (see Fig. 58) is equal to 0 1 A 1 or 0 1 B 1, and the second - 0 2 A 2 or O 2 B 2. The amplitude is denoted by the letter A and in SI it is measured in units of length - meters (m), centimeters (cm), etc. The amplitude can also be measured in units of a flat angle, for example, in degrees, since a certain central angle corresponds to the arc of a circle, i.e. angle with a vertex at the center of the circle (in this case at point O).

Rice. 59. For oscillations with a small amplitude, the length of the arc AB is equal to the segment AB

The amplitude of oscillation of the spring pendulum (see Fig. 53) is equal to the length of the segment OB or OA.

An oscillating body makes one complete oscillation if a path equal to four amplitudes passes from the beginning of the oscillations. For example, having moved from point O 1 to point B 1, then to point A 1 and again to point O 1 (see Fig. 58), the ball makes one complete oscillation.

The period of time during which the body makes one complete oscillation is called the period of oscillation.

The period of oscillation is denoted by the letter T and in SI is measured in seconds (s).

We hang two identical balls on threads of different lengths and bring them into oscillatory motion. We will see that in the same period of time a short pendulum will make more oscillations than a long one.

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

The frequency is denoted by the Greek letter v (“nu”). The unit of frequency is one oscillation per second. This unit is named hertz (Hz) in honor of the German scientist Heinrich Hertz.

Let's say that in one second the pendulum makes two oscillations, i.e., the frequency of its oscillations is 2 Hz. To find the oscillation period, it is necessary to divide one second by the number of oscillations in this second, i.e., by the frequency:

Thus, the oscillation period T and the oscillation frequency v are related by the following relationship:

Using the example of oscillations of pendulums of different lengths, we come to the conclusion that the frequency and period of free oscillations of a filament pendulum depend on the length of its filament. The longer the pendulum thread, the longer the period of oscillation and the lower the frequency.

Free oscillations in the absence of friction and air resistance are called natural oscillations, and their frequency is the natural frequency of the oscillatory system

Not only a filament pendulum, but also any other oscillatory system has a certain natural frequency, which depends on the parameters of this system. For example, the natural frequency of a spring pendulum depends on the mass of the load and the stiffness of the spring.

Consider the oscillations of two identical pendulums (Fig. 60). At the same time, the left pendulum from the leftmost position starts moving to the right, and the right pendulum from the rightmost position moves to the left. Both pendulums oscillate with the same frequency (because the lengths of their threads are equal) and with the same amplitudes. However, these oscillations differ from each other: at any moment of time, the speeds of the pendulums are directed in opposite directions. In this case, the pendulums are said to oscillate in opposite phases.

Rice. 60. Oscillations of pendulums occurring in opposite phases

The pendulums shown in Figure 58 also oscillate with the same frequencies. The velocities of these pendulums are directed in the same direction at any moment of time. In this case, the pendulums are said to oscillate in the same phases.

Let's consider one more case. At the moment shown in Figure 61, a, the velocities of both pendulums are directed to the right. But after a while (Fig. 61, b) they will be directed in different directions. In this case, the oscillations are said to occur with a certain phase difference.

Rice. 61. Oscillations of pendulums occurring with a certain phase difference

A physical quantity called phase is used not only when comparing the vibrations of two or more bodies, but also to describe the vibrations of one body.

The formula for determining the phase at any given time will be covered in high school.

Thus, oscillatory motion is characterized by amplitude, frequency (or period) and phase.

Questions

What is called the amplitude of oscillations; period of oscillation; oscillation frequency? In what units is each of these quantities measured?
What is the mathematical relationship between the period and frequency of oscillations?
How do they depend: a) frequency; b) the period of free oscillations of the pendulum on the length of its thread?
What vibrations are called natural?
What is the natural frequency of an oscillating system called?

Exercise 24

Figure 62 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 oscillation frequency of a hundred-meter railway bridge is 2 Hz. Determine the period of these oscillations.
The period of vertical oscillations of a railway car is 0.5 s. Determine the oscillation frequency of the car.
The needle of the sewing machine makes 600 complete oscillations per minute. What is the oscillation frequency of the needle?
The amplitude of oscillations of the load on the spring is 3 cm. Which way from the equilibrium position will the load pass in a time equal to - ¼T; - ½T; - ¾T; - T?
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 seconds?

Exercise

Design an experiment involving magnetic forces that simulate an increase in free fall acceleration and act on an oscillating filament pendulum. Carry out this experiment and draw a conclusion about the qualitative dependence of the oscillation period on the free fall acceleration.

Consider the following figure:

It features two identical pendulums. As can be seen from the figure, the first pendulum oscillates with a larger swing than the second. That is, in other words, the extreme positions that the first pendulum occupies are at a greater distance from each other than that of the second pendulum.

Amplitude

Oscillation amplitude- the largest deviation of the oscillating body from the equilibrium position in absolute value.

Usually, the letter A is used to denote the amplitude of vibrations. The units of measurement of the amplitude are the same as the units of measurement of length, that is, they are meters, centimeters, etc. In principle, the amplitude can be written in units of a plane angle, since each arc of a circle will correspond to a single central angle.

It is said that an oscillating body makes one complete oscillation when it travels a path equal to four amplitudes.

Oscillation period

Oscillation period is the time it takes for the body to make one complete oscillation.

The oscillation period is denoted by the letter T. The units of the oscillation period T are seconds.

If we hang two identical balls on threads of different lengths, and bring them into oscillatory motion, we will notice that in the same intervals of time, they will make a different number of oscillations. A ball suspended from a short string will oscillate more than a ball suspended from a long string.

Oscillation frequency

Oscillation frequency called the number of oscillations that were made in a unit of time.

The oscillation frequency is denoted by the letter ν (read as "nu"). The units of oscillation frequency are called Hertz. One hertz means one oscillation per second.

The period and frequency of oscillations are interconnected by the following relationship:

The frequency of free oscillations is called the natural frequency of the oscillatory system. Each system has its own oscillation frequency.

Oscillation phase

There is also such a thing as the phase of oscillations. Two pendulums can have the same oscillation frequency, but at the same time they can oscillate in different phases, that is, their speeds at any time will be directed in opposite directions.

If the speeds of the pendulums at any moment of time are directed in the same direction, then they say that the pendulums oscillate in the same phases of oscillation.

Pendulums can also oscillate with a certain phase difference, in which case at some points in time the direction of their velocities will coincide, and at others not.