Open physics lesson harmonic oscillations. Physics lesson plan. Harmonic oscillations Lesson oscillatory motion harmonic oscillations

Team _________________________________

1 option

Surname:

Through

A=50 cm, ω= 2 rad/s, 0 =

Checked by the student:

Physics Score:

Math score:

Option 2

Surname:

Write down the equation of harmonic vibration:

Through

Create an equation for harmonic vibration from these quantities

A=30 cm, ω= 3 rad/s, 0 =

Construct a graph of harmonic vibration using the compiled equation

Checked by the student: .

One of the most striking evidence was found by a French physicist and astronomerJean Foucault V 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 several hours in a row. The ball had a point at the bottom, 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 was drawing furrows in another part of the bed. The plane of oscillation of the pendulum seemed to rotate clockwise. In fact, the plane of oscillation of the pendulum was preserved. The planet rotated, carrying with it the Pantheon with its dome and sand bed.(On the screen is a photo of a 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 . The Kiev Foucault pendulum is considered the largest in the CIS and one of the largest in Europe.

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

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

A working Foucault pendulum, weighing 12 kilograms and thread length 8.5 meters, available in Volgograd Planetarium .

A functioning Foucault pendulum is currently available inSt. Petersburg Planetarium . The length of its thread is 8 meters.

Foucault's experience was repeated in St. Isaac's Cathedral in St. 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.

Federal State Budgetary Educational Institution

"Ural State University means of communication"
PERM INSTITUTE OF RAILWAY TRANSPORT
branch of the federal state budgetary educational institution
higher professional education
"Ural State Transport University" in Perm
(PIZHT UrGUPS)

Methodological development of an integrated lesson
algebra and physics on the topic:
"Harmonic vibrations"
for specialty 220415 Automation and telemechanics in transport (on
railway transport)
V.I. Dolgintseva,
mathematics teacher highest category

Perm, 2017
Explanatory note
Brief description of the lesson. Topic: “Harmonic oscillation graph”
is considered in the 1st year in the process of mastering the academic discipline “Algebra”
and start the analysis." This topic ends the discussion of the chapter.
“Trigonometric functions.” The purpose of this lesson is not only to
to learn how to plot a harmonic oscillation graph, and also show
connection of a given mathematical object with phenomena of the real world.
At the beginning of the lesson, students remember physical processes and phenomena, in
which there are fluctuations (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
is to repeat the physical meaning of the quantities included in the equation
harmonic vibration, and then the mathematical rules are repeated
converting graphs of trigonometric functions using compression
(stretching) and parallel transfer. At the end of the lesson there is
independent work of a training nature followed by
mutual verification. The lesson ends with a message from the student who
Using a video clip, he introduces students to the Foucault pendulum.
Lesson objectives:
educational: summarize and systematize students’ knowledge about
harmonic vibrations. Teach students to obtain equations and
build graphs of the obtained functions. Create a mathematical model
harmonic vibrations.
developmental: develop memory, logical thinking; form
communication skills, develop oral speech;
educational: to create a culture of mental work; create
a situation of success for each student; develop the ability to work in
team.
Lesson type: generalization and systematization of knowledge.
Lesson methods: partially search, explanatory and illustrative.
Interdisciplinary connections: physics, mathematics, history.
Visualization and TSO: computer, lesson presentation, video “Pendulum
Foucault", cards with tasks for the game "One for all and all for one", cards
to do independent work.
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 1011 classes.
3. Myakishev G.Ya., Bukhovtsev B.B. Physics 10. Textbook.

4. Stepanova G.I. Collection of physics problems for 1011th grades.
During the classes
1. Organizational moment.
2. Motivation and stimulation of cognitive activity.
Slide number 1
Teacher: I would like to start today’s lesson with the epigraph: “All
our previous experience leads us to believe that nature is
the implementation of what is mathematically easiest to imagine” A.
Einstein.
The task of physics is to identify and understand the connection between observed phenomena and
characterizing them.
establish a relationship between quantities,
A quantitative description of the physical world is impossible without mathematics.
Mathematics creates methods of description that correspond to character
physical problem, gives ways to solve physics equations.
Back in the 18th century, A. Volta (Italian physicist, chemist and physiologist, one of
the founders of the doctrine of electricity; Count Alessandro Giuseppe
Ant nio Anast sio Jerol mo Umberto V lta
) said: “What can be done
good, especially in physics, if you don’t reduce everything to measure and degree?”
oa
oa
ahh
ahh
ahh
Mathematical constructions themselves have no relation to properties
the surrounding world, these are purely logical constructions. They make sense
only when applied to real physical processes.
A mathematician obtains relations without being interested in what physical
values ​​they will be used. The same mathematical equation
can be used to describe many physical objects. Exactly this
remarkable generality makes mathematics a universal tool for
studying natural sciences. We will use this feature of mathematics
use in our lesson.
In the last lesson, basic definitions on the topic were formulated
“Mechanical vibrations”, but there was no analytical and graphic
descriptions of the oscillatory process.
Slide number 2
3. Statement of the topic and purpose of the lesson.
Teacher. Let's try to formulate the topic and purpose of the lesson.
(The teacher draws attention to the fact that each correct answer
marked with a point that will be taken into account when grading for
work in class.)
We studied the topic: “Graphs of trigonometric functions and their
transformation." And trigonometric functions are used to describe

oscillatory processes. Today in the lesson we will create
mathematical model of harmonic oscillations.
Algebra is concerned with describing real processes in
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. Updating basic knowledge in physics.
Slide number 3
What are fluctuations? (this is a real physical process).
What are harmonic oscillations?
Give examples of oscillatory processes.
Slide number 4
What is the amplitude of oscillations called?
Determine the amplitude of oscillations from the graph of the coordinate versus
time.
Slide number 5
What is the period of oscillation?
Determine the oscillation period from the graph of the coordinate versus
time.
Slide number 6
What is the oscillation frequency?
Determine the oscillation frequency from the graph of the coordinate versus
time.
Slide number 7
What is the cyclic frequency of oscillation?
Determine the cyclic frequency of oscillations from the dependence graph
coordinates versus time.
Slide number 8
Determine the initial phases of oscillations for each of the four patterns.
Slide number 9
Teacher:
 formulates the definition of harmonic vibrations;
 reminds that such free vibrations do not exist in nature;
 clarifies that in cases where friction is low, free vibrations
can be considered harmonic;
 shows the equation of harmonic vibrations.
5. Consolidation of knowledge.
Game “One for all and all for one” (Appendix 1)
Students sitting at the first desk are given a card with blank
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
person, then the student from the first desk moves to the end of the row and writes the answer in
the right window.
For those students who finish filling out the card first,
extra point is given.
Slide number 10 (check)
Slide number 11
6. Updating basic knowledge in mathematics.
Teacher. Slide number 12
“There is not a single branch of mathematics that will someday not turn out to be
applicable to the phenomena of the real world" N.I. Lobachevsky.
Today in class we must learn how to graph functions
harmonic oscillations, using the ability to construct a sine wave and knowledge of the rules
compression (tension) and parallel transfer along the coordinate axes. For this
Let us recall the transformations of graphs of trigonometric functions.
Slide number 13
What should you do with the graph of a trigonometric function if
y=sinx y=3sinx stretching from the X axis with a factor of 3.
Slide number 14
y=1/2sinx – compression to the X axis with a factor of ½.
Slide number 15
y=sin0.5x stretching from the Y axis with a factor of 2.
Slide number 16
y=sin2x compression to the Y axis with a factor of 2.
Slide number 17
What transformations were made to the graph y = sinx?
Slide number 18
Match.

6. Consolidation of knowledge.
Independent work. (Appendix 2)
Teacher. The equations you get are the equations
(laws) of harmonic vibrations (algebraic model), and the constructed
graph – graphical model of harmonic oscillations. Thus,

By modeling harmonic oscillations, we created two
mathematical models of harmonic vibrations:
algebraic and
graphic. Of course, these models are “ideal” (smoothed) models
harmonic vibrations. Oscillations are a more complex process. For building
a more accurate model needs to take into account more parameters that influence
this process.
What oscillatory systems do you know?
Who knows how the mathematical pendulum was used to prove
rotation of the Earth?
Student's message about the Foucault pendulum. (Appendix 3)
Video.
7. Summing up the lesson. Grading.
Slide number 19
Teacher. We would like to end the lesson with the words of F. Bacon: “Everything
information about natural bodies and their properties must contain precise instructions
on number, weight, volume, dimensions... Practice is born only from close
connections of physics and mathematics". F. Bacon
Today in class we looked at free vibrations, using the example
solving problems, we were convinced that all physical quantities describing
harmonic oscillations change according to the harmonic law. But free
the oscillations are damped. Along with free vibrations,
there are forced oscillations. By studying forced oscillations we
We'll do it in the next lesson.
8. Homework.
Message “Forced vibrations”.
9. Reflection.

The purpose of the lesson: to form in students an idea of ​​harmonic vibrations as harmonic changes in coordinates and other physical quantities; introduce the concept of amplitude, period, frequency, cyclic frequency; obtain a formula for calculating the period of free oscillations.

During the classes

Examination homework by individual survey method

1. Explain, using a drawing, what forces make a mathematical pendulum oscillate.

2. Obtain the equation of motion for spring pendulum. at the blackboard)

3. Obtain the equation of motion of a mathematical pendulum. (at the blackboard)

Learning new material

1. Having studied the dependence of acceleration on the coordinates of an oscillating body, we find Dependence of coordinates on time.

2. Acceleration is the second derivative of a coordinate with respect to time.

A = – k x/m; x“= – k x/m; where x“ is the second derivative of the coordinate with respect to time.

If the oscillations are free, then the x coordinate changes over time so that the second derivative of the coordinate with respect to time is directly proportional to the coordinate itself and is opposite in sign.

3. Harmonic vibrations

The x coordinate changes periodically over time. We know two periodic functions: sine and cosine

As the argument increases from zero, the cosine changes slowly, and as it approaches zero, its changes occur faster and faster.

A spring pendulum, removed from its equilibrium position, behaves in exactly the same way. Sine and cosine have the property that the second derivative of these functions is proportional to the functions themselves, taken with the opposite sign.

Based on this, it can be argued that the coordinate of a body performing free oscillations changes over time according to the law of cosine or sine.

Periodic changes in a physical quantity depending on time, occurring according to the law of sine or cosine, are called harmonic oscillations.

4. Oscillation amplitude

The modulus of the greatest displacement of a body from the equilibrium position is called Amplitude harmonic vibrations.

Amplitude is a characteristic of oscillatory motion; it shows how the body is displaced from its equilibrium position.

5. Solution of the equation of motion describing free oscillations. Let's write down the solution to the equation; x“= – k x/m; — X= xm QUOTE t; The first derivative will have the form: Xʹ= – QUOTE xm QUOTE ·t;

The second derivative will be equal to: X“= – QUOTE xm QUOTE ·t = – k x/m; that is, we got the original equation. The solution to this equation will also be a function; QUOTE t

From the experiments we received

А= – k x/m a= – g x/L

For a spring pendulum for a mathematical pendulum

LET'S DEnote

We have equations of motion

A= – ω02x Subject to one pattern a= – ω02x

A ~x x~x“ x “= – ω02x – solution of this differential equation

Is: X = xm QUOTE . The graph of coordinates versus time is Cosine. Harmonic vibrations occur according to this law.

6. Period and frequency of harmonic oscillations

Lesson type: lesson in the formation of new knowledge.

Lesson objectives:

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

Equipment: spring and mathematical pendulums, projector, computer, teacher’s presentation, CD “Library of Visual Aids”, student learning sheet, cards with symbols of physical quantities, text “Resonance Phenomenon”.

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

During the classes

I. Motivation.

Teacher: So that you understand what will be discussed in the lesson today, read an excerpt from the poem “Morning” by N.A. Zabolotsky

Born of the desert
The sound fluctuates
Blue wavers
There's a spider on a thread.
The air vibrates
Transparent and clean
In the shining stars
The leaf sways.

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

Students call different examples fluctuations(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 have to:

1. Find out what oscillation is?
2. Conditions for the occurrence of oscillations.
3. Types of vibrations.
4. Harmonic vibrations.
5. Characteristics of harmonic vibration.
6. Resonance.
7. Problem solving (slide 5).

Teacher: Look at the oscillations of the mathematical and spring pendulums (oscillations are demonstrated). Are the oscillations absolutely repeatable?

Students: No.

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

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

Teacher: Why does the oscillation continue for so long? (slide 7) Using spring and mathematical pendulums, 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 for oscillations to begin?

Students: You need to push the body, apply force to it. To make the oscillations last a long time, you need to reduce the friction force (slide 8), the conditions are written down in a notebook.

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

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

Teacher: Oscillations can also be damped or undamped (slide 11 with a mouse click). Damped oscillations are oscillations that, under the influence 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 over time; There are no frictional forces or resistance. To maintain undamped oscillations, an energy source is required (slide 13); these oscillations are shown on the graph on the slide.

Examples of oscillations are given (slide 14).

1 option writes out examples damped oscillations.

Option 2 writes out examples undamped oscillations.

1. vibrations of leaves on trees during the wind;
2. heartbeat;
3. swing vibrations;
4. oscillation of the load on the spring;
5. rearrangement of legs when walking;
6. vibration of the string after it is removed from its equilibrium position;
7. vibrations of the piston in the cylinder;
8. vibration of a ball on a thread;
9. the swaying of grass in a field in the wind;
10. vibration of the vocal cords;
11. vibrations of windshield wiper blades (windshield wipers in a car);
12. vibrations of the janitor's broom;
13. vibrations of the sewing machine needle;
14. vibrations of the ship on the waves;
15. swinging arms while walking;
16. vibrations of the phone membrane.

Students Among the given oscillations, they write down examples of free and forced oscillations according to the options, then exchange information and 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 oscillations are damped, and forced oscillations are undamped. Find automatic oscillations among the examples given. Students give themselves a grade on the knowledge mastery sheet in point 1 of the knowledge mastery sheet ( Annex 1)

Teacher: Among all types of vibrations there are special kind vibrations are harmonic.

The manual “Library of Visual Aids” demonstrates a model of harmonic oscillations (mechanics, model 4 harmonic oscillations) (slide 16).

What is the schedule? mathematical function does it work on the model?

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

Students write down the equations of harmonic vibrations in a notebook.

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

Students: Meter (slide 17, mouse click).

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

Students: Maximum displacement.

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

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

Teacher: Before considering the next quantity, let us recall the concepts of quantities studied in 1st year. 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 a 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 in different colors on the graph.

Students The period is determined on the graph by finding it between points of different colors.

Teacher demonstrates on a mathematical pendulum different frequency for different pendulum lengths. Frequency ν– 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/Т Т=1/ν (slide 20 mouse click).

Teacher: The sine and cosine function is repeated 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 a quantity under the sine or cosine sign. It is measured in radians (rad) (slide 22).

The oscillation phase at the initial time (t=0) is called initial phase – φ 0. It is measured in radians (rad) (slide 21, click).

Teacher: Now let's repeat the material.

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

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

c) Each four students are given a card with a value; they need to tell everything about it according to the plan on slide 23. Then the groups exchange cards with values ​​and complete the same task.

Students give themselves grades on their report card (clause 2, 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, he 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 in inverse proportion (slide 25)

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

Students: You need to rock the car together on command.

Teacher: Right. In doing so, we use a 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 manual “Library of Visual Aids” demonstrates a resonance model (mechanics, model 27 “Swinging of a spring pendulum” at a frequency of >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 Waltz of the Flowers are playing ( Appendix 4). The text is marked with the following signs (they are located on the stand in the office): V – interested; + knew; - did not know; ? – I would like to know more. The text remains in each student's notebook. Next lesson, you should come back to it and answer students' questions if they don't find the answers at home.

III. Fixing the material.

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

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 summary.

Teacher: What new did you learn in class today?

V. Homework.

Everyone learn the lesson notes. Solve the problem: using the equation of harmonic vibration, find everything you can (slide 29). Find answers to questions when marking the text. Those who wish can find material about the benefits of resonance and the dangers of resonance (you can make a message, an abstract, or prepare a presentation).