Presentation on physics on the topic “Free and forced vibrations. Dynamics of oscillatory motion". Bank of ready-made tasks

LECTURE No. 8

Mechanics

Oscillations

Oscillatory movement. Kinematic and dynamic characteristics of oscillatory motion. Mathematical, physical and spring pendulum.

We live in a world where oscillatory processes are an integral part of our world and are found everywhere.

An oscillatory process or oscillation is a process characterized by varying degrees of repeatability.

If a oscillating quantity repeats its values ​​at equal intervals of time, then such oscillations are called periodic, and these time intervals are called the period of oscillation.

Depending on the physical nature of the phenomenon, vibrations are distinguished: mechanical, electromechanical, electromagnetic, etc.

Oscillations are widespread in nature and technology. Oscillatory processes underlie some branches of mechanics. In this course of lectures we will only talk about mechanical vibrations.

Depending on the nature of the impact on the oscillatory system, vibrations are distinguished: 1. Free or natural, 2. Forced vibrations, 3. Self-oscillations, 4. Parametric vibrations.

Free vibrations are vibrations that occur without external influence and are caused by an initial “push”.

Forced oscillations occur under the influence of a periodic external force

Self-oscillations also occur under the influence of an external force, but the moment of influence of the force on the system is determined by the oscillatory system itself.

With parametric oscillations, due to external influences, a periodic change in the parameters of the system occurs, which causes this type of oscillation.

The simplest form is harmonic vibrations

Harmonic oscillations are vibrations that occur according to the lawsin orcos . An example of harmonic oscillations is the oscillation of a mathematical pendulum

The maximum deviation of an oscillating quantity during the oscillation process is called amplitude of oscillations(A) . The time it takes to complete one complete oscillation is called period of oscillation(T) . The reciprocal of the oscillation period is called vibration frequency(). Often vibrations multiplied by 2 are called cyclic frequency

(). Thus, harmonic vibrations are described by the expression ( Here+  0 ) t  0 oscillation phase, and

– initial phase

The simplest mechanical oscillatory systems are the so-called mathematical, spring and physical pendulums. Let's look at these pendulums in more detail

8.1. Math pendulum

A mathematical pendulum is an oscillatory system consisting of a massive point body suspended in a field of gravity on an inextensible weightless thread.  At the bottom point the pendulum has a minimum of potential energy. Let's deflect the pendulum by an angle . The center of gravity of a massive point body will rise to a height h and at the same time the potential energy of the pendulum will increase by the amount At the bottom point the pendulum has a minimum of potential energy. Let's deflect the pendulum by an angle . The center of gravity of a massive point body will rise to a height mg

Thus, oscillatory mechanical systems are characterized by the fact that when they deviate from the equilibrium position, a restoring force arises in the system, tending to return the system to the equilibrium position. In this case, oscillations occur, accompanied by a periodic transition of the potential energy of the system into its kinetic energy and vice versa.

Let's calculate the oscillatory process. moment of force M acting on the pendulum is obviously equal to - mglsin The minus sign reflects the fact that the moment of force tends to return the load to the equilibrium position. On the other hand, according to the basic law of rotational motion M=ID 2  / dt 2 . Thus, we obtain the equality

B
We will consider only small angles of deviation of the pendulum from the equilibrium position. Then sin ≈  . And our equality will take the form:

D
For a mathematical pendulum it is true I= ml 2 . Substituting this equality into the resulting expression, we obtain an equation describing the process of oscillation of a mathematical pendulum:

This differential equation describes the oscillatory process. The solution to this equation is the harmonic functions sin( Here+  0 ) or cos ( Here+  0 ) Indeed, we substitute any of these functions into the equation and get:  2 = g/ l. Thus, if this condition is met, then the functions sin( Here+  0 ) or cos( Here+  0 ) transform the differential equation of oscillations into an identity.

ABOUT
Here the cyclic frequency and period of oscillation of a harmonic pendulum is expressed as:

The amplitude of oscillations is found from the initial conditions of the problem.

As we can see, the frequency and period of oscillation of a mathematical pendulum does not depend on the mass of the load and depends only on the acceleration of free fall and the length of the suspension thread, which allows the pendulum to be used as a simple but very accurate device for determining the acceleration of free fall.

Another type of pendulum is any physical body suspended from some point of the body and having the ability to perform an oscillatory movement.

8.2. Physical pendulum

IN Let's take an arbitrary body, pierce it at some point with an axis that does not coincide with its center of mass, around which the body can freely rotate. Let us suspend the body on this axis and deflect it from the equilibrium position by a certain angle  .

T
when on a body with a moment of inertia I relative to the axis ABOUT there will be a moment returning to the equilibrium position M = - mglsin and the oscillations of a physical pendulum, like a mathematical one, will be described by the differential equation:

Since for different physical pendulums the moment of inertia will be expressed differently, we will not describe it as in the case of a mathematical pendulum. This equation also has the form of an oscillation equation, the solution of which is the functions describing harmonic oscillations. In this case, the cyclic frequency ( ) , oscillation period (T) are defined as:

We see that in the case of a physical pendulum, the period of oscillation depends on the geometry of the pendulum body, and not on its mass, as in the case of a mathematical pendulum. Indeed, the expression for the moment of inertia includes the mass of the pendulum to the first power. The moment of inertia in the expression for the period of oscillation is in the numerator, while the mass of the pendulum is in the denominator and also to the first power. Thus, the mass in the numerator cancels with the mass in the denominator.

A physical pendulum has one more characteristic: reduced length.

The reduced length of a physical pendulum is the length of a mathematical pendulum, the period of which coincides with the period of the physical pendulum.

This definition makes it easy to define an expression for the given length.

Comparing these expressions we get

If on a line drawn from the point of suspension through the center of mass of the physical pendulum we plot (starting from the point of suspension) the reduced length of the physical pendulum, then at the end of this segment there will be a point that has a remarkable property. If a physical pendulum is suspended from this point, then its period of oscillation will be the same as in the case of hanging the pendulum at the previous suspension point. These points are called the swing centers of the physical pendulum.

Let's consider another simple oscillatory system that performs harmonic oscillations

8.3. Spring pendulum

P Let us imagine that at the end of a spring with a stiffness coefficient k attached mass m.

If we move the load along the x-axis by stretching the spring, then a force returning to the equilibrium position will act on the load F return = - kx. If the load is released, this force will cause acceleration d 2 x / dt 2 . According to Newton's second law we get:

md 2 x / dt 2 = - kx from this equation we obtain the equation for the oscillation of a load on a spring in its final form: d 2 x / dt 2 + (k/ m) x = 0

E
then the equation of oscillations has the same form as the equations of oscillations in the cases already considered, which means that the solution to this equation will be the same harmonic functions. The frequency and period of oscillations will be respectively equal

Moreover, gravity in no way affects the oscillations of the spring pendulum. Since in this case it is a constantly acting factor, acting all the time in one direction and having nothing to do with the restoring force.

Thus, as we see the oscillatory process in a mechanical oscillatory system, it is characterized primarily by the presence in the system restoring force acting on the system, and the oscillations themselves are characterized by: amplitude of oscillations, their period, frequency and phase of oscillations.

In order to quantitatively describe the vibrations of a body under the action of the elastic force of a spring or the vibrations of a ball suspended on a thread, we use Newton’s laws of mechanics

Equation of motion of a body oscillating under the action of an elastic force. According to Newton’s second law, the product of a body’s mass m and its acceleration is equal to the resultant of all forces applied to the body:

This is the equation of motion. Let us write the equation of motion for a ball moving rectilinearly along the horizontal under the action of the elastic force of a spring (see Fig. 3.3). Let's direct the OX axis to the right. Let the origin of the coordinates correspond to the equilibrium position of the ball (see Fig. 3.3, a).

In projection onto the OX axis, the equation of motion (3.1) can be written as follows: ma x = F x control, where a x and F x control, respectively projections of acceleration and elastic force of the spring onto this axis.

According to Hooke's law, the projection F x ynp is directly proportional to the displacement of the ball from its equilibrium position. The displacement is equal to the x coordinate of the ball, and the projection of the force and the coordinate have opposite signs (see Fig. 3.3, b, c). Hence,

F x control = -kx (3.2)

where k is the spring stiffness.

The equation of motion of the ball will then take the form

ma x = -kx. (3.3)

Dividing the left and right sides of equation (3.3) by m, we obtain

Since mass m and stiffness k are constant values, their ratio is also a constant value.

We have obtained an equation that describes the vibrations of a body under the action of elastic force. It is very simple: the projection a x of the body’s acceleration is directly proportional to its x coordinate, taken with the opposite sign.

Equation of motion of a mathematical pendulum. When a ball oscillates on an inextensible thread, it constantly moves along an arc of a circle, the radius of which is equal to the length of the thread l. Therefore, the position of the ball at any time is determined by one value - the angle of deviation of the thread from the vertical. We will consider the angle positive if the pendulum is tilted to the right from the equilibrium position, and negative if it is tilted to the left (see Fig. 3.5). The tangent to the trajectory will be considered directed towards the positive angle reference.

Let us denote the projection of gravity onto the tangent to the trajectory of the pendulum by F t. This projection at the moment when the pendulum thread is deviated from the equilibrium position by an angle is equal to:

The “-” sign is here because the values ​​F t and have opposite signs. When the pendulum deviates to the right (> 0), the gravity component t is directed to the left and its projection is negative: F t< 0. При отклонении маятника влево ( < 0) эта проекция положительна: F t > 0.

Let us denote the projection of the acceleration of the pendulum onto the tangent to its trajectory by t.. This projection characterizes the rate of change in the modulus of the pendulum's velocity.

According to Newton's second law

Dividing the left and right sides of this equation by m, we get

Previously it was assumed that the angles of deflection of the pendulum thread from the vertical could be any. In the future we will consider them small. For small angles, if the angle is measured in radians,

If the angle is small, then the acceleration projection is approximately equal to the acceleration projection on the OX axis: (see Fig. 3.5). From triangle ABO for small angle a we have:

Substituting this expression into equality (3.8) instead of the angle , we obtain

This equation has the same form as equation (3.4) for the acceleration of a ball attached to a spring. Consequently, the solution to this equation will have the same form as the solution to equation (3.4). This means that the movement of the ball and the oscillations of the pendulum occur in the same way. The displacements of the ball on the spring and the pendulum body from the equilibrium positions change over time according to the same law, despite the fact that the forces causing the oscillations have a different physical nature. By multiplying equations (3.4) and (3.10) by m and remembering Newton’s second law ma x = Fх res, we can conclude that oscillations in these two cases occur under the influence of forces, the resultant of which is directly proportional to the displacement of the oscillating body from the equilibrium position and is directed towards the side opposite to this displacement.

Equation (3.4), like (3.10), is apparently very simple: acceleration is directly proportional to the coordinate (displacement from the equilibrium position).

In § 27 we found out that during oscillatory motion the acceleration is variable. Consequently, this movement is due to the action of a variable force. Let, under the action of a variable force, a material point with mass perform a harmonic oscillation with acceleration a. Then, taking into account formula (5), we can write

Thus, the force causing a harmonic oscillation is proportional to the displacement and directed against the displacement. In this regard, we can give the following definition of a harmonic oscillation (except for that given in § 27): a oscillation is called harmonic,

caused by a force proportional to the displacement and directed against the displacement. This force tends to return the point to its equilibrium position, which is why it is called the restoring force. The restoring force can be, for example, the elastic force, since it is also proportional to the displacement and opposite in sign (see § 10). The restoring forces can also have a different, non-elastic nature. In these cases they are called quasi-elastic forces.

If the mass of the material point and the coefficient are known, then from formula (10) we can determine the circular frequency and period of oscillation:

Let us now consider a mechanical oscillatory system called a physical pendulum; This is a solid body that oscillates under the influence of gravity about a horizontal axis. Typically a physical pendulum is a rod with a weighted end; its other end is movably connected to the horizontal axis B, perpendicular to the rod (Fig. 51). Deflected from the equilibrium position by an angle a, the pendulum, under the influence of gravity, returns to this position, passes it by inertia, deviates in the opposite direction, then again passes the equilibrium position, etc. If the friction in the suspension is small, then the pendulum will oscillate for a very long time . The center of gravity of the pendulum C will describe an arc of a circle. Let us agree to consider angle a positive when the pendulum deviates to the right from the equilibrium position and negative when it deviates to the left.

restoring force

where is the mass of the pendulum. The minus sign is due to the fact that the directions of the force and the angle of deflection are always opposite. For small deviations rad a a. Then

where is the arc displacement of the center of gravity of the pendulum from the equilibrium position, the length of the pendulum (the distance from the point of suspension to the center of gravity). Thus, the restoring force turns out to be proportional to the displacement and opposite in sign (i.e., it is a quasi-elastic force). Therefore, the oscillations of the pendulum are harmonic.

In accordance with the basic law of rotation dynamics (see § 21), the moment of the restoring force will be expressed by the relation:

where is the moment of inertia of the pendulum relative to the suspension axis, and is the angular acceleration. Then

Since (see § 6), then, taking into account formula (5), we can write

where (o is the circular frequency of oscillations of the pendulum. Comparing formulas (13) and (14), we obtain

from where we find expressions for the circular frequency and period of oscillation of a physical pendulum:

In practice, it is often possible to consider a physical pendulum as a mathematical one. A mathematical pendulum is a material point that oscillates on a weightless and indeformable thread (Fig. 52). According to the definition of the moment of inertia of a material point (see § 21), the moment of inertia of a mathematical pendulum

where is the mass of the material point, the length of the thread. Substituting this value into formula (16), we obtain the final expression for the oscillation period of a mathematical pendulum:

From formula (17) it follows that

for small deviations a, the period of oscillation of a mathematical pendulum is proportional to the square root of the length of the pendulum, inversely proportional to the square root of the acceleration of gravity and does not depend on the amplitude of oscillations and the mass of the pendulum.

In order to quantitatively describe the vibrations of a body under the action of the elastic force of a spring or the vibrations of a ball suspended on a thread, we will use Newton’s laws of mechanics. Equation of motion of a body oscillating under the action of elastic forces. According to Newton’s second law, the product of body mass m and acceleration a is equal to the resultant F of all forces applied to the body: Let us write the equation of motion of a ball moving rectilinearly along the horizontal under the action of the elastic force F of the spring (see Fig. 56). Let's direct the Ox axis to the right. Let the origin of the coordinates correspond to the equilibrium position (see Fig. 56, a). In projections onto the Ox axis, equation (3.1) will be written as follows: max = Fxynp, where ax and Fxyn are respectively projections of acceleration and elastic force. According to Hooke's law, the projection Fx is directly proportional to the displacement of the ball from its equilibrium position. The displacement is equal to the x coordinate of the ball, and the projection of the force and the coordinate have opposite signs (see Fig. 56, b, c). Consequently, Fx m=~kx, (3.2) where k is the spring stiffness. The equation of motion of the ball will then take the form: max=~kx. (3.3) Dividing the left and right sides of equation (3.3) by m, we obtain a = - - x. + (3.4) x m v " Since mass m and stiffness k are constant quantities, their ratio - " k ratio is also a constant quantity. t We have obtained the equation of motion of a body oscillating under the action of an elastic force. It is very simple: the projection ax of the acceleration of a body is directly proportional to its coordinate x, taken with the opposite sign. Equation of motion of a mathematical pendulum. When a ball oscillates on an inextensible thread, it constantly moves along an arc of a circle, the radius of which is equal to the length of the thread /. Therefore, the position of the ball at any moment in time is determined by one quantity - the angle a of the deviation of the thread from the vertical. We will consider angle a to be positive if the pendulum is tilted to the right from the equilibrium position, and negative if it is tilted to the left (see Fig. 58). The tangent to the trajectory will be considered directed towards the positive angle reference. Let us denote the projection of gravity onto the tangent to the trajectory of the pendulum by Fz. This projection at the moment when the pendulum thread is deflected from the equilibrium position by an angle a is expressed as follows: Fl=-Fs\na=-mgs"ma. (3.5) Here the sign “-” is because Fx and a have opposite signs. When the pendulum deviates to the right (a>0), the component Fx of the force of gravity is directed to the left and its projection is negative: Fx 0. Let us denote the projection of the acceleration of the pendulum onto the tangent to its trajectory through aT. This projection characterizes the speed of change in the modulus of the pendulum's velocity. According to Newton's second law, dividing the left and right sides of this equation by m, we get jf. ax~-g sin a. (3.7) Until now it was assumed that the angles of deviation of the pendulum thread from the vertical can be any. In what follows we will consider them small. At small angles, if the angle is measured in radians, sin a~a. Therefore, we can accept a=~ga. (3.8) Denoting the length of the arc OA by s (see Fig. 58), we can write s=al, from which a=y. (3.9) Substituting this expression into equality (3.8) instead of angle a, we obtain ax = - js. (3.10) This equation has the same form as equation (3.4) for the motion of a ball attached to a spring. Here, only instead of the projection ax of the acceleration there is a projection aT of the acceleration and instead of the coordinate x there is the value s. And the proportionality coefficient no longer depends on the stiffness of the spring and the mass of the ball, but on the acceleration of free fall and the length of the thread. But as before, the acceleration is directly proportional to the displacement (determined by the arc) of the ball from the equilibrium position. We have come to a remarkable conclusion: the equations of motion that describe the oscillations of such different systems as a ball on a spring and a pendulum are the same. This means that the movement of the ball and the oscillations of the pendulum occur in the same way. The displacements of the ball on the spring and the pendulum ball from the equilibrium positions change over time according to the same law, despite the fact that the forces causing the oscillations have a different physical nature. In the first case, this is the elastic force of the spring, and in the second, it is the component of gravity. The equation of motion (3.4), like equation (3.10), appears to be very simple: acceleration is directly proportional to the coordinate. But solving it, that is, determining how the position of an oscillating body in space changes over time, is far from easy.

DEPARTMENT OF EDUCATION OF THE CITY OF MOSCOW

State budgetary professional

educational institution in Moscow

“Polytechnic College No. 47 named after V.G. Fedorov"

(GBPOU PT No. 47)

Methodological development

physics lesson for 1st year students

on this topic: "Mathematical pendulum.

Dynamics of oscillatory motion"

physics teacher at VKK

Moscow, 2016

The methodological development of the lesson is compiled in accordance with the requirements of the Federal State Educational Standard for Special Education and Special Education. The lesson scenario implements elements of information and communication technology and a problem-based activity method for the formation and systematization of knowledge in the process of subject teaching.

Lesson type : combined.

The purpose of the lesson : formation of universal educational actions in the lesson of discovering new knowledge in the technology of the activity method.

Lesson objectives:

1. About educational: contribute to the acquisition of knowledge about the physical foundations of mechanical vibrations, to form such concepts as a mathematical pendulum, period, frequency of oscillations; experimentally establish the laws of oscillation of mathematical and spring pendulums; consider the causes and features of pendulum oscillations.

2. B indoctrinated: create conditions for positive motivation for learning activities, in order to identify the quality and level of mastery of knowledge and skills by students; develop communication skills to speak publicly on a topic and conduct a dialogue; maintain interest in scientific knowledge and the subject “Physics”.

3. Developmental: continue to develop the ability to analyze, systematize, generalize theoretical educational knowledge and data obtained experimentally; contribute to the acquisition of the skill of independent work with a large amount of information, the ability to formulate a hypothesis and outline ways to solve it in the process of group project activities.

Equipment and materials : computer, multimedia projector, screen, presentation for the lesson, video lesson, laboratory equipment for students: tripod, thread pendulum, spring pendulum, weights of different masses, springs of different stiffness, rulers, stopwatch, handouts, textbook (basic and specialized levels) on Physics_11th grade (authors: G.Ya. Myakishev, B.B. Bukhovtsev, V.M. Charugin, edited by N.A. Parfentieva, M. Prosveshchenie, 2015).

Lesson time: 90 minutes (pair).

Lesson structure

planning educational cooperation

There's a song playing "Winged swing". Introductory speech by the teacher. Lesson motto: “Abilities are like muscles, they grow with training.” (Soviet geologist and geographer V.A. Obruchev)

Students greet the teacher, sit down and listen to the teacher.

2. Motivation for learning activities

1) Organize the updating of the requirements of educational activities for the student (“ necessary»).

2) Organize student activities to establish thematic frameworks (“ Can»).

3) Create conditions for the student to experience a situation of success and an internal need for inclusion in educational activities (“ Want»).

Regulatory: volitional self-regulation.

Personal: the action of meaning-making.

1) The teacher suggests finding a connection between the song and the topic of the lesson.

2) On the board there is a crossword puzzle for guessing the concept that determines the topic of the lesson.

3) The teacher writes the date and topic of the lesson on the board.

4) The teacher voices the purpose and objectives of the lesson.

1) Students find an association between the movement of a swing and a pendulum.

2) guess the crossword clue “oscillation”.

3) Write down the date and topic of the lesson in your notebooks.

3. Updating basic knowledge and fixing difficulties in a problem-based learning activity

1) Organize the updating of the studied methods of action sufficient to build new knowledge.

2) Record updated methods of action in speech.

3) Record updated methods of action in signs (standards).

4) Organize a generalization of updated methods of action.

5) Organize the actualization of mental operations sufficient to construct new knowledge.

6) Motivate for problem-based learning activities (“need-can-want”).

7) Organize your own (group) performing a problem-based learning activity.

8) Organize recording of individual difficulties in students’ performance of a trial educational action or in justifying it.

Cognitive:

general education: the ability to structure knowledge, control and evaluate the process and results of activities;

brain teaser: analysis, synthesis, selection of bases for comparison.

Regulatory:

forecasting(when analyzing a trial action before performing it); control, correction(when checking an independent assignment)

1) In the table on the board " KNEW - I LEARNED - I WANT TO KNOW” the teacher fills out first column

2) Demonstration video lesson (9:20) « Free and forced vibrations."

3) In the table on the board “KNEW - LEARNED - I WANT TO KNOW” the teacher fills out second column tables of student responses.

1. What is mechanical vibration.

2. Oscillatory systems and pendulum.

3. Free and forced vibrations.

4. Conditions for the existence of oscillations.

4) In the table on the board “KNEW - LEARNED - I WANT TO KNOW » teacher fills in third column tables of student responses using:

slide “Using a pendulum” from the lesson presentation;

video demonstration "Thermal compensation pendulums" avi. (2 minutes)

1) Students offer previously acquired knowledge on the topic for recording.

2) Students watch a video lesson.

3) Students discuss in pairs and offer the knowledge gained on the topic for recording.

4) Students offer their acquired knowledge on the topic for recording.

4. Identifying the location and cause of the difficulty

1) Organize the restoration of completed operations.

2) Organize recording of the place (step, operation) where the difficulty arose.

3) Organize the correlation of your actions with the standards used (algorithm, concept).

4) Organize the identification and recording in external speech of the cause of the difficulty - that specific knowledge, skills, abilities that are lacking to solve the initial problem of this type.

Cognitive: setting and formulating an educational problem.

1) The teacher suggests opening the textbook Physics 11th grade, p. 58 p. 20 “Mathematical pendulum”.

slide "Mathematical pendulum".

The teacher asks questions:

1. What is called a mathematical pendulum?

2. What forces act on the pendulum in motion?

3. What is the work done by these forces?

4. Where is it directed?

centripetal acceleration of a pendulum?

5. How does the speed of the load on the thread change in magnitude and direction?

6. Under what conditions does a pendulum oscillate freely?

2) On screen demo from presentation slide "Dynamics of oscillatory motion" . Teacher's explanation.

1. Equation of motion of a body oscillating on a spring.

ma x = - kx;

a x = - (k/m) x X (1)

2. Equation of motion of a body oscillating on a thread.

ma Here = - mg x sina; a Here = - g x sina;

a Here = - ( g / L ) X X (2)

3. Draw a conclusion if you multiply (1) and (2) by m , then the resultant force in two cases…..(continue answer)

4. Write down the formulas to calculate (Physics 11th grade, pp. 64-65)

period, frequency, cyclic frequency.

Huygens formula (valid only for small deflection angles).

1) Students work independently with educational material, read, discuss answers to questions in pairs and answer out loud.

2) Students listen and write equations in a notebook.

3. Answer: will be directly proportional to the displacement of the oscillating body from the equilibrium position and directed in the direction opposite to this displacement.

4. Students write in a notebook (work with a textbook).

5. Constructing a project for getting out of a difficulty

Organize the construction of a project to get out of the difficulty:

1) Students set the goal of the project(the goal is always to eliminate the cause of the problem).

2) Students clarify and agree on the topic and purpose of the project.

3) Students determine the means(algorithms, models, reference books, etc.).

4) Students formulate steps that need to be done to implement the project.

Regulatory:

goal setting as setting an educational task, planning, forecasting

Cognitive:

general education: sign-symbolic-modeling; choosing the most effective ways to solve problems depending on specific conditions.

1. Teacher divides a group of students into 6 subgroups to carry out mini-projects in order to study the dependence of the quantities of the oscillatory system.

2. Safety precautions:

Persons familiar with its structure and operating principle are allowed to work with the installation.

To prevent the unit from tipping over, it must be placed only on a horizontal surface.

3. Show slides with tasks for subgroups on the screen in the presentation.

Group No. 1 "Study of the dependence of the period of oscillation of a mathematical pendulum on the amplitude." Draw a graph of this relationship.

Group No. 2 “Investigation of the dependence of the period of oscillation of a mathematical pendulum on the mass of the load.” Draw a graph of this relationship.

Group No. 3 "Investigation of the dependence of the period of oscillation of a mathematical pendulum on the length of the thread." Draw a graph of this relationship.

Group No. 4 "Study of the dependence of the oscillation period of a spring pendulum on the amplitude." Draw a graph of this relationship.

Group No. 5 “Study of the dependence of the period of oscillation of a spring pendulum on the mass of the load.” Draw a graph of this relationship.

Group No. 6 “Study of the dependence of the period of oscillation of a spring pendulum on the stiffness of the spring.” Draw a graph of this relationship.

Do tasks in groups according to plan:

- put forward a hypothesis;

- to conduct an experiment;

- record the received data;

- analyze the result;

- construct a graph of the dependence of the parameters of the oscillatory system;

- draw a conclusion.

6. Implementation of the completed project

1) Organize the fixation of a new method of action in accordance with the plan.

2) Organize the recording of a new method of action in speech.

3) Organize the fixation of a new method of action in signs (using a standard).

4) Organize a record of overcoming the difficulty.

5) Organize clarification of the general nature of the new knowledge (the possibility of using a new method of action to solve all tasks of this type).

Communicative:

planning educational cooperation with peers, proactive cooperation in searching and collecting information; partner behavior management; the ability to express one's thoughts.

Cognitive:

general education:

application of information retrieval methods, semantic reading of a scientific text, the ability to consciously and voluntarily construct a speech statement.

brain teaser:

building a logical chain of reasoning, analysis, synthesis. putting forward hypotheses and their substantiation.

UUD for setting and solving problems:

independent creation of methods for solving search problems.

1) The teacher controls and corrects the progress of research in groups.

2) The teacher, approaching each group, asks questions:

Which physical quantities will you keep constant?

What physical quantities will you change?

Which ones to measure?

Which ones should I calculate?

T mm . = 2
;

T pr.m .= 2
.

Answers:

Group No. 1: Period m.m. does not depend on amplitude.

Group No. 2: Period m.m. does not depend on the mass of the load.

Group No. 3: Period m.m. depends directly proportionally to sq. root of the length of the thread. T ~

Group No. 4: Period pr.m. does not depend on amplitude.

Group No. 5: Period pr.m. depends directly proportionally to sq. root of the load mass. T~

Group No. 6: Period pr.m. depends inversely on sq. root of the spring stiffness. T~

7. Primary consolidation in external speech

Organize students’ assimilation of the method of action when solving this type of problem with their pronunciation in external speech:

Frontal;

- in pairs or groups.

Communicative:

Managing the behavior of the partner(s);

The ability to express your thoughts.

1) On the screen in a presentation on slides verification of the obtained experimental data with the reference answer.

2) Will the period and frequency of oscillation of a mathematical pendulum change when it is transferred to the Moon, where the acceleration of free fall is 6 times less than on Earth? If it changes, how? Explain.

1) Students correct notes and graphs in notebooks.

2) Period mm. increase, because the period is inversely proportional g , A the frequency will decrease, because frequency is directly proportional g .

8.Independent work with self-test according to the standard

1) Organize students’ independent completion of standard tasks for a new method of action.

2) Organize correlation of work with a standard for self-test.

3) Organize verbal comparison of work with a standard for self-test(organization of step-by-step inspection).

4) Based on the results of independent work organize activity reflection on the use of a new method of action.

Regulatory:

control in the form of comparison of the method of action and its result with a given standard; assessing the quality and level of learning; correction.

1) Qualitative questions on the topic (see presentation slides).

2) Solving calculation problems(see presentation slides) - on one's own:

First level- familiarization (recognition of previously studied);

Enough level- reproductive (execution according to the model);

High level-productive (independent solution of a problem task).

3) Presentation slides on the screen to check assignments out loud.

1) Answer verbally out loud.

2) Students choose the level of the task for themselves and complete it independently.

9. Inclusion in the knowledge system and repetition

1) Organize identifying the types of tasks where the method of action is used.

2) Organize the repetition of educational content necessary to ensure meaningful continuity.

Regulatory:

forecasting

On the screen are presentation slides with a supporting outline of the lesson. The teacher repeats the material studied. Corrects errors in students' answers. Aims students to resolve difficulties that arise in learning activities in the following lessons.

Slide "Test yourself"

Students listen and briefly answer questions as they repeat. Summarizing the results obtained, students independently formulate conclusions:

- for m.m. the period depends on the length of the thread and the acceleration of gravity and does not depend on the amplitude of oscillations of the mass of the load;

- for pr.m. the period depends on the mass of the load and the stiffness of the spring and does not depend on the amplitude of oscillations.

10. Reflection on learning activities

1) Organize fixation of new content learned in the lesson.

2) Organize reflective analysis of educational activities from the point of view of fulfilling the requirements known to students.

3) Organize students' assessment of their own activities at the lesson.

4) Organize fixing unresolved difficulties in the lesson as directions for future educational activities.

5) Organize recording and discussing homework.

Cognitive:

general educational: the ability to structure knowledge, assessment of the process and results of activities.

Communicative:

the ability to express one's thoughts.

Regulatory:

volitional self-regulation, assessment - highlighting and awareness of what has already been learned and what still needs to be learned, forecasting.

1) Analysis and practical use of acquired knowledge.

Where is this dependency used?

(see Slide “This is interesting”)

Reflection is organized at the end of the lesson using a model"Clock face" - students are asked to draw an arrow in that sector(4 sectors of the dial – “I understand well, I can explain to others”, “I understand, but solving problems causes difficulties”, “Not everything is clear, solving problems causes difficulties”, “I understood almost nothing”) , which, in their opinion, most corresponds to their level of knowledge of new material.(This method can be carried out on a piece of notebook paper.)

3) The teacher summarizes the large percentage of filling 1-2 sectors of the dial!

4) Grades for the lesson.

5) Recording and discussing homework.

D/Z: Physics 11th grade, pp. 53-66, paragraphs 18-22, questions.

Exercise 1: Measure your heart rate in 30 seconds. Determine the period and frequency of your heartbeat.

Task 2 : Make a mathematical pendulum from available materials and determine its period and frequency of oscillation.

Answer: The design of the first clock was based on the action of a mathematical pendulum. The movement of these watches was regulated by the length of the suspension thread. Using a mathematical pendulum, it is very easy to measure the acceleration of gravity. The value of g varies depending on the structure of the earth’s crust, on the presence of certain minerals in it, so geologists for exploration of deposits still use a device based on the dependence of the oscillation period of a mathematical pendulum on the value of g. The pendulum was used to prove the daily rotation of the Earth.

Students write down the D/Z.

11. Summing up the lesson

Commit a positive tendency to acquire new knowledge.

Guys, learn physics and try to put your knowledge into practice in life. I wish you success!

V.F. Dmitrieva PHYSICS for professions and technical specialties, M., “Academy”, 2010;

Glazunov A.T., Kabardin O.F., Malinin A.N., edited by A.A. Pinsky PHYSICS_textbook for grade 11 with in-depth study of physics, M., “Enlightenment”, 2008;

L.E. Gendenshtein, Yu.I.Dick PHYSICS_textbook for 11th grade basic level, M., “Ilexa”, 2008;

G.Ya. Myakishev, B.B. Bukhovtsev, V.M. Charugin _PHYSICS_textbook for grade 11 at basic and specialized levels, M., "Prosveshchenie", 2015.