I. Mechanics

Alexandrova Zinaida Vasilievna, teacher of physics and computer science

Educational institution: MBOU secondary school No. 5 Pechenga village, Murmansk region.

Item: physics

Class : 9th grade

Lesson topic : Movement of a body in a circle with a constant absolute speed

The purpose of the lesson:

give an idea of ​​curvilinear motion, introduce the concepts of frequency, period, angular velocity, centripetal acceleration and centripetal force.

Lesson objectives:

Educational:

Review the types of mechanical motion, introduce new concepts: circular motion, centripetal acceleration, period, frequency;

Reveal in practice the relationship between period, frequency and centripetal acceleration with the radius of circulation;

Use educational laboratory equipment to solve practical problems.

Developmental :

Develop the ability to apply theoretical knowledge to solve specific problems;

Develop a culture of logical thinking;

Develop interest in the subject; cognitive activity when setting up and conducting an experiment.

Educational :

Form a worldview in the process of studying physics and justify your conclusions, cultivate independence and accuracy;

Foster the communicative and information culture of students

Lesson equipment:

computer, projector, screen, presentation for lesson "Movement of a body in a circle", printing out cards with tasks;

tennis ball, badminton shuttlecock, toy car, ball on a string, tripod;

sets for the experiment: stopwatch, tripod with coupling and foot, ball on a string, ruler.

Form of training organization: frontal, individual, group.

Lesson type: study and primary consolidation of knowledge.

Educational and methodological support: Physics. 9th grade. Textbook. Peryshkin A.V., Gutnik E.M. 14th ed., erased. - M.: Bustard, 2012.

Lesson implementation time : 45 minutes

1. Editor in which the multimedia resource is created:MSPowerPoint

2. Type of multimedia resource: visual presentation of educational material using triggers, embedded video and interactive test.

Lesson Plan

Organizing time. Motivation for learning activities.

Updating basic knowledge.

Learning new material.

Conversation on issues;

Problem solving;

Carrying out practical research work.

Summing up the lesson.

During the classes

Lesson steps

Temporary implementation

Organizing time. Motivation for learning activities.

Slide 1. ( Checking readiness for the lesson, announcing the topic and objectives of the lesson.)

Teacher. Today in the lesson you will learn what acceleration is during uniform motion of a body in a circle and how to determine it.

2 minutes

Updating basic knowledge.

Slide 2.

Fphysical dictation:

Changes in body position in space over time.(Movement)

A physical quantity measured in meters.(Move)

A physical vector quantity characterizing the speed of movement.(Speed)

The basic unit of length in physics.(Meter)

A physical quantity whose units are year, day, hour.(Time)

A physical vector quantity that can be measured using an accelerometer device.(Acceleration)

Path length. (Path)

Acceleration units(m/s 2 ).

(Conducting a dictation followed by testing, self-assessment of work by students)

5 minutes

Learning new material.

Slide 3.

Teacher. We quite often observe a movement of a body in which its trajectory is a circle. For example, a point on the rim of a wheel moves along a circle as it rotates, points on rotating parts of machine tools, or the end of a clock hand.

Demonstrations of experiments 1. The fall of a tennis ball, the flight of a badminton shuttlecock, the movement of a toy car, the vibrations of a ball on a string attached to a tripod. What do these movements have in common and how do they differ in appearance?(Students' answers)

Teacher. Rectilinear movement is movement whose trajectory is a straight line, curvilinear movement is a curve. Give examples of rectilinear and curvilinear motion that you have encountered in life.(Students' answers)

The movement of a body in a circle isa special case of curvilinear motion.

Any curve can be represented as the sum of circular arcsdifferent (or the same) radius.

Curvilinear motion is a movement that occurs along circular arcs.

Let us introduce some characteristics of curvilinear motion.

Slide 4. (watch video " speed.avi" (link on slide)

Curvilinear motion with a constant modulus speed. Movement with acceleration, because speed changes direction.

Slide 5 . (watch video “Dependence of centripetal acceleration on radius and speed. avi » via the link on the slide)

Slide 6. Direction of velocity and acceleration vectors.

(working with slide materials and analyzing drawings, rational use of animation effects embedded in the elements of drawings, Fig. 1.)

Fig.1.

Slide 7.

When a body moves uniformly in a circle, the acceleration vector is always perpendicular to the velocity vector, which is directed tangentially to the circle.

A body moves in a circle provided that that the linear velocity vector is perpendicular to the centripetal acceleration vector.

Slide 8. (working with illustrations and slide materials)

Centripetal acceleration - the acceleration with which a body moves in a circle with a constant absolute speed is always directed along the radius of the circle towards the center.

a ts =

Slide 9.

When moving in a circle, the body will return to its original point after a certain period of time. Circular motion is periodic.

Circulation period - this is a period of timeT , during which the body (point) makes one revolution around the circle.

Period unit -second

Rotational speed  – number of full revolutions per unit time.

[  ] = s -1 = Hz

Frequency unit

Student message 1. A period is a quantity that is often found in nature, science and technology. The earth rotates around its axis, the average period of this rotation is 24 hours; a complete revolution of the Earth around the Sun occurs in approximately 365.26 days; a helicopter propeller has an average rotation period of 0.15 to 0.3 s; The period of blood circulation in humans is approximately 21 - 22 s.

Student message 2. Frequency is measured with special devices - tachometers.

Rotation speed of technical devices: the gas turbine rotor rotates at a frequency of 200 to 300 1/s; a bullet fired from a Kalashnikov assault rifle rotates at a frequency of 3000 1/s.

Slide 10. Relationship between period and frequency:

If during time t the body has made N full revolutions, then the period of revolution is equal to:

Period and frequency are reciprocal quantities: frequency is inversely proportional to the period, and period is inversely proportional to frequency

Slide 11. The speed of rotation of a body is characterized by angular velocity.

Angular velocity(cyclic frequency) - the number of revolutions per unit of time, expressed in radians.

Angular velocity is the angle of rotation through which a point rotates in timet.

Angular velocity is measured in rad/s.

Slide 12. (watch video "Path and displacement in curved motion.avi" (link on slide)

Slide 13 . Kinematics of motion in a circle.

Teacher. With uniform motion in a circle, the magnitude of its speed does not change. But speed is a vector quantity, and it is characterized not only by its numerical value, but also by its direction. With uniform motion in a circle, the direction of the velocity vector changes all the time. Therefore, such uniform motion is accelerated.

Linear speed: ;

Linear and angular velocities are related by the relation:

Centripetal acceleration: ;

Angular velocity: ;

Slide 14. (working with illustrations on the slide)

Direction of the velocity vector.Linear (instantaneous speed) is always directed tangentially to the trajectory drawn to the point where the physical body in question is currently located.

The velocity vector is directed tangentially to the circumscribed circle.

Uniform motion of a body in a circle is motion with acceleration. With uniform motion of a body in a circle, the quantities υ and ω remain unchanged. In this case, when moving, only the direction of the vector changes.

Slide 15. Centripetal force.

The force that holds a rotating body on a circle and is directed towards the center of rotation is called centripetal force.

To obtain a formula for calculating the magnitude of the centripetal force, you need to use Newton's second law, which applies to any curvilinear motion.

Substituting into the formula centripetal acceleration valuea ts = , we obtain the formula for centripetal force:

F=

From the first formula it is clear that at the same speed, the smaller the radius of the circle, the greater the centripetal force. So, at road turns, a moving body (train, car, bicycle) should act towards the center of the curve, the greater the force, the sharper the turn, i.e., the smaller the radius of the curve.

Centripetal force depends on linear speed: as speed increases, it increases. This is well known to all skaters, skiers and cyclists: the faster you move, the more difficult it is to make a turn. Drivers know very well how dangerous it is to turn a car sharply at high speed.

Slide 16.

Summary table of physical quantities characterizing curvilinear motion(analysis of dependencies between quantities and formulas)

Slides 17, 18, 19. Examples of movement in a circle.

Circular traffic on the roads. The movement of satellites around the Earth.

Slide 20. Attractions, carousels.

Student message 3. In the Middle Ages, knightly tournaments were called carousels (the word then had a masculine gender). Later, in the 18th century, to prepare for tournaments, instead of fights with real opponents, they began to use a rotating platform, the prototype of the modern entertainment carousel, which then appeared at city fairs.

In Russia, the first carousel was built on June 16, 1766 in front of the Winter Palace. The carousel consisted of four quadrilles: Slavic, Roman, Indian, Turkish. The second time the carousel was built in the same place, on July 11th of the same year. A detailed description of these carousels is given in the newspaper St. Petersburg Gazette of 1766.

A carousel, common in courtyards during Soviet times. The carousel can be driven either by a motor (usually electric) or by the forces of the spinners themselves, who spin it before sitting on the carousel. Such carousels, which need to be spun by the riders themselves, are often installed on children's playgrounds.

In addition to attractions, carousels are often called other mechanisms that have similar behavior - for example, in automated lines for bottling drinks, packaging bulk substances or producing printed materials.

In a figurative sense, a carousel is a series of rapidly changing objects or events.

18 min

Consolidation of new material. Application of knowledge and skills in a new situation.

Teacher. Today in this lesson we learned about the description of curvilinear motion, new concepts and new physical quantities.

Conversation on questions:

What is a period? What is frequency? How are these quantities related to each other? In what units are they measured? How can they be identified?

What is angular velocity? In what units is it measured? How can you calculate it?

What is angular velocity called? What is the unit of angular velocity?

How are the angular and linear velocities of a body related?

What is the direction of centripetal acceleration? What formula is it calculated by?

Slide 21.

Exercise 1. Fill out the table by solving problems using the source data (Fig. 2), then we will compare the answers. (Students work independently with the table; it is necessary to prepare a printout of the table for each student in advance)

Fig.2

Slide 22. Task 2.(orally)

Pay attention to the animation effects of the drawing. Compare the characteristics of uniform motion of a blue and red ball. (Working with the illustration on the slide).

Slide 23. Task 3.(orally)

The wheels of the presented modes of transport make an equal number of revolutions at the same time. Compare their centripetal accelerations.(Working with slide materials)

(Work in a group, conduct an experiment, print out instructions for conducting the experiment are on each table)

Equipment: stopwatch, ruler, ball attached to a thread, tripod with coupling and foot.

Target: researchdependence of period, frequency and acceleration on the radius of rotation.

Work plan

Measuretime t 10 full revolutions of rotational motion and radius R of rotation of the ball attached to a thread in a tripod.

Calculateperiod T and frequency, rotation speed, centripetal acceleration. Formulate the results in the form of a problem.

Changeradius of rotation (length of the thread), repeat the experiment 1 more time, trying to maintain the same speed,applying the same effort.

Draw a conclusionon the dependence of the period, frequency and acceleration on the radius of rotation (the smaller the radius of rotation, the shorter the period of revolution and the greater the frequency value).

Slides 24 -29.

Frontal work with an interactive test.

You must select one answer out of three possible ones; if the correct answer was selected, it remains on the slide and the green indicator begins to blink; incorrect answers disappear.

A body moves in a circle with a constant absolute speed. How will its centripetal acceleration change when the radius of the circle decreases by 3 times?

In the centrifuge of a washing machine, during spinning, the laundry moves in a circle with a constant modulus speed in the horizontal plane. What is the direction of its acceleration vector?

A skater moves at a speed of 10 m/s in a circle with a radius of 20 m. Determine his centripetal acceleration.

Where is the acceleration of a body directed when it moves in a circle with a constant velocity?

A material point moves in a circle with a constant absolute speed. How will the modulus of its centripetal acceleration change if the speed of the point is tripled?

A car wheel makes 20 revolutions in 10 s. Determine the period of revolution of the wheel?

Slide 30. Problem solving(independent work if there is time in class)

Option 1.

With what period must a carousel with a radius of 6.4 m rotate so that the centripetal acceleration of a person on the carousel is equal to 10 m/s 2 ?

In the circus arena, a horse gallops at such a speed that it runs 2 circles in 1 minute. The radius of the arena is 6.5 m. Determine the period and frequency of rotation, speed and centripetal acceleration.

Option 2.

Carousel rotation frequency 0.05 s -1 . A person spinning on a carousel is at a distance of 4 m from the axis of rotation. Determine the man's centripetal acceleration, period of revolution, and angular velocity of the merry-go-round.

A point on the rim of a bicycle wheel makes one revolution in 2 s. The radius of the wheel is 35 cm. What is the centripetal acceleration of the wheel rim point?

18 min

Summing up the lesson.

Grading. Reflection.

Slide 31 .

D/z: paragraphs 18-19, Exercise 18 (2.4).

http:// www. stmary. ws/ highschool/ physics/ home/ lab/ labGraphic. gif

1.Uniform movement in a circle

2. Angular speed of rotational motion.

3. Rotation period.

4. Rotation speed.

5. Relationship between linear speed and angular speed.

6.Centripetal acceleration.

7. Equally alternating movement in a circle.

8. Angular acceleration in uniform circular motion.

9.Tangential acceleration.

10. Law of uniformly accelerated motion in a circle.

11. Average angular velocity in uniformly accelerated motion in a circle.

12. Formulas establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.

1.Uniform movement around a circle– movement in which a material point passes equal segments of a circular arc in equal time intervals, i.e. the point moves in a circle with a constant absolute speed. In this case, the speed is equal to the ratio of the arc of a circle traversed by the point to the time of movement, i.e.

and is called the linear speed of movement in a circle.

As in curvilinear motion, the velocity vector is directed tangentially to the circle in the direction of motion (Fig. 25).

2. Angular velocity in uniform circular motion– ratio of the radius rotation angle to the rotation time:

In uniform circular motion, the angular velocity is constant. In the SI system, angular velocity is measured in (rad/s). One radian - a rad is the central angle subtending an arc of a circle with a length equal to the radius. A full angle contains radians, i.e. per revolution the radius rotates by an angle of radians.

3. Rotation period– time interval T during which a material point makes one full revolution. In the SI system, the period is measured in seconds.

4. Rotation frequency– the number of revolutions made in one second. In the SI system, frequency is measured in hertz (1Hz = 1). One hertz is the frequency at which one revolution is completed in one second. It's easy to imagine that

If during time t a point makes n revolutions around a circle then .

Knowing the period and frequency of rotation, the angular velocity can be calculated using the formula:

5 Relationship between linear speed and angular speed. The length of an arc of a circle is equal to where is the central angle, expressed in radians, the radius of the circle subtending the arc. Now we write the linear speed in the form

It is often convenient to use the formulas: or Angular velocity is often called cyclic frequency, and frequency is called linear frequency.

6. Centripetal acceleration. In uniform motion around a circle, the velocity module remains unchanged, but its direction continuously changes (Fig. 26). This means that a body moving uniformly in a circle experiences acceleration, which is directed towards the center and is called centripetal acceleration.

Let a distance travel equal to an arc of a circle in a period of time. Let's move the vector, leaving it parallel to itself, so that its beginning coincides with the beginning of the vector at point B. The modulus of change in speed is equal to , and the modulus of centripetal acceleration is equal

In Fig. 26, the triangles AOB and DVS are isosceles and the angles at the vertices O and B are equal, as are the angles with mutually perpendicular sides AO and OB. This means that the triangles AOB and DVS are similar. Therefore, if, that is, the time interval takes arbitrarily small values, then the arc can be approximately considered equal to the chord AB, i.e. . Therefore, we can write Considering that VD = , OA = R we obtain Multiplying both sides of the last equality by , we further obtain the expression for the modulus of centripetal acceleration in uniform motion in a circle: . Considering that we get two frequently used formulas:

So, in uniform motion around a circle, the centripetal acceleration is constant in magnitude.

It is easy to understand that in the limit at , angle . This means that the angles at the base of the DS of the ICE triangle tend to the value , and the speed change vector becomes perpendicular to the speed vector, i.e. directed radially towards the center of the circle.

7. Equally alternating circular motion– circular motion in which the angular velocity changes by the same amount over equal time intervals.

8. Angular acceleration in uniform circular motion– the ratio of the change in angular velocity to the time interval during which this change occurred, i.e.

where the initial value of angular velocity, the final value of angular velocity, angular acceleration, in the SI system is measured in . From the last equality we obtain formulas for calculating the angular velocity

And if .

Multiplying both sides of these equalities by and taking into account that , is the tangential acceleration, i.e. acceleration directed tangentially to the circle, we obtain formulas for calculating linear speed:

And if .

9. Tangential acceleration numerically equal to the change in speed per unit time and directed along the tangent to the circle. If >0, >0, then the motion is uniformly accelerated. If<0 и <0 – движение.

10. Law of uniformly accelerated motion in a circle. The path traveled around a circle in time in uniformly accelerated motion is calculated by the formula:

Substituting , , and reducing by , we obtain the law of uniformly accelerated motion in a circle:

Or if.

If the movement is uniformly slow, i.e.<0, то

11.Total acceleration in uniformly accelerated circular motion. In uniformly accelerated motion in a circle, centripetal acceleration increases over time, because Due to tangential acceleration, linear speed increases. Very often, centripetal acceleration is called normal and is denoted as. Since the total acceleration at a given moment is determined by the Pythagorean theorem (Fig. 27).

12. Average angular velocity in uniformly accelerated motion in a circle. The average linear speed in uniformly accelerated motion in a circle is equal to . Substituting here and and reducing by we get

If, then.

12. Formulas establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.

Substituting the quantities , , , , into the formula

and reducing by , we get

Lecture-4. Dynamics.

1. Dynamics

2. Interaction of bodies.

3. Inertia. The principle of inertia.

4. Newton's first law.

5. Free material point.

6. Inertial reference system.

7. Non-inertial reference system.

8. Galileo's principle of relativity.

9. Galilean transformations.

11. Addition of forces.

13. Density of substances.

14. Center of mass.

15. Newton's second law.

16. Unit of force.

17. Newton's third law

1. Dynamics there is a branch of mechanics that studies mechanical motion, depending on the forces that cause a change in this motion.

2.Interactions of bodies. Bodies can interact both in direct contact and at a distance through a special type of matter called a physical field.

For example, all bodies are attracted to each other and this attraction is carried out through the gravitational field, and the forces of attraction are called gravitational.

Bodies carrying an electric charge interact through an electric field. Electric currents interact through a magnetic field. These forces are called electromagnetic.

Elementary particles interact through nuclear fields and these forces are called nuclear.

3.Inertia. In the 4th century. BC e. The Greek philosopher Aristotle argued that the cause of the movement of a body is the force acting from another body or bodies. At the same time, according to Aristotle’s movement, a constant force imparts a constant speed to the body and, with the cessation of the action of the force, the movement ceases.

In the 16th century Italian physicist Galileo Galilei, conducting experiments with bodies rolling down an inclined plane and with falling bodies, showed that a constant force (in this case, the weight of a body) imparts acceleration to the body.

So, based on experiments, Galileo showed that force is the cause of the acceleration of bodies. Let us present Galileo's reasoning. Let a very smooth ball roll along a smooth horizontal plane. If nothing interferes with the ball, then it can roll for as long as desired. If a thin layer of sand is poured on the path of the ball, it will stop very soon, because it was affected by the frictional force of the sand.

So Galileo came to the formulation of the principle of inertia, according to which a material body maintains a state of rest or uniform rectilinear motion if no external forces act on it. This property of matter is often called inertia, and the movement of a body without external influences is called motion by inertia.

4. Newton's first law. In 1687, based on Galileo's principle of inertia, Newton formulated the first law of dynamics - Newton's first law:

A material point (body) is in a state of rest or uniform linear motion if other bodies do not act on it, or the forces acting from other bodies are balanced, i.e. compensated.

5.Free material point- a material point that is not affected by other bodies. Sometimes they say - an isolated material point.

6. Inertial reference system (IRS)– a reference system relative to which an isolated material point moves rectilinearly and uniformly, or is at rest.

Any reference system that moves uniformly and rectilinearly relative to the ISO is inertial,

Let us give another formulation of Newton's first law: There are reference systems relative to which a free material point moves rectilinearly and uniformly, or is at rest. Such reference systems are called inertial. Newton's first law is often called the law of inertia.

Newton's first law can also be given the following formulation: every material body resists a change in its speed. This property of matter is called inertia.

We encounter manifestations of this law every day in urban transport. When the bus suddenly picks up speed, we are pressed against the back of the seat. When the bus slows down, our body skids in the direction of the bus.

7. Non-inertial reference system – a reference system that moves unevenly relative to the ISO.

A body that, relative to the ISO, is in a state of rest or uniform linear motion. It moves unevenly relative to a non-inertial reference frame.

Any rotating reference system is a non-inertial reference system, because in this system the body experiences centripetal acceleration.

There are no bodies in nature or technology that could serve as ISOs. For example, the Earth rotates around its axis and any body on its surface experiences centripetal acceleration. However, for fairly short periods of time, the reference system associated with the Earth’s surface can, to some approximation, be considered ISO.

8.Galileo's principle of relativity. ISO can be as much salt as you like. Therefore, the question arises: what do the same mechanical phenomena look like in different ISOs? Is it possible, using mechanical phenomena, to detect the movement of the ISO in which they are observed.

The answer to these questions is given by the principle of relativity of classical mechanics, discovered by Galileo.

The meaning of the principle of relativity of classical mechanics is the statement: all mechanical phenomena proceed exactly the same way in all inertial frames of reference.

This principle can be formulated as follows: all laws of classical mechanics are expressed by the same mathematical formulas. In other words, no mechanical experiments will help us detect the movement of the ISO. This means that trying to detect ISO movement is meaningless.

We encountered the manifestation of the principle of relativity while traveling on trains. At the moment when our train is standing at the station, and the train standing on the adjacent track slowly begins to move, then in the first moments it seems to us that our train is moving. But it also happens the other way around, when our train smoothly picks up speed, it seems to us that the neighboring train has started moving.

In the above example, the principle of relativity manifests itself over small time intervals. As the speed increases, we begin to feel shocks and swaying of the car, i.e. our reference system becomes non-inertial.

So, trying to detect ISO movement is pointless. Consequently, it is absolutely indifferent which ISO is considered stationary and which is moving.

9. Galilean transformations. Let two ISOs move relative to each other with a speed. In accordance with the principle of relativity, we can assume that the ISO K is stationary, and the ISO moves relatively at a speed. For simplicity, we assume that the corresponding coordinate axes of the systems and are parallel, and the axes and coincide. Let the systems coincide at the moment of beginning and the movement occurs along the axes and , i.e. (Fig.28)

11. Addition of forces. If two forces are applied to a particle, then the resulting force is equal to their vector force, i.e. diagonals of a parallelogram built on vectors and (Fig. 29).

The same rule applies when decomposing a given force into two force components. To do this, a parallelogram is constructed on the vector of a given force, as on a diagonal, the sides of which coincide with the direction of the components of the forces applied to a given particle.

If several forces are applied to the particle, then the resulting force is equal to the geometric sum of all forces:

12.Weight. Experience has shown that the ratio of the modulus of force to the modulus of acceleration, which this force imparts to the body, is a constant value for a given body and is called the mass of the body:

From the last equality it follows that the greater the mass of the body, the greater the force must be applied to change its speed. Consequently, the greater the mass of a body, the more inert it is, i.e. mass is a measure of the inertia of bodies. The mass determined in this way is called inertial mass.

In the SI system, mass is measured in kilograms (kg). One kilogram is the mass of distilled water in a volume of one cubic decimeter taken at a temperature

13. Density of matter– the mass of a substance contained in a unit volume or the ratio of body mass to its volume

Density is measured in () in the SI system. Knowing the density of a body and its volume, you can calculate its mass using the formula. Knowing the density and mass of a body, its volume is calculated using the formula.

14.Center of mass- a point of a body that has the property that if the direction of the force passes through this point the body moves translationally. If the direction of action does not pass through the center of mass, then the body moves while simultaneously rotating around its center of mass

15. Newton's second law. In ISO, the sum of forces acting on a body is equal to the product of the mass of the body and the acceleration imparted to it by this force

16.Unit of force. In the SI system, force is measured in newtons. One newton (n) is a force that, acting on a body weighing one kilogram, imparts acceleration to it. That's why .

17. Newton's third law. The forces with which two bodies act on each other are equal in magnitude, opposite in direction and act along one straight line connecting these bodies.

Movement of a body in a circle with a constant absolute speed- this is a movement in which a body describes identical arcs at any equal intervals of time.

The position of the body on the circle is determined radius vector\(~\vec r\) drawn from the center of the circle. The modulus of the radius vector is equal to the radius of the circle R(Fig. 1).

During time Δ t body moving from a point A exactly IN, makes a displacement \(~\Delta \vec r\) equal to the chord AB, and travels a path equal to the length of the arc l.

The radius vector rotates by an angle Δ φ . The angle is expressed in radians.

The speed \(~\vec \upsilon\) of a body's movement along a trajectory (circle) is directed tangent to the trajectory. It is called linear speed. The modulus of linear velocity is equal to the ratio of the length of the circular arc l to the time interval Δ t for which this arc is completed:

\(~\upsilon = \frac(l)(\Delta t).\)

A scalar physical quantity, numerically equal to the ratio of the angle of rotation of the radius vector to the period of time during which this rotation occurred, is called angular velocity:

\(~\omega = \frac(\Delta \varphi)(\Delta t).\)

The SI unit of angular velocity is radian per second (rad/s).

With uniform motion in a circle, the angular velocity and the linear velocity module are constant quantities: ω = const; υ = const.

The position of the body can be determined if the modulus of the radius vector \(~\vec r\) and the angle φ , which it composes with the axis Ox(angular coordinate). If at the initial moment of time t 0 = 0 angular coordinate is φ 0 , and at time t it is equal φ , then the rotation angle Δ φ radius vector for time \(~\Delta t = t - t_0 = t\) is equal to \(~\Delta \varphi = \varphi - \varphi_0\). Then from the last formula we can get kinematic equation of motion of a material point along a circle:

\(~\varphi = \varphi_0 + \omega t.\)

It allows you to determine the position of the body at any time t. Considering that \(~\Delta \varphi = \frac(l)(R)\), we obtain\[~\omega = \frac(l)(R \Delta t) = \frac(\upsilon)(R) \Rightarrow\]

\(~\upsilon = \omega R\) - formula for the relationship between linear and angular speed.

Time interval Τ during which the body makes one full revolution is called rotation period:

\(~T = \frac(\Delta t)(N),\)

Where N- number of revolutions made by the body during time Δ t.

During time Δ t = Τ the body travels the path \(~l = 2 \pi R\). Hence,

\(~\upsilon = \frac(2 \pi R)(T); \ \omega = \frac(2 \pi)(T) .\)

Magnitude ν , the inverse of the period, showing how many revolutions a body makes per unit time, is called rotation speed:

\(~\nu = \frac(1)(T) = \frac(N)(\Delta t).\)

Hence,

\(~\upsilon = 2 \pi \nu R; \\omega = 2 \pi \nu .\)

Literature

Aksenovich L. A. Physics in secondary school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyakhavanne, 2004. - P. 18-19.

Topics of the Unified State Examination codifier: motion in a circle with a constant absolute speed, centripetal acceleration.

Uniform movement around a circle - This is a fairly simple example of motion with an acceleration vector that depends on time.

Let the point rotate along a circle of radius . The speed of the point is constant in absolute value and equal to . Speed ​​is called linear speed points.

Circulation period - this is the time of one full revolution. For the period we have an obvious formula:

. (1)

Frequency is the reciprocal of the period:

Frequency shows how many full revolutions a point makes per second. The frequency is measured in rps (revolutions per second).

Let, for example, . This means that during the time the point makes one complete
turnover The frequency is then equal to: r/s; per second the point makes 10 full revolutions.

Angular velocity.

Let's consider the uniform rotation of a point in a Cartesian coordinate system. Let's place the origin of coordinates at the center of the circle (Fig. 1).

Rice. 1. Uniform movement in a circle

Let be the initial position of the point; in other words, at the point had coordinates . Let the point turn through an angle and take position .

The ratio of the angle of rotation to time is called angular velocity point rotation:

. (2)

Angle is typically measured in radians, so angular velocity is measured in rad/s. In a time equal to the rotation period, the point rotates through an angle. That's why

. (3)

Comparing formulas (1) and (3), we obtain the relationship between linear and angular velocities:

. (4)

Law of motion.

Let us now find the dependence of the coordinates of the rotating point on time. We see from Fig. 1 that

But from formula (2) we have: . Hence,

. (5)

Formulas (5) are the solution to the main problem of mechanics for the uniform motion of a point along a circle.

Centripetal acceleration.

Now we are interested in the acceleration of the rotating point. It can be found by differentiating relations (5) twice:

Taking into account formulas (5) we have:

(6)

The resulting formulas (6) can be written as one vector equality:

(7)

where is the radius vector of the rotating point.

We see that the acceleration vector is directed opposite to the radius vector, i.e., towards the center of the circle (see Fig. 1). Therefore, the acceleration of a point moving uniformly around a circle is called centripetal.

In addition, from formula (7) we obtain an expression for the modulus of centripetal acceleration:

(8)

Let us express the angular velocity from (4)

and substitute it into (8). Let's get another formula for centripetal acceleration.