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DEPARTMENT OF EDUCATION OF THE CITY OF MOSCOW
State budget professional
educational institution Moscow city
"Polytechnic College No. 47 named after V.G. Fedorov"
(GBPOU PT No. 47)
physics lesson for 1st year students
on this topic: “Mathematical pendulum.
Dynamics oscillatory motion»
physics teacher VKK
Moscow, 2016
The methodological development of the lesson was compiled in accordance with the requirements of the Federal State Educational Standard of the SOO and SPO. In the scenario of the lesson, elements of information and communication technology and the problem-activity method of forming and systematizing knowledge in the process of subject education are implemented.
Lesson type : combined.
The purpose of the lesson : the formation of universal educational actions at the lesson of discovering new knowledge in the technology of the activity method.
Lesson objectives:
1. About educational: promote knowledge about physical foundations mechanical oscillations, 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. In nurturing: create conditions for positive motivation to learning activities, in order to identify the quality and level of mastery of knowledge and skills by students; to form communication skills to speak publicly on the topic, to conduct a dialogue; maintain interest in scientific knowledge and to the subject "Physics".
3. Developing: continue the formation of the ability to analyze, systematize, generalize theoretical educational knowledge and data obtained experimentally; to promote 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, lesson presentation, video lesson, laboratory equipment for students: tripod, thread pendulum, spring pendulum, weights different weight, springs of different stiffness, rulers, stopwatch, handouts, textbook (basic and profile levels) in physics_11 class (authors: G.Ya. Myakishev, B.B. Bukhovtsev, V.M. Charugin, edited by N.A. Parfentyeva , M. Education, 2015).
Lesson time: 90 minutes (pair).
Lesson structure
Personal:planning of educational cooperation
The song sounds "Winged swing". Introductory speech of the teacher. Lesson motto: "Abilities are like muscles, they grow with training" (Soviet geologist and geographer Obruchev V.A.)
Students greet the teacher, sit down and listen to the teacher.
2. Motivation for learning activities
1) Organize the actualization of the requirements of educational activities for the student (“ necessary»).
2) Organize the activities of students to set the thematic framework (" can»).
3) Create conditions for the student to have a situation of success and an internal need for inclusion in educational activities (“ want»).
Regulatory: volitional self-regulation.
Personal: sense-making action.
1) The teacher suggests finding a connection between the song and the topic of the lesson.
2) On the board 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 the association of the movement of a swing with a pendulum.
2) guess keyword crossword "fluctuation".
3) Write down the date and topic of the lesson in notebooks.
3. Actualization of basic knowledge and fixation of difficulties in a problematic educational action
1) Organize the actualization of the studied methods of action, sufficient to build new knowledge.
2) Fix the updated methods of action in speech.
3) Fix the updated methods of action in signs (standards).
4) Organize a generalization of the updated methods of action.
5) Organize the actualization of mental operations sufficient to build new knowledge.
6) Motivate for problem learning action ("need-can-want").
7) Organize independent (group) performance of a problem learning activity.
8) Organize the fixation of individual difficulties in the implementation by students of a trial educational action or in its justification.
Cognitive:
general education: the ability to structure knowledge, control and evaluation of the process and results of activities;
brain teaser: analysis, synthesis, choice of bases for comparison.
Regulatory:
forecasting(when analyzing a trial action before its execution); control, correction(when checking an independent task)
1) In the table on the board " KNEW -I LEARNED-I WANT TO LEARN" the teacher fills in first column
2) Demo video lesson (9:20) « Free and forced vibrations.
3) In the table on the board “KNOW - LEARNED - I WANT TO KNOW" the teacher fills out second column tables according to the answers of students.
1. What is mechanical oscillation.
2. Oscillatory systems and pendulum.
3. Free and forced vibrations.
4. Conditions for the existence of oscillations.
4) In the table on the board “KNOW - LEARNED - I WANT TO KNOW » teacher fills out third column tables on the answers of students using:
slide "Application of the pendulum" from the presentation to the lesson;
video demonstration "Thermal compensation pendulums" avi. (2 minutes)
1) Students offer to record the knowledge on the topic received earlier.
2) Viewing a video lesson by students.
3) Students discuss in pairs and offer for recording the acquired knowledge on the topic.
4) Students offer to record their knowledge on the topic.
4. Identification of the place and cause of the difficulty
1) Organize the restoration of completed operations.
2) Organize the fixation of the place (step, operation) where the difficulty arose.
3) Organize the correlation of their actions with the standards used (algorithm, concept).
4) Organize the identification and fixation during external speech the reasons for the difficulty are those specific knowledge, skills, which are not enough to solve the initial problem of this type.
Cognitive: formulation and formulation of the educational problem.
1) The teacher offers to open the textbook Physics class 11, 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 of these forces?
4. Where directed
centripetal acceleration of a pendulum?
5. How does the speed of the load on the thread change in absolute value and direction?
6. Under what conditions does the pendulum swing freely?
2) On-screen demonstration from the 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 t = - mg x sina; a t = -g x sina;
a t = - ( 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 the answer)
4. Write formulas for calculation (Physics grade 11, pp. 64-65)
period, frequency, cyclic frequency.
Huygens formula (valid only for small deflection angles).
1) Students work independently with educational material, read, discuss in pairs the answers to the questions and answer aloud.
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. Building 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 difficulty).
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 a learning task, planning, forecasting
Cognitive:
general education: sign-symbolic-modeling; selection of the most effective ways solving problems depending on specific conditions.
1. Teacher divides a group of students into 6 subgroups for the implementation of mini-projects, in order to study the dependence of the values of the oscillatory system.
2. Safety precautions:
Persons who are familiar with its structure and operation principle are allowed to work with the installation.
To prevent the unit from tipping over, it must be positioned only on a horizontal surface.
3 . On the screen in the presentation, show slides with tasks for subgroups.
Group #1 "Investigation of the dependence of the period of oscillation of a mathematical pendulum on the amplitude". Draw a graph of this relationship.
Group #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 #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 #4 "Investigation of the dependence of the period of oscillation of a spring pendulum on the amplitude". Draw a graph of this relationship.
Group #5 "Investigation 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 #6 "Investigation of the dependence of the period of oscillation of a spring pendulum on the stiffness of the spring." Draw a graph of this relationship.
Perform tasks in groups according to plan:
- put forward a hypothesis;
- to make an experiment;
- write down the received data;
- analyze the result;
- build a graph of the dependence of the parameters of the oscillatory system;
- draw a conclusion.
6. Implementation of the constructed project
1) Organize the fixation of a new mode of action in accordance with the plan.
2) Organize the fixation of a new mode of action in speech.
3) Organize the fixation of a new mode of action in signs (with the help of a standard).
4) Organize the fixation of overcoming the difficulty.
5) Arrange clarification general new knowledge (the ability to apply a new method of action to solve all tasks of a given type).
Communicative:
planning educational cooperation with peers, proactive cooperation in the search and collection of information; partner behavior management; the ability to express one's thoughts.
Cognitive:
general education:
application of information retrieval methods, semantic reading scientific text, the ability to consciously and voluntarily build a speech statement.
brain teaser:
construction of a logical chain of reasoning, analysis, synthesis. hypotheses and their justification.
UUD for setting and solving problems:
independent creation of ways to solve search problems.
1) The teacher controls and corrects the course of research in groups.
2) The teacher, approaching each group, asks questions:
What kind physical quantities will you keep constant?
What physical quantities will you change?
Which ones to measure?
What - to calculate?
T
mm
. = 2
;
T
pr.m
.= 2
.
Answers:
Group #1: Period m.m. does not depend on amplitude.
Group #2: Period m.m. does not depend on the weight of the load.
Group #3: Period m.m. depends directly on sq. root of the thread length.
T
~
Group #4: R.m. period does not depend on amplitude.
Group #5: R.m. period depends directly on sq. the root of the weight of the load.
T ~
Group #6: R.m. period depends inversely on sq. the root of the spring constant.
T~
7. Primary consolidation in external speech
To organize the assimilation by students of the method of action in 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 screen in presentation on slides verification of the obtained experimental data with a reference answer.
2) Will the period and frequency of oscillations of the mathematical pendulum change when it is transferred to the Moon, where the acceleration free fall less than 6 times than on Earth? If it changes, how? Explain.
1) Students in notebooks correct notes and graphs.
2) Period mm. increase, since 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 independent execution students typical tasks on the new way actions.
2) Organize correlation of work with the standard for self-examination.
3) Organize verbal comparison of work with a standard for self-examination(organization of step-by-step verification).
4) Based on the results of independent work organize the reflection of activities on the application of a new mode of action.
Regulatory:
control in the form of comparing the method of action and its result with a given standard; assessment of the quality and level of assimilation; correction.
1) Qualitative questions on the topic (see presentation slides).
2) Solution of calculation problems(see presentation slides) - on one's own:
First level- introductory (recognition of previously studied);
Enough level- reproductive (execution according to the model);
High level- productive ( independent solution problem task).
3) Presentation slides on screen to check assignments aloud.
1) Orally answer aloud.
2) The students themselves choose the level of the task for themselves and complete it on their own.
9. Inclusion in the knowledge system and repetition
1) Organize identification of types of tasks where the mode of action is used.
2) Organize the repetition of educational content necessary to ensure meaningful continuity.
Regulatory:
forecasting
Presentation slides with a basic outline of the lesson on the screen. The teacher repeats the studied material. Corrects errors in student responses. Aims students to resolve the difficulties that have arisen in educational activities in the following lessons.
"Check Yourself" slide
The students listen and briefly answer the questions as they go through the repetition. Summarizing the results obtained, students independently formulate conclusions:
- for mm the period depends on the length of the thread and the acceleration of free fall and does not depend on the amplitude of fluctuations in 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 the oscillations.
10. Reflection of educational activity
1) Organize fixing new content learned in the lesson.
2) Organize reflective analysis of learning activities in terms of fulfilling the requirements known to the learners.
3) Organize assessment by students of their own activities at the lesson.
4) Organize fixing unresolved problems in the lesson as directions for future educational activities.
5) Organize writing and discussing homework.
Cognitive:
general educational: the ability to structure knowledge, evaluation of the process and results of activities.
Communicative:
the ability to express one's thoughts.
Regulatory:
volitional self-regulation, assessment - selection and awareness of what has already been learned and what is still to be learned, forecasting.
1) Analysis and practical use acquired knowledge.
Where is this dependency used?
(see slide "It's interesting")
Reflection is organized at the end of the lesson using a model"Clock face" - students are invited to draw an arrow in that sector(4 sectors of the dial - “Understood well, I can explain to others”, “Understood, but solving problems causes difficulties”, “Not everything is clear, solving problems causes difficulties”, “Almost nothing understood”) , which, in their opinion, most of all corresponds to their level of knowledge of new material.(This method can be carried out on a sheet of notebook).
3) The teacher summarizes the large percentage of filling 1-2 sectors of the dial!
4) Grades for the lesson.
5) Recording and discussion of homework.
D / W: Physics 11 cells, pp. 53-66, p. 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 improvised means and determine its period and frequency of oscillation.
Answer: The device of the first clock was based on the action of a mathematical pendulum. The course of this clock was regulated by the length of the suspension thread. Using a mathematical pendulum, it is very easy to measure the acceleration of free fall. The value of g varies depending on the structure earth's crust, from the presence of certain minerals in it, therefore, for exploration of deposits, geologists 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 daily rotation Earth.
Students write down D / Z.
11. Summing up the lesson
Fix a positive tendency to acquire new knowledge.
Guys, learn physics and try to apply your knowledge in life in practice. I wish you success!
www . chrono . info / biograph / imena . html - biographies of scientists;V.F. Dmitrieva PHYSICS for professions and specialties of a technical profile, 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. Gendenstein, Yu.I.Dik PHYSICS_textbook for grade 11 of the basic level, M., "Ileksa", 2008;
G.Ya. Myakishev, B.B. Bukhovtsev, V.M. Charugin _PHYSICS_textbook for grade 11 basic and profile level, M., "Enlightenment", 2015.
GOU DOD "POISK"
yov
Dynamics
Lab #9.7
DYNAMICS OF VIBRATIONAL MOTION
Instruction
to perform measurements and research.
Report Form
Filled in with a simple pencil.
The most accurate and legible.
I've done the work
“……” …………….20..….g.
Work checked
.....................................................
Grade
...............%
“……” …………….20..….g.
Stavropol 2011
Objective:
Deepen your understanding of the theory of harmonic oscillations. Master the methodology of experimental observations and test the laws of undamped harmonic oscillations using the example of a mathematical and physical pendulum.
Equipment:a stand for observing the oscillations of various pendulums, a stopwatch, a ruler.
1. Theoretical part
Mechanical vibrations - this is a type of movement when the coordinates, velocities and accelerations of the body are repeated many times.
Free oscillations occurring under the action of internal forces of a system of bodies are called. If, when the system is removed from the equilibrium position, a force arises that is directed towards the equilibrium position and is proportional to the displacement, then in such a system, harmonic vibrations. Here coordinates, velocities and accelerations occur according to the law of cosine (sine)
x=Acos(w0 t+a0 ); v=-v0sin(w0 t+a0 ); a=a0 Acos(w0 t+a0 ) (1)
where A– amplitude,w0 is the cyclic frequency,a0 is the initial phase of oscillations. The cyclic frequency is related to the oscillation period T
(2)
Free vibrations are harmonic only when there is no friction, or it is negligibly small.
font-size:16.0pt"> Systems of bodies in which free vibrations occur are often called pendulums.
physical pendulum called solid which, under the action of gravity, oscillates around a fixed axis O, not passing through the center of mass WITH body (Fig. 1).
When removing the pendulum from the equilibrium position at a certain anglej, component fn gravity mg balanced by the reaction force N axes O, and the component F tseeks to return the pendulum to its equilibrium position. All forces are applied to the center of mass of the body.
Wherein
Ft =-mgsinj (3)
The minus sign means that the angular displacementj and restoring force F t have opposite directions. For sufficiently small deflection angles of the pendulum ( 5-6 ° ) sin j » j (j in radians ) and F t » - mgj, i.e. the restoring force is proportional to the angle of deflection and directed towards the equilibrium position, which is required to obtain harmonic oscillations.
The pendulum in the process of oscillation makes a rotational movement about the axis O, which is described by the basic equation of the dynamics of rotational motion
M=Je , ( 4)
where M- moment of power F tabout the axis O, J is the moment of inertia of the pendulum about the same axis, ε is the angular acceleration of the pendulum.
moment of force in F tabout the axis O equals:
M=Ft× l = - mgj× l, (5)
where l- arm of strengthFt- the shortest distance between the point of suspension and the center of gravity of the pendulum.
From equations (4) and (5) , compiled in differential form, a solution is obtained in the form
j = jm× cos(w0 t+j0 ) , (6)
where . (7)
It follows from this solution that for small oscillation amplitudes (j<5-6 ° ) a physical pendulum performs harmonic oscillations with an angular amplitude of oscillationsjm, cyclic frequency and period T
font-size:16.0pt; font-weight:normal"> .(8)
Analysis of formula (8) makes it possible to formulate the following patterns of oscillations of a physical pendulum (at a small amplitude and in the absence of friction forces):
· The oscillation period of a physical pendulum at small displacements does not depend on the oscillation amplitude.
· The period of oscillation of a physical pendulum depends on the moment of inertia of the pendulum about the axis of rotation (swing).
· The oscillation period of a physical pendulum depends on the position of the center of mass of the pendulum relative to the suspension point.
The simplest physical pendulum is a massive weight on a suspension, located in the field of gravity. If the suspension is inextensible, the dimensions
loads are negligible compared to the length of the suspension and the mass of the thread is negligible compared to the mass of the load, then the load can be considered as a material point located at a constant distance l from suspension point O. Such an idealized pendulum model is called mathematical pendulum(Fig. 2).
Oscillations of such a pendulum occur according to the harmonic law (6). Since the moment of inertia of a material point about the axis passing through the point O, is equal to J=ml2, then the period of oscillation of the mathematical pendulum is equal to
.
(9)
Analysis of formula (9) makes it possible to formulate the following patterns of oscillations of a mathematical pendulum (at a small amplitude and in the absence of friction forces):
· The period of oscillation of a mathematical pendulum does not depend on the mass of the pendulum (which was verified in the previous series of laboratory work).
· The oscillation period of a mathematical pendulum at small oscillation angles does not depend on the oscillation amplitude (which was also verified earlier).
· The period of oscillation of a mathematical pendulum is directly proportional to the square root of its length.
2. experimental part
Wassignment 1.Study of oscillations of a physical pendulum
Target.Check the correctness of dependence (8) of the oscillation period of a physical pendulum on its characteristics. To do this, it is necessary to construct the corresponding experimental graphs.
The physical pendulum used in this work is a straight homogeneous rod. The distance from the center of gravity of the rod, i.e. its middle, to the suspension point can be changed. Moment of inertia of the rod about the axis of rotation (rocking) font-size:16.0pt;font-weight:normal">font-size:16.0pt; font-weight:normal"> (10)
where d- rod length, l is the distance from the center of gravity (center of the rod) to the swing axis.
dependency graph T=f(l) is a curve complex shape. For further processing, it should be linearized. To do this, we transform formula (10) to the form
font-size:16.0pt; font-weight:normal"> (11)
It can be seen from this that if we construct a dependency graph (T2l) = f(l2), then you should get a straight line y=kx+b, the slope of which is equal to https://pandia.ru/text/79/432/images/image012_32.gif" width="95" height="53 src=">.
1. Fix the gimbal in extreme position. measure distance l from center of gravity to axis
2. Measure the period of oscillation T pendulum. To do this, it must be deflected by a small angle and measure the time 10-15 full swing.
4. Consistently reducing the distance l , measure the periods of oscillation of the pendulum in each of these positions.
5. You should build two graphs. First dependency graph T=f(l) displays a complex non-linear dependence of the oscillation period of a physical pendulum on the distance to the swing axis. The second graph is the linearization of the same dependence. If the points on the second graph lie on a straight line with a small spread (which can be explained by measurement errors), then we can conclude that general formula(8) and, in this case, formulas (10) for the oscillation period of a physical pendulum.
6. Using the resulting dependency graph(T2l) = f(l2), determine the free fall acceleration and the length of the rod used in the experiment. To do this, you must first determine the slope of the straight line and the value of the segment b cut off by a straight line from the vertical axis (Fig. 3). Then
(12)
When calculating the length of the rod, use the experimentally obtained value of the gravitational acceleration.
In conclusion, compare the obtained values g and d with their actual values.
Report
Table 1
No. p / p | l, m | t, s | T, s | l2,m2 | T2l, c2 × m |
|
|
|
|
dependency graph T = f(l).
|
|
dependency graph T2l =f(l2)
The results of the experiment: ……………………………………………………….
Conclusions: …………………………………………………………………………….
……..………………………………………………………………………………..
font-size:16.0pt; line-height:150%"> ……… m/s2 ………m
Conclusion : ……………………………………………………………………
……………………………………………………………………………
Task 2. Study oscillations of a mathematical pendulum
1. Hang a lead ball on the thread, which best imitates a material point. Change the length of the suspension in increments of approximately 10 cm so as to get 5-6 experimental points. The number of oscillations in each experiment is not less than. The angle of deviation of the pendulum from the equilibrium position should not exceed 5-6°.
2. Addiction T=f(l) non-linear. Therefore, for the convenience of experimental verification, this dependence should be linearized. To do this, plot the dependence of the square of the period of oscillation on the length of the pendulum T2=f(l). If the experimental points lie on a straight line with a small spread (which can be explained by measurement errors), then we can conclude that formula (9) is satisfied. If the spread is large, then the entire series of measurements should be repeated.
3. Using the resulting graph, determine the free fall acceleration. First, you should obtain the exact equation of the experimental line: y=kx+b. To do this, apply the least squares method (LSM) (table 3) and determine the slope of the straight line k. Based on the obtained value of the angular coefficient, calculate the acceleration of free fall.
k=DT2/Dl = 4p2 /g, where g=4 p2 /k. (13)
Report
Initial deviationj = ................
table 2
No. p / p | l, m | N | t, c | T, c | T2 , c2 |
|
|
OLS Table 3
Designations: l = x, T2 =y
No. p / p | (xi- | (xi- | (yi- | (yi- | (xi- |
||
S= | S= | S= | S= | ........................................................................................................................ Conclusion:……………………………………………………………………… ……………………………………………………… ……………………… ……………………………………………………………………………………………………………………………………………………………… Free Fall Acceleration Calculation and its measurement errors font-size:16.0pt; font-style:normal">……… m/s2; △ g =………. m/s2 g = ……… ± ……… m/s2, d = …… % Conclusion:……………………………………………………………………… ….. ……………………………………………………………………………………… …………………………………………………………………………… ……………………………………………………………………………………………………………………………………………………………………………… Additional tasks 1. dependency graphT2 = f( l) in the third task, most likely, does not pass through zero. How can this be explained? 2. Why, in order to obtain harmonic oscillations of pendulums, it is necessary to fulfill the requirementj < 5-6 ° ? Answers DYNAMICS OF VIBRATIONAL MOTION. Terms, laws, ratios (know Tooffset) 1. What are fluctuations? harmonic vibrations? periodic processes? 2. Give definitions of amplitude, period, frequency, phase, cyclic frequency of oscillation. 3. Derive formulas for the speed and acceleration of a harmonically oscillating point as a function of time. 4. What determines the amplitude and initial phase of harmonic mechanical oscillations? 5. Derive and comment formulas for kinetic, potential and total energy of harmonic oscillations. 6. How can the masses of bodies be compared with each other by measuring the vibration frequencies when these bodies are suspended from a spring? 7. Derive formulas for the periods of oscillation of a spring, physical and mathematical pendulum. 8. What is the reduced length of a physical pendulum? When constructing this graph, the vertical axis does not have to start from scratch. It is better to choose the scale so that the vertical axis starts from the minimum value of the period of oscillation of the pendulum. |
A mathematical pendulum is a model of an ordinary pendulum. Under the mathematical pendulum - is understood material point, which is suspended on a long weightless and inextensible thread.
Bring the ball out of equilibrium and release it. There are two forces acting on the ball: gravity and tension in the string. When the pendulum moves, the force of air friction will still act on it. But we will consider it very small.
Let us decompose the force of gravity into two components: the force directed along the thread, and the force directed perpendicular to the tangent to the trajectory of the ball.
These two forces add up to gravity. The elastic forces of the thread and the component of gravity Fn impart centripetal acceleration to the ball. The work of these forces will be equal to zero, and therefore they will only change the direction of the velocity vector. At any point in time, it will be tangent to the arc of the circle.
Under the action of the gravity component Fτ, the ball will move along the arc of a circle with a speed increasing in absolute value. The value of this force always changes in absolute value; when passing through the equilibrium position, it is equal to zero.
Dynamics of oscillatory motion
The equation of motion of a body oscillating under the action of an elastic force.
General equation of motion:
Oscillations in the system occur under the action of an elastic force, which, according to Hooke's law, is directly proportional to the displacement of the load
Then the equation of motion of the ball will take the following form:
Divide this equation by m, we get the following formula:
And since the mass and coefficient of elasticity are constant values, then the ratio (-k / m) will also be constant. We have obtained an equation that describes the vibrations of a body under the action of an elastic force.
The projection of the acceleration of the body will be directly proportional to its coordinate, taken with the opposite sign.
The equation of motion of a mathematical pendulum
The equation of motion of a mathematical pendulum is described by the following formula:
This equation has the same form as the equation for the movement of a load on a spring. Consequently, the oscillations of the pendulum and the movement of the ball on the spring occur in the same way.
The displacement of the ball on the spring and the displacement of the pendulum body from the equilibrium position change with time according to the same laws.
In order to describe quantitatively 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 the laws of Newtonian mechanics.
The equation of motion of a body oscillating under the action of an elastic force. According to Newton's second law, the product of the body's mass m and its acceleration is equal to the resultant F of all forces applied to the body:
This is the equation of motion. Let us write the equation of motion for a ball moving in a straight line along the horizontal under the action of the elastic force of the spring (see Fig. 3.3). Let's direct the x-axis to the right. Let the origin of the coordinates correspond to the equilibrium position of the ball (see Fig. 3.3, a).
In the 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, are the projections of the acceleration and the elastic force of the spring on this axis.
According to Hooke's law, the projection F x ynp is directly proportional to the displacement of the ball from the 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 ypr = -kх, (3.2)
Dividing the left and right sides of equation (3.3) by m, we obtain
Since the mass m and stiffness k are constants, their ratio is also a constant.
We have obtained an equation describing the vibrations of a body under the action of an elastic force. It is very simple: the projection a x of the acceleration of the body is directly proportional to its coordinate x, taken with the opposite sign.
The equation of motion of a mathematical pendulum. When a ball oscillates on an inextensible thread, it always 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 moment of time is determined by one value - the angle α of deviation of the thread from the vertical. We will consider the angle α positive if the pendulum is deflected to the right from the equilibrium position, and negative if it is deflected to the left (see Fig. 3.5). We will assume that the tangent to the trajectory is directed towards the positive reading of the angles.
Let us denote the projection of gravity on the tangent to the trajectory of the pendulum as F τ . This projection at the moment when the pendulum thread is deflected from the equilibrium position by the angle α is equal to:
F τ \u003d -mg sin α. (3.5)
The “-” sign is here because the quantities F τ and a have opposite signs. When the pendulum deviates to the right (α > 0), the gravity component τ is directed to the left and its projection is negative: F τ< 0. При отклонении маятника влево (α < 0) эта проекция положительна: F τ > 0.
Let us denote the projection of the pendulum's acceleration on the tangent to its trajectory as а τ . This projection characterizes the rate of change of the pendulum velocity modulus.
According to Newton's second law
ma τ = -mg sin α. (3.6)
Dividing the left and right sides of this equation by m, we get
and τ \u003d -g sin α. (3.7)
Previously, 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. For small angles, if the angle is measured in radians,
Therefore, one can take
and τ = -gα. (3.8)
If the angle α is small, then the acceleration projection is approximately equal to the acceleration projection on the OX axis: a τ ≈ a x (see Fig. 3.5). From the triangle ABO for a small angle a we have:
Substituting this expression into equality (3.8) instead of the angle α, 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 of this equation will also have the same form as the solution of equation (3.4). This means that the movement of the ball and the oscillation of the pendulum occur in the same way. The displacements of the ball on the spring and the body of the pendulum from the equilibrium positions change with time according to the same law, despite the fact that the forces causing oscillations have a different physical nature. Multiplying equations (3.4) and (3.10) by m and remembering Newton's second law ma x = F x res, we can conclude that oscillations in these two cases are performed under the action of forces whose resultant is directly proportional to the displacement of the oscillating body from the equilibrium position and is directed in the direction opposite to this displacement.
Equation (3.4), like (3.10), looks very simple: acceleration is directly proportional to the coordinate (displacement from the equilibrium position).