PHYSICAL QUANTITIES CHARACTERIZING THE CIRCULAR MOTION OF A BODY.
1. PERIOD (T) - the period of time during which the body makes one full revolution.
, where t is the time during which N revolutions are completed.
2. FREQUENCY () - the number of revolutions N made by a body per unit of time.
(hertz)
3. RELATIONSHIP OF PERIOD AND FREQUENCY:
4. MOVE () is directed along chords.
5. ANGULAR MOVEMENT (angle of rotation).
UNIFORM CIRCULAR MOTION is a movement in which the speed module does not change.
6. LINEAR SPEED (directed tangentially to the circle.
7. ANGULAR SPEED 
8. RELATIONSHIP OF LINEAR AND ANGULAR SPEED
Angular velocity does not depend on the radius of the circle along which the body moves. If the problem considers the movement of points located on the same disk, but at different distances from its center, then we must keep in mind that the ANGULAR VELOCITY OF THESE POINTS IS THE SAME.
9. CENTRIPETAPAL (normal) ACCELERATION ().
Since when moving in a circle, the direction of the velocity vector constantly changes, the movement in the circle occurs with acceleration. If a body moves uniformly around a circle, then it has only centripetal (normal) acceleration, which is directed radially towards the center of the circle. Acceleration is called normal, since at a given point the acceleration vector is located perpendicular (normal) to the linear velocity vector. .
If a body moves in a circle with a speed varying in absolute value, then along with normal acceleration, which characterizes the change in speed in direction, TANGENTIAL ACCELERATION appears, which characterizes the change in speed in absolute value (). The tangential acceleration is directed tangent to the circle. The total acceleration of a body during uneven circular motion is determined by the Pythagorean theorem:
RELATIVITY OF MECHANICAL MOTION
When considering the motion of a body relative to different systems the reference trajectory, path, speed, movement turn out to be different. For example, a person is sitting on a moving bus. Its trajectory relative to the bus is a point, and relative to the Sun - an arc of a circle, path, speed, displacement relative to the bus are equal to zero, and relative to the Earth they are different from zero. If the motion of a body relative to a moving and stationary reference system is considered, then according to the classical law of addition of velocities, the speed of a body relative to a stationary reference system is equal to the vector sum of the body’s speed relative to a moving reference system and the speed of a moving reference system relative to a stationary one:
Likewise
SPECIAL CASES OF USING THE LAW OF SPEED ADDITION
1) Movement of bodies relative to the Earth
b) bodies move towards each other
2) Movement of bodies relative to each other
a) bodies move in one direction
b) bodies move in different directions(towards each other)
3) Speed of a body relative to the shore when moving
a) downstream
b) against the current, where is the speed of the body relative to the water, is the speed of the current.
4) The velocities of the bodies are directed at an angle to each other.
For example: a) a body swims across a river, moving perpendicular to the flow
b) the body swims across the river, moving perpendicular to the shore
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c) the body simultaneously participates in translational and rotational motion, for example, the wheel of a moving car. Each point of the body has a translational velocity directed in the direction of the body’s movement and a rotational velocity directed tangentially to the circle. Moreover, to find the speed of any point relative to the Earth, it is necessary to vectorially add the speed of translational and rotational motion:
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DYNAMICS
NEWTON'S LAWS
NEWTON'S FIRST LAW (LAW OF INERTIA)
There are such reference systems relative to which the body is at rest or moves rectilinearly and uniformly, if other bodies do not act on it or the actions of the bodies are compensated (balanced).
The phenomenon of maintaining the speed of a body in the absence of the action of other bodies on it or when compensating for the action of other bodies is called inertia.
The reference systems in which Newton's laws are satisfied are called inertial reference systems (IRS). ISO refers to reference systems associated with the Earth or not having acceleration relative to the Earth. Frames of reference moving with acceleration relative to the Earth are non-inertial, and Newton's laws are not satisfied in them. According to Galileo’s classical principle of relativity, all ISOs are equal in rights, the laws of mechanics have the same form in all ISOs, all mechanical processes proceed in the same way in all ISOs (no mechanical experiments carried out inside an ISO can determine whether it is at rest or moving rectilinearly and uniformly).
NEWTON'S SECOND LAW
The speed of a body changes when a force is applied to the body. Any body has the property of inertia . Inertia – This is a property of bodies, consisting in the fact that it takes time to change the speed of a body; the speed of a body cannot change instantly. The body that changes its speed more under the action of the same force is less inert. The measure of inertia is body mass.
The acceleration of a body is directly proportional to the force acting on it and inversely proportional to the mass of the body.
Force and acceleration are always co-directional. If several forces act on a body, then the acceleration imparts to the body resultant these forces (), which is equal to the vector sum of all forces acting on the body: 
If the body does uniformly accelerated motion, then a constant force acts on it.
NEWTON'S THIRD LAW
Forces arise when bodies interact.
Bodies act on each other with forces directed along the same straight line, equal in magnitude and opposite in direction.
Features of forces arising during interaction:
1. Forces always arise in pairs.
2 The forces arising during interaction are of the same nature.
3. Forces do not have a resultant, because they are applied to different bodies.
FORCES IN MECHANICS
UNIVERSAL GRAVITATION is the force with which all bodies in the Universe are attracted.
LAW OF UNIVERSAL GRAVITY: bodies attract each other with forces directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
(the formula can be used to calculate the attraction of point bodies and balls), where G is the gravitational constant (universal gravity constant), G = 6.67·10 -11, is the mass of the bodies, R is the distance between the bodies, measured between the centers of the bodies. 
GRAVITY – the force of attraction of bodies towards the planet. Gravity is calculated using the formulas:
1) , where is the mass of the planet, is the mass of the body, is the distance between the center of the planet and the body.
2) , where is the acceleration of free fall,
The force of gravity is always directed towards the center of gravity of the planet.
The radius of the orbit of an artificial satellite, - the radius of the planet, - the height of the satellite above surface of the planet,
A body becomes an artificial satellite if it is given the required speed in the horizontal direction. The speed required for a body to move in a circular orbit around a planet is called first escape velocity. To obtain a formula for calculating the first cosmic velocity, it is necessary to remember that all cosmic bodies, including artificial satellites, move under the influence of universal gravity, in addition, velocity is a kinematic quantity; the formula following from Newton's second law Equating the right-hand sides of the formulas, we obtain: or Considering that the body moves in a circle and therefore has centripetal acceleration, we obtain: or. From here - formula for calculating the first escape velocity. Considering that the formula for calculating the first escape velocity can be written as: .Similarly, using Newton’s second law and formulas curvilinear movement, you can determine, for example, the period of revolution of a body in orbit.
ELASTIC FORCE is a force acting on the part of a deformed body and directed in the direction opposite to the displacement of particles during deformation. The elastic force can be calculated using Hooke's law: elastic force is directly proportional to elongation: where is the elongation,
Hardness, . Rigidity depends on the material of the body, its shape and size.
SPRING CONNECTION
Hooke's law is valid only for elastic deformations of bodies. Elastic deformations are those in which, after the cessation of the force, the body acquires its previous shape and size.
1. Quite often one can 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, the end of a clock hand, a child sitting on some figure of a rotating carousel.
When moving in a circle, not only the direction of the body’s velocity can change, but also its modulus. Movement is possible in which only the direction of velocity changes, and its magnitude remains constant. This movement is called uniform movement of the body in a circle. Let us introduce the characteristics of this movement.
2. The circular motion of a body is repeated at certain intervals equal to the period of revolution.
The period of revolution is the time during which a body completes one full revolution.
The circulation period is designated by the letter T. The unit of circulation period in SI is taken to be second (1 s).
If during the time t the body has committed N full revolutions, then the period of revolution is equal to:
T = .
The rotation frequency is the number of complete rotations of a body in one second.
The frequency of circulation is indicated by the letter n.
n = .
The unit of circulation frequency in SI is taken to be second to the minus first power (1 s– 1).
Frequency and period of revolution are related as follows:
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n = . |
3. Let's consider a quantity characterizing the position of a body on a circle. Let at the initial moment of time the body be at the point A, and in time t it moved to a point B(Fig. 38).
Let's draw a radius vector from the center of the circle to the point A and radius vector from the center of the circle to the point B. When a body moves in a circle, the radius vector will rotate in time t at angle j. Knowing the angle of rotation of the radius vector, you can determine the position of the body on the circle.
Unit of rotation angle of the radius vector in SI - radian (1 rad).
At the same angle of rotation of the radius vector of the point A And B, located at different distances from its center of a uniformly rotating disk (Fig. 39), will travel different paths.
4. When a body moves in a circle, the instantaneous speed is called linear speed.
The linear speed of a body moving uniformly in a circle, while remaining constant in magnitude, changes in direction and at any point is directed tangentially to the trajectory.
The linear velocity module can be determined by the formula:
v = .
Let a body moving in a circle with a radius R, made one full revolution, Then the path it traveled equal to length circles: l= 2p R, and time is equal to the revolution period T. Therefore, the linear speed of the body:
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v = . |
Because T= , then we can write
v= 2p Rn.
The speed of rotation of a body is characterized by angular velocity.
Angular velocity is a physical quantity equal to the ratio the angle of rotation of the radius vector to the time period during which this rotation occurred.
Angular velocity is denoted by w.
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w = . |
The SI unit of angular velocity is radians per second (1 rad/s):
[w] == 1 rad/s.
For a time equal to the circulation period T, the body makes a full revolution and the angle of rotation of the radius vector j = 2p. Therefore, the angular velocity of the body is:
w =or w = 2p n.
Linear and angular velocities are related to each other. Let's write down the ratio of linear speed to angular speed:
== R.
Thus,
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v=w R. |
At the same angular velocity of points A And B, located on a uniformly rotating disk (see Fig. 39), the linear speed of the point A greater than the linear speed of the point B: v A > vB.
5. When a body moves uniformly in a circle, the magnitude of its linear velocity remains constant, but the direction of the velocity changes. Since speed is a vector quantity, a change in the direction of speed means that the body is moving in a circle with acceleration.
Let's find out how this acceleration is directed and what it is equal to.
Let us recall that the acceleration of a body is determined by the formula:
a == ,
where D v- vector of change in body speed.
Acceleration vector direction a coincides with the direction of vector D v.
Let a body moving in a circle with radius R, for a short period of time t moved from point A to the point B(Fig. 40). To find the change in body speed D v, to the point A move the vector parallel to itself v and subtract from it v 0, which is equivalent to adding the vector v with vector – v 0 . Vector directed from v 0 k v, and there is a vector D v.
Consider triangles AOB And ACD. Both of them are isosceles ( A.O. = O.B. And A.C. = A.D. because v 0 = v) and have equal angles: _AOB = _CAD(like angles with mutual perpendicular sides: A.O. B v 0 , O.B. B v). Therefore, these triangles are similar and we can write the ratio of the corresponding sides: = .
Since the points A And B located close to each other, then the chord AB is small and can be replaced with an arc. Arc length is the path traveled by a body in time t With constant speed v: AB = vt.
Besides, A.O. = R, DC= D v, AD = v. Hence,
= ;= ;= a.
Where does the acceleration of the body come from?
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a = . |
From Figure 40 it is clear that the smaller the chord AB, the more accurate the direction of vector D v coincides with the radius of the circle. Therefore, the velocity change vector D v and acceleration vector a directed radially towards the center of the circle. Therefore, the acceleration during uniform motion of a body in a circle is called centripetal.
Thus,
When a body moves uniformly in a circle, its acceleration is constant in magnitude and at any point is directed along the radius of the circle towards its center.
Considering that v=w R, we can write another formula for centripetal acceleration:
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a= w 2 R. |
6. Example of problem solution
The rotation frequency of the carousel is 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.
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Given: |
Solution |
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n= 0.05 s– 1 R= 4 m |
Centripetal acceleration is equal to: a= w2 R=(2p n)2R=4p2 n 2R. Treatment period: T = . Angular speed of carousel: w = 2p n. |
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a? T? |
a= 4 (3.14) 2 (0.05s–1) 2 4 m 0.4 m/s 2 ;
T== 20 s;
w = 2 3.14 0.05 s– 1 0.3 rad/s.
Answer: a 0.4 m/s 2 ; T= 20 s; w 0.3 rad/s.
Self-test questions
1. What kind of motion is called uniform circular motion?
2. What is the orbital period called?
3. What is called frequency of circulation? How are period and frequency related?
4. What is linear speed called? How is it directed?
5. What is angular velocity called? What is the unit of angular velocity?
6. How are the angular and linear velocities of a body related?
7. What is the direction of centripetal acceleration? What formula is it calculated by?
Task 9
1. What is the linear speed of a point on the wheel rim if the radius of the wheel is 30 cm and it makes one revolution in 2 s? What is the angular velocity of the wheel?
2. The car speed is 72 km/h. What are the angular speed, frequency and period of rotation of a car wheel if the wheel diameter is 70 cm? How many revolutions will the wheel make in 10 minutes?
3. What is the distance traveled by the end of the minute hand of the alarm clock in 10 minutes, if its length is 2.4 cm?
4. What is the centripetal acceleration of a point on the rim of a car wheel if the diameter of the wheel is 70 cm? The car speed is 54 km/h.
5. 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?
Since linear speed uniformly changes direction, the circular motion cannot be called uniform, it is uniformly accelerated.
Angular velocity
Let's choose a point on the circle 1 . Let's build a radius. In a unit of time, the point will move to point 2 . In this case, the radius describes the angle. Angular velocity is numerically equal to the angle of rotation of the radius per unit time.
Period and frequency
Rotation period T- this is the time during which the body makes one revolution.
Rotation frequency is the number of revolutions per second.
Frequency and period are interrelated by the relationship
Relationship with angular velocity
Linear speed
Each point on the circle moves at a certain speed. This speed is called linear. The direction of the linear velocity vector always coincides with the tangent to the circle. For example, sparks from under a grinding machine move, repeating the direction of instantaneous speed.
Consider a point on a circle that makes one revolution, the time spent is the period T. The path that a point travels is the circumference.
Centripetal acceleration
When moving in a circle, the acceleration vector is always perpendicular to the velocity vector, directed towards the center of the circle.
Using the previous formulas, we can derive the following relationships
Points lying on the same straight line emanating from the center of the circle (for example, these could be points that lie on the spokes of a wheel) will have the same angular velocities, period and frequency. That is, they will rotate the same way, but with different linear speeds. The further a point is from the center, the faster it will move.
The law of addition of velocities is also valid for rotational motion. If the motion of a body or frame of reference is not uniform, then the law applies to instantaneous velocities. For example, the speed of a person walking along the edge of a rotating carousel is equal to the vector sum of the linear speed of rotation of the edge of the carousel and the speed of the person.
The Earth participates in two main rotational movements: diurnal (around its axis) and orbital (around the Sun). The period of rotation of the Earth around the Sun is 1 year or 365 days. The Earth rotates around its axis from west to east, the period of this rotation is 1 day or 24 hours. Latitude is the angle between the plane of the equator and the direction from the center of the Earth to a point on its surface.
According to Newton's second law, the cause of any acceleration is force. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration may be different. For example, if a body moves in a circle on a rope tied to it, then acting force is the elastic force.
If a body lying on a disk rotates with the disk around its axis, then such a force is the friction force. If the force stops its action, then the body will continue to move in a straight line
Consider the movement of a point on a circle from A to B. The linear speed is equal to v A And vB respectively. Acceleration is the change in speed per unit time. Let's find the difference between the vectors.
Law. All movements occur equally in reference systems at rest, or moving relative to each other at a constant speed. This is the principle of sameness or equivalence of inertial frames of reference or Galileo's principle of independence.
General laws movement
1 Law. If the body is not acted upon by other bodies, it maintains a state of rest or uniform rectilinear motion. This is the law of inertia, Newton's first law.
3 Law. All movements of a material body occur independently of each other and add up as vector quantities. So, any body on earth simultaneously participates in the movement of the Sun with the planets around the Galaxy Center at a speed of about 200 km/sec, in the movement of the Earth in orbit at a speed of about 30 km/sec, in the rotation of the Earth around its axis at a speed of up to 400 m/sec and possibly in other movements. The result is a very intricate curvilinear trajectory!
If a body is thrown with an initial speed Vo, at an angle a to the horizon, then the flight range –S is calculated by the formula:
S = 2 V*SIN(a) * COS(a) / g = V*SIN(2a) / g
Maximum range at a =45 degrees. The maximum flight altitude –h is calculated by the formula:
h = V* SIN(a)/2g
Both of these formulas can be obtained by taking into account that the vertical component Vo*SIN(a), and horizontal Vo * COS(a), V =g*t, t =V/g.
Let's make a substitution in the basic formula for height
h = g t/2 = g* (V/g)/2 = V/2g = V* SIN(a)/2g.
This is the required formula. The maximum height when thrown vertically upwards, while
a =90 degrees, SIN(a) =1; h = V*/2g
To derive the formula for flight range, you need to multiply the horizontal component by twice the time of falling from a height h. If you take into account air resistance, the path will be shorter. For a projectile, for example, almost twice. The same range will correspond to two different angles throwing.
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Fig. 11 Flight trajectories of a body thrown at an angle to the horizon. The drawing on the right is a movement in a circle.
w- Angular velocity of a rotating body; radian/sec
b - Angular position of the rotating body; radians or degrees relative to an axis. Radian is the angle at which an arc equal to the radius of the circle is visible from the center of the circle, respectively rad = 360/6.28 = 57.32 degrees
a-angular acceleration is measured in rad/sec 2
b = bo + w * t, Angular movement from bo.
S = b *R - Linear movement along a circle of radius R.
w =(b - bo)/(t –to); - Angular velocity . V = w* R – Circumferential speed
T = 2*p/w =2*p*R/V Hence V = 2*p*R/T
a =ao + w/t –Angular acceleration. Angular acceleration is determined by tangential force and in its absence there will be uniform motion of the body in a circle. In this case, the body is affected by centripetal acceleration, which changes the speed by a factor of 2*p during a revolution. Its value is determined by the formula. a =DV/T =2*p*V/2*p*R/V =V/R
Average values of speed and acceleration do not allow one to calculate the position of a body during uneven movement. To do this, it is necessary to know the values of speed and acceleration in short periods of time or instantaneous values. Instantaneous values are determined through derivatives or differentials.
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
Objective 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;
Develop communication and information culture 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 educational material using triggers, embedded videos and an interactive test.
Lesson Plan
Organizational moment. Motivation for learning activities.
Updating basic knowledge.
Learning new material.
Conversation on issues;
Problem solving;
Carrying out practical research work.
Summing up the lesson.
Lesson progress
Lesson steps
Temporary implementation
Organizational moment. 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 min
Updating basic knowledge.
Slide 2.
Fphysical dictation:
Changes in body position in space over time.(Movement)
A physical quantity measured in meters.(Move)
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)
Trajectory length. (Path)
Acceleration units(m/s 2 ).
(Conducting a dictation followed by testing, self-assessment of work by students)
5 min
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 absolute 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 animation effects embedded in the elements of the 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 instruments - 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 tangent to the trajectory drawn to the point where at at the moment the physical body in question is 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: what with higher speed 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, carousels (the word then had masculine) were called knightly tournaments. Later, in the 18th century, to prepare for tournaments, instead of fighting 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. Detailed Description of these carousels are given in the newspaper St. Petersburg Gazette of 1766.
Carousel, common in courtyards in Soviet era. 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.
Task 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 period, frequency and acceleration on the radius of rotation (the smaller the radius of rotation, the shorter the period of revolution and more value frequencies).
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).
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