Lesson topic: Oscillatory motion. Harmonic vibrations. Amplitude, period, frequency, phase of oscillations. The equation of harmonic oscillations. Plan-summary of a lesson in physics. Harmonic vibrations Lesson oscillatory motion harmonic vibrations

The purpose and objectives of the lesson:

educational : the formation of students' knowledge about oscillatory motion, harmonic oscillation, the equation of harmonic oscillations; concepts: amplitude, period, frequency, phase of oscillations;

educational: contribute to the formation cognitive interest, the scientific outlook of students through the study of concepts oscillating motion, harmonic oscillation, amplitude, period, frequency, phase of oscillations;

developing: the development of logical thinking of students to operate with the concepts of oscillatory movement, harmonic oscillation, amplitude, period, frequency, phase of oscillations.

Leading idea of the lesson: any process that has the property of repeatability in time is called.

periodic motionis called a movement in which physical quantities, describing this movement, take the same values at regular intervals. fluctuations

Lesson type: learning lesson.

Lesson form: rock lecture.

Teaching methods: verbal.

Used literature, electronic sources:

1) . Collection of problems in physics. M. "Enlightenment", 1994

For example, a mechanical oscillatory movement is the movement of a small body suspended on a thread, a load on a spring, a piston in a car engine cylinder. Fluctuations can be not only mechanical, but also electromagnetic (periodic changes in voltage and current in the circuit), thermodynamic (temperature fluctuations day and night).

In this way, fluctuations- this is a special form of motion, in which physical processes that are heterogeneous in nature are described by the same dependences of physical quantities on time.

Necessary conditions for the existence of oscillations in the system:

Quantities characterizing mechanical vibrations:

1) x(t) - coordinate of the body (displacement of the body from the equilibrium position) at time t:

x= f(t), f(t)= f(t + T),

where f(t) - given periodic function of time t,

T is the period of this function.

2) A (A >0) xmax

3) T- period - the duration of one complete oscillation, i.e., the smallest period of time after which the values of all physical quantities characterizing the oscillation are repeated.

4) ν - frequency - the number of complete oscillations per unit time.

[ν] = 1 s-1 = 1 Hz.

t, equal to 2π seconds:

ω= 2πν= 2π/T,

[ω] = 1 rad/s.

6) φ= ωt+ φ0 - phase - an argument of a periodic function that determines the value of a changing physical quantity at a given time t.

[φ] = 1 rad ( radian)

Oscillations are called harmonic, in which the dependence of the coordinate (displacement) of the body on time is described by the formulas:

The kinematic law of harmonic oscillations (the law of motion) is the dependence of the coordinate on time x(t) , allows you to determine the position of the body, its speed, acceleration at an arbitrary point in time.

A harmonic oscillatory system or a one-dimensional harmonic oscillator is a system (body) that performs harmonic oscillations described by the equation:

ax(t) + ω2х(t) = 0.

With harmonic oscillations, the projection of the acceleration of a point is directly proportional to its displacement from the equilibrium position and is opposite in sign to it.

fluctuations material point are harmonic if they occur under the action of a restoring force, the modulus of which is directly proportional to the displacement of the point from the equilibrium position:

where k is a constant coefficient.

The "-" sign in the formula reflects the return nature of the force.

The equilibrium position corresponds to the point x=0, while the restoring force is equal to zero ().

Homework 1 minute.

Lesson summary 2 min.

It should be noted Good work individual students, point to difficult moments that emerged during the explanation new topic. Based on the results of the work, draw a conclusion about the formed knowledge, set marks .

Student abstract.

Lesson topic: Oscillatory motion. Harmonic vibrations. Amplitude, period, frequency, phase of oscillations. The equation of harmonic oscillations.

Oscillatory movement (oscillations) any process that has the property of repeatability in time is called.

Periodic motion - this is a movement in which the physical quantities describing this movement take on the same values at regular intervals.

fluctuations- this is a special form of motion, in which physical processes that are heterogeneous in nature are described by the same dependences of physical quantities on time.

1) the presence of a force tending to return the body to the equilibrium position with a small displacement from this position;

2) the smallness of friction that prevents vibrations.

1) x(t) - coordinate of the body (displacement of the body from the equilibrium position) at time t. x= f(t), f(t)= f(t + T).

2) A (A >0) - amplitude - maximum displacement of the body xmax or system of bodies from the equilibrium position.

3) T- period - the duration of one complete oscillation. [T] = 1s.

4) ν - frequency - the number of complete oscillations per unit time. [ν] = 1 s-1 = 1 Hz.

5) ω - cyclic frequency - the number of complete oscillations over a period of time Δ t, equal to 2π seconds: ω= 2πν= 2π/T,

[ω] = 1 rad/s.

6) φ= ωt+ φ0 - phase - an argument of a periodic function that determines the value of a changing physical quantity at time t. [φ] = 1 rad.

7) φ0 - the initial phase, which determines the position of the body at the initial moment of time (t0 = 0).

Harmonic oscillations are called, in which the dependence of the coordinate (displacement) of the body on time is described by the formulas:

x(t) = xmaxcos(ωt + φ0) or x(t) = xmaxsin(ωt + φ0).

or one-dimensional harmonic oscillator called a system (body) that performs harmonic oscillations described by the equation:

ax(t) + ω2х(t) = 0.

Board.

Lesson topic: Oscillatory motion. Harmonic vibrations. Amplitude, period, frequency, phase of oscillations. The equation of harmonic oscillations.

Oscillatory movement (oscillations)

Periodic motion - this

fluctuations- this

Necessary conditions for the existence of oscillations in the system:

Quantities characterizing mechanical vibrations:

1) x(t) - x= f(t), f(t)= f(t + T).

2) A (A >0) - amplitude -

3) T- period -

4) ν - frequency -

[ν] = 1 s-1 = 1 Hz.

5) ω - cyclic frequency -

ω= 2πν= 2π/T,

[ω] = 1 rad/s.

6) φ= ωt+ φ0 - phase -

[φ] = 1 rad.

7) φ0 - initial phase -

Harmonic are called fluctuations

x(t) = xmaxcos(ωt + φ0) or x(t) = xmaxsin(ωt + φ0).

Harmonic oscillatory system or one-dimensional harmonic oscillator

ax(t) + ω2х(t) = 0.

Lesson type: a lesson in the formation of new knowledge.

Lesson Objectives:

formation of ideas about oscillations as physical processes;
clarification of the conditions for the occurrence of oscillations;
formation of the concept of harmonic oscillation, characteristics of the oscillatory process;
formation of the concept of resonance, its application and methods of dealing with it;
the formation of a sense of mutual assistance, the ability to work in groups, pairs;
development of independent thinking

Equipment: spring and mathematical pendulum and, a projector, a computer, a teacher's presentation, a disk "Library of visual aids", a sheet of knowledge acquisition by students, cards with symbols of physical quantities, the text "Resonance Phenomenon".

On each table is a sheet of learning for each student, a text about the phenomenon of resonance.

During the classes

I. Motivation.

Teacher: To understand what the lesson will be about today, read an excerpt from the poem “Morning” by N.A. Zabolotsky

Born of the desert
The sound oscillates
fluctuates blue
Spider on a thread.
The air oscillates
Transparent and pure
In shining stars
The leaf is shaking.

So today we're going to talk about fluctuations. Think and name where fluctuations occur in nature, in life, in technology.

Students name different examples of vibrations(slide 2).

Teacher: What do all these movements have in common?

Students: These movements are repeated (slide 3).

Teacher: Such movements are called oscillations. Today we will talk about them. Write down the topic of the lesson (slide 4).

II. Updating knowledge and learning new material.

Teacher: We should:

Find out what is fluctuation?
Conditions for the occurrence of oscillations.
Types of vibrations.
Harmonic vibrations.
Characteristics of harmonic oscillation.
Resonance.
Problem solving (slide 5).

Teacher: Look at the oscillations of the mathematical and spring pendulums (oscillations are demonstrated). Are the vibrations exactly repeated?

Students: No.

Teacher: Why? It turns out that the friction force interferes. So what is hesitation? (slide 6)

Students: Oscillations are movements that repeat exactly or approximately over time.(slide 6, click). The definition is written in a notebook.

Teacher: Why do the fluctuations continue for so long? (slide 7) On a spring and mathematical pendulum, the transformation of energy during oscillations is explained with the help of students.

Teacher: Let us find out the conditions for the occurrence of oscillations. What does it take to start fluctuations?

Students: You need to push the body, apply force to it. In order for the oscillations to last for a long time, it is necessary to reduce the friction force (slide 8), the conditions are written in a notebook.

Teacher: There are a lot of fluctuations. Let's try to classify them. Forced oscillations are demonstrated, on spring and mathematical pendulums - free oscillations (slide 9). Students write down the types of vibrations in a notebook.

Teacher: If the external force is constant, then the oscillations are called automatic (mouse click). Students in a notebook write down the definitions of free (slide 10), forced (slide 10, mouse click), automatic oscillations (slide 10 with a mouse click).

Teacher: There are also damped and undamped oscillations (slide 11 with a mouse click). Damped oscillations are oscillations that, under the action of friction or resistance forces, decrease over time (slide 12), these oscillations are shown on the graph on the slide.

Continuous oscillations are oscillations that do not change with time; friction forces, no resistance. To maintain undamped oscillations, an energy source is needed (slide 13), these oscillations are shown on the graph on the slide.

Examples of fluctuations are given (slide 14).

1 option writes out examples damped vibrations.

Option 2 writes out examples undamped vibrations.

fluctuations of leaves on trees during the wind;
heartbeat;
swing swings;
fluctuation of the load on the spring;
rearrangement of legs when walking;
the vibration of the string after it is taken out of equilibrium;
vibrations of the piston in the cylinder;
oscillation of a ball on a thread;
swaying grass in a field in the wind;
hesitation vocal cords;
vibrations of the wiper blades (wipers in the car);
swings of the sweeper's broom;
vibrations of the sewing machine needle;
vibrations of the ship on the waves;
swinging arms while walking;
phone membrane vibrations.

students among the given oscillations, examples of free and forced oscillations are written out according to the options, then they exchange information, work in pairs (slide 15). They also perform tasks on dividing into damped and undamped oscillations in the same examples, then exchange information, work in pairs.

Teacher: You see that all free vibrations are damped, and forced vibrations are undamped. Find automatic oscillations among the given examples. Students rate themselves on the learning sheet in paragraph 1 of the learning sheet ( Attachment 1)

Teacher: Among all types of oscillations, a special type of oscillations is distinguished - harmonic.

The manual "Library of visual aids" demonstrates a model of harmonic oscillations (mechanics, model 4 harmonic oscillations) (slide 16).

What mathematical function is plotted on the model?

Students: This is a graph of the sine and cosine function (slide 16 with a mouse click).

students write down the equations of harmonic oscillations in a notebook.

Teacher: Now we need to consider each quantity in the harmonic equation. (Displacement X is shown on the mathematical and spring pendulums) (slide 17). X-displacement - deviation of the body from the equilibrium position. What is the unit of displacement?

Students: Meter (slide 17, mouse click).

Teacher: On the oscillation graph, determine the offset at times 1 s, 2 s, 3 s, 4 s, 5 s, 6 s, and so on. (slide 17, click). The next value is X max. What's this?

Students: Maximum offset.

Teacher: The maximum offset is called the amplitude (slide 18, mouse click).

students on the graphs, the amplitude of damped and undamped oscillations is determined (slide 18, mouse click).

Teacher: Before considering the next value, let us recall the concepts of quantities studied in the 1st course. Let's count the number of oscillations on a mathematical pendulum. Is it possible to determine the time of one oscillation?

Students: Yes.

Teacher: The time of one complete oscillation is called the period - T (slide 19, mouse click). Measured in seconds (slide 19, mouse click). You can calculate the period using the formula if it is very small (slide 19, mouse click). Points are marked with different colors on the graph.

students on the chart, the period is determined by finding it between points of different colors.

Teacher on a mathematical pendulum demonstrates different frequencies for different lengths of the pendulum. Frequency v- the number of complete oscillations per unit of time (slide 20).

The unit of measurement is Hz (slide 20 mouse click). There are relationship formulas between period and frequency. ν=1/T T=1/ν (slide 20 mouse click).

Teacher: The sine and cosine function repeats through 2π. Cyclic (circular) frequency ω(omega) oscillations is the number of complete oscillations that occur in 2π units of time (slide 21). Measured in rad/s (slide 21, mouse click) ω=2 πν (slide 21, click).

Teacher: Oscillation phase- (ωt + φ 0) is the value under the sine or cosine sign. Measured in radians (rad) (slide 22).

The oscillation phase at the initial time (t=0) is called initial phase - φ 0 . Measured in radians (rad) (slide 21, mouse click).

Teacher: And now we repeat the material.

a) Students are shown cards with values, they name these values. ( Appendix 2)

b) Students are shown cards with units of measurement of physical quantities. You need to name these values.

c) Each four students are given a card with some value, you need to tell everything about it according to the plan on slide 23. Then the groups change cards with values and perform the same task.

students give themselves grades on the progress sheet (paragraph 2 of Appendix 1)

Teacher: Today we worked with spring and mathematical pendulums, the formulas for the periods of these pendulums are calculated using formulas. On a mathematical pendulum, it demonstrates periods of oscillation at different lengths of the pendulum.

students find out that the period of oscillation depends on the length of the pendulum (slide 24)

Teacher on a spring pendulum demonstrates the dependence of the period of oscillation on the mass of the load and the stiffness of the spring.

students find out that the period of oscillation depends on the mass in direct proportion and on the stiffness of the spring inversely proportional (slide 25)

Teacher: How do you push a car out if it's stuck?

Students: It is necessary to rock the car together on command.

Teacher: Right. In doing so, we use physical phenomenon called resonance. Resonance occurs only when the frequency of natural oscillations coincides with the frequency of the driving force. Resonance is a sharp increase in the amplitude of forced oscillations (slide 26). The Visual Aids Library demonstrates a resonance model (Mechanics, Model 27 "Swinging a Spring Pendulum" at >2Hz).

For students it is proposed to mark the text about the influence of resonance. While the work is being done, Beethoven's Moonlight Sonata and Tchaikovsky's Flower Waltz ( Appendix 4). The text is marked with the following signs (they are on the stand in the office): V - interested; + knew; - did not know; ? - I would like to know more. The text remains with each student in a notebook. In the next lesson, you need to return to it and answer students' questions if they do not find answers at home.

III. Fixing the material.

takes place in the form of tasks (slide 27). The problem is discussed at the blackboard.

For students it is proposed to independently solve problems according to the options on the progress sheets (slide 28) As a result of work in the lesson, the teacher gives an overall grade.

IV. Lesson results.

Teacher: What new did you learn at the lesson today?

V. Homework.

Everyone learn the lesson summary. Solve the problem: according to the equation of harmonic oscillation, find everything that is possible (slide 29). Find answers to questions while marking text. Those who wish can find material about the benefits of resonance and the dangers of resonance (you can make a message, an abstract, prepare a presentation).

LESSON 2/24

Topic. Harmonic vibrations

The purpose of the lesson: to acquaint students with the concept of harmonic oscillations.

Type of lesson: lesson learning new material.

LESSON PLAN

Knowledge control		1. Mechanical vibrations. 2. Main characteristics of vibrations. 3. Free vibrations. Conditions for the occurrence of free oscillations
Demonstrations		1. Free vibrations of a load on a spring. 2. Recording of oscillatory motion
Learning new material		1. The equation of the oscillatory motion of a load on a spring. 2. Harmonic vibrations
Consolidation of the studied material		1. Qualitative questions. 2. Learn to solve problems

STUDY NEW MATERIAL

In many oscillatory systems, with small deviations from the equilibrium position, the modulus of rotational force, and hence the modulus of acceleration, is directly proportional to the modulus of displacement relative to the equilibrium position.

Let us show that in this case the displacement depends on time according to the cosine (or sine) law. To this end, we analyze the oscillations of the load on the spring. Let us choose as the origin the point where the center of mass of the load on the spring is in the equilibrium position (see figure).

If a load of mass m is displaced from the equilibrium position by x (for the equilibrium position x = 0), then the elastic force Fx = - kx acts on it, where k is the spring stiffness (the “-” sign means that the force is directed at any time in the direction opposite to the offset).

According to Newton's second law Fx = m ah. Thus, the equation describing the movement of the load has the form:

Denote ω2 = k / m . Then the equation of the movement of the load will look like:

An equation of this kind is called differential equation. The solution to this equation is the function:

Thus, for the vertical displacement of the load on the spring from the equilibrium position, it will oscillate freely. The coordinate of the center of mass in this case changes according to the cosine law.

It is possible to verify that oscillations occur according to the law of cosine (or sine) by experiment. It is advisable for students to show a record of the oscillatory movement (see figure).

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

Free vibrations of a load on a spring are an example of mechanical harmonic vibrations.

Let at some point in time t 1 the coordinate of the oscillating load be x 1 = xmax cosωt 1 . According to the definition of the oscillation period, at time t 2 \u003d t 1 + T, the coordinate of the body must be the same as at time t 1, that is, x2 \u003d x1:

The period of the function cosωt is equal to 2, therefore, ωТ = 2, or

But since T \u003d 1 / v, then ω \u003d 2 v, that is, the cyclic oscillation frequency ω is the number of complete oscillations made in 2 seconds.

QUESTION TO STUDENTS DURING THE PRESENTATION OF NEW MATERIAL

First level

1. Give examples of harmonic oscillations.

2. The body performs undamped oscillations. Which of the quantities characterizing this movement are constant, and which ones change?

Second level

How do the force acting on the body, its acceleration and speed change during the implementation of harmonic oscillations?

CONFIGURATION OF THE STUDYED MATERIAL

1. Write the equation of a harmonic oscillation if its amplitude is 0.5 m and the frequency is 25 Hz.

2. Fluctuations of the load on the spring are described by the equation x \u003d 0.1 sin 0.5. Determine the amplitude, circular frequency and oscillation frequency.

The topic "Graph of harmonic oscillation" is considered in the 1st course in the process of development academic discipline"Algebra and the Beginnings of Analysis". This topic ends the consideration of the chapter “Trigonometric functions”. The purpose of this lesson is not only to learn how to plot a harmonic oscillation, but also to show the connection of this mathematical object with the phenomena of the real world. Therefore, it is advisable to consider this topic together with a physics teacher.

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Ministry of Education, Science and Youth Policy

Trans-Baikal Territory

State educational institution

initial vocational education

"Professional School No. 1"

Methodological development of an integrated lesson

algebra and physics on the topic:

"Harmonic vibrations"

Compiled by:

physics teacher M.G. Greshnikov

Mathematics teacher L.G. Izmailova

Chita, 2014

Explanatory note

Brief description of the lesson.The topic “Graph of harmonic oscillation” is considered in the 1st year in the process of mastering the academic discipline “Algebra and the beginning of analysis”. This topic ends the consideration of the chapter “Trigonometric functions”. The purpose of this lesson is not only to learn how to plot a harmonic oscillation, but also to show the connection of this mathematical object with the phenomena of the real world. Therefore, it is advisable to consider this topic together with a physics teacher.

At the beginning of the lesson, students recall physical processes and phenomena in which oscillations occur (the work is accompanied by a presentation). Consolidation of knowledge in physics is offered in the form of a game, the purpose of which is to repeat physical meaning quantities included in the harmonic oscillation equation, and then the mathematical rules for transforming the graphs of trigonometric functions are repeated using compression (stretching) and parallel transfer. At the end of the lesson there is independent work educational nature with subsequent peer review. The lesson ends with a message from the student, who, using a video clip, introduces students to the Foucault pendulum.

Lesson Objectives:

- educational:generalize and systematize the knowledge of students about harmonic oscillations; to teach students to obtain equations and build graphs of the resulting functions; create a mathematical model of harmonic oscillations;

Developing: develop memory, logical thinking; to form communication skills, develop oral speech;

Educational:to form a culture of mental work; create a situation of success for each student; develop the ability to work in a team.

Lesson type: generalization and systematization of knowledge.

Lesson methods: partially exploratory, explanatory and illustrative.

Interdisciplinary connections:physics, mathematics, history.

Visibility and TCO:laptop, projector and screen, presentation for the lesson, task cards for the game "One for all and all for one",cards to complete independent work.

The relevance of using ICT in the classroom:

visibility;
little time spent explaining;
novelty of presentation of information;
optimization of the teacher's work in preparation for the lesson;
establishment of interdisciplinary connections;
involvement of students in the presentation of the practical side of the lesson in question;
the possibility of showing the experiments conducted by students in preparation for the lesson in the recording.

Time: 90 minutes.

Literature:

1. Maron A.E., Maron E.A. Physics. Didactic materials. -

2. Mordkovich A.G. Algebra and the beginnings of analysis. Textbook for 10-11 grades. -

3. Myakishev G.Ya., Bukhovtsev B.B. Physics 10. Textbook. -

4. Stepanova G.I. Collection of problems in physics for grades 10-11. -

During the classes

1. Organizational moment.

2. Motivation and stimulation of cognitive activity.

slide 1

Physics teacher.I would like to start today's lesson with an epigraph: "All our previous experience leads to the conviction that nature is the realization of what is mathematically easiest to imagine" A. Einstein.

Slide 2. The task of physics is to reveal and understand the connection between the observed phenomena and to establish the relationship between the quantities that characterize them. A quantitative description of the physical world is impossible without mathematics.

Mathematics teacher.Mathematics creates methods of description corresponding to the nature of the physical problem, gives ways to solve the equations of physics.

Physics teacher.Back in the 18th century A. Volta (Italian physicist , chemist And physiologist , one of the founders of the doctrine ofelectricity ; Count Alessandro Giuseppe Antonio Anastasio Gerolamo Umberto Volta) said: “What good can be done, especially in physics, if not to reduce everything to measure and degree?”

Mathematics teacher. Mathematical constructions in themselves are not related to the properties of the surrounding world, these are purely logical constructions. They become meaningful only when applied to real physical processes. The mathematician receives ratios without being interested in what physical quantities they will be used for. The same mathematical equation can be used to describe many physical objects. It is this remarkable generality that makes mathematics a universal tool for studying natural sciences. We will use this feature of mathematics in our lesson.

Physics teacher.At the last lesson, the main definitions on the topic “Mechanical vibrations” were formulated, but there was no analytical and graphical description of the oscillatory process.

Clip.

slide 4.

3. Communication of the topic and purpose of the lesson.

Physics teacher.Let's try to formulate the topic and purpose of the lesson.

(The teacher draws attention to the fact that each correct answer is marked with a point, which will be taken into account when grading the work in the lesson.)

Slide 5.

Mathematics teacher.We studied the topic: "Graphs of trigonometric functions and their transformations." And trigonometric functions are used to describe oscillatory processes. Today in the lesson we will create a mathematical model of harmonic oscillations.

Algebra is concerned with describing real processes on mathematical language in the form of mathematical models, and then deals not with real processes, but with these models, using various rules, properties, laws developed in algebra.

4. Actualization of basic knowledge in physics.

slide 6

What are fluctuations?(this is a real physical process).

What is called harmonic vibration?

Give examples of oscillatory processes.

Slide 7

What is called the amplitude of oscillations?

Determine the amplitude of oscillations according to the graph of coordinates versus time.

Slide 8

What is called the period of oscillation?

Determine the period of oscillation from the graph of coordinates versus time.

Slide 9

What is the oscillation frequency?

Determine the oscillation frequency from the graph of coordinates versus time.

Slide 10

What is called the cyclic frequency?

Determine the cyclic oscillation frequency from the graph of coordinates versus time.

slide 11

Determine initial phases oscillations for each of the four patterns.

slide 12

Physics teacher:

formulates the definition of harmonic oscillations;
recalls that such free oscillations do not exist in nature;
clarifies that in cases where friction is small, free vibrations can be considered harmonic;
shows the equation of harmonic vibrations.

slide 13

5. Consolidation of knowledge.

A game "One for all and all for one"(Attachment 1)

Students sitting at the first desk are given a card with empty windows for recording answers. Each student writes the answer in the first window and passes the card to the second desk to the student sitting behind him. The student sitting at the second desk writes the answer in the second window and passes the card on, etc. If there are less than six students in a row, then the student from the first desk goes to the end of the row and writes the answer in the right box.

Those students who are the first to complete the card are given an additional point.

slide 13 (check)

Slide 14

6. Actualization of basic knowledge in mathematics.

Mathematics teacher.“There is not a single area of mathematics that will someday not be applicable to the phenomena of the real world” N.I. Lobachevsky.

Today in the lesson we must learn how to plot the functions of harmonic oscillations using the ability to build a sinusoid and knowledge of the rules of compression (stretching) and parallel translation along the coordinate axes. To do this, we recall the transformations of the graphs of trigonometric functions.

slide 15

What to do with the schedule trigonometric function, if

y=sin x y=sin x+2 y=sin x-2

y=sinx y=sin(x+a) y=sin(x-a)

y=sinx y=2sinx y=1/2sinx

y=cosx y=cos2x y=cos(1/2x)

Slides 15-19

6. Consolidation of knowledge.

Independent work.(Annex 2)

Mathematics teacher.The equations you have received are the equations (laws) of harmonic oscillations (algebraic model), and the graph you have constructed is a graphical model of harmonic oscillations. Thus, by modeling harmonic oscillations, we have created two mathematical models of harmonic oscillations: algebraic and graphic. Of course, these models are “ideal” (smoothed) models of harmonic oscillations. Fluctuations are a more complex process. To build a more accurate model, it is necessary to take into account more parameters that affect this process.

Physics teacher:

What oscillatory systems do you know?

Who knows how the mathematical pendulum was used to prove the rotation of the Earth?

Slides 20-21

Student's report about the Foucault pendulum. (Annex 3)

Clip

slide 22

7. Summing up the lesson. Grading.

slide 23

Mathematics teacher.We would like to finish the lesson with the words of F. Bacon: “All information about natural bodies and their properties must contain precise indications of number, weight, volume, dimensions ... Practice is born only from a close connection of physics and mathematics.”

Physics teacher.Today in the lesson we examined free oscillations, using the example of solving problems, we were convinced that all physical quantities describing harmonic oscillations change according to a harmonic law. But free vibrations are damped. Along with free vibrations, there are forced vibrations. We will study forced oscillations in the next lesson.

8. Homework.

slide 24

9. Reflection.

Command _________________________________

Appendix 2

Independent work

1 option

Surname:

Across

A=50 cm, ω= 2 rad/s, 0=

Student checked:

Physics score:

Math score:

Independent work

Option 2

Surname:

Write the harmonic oscillation equation:

Across

Compose an equation for a harmonic oscillation from these quantities

A=30 cm, ω= 3 rad/s, 0=

Plot a harmonic oscillation graph according to the equation

Student checked: .

One of the most striking evidence was found by a French physicist and astronomerJean Foucault in g., he hung a huge pendulum in the Parisian Pantheon-hall with a very high dome. The length of the suspension was 67 m. The mass of the ball was 28 kg. The pendulum swung for hours on end. From below, the ball had a point, and a bed of sand was poured on the floor in a ring with a diameter of 6 meters. The pendulum was swinging. The point began to leave grooves in the sand. A few hours later he drew grooves in another part of the bed. The plane of oscillation of the pendulum seemed to turn clockwise. In fact, the plane of oscillation of the pendulum was preserved. The planet rotated, dragging the Pantheon with its dome and sand bed.(On the screen is a photo of the Foucault pendulum)

In February 2011, the pendulum model appeared inKyiv . It is installed in. The bronze ball weighs 43 kilograms, and the length of the thread is 22 meters . Kyiv Foucault pendulum is considered the largest in the CIS and one of the largest in Europe.

Active Foucault pendulum with thread length 20 meters available in Siberian Federal University , which includes the Foucault tower with a pendulum, the length of the thread of which is 15 meters.

In September 2013 in the atrium of the 7th floor of the Fundamental LibraryMoscow State University launched a Foucault pendulum with a mass of 18 kg and a length 14 meters.

The current Foucault pendulum, weighing 12 kilograms and the length of the thread 8.5 meters available in Volgograd planetarium .

The current Foucault pendulum is currently inSt. Petersburg Planetarium . Its thread length is 8 meters.

Foucault's experiment was repeated in St. Isaac's Cathedral In Petersburg. The pendulum made 3 swings per minute. Based on these data, you can estimate the length of the pendulum, and, consequently, the height of St. Isaac's Cathedral.