In the course of the lesson, we will look at curvilinear movement, circular movement and some other examples. We will also discuss the cases in which it is necessary to apply various models for describing body motion.
Are there really straight lines? They seem to surround us everywhere. But let's take a closer look at the edge of the table, the case or the monitor screen: there is always a notch, roughness of the material in them. Let's look through a microscope, and doubts about the curvature of these lines will disappear.
It turns out that the straight line is really an abstraction, something ideal and non-existent. But with the help of this abstraction, it is possible to describe many real objects, if, when considering them, their small irregularities are not important to us and we can consider them straight.
We have considered the simplest movement - a uniform rectilinear movement. This is the same idealization as the straight line itself. V real world real objects are moving, and their trajectory cannot be perfectly straight. The car moves from city A to city B: there can be no absolutely flat road between cities and it will not be possible to keep a constant speed. Nevertheless, using the model of uniform rectilinear motion, we can describe even such a motion.
This model is not always applicable for describing motion.
1) Movement may be uneven.
2) For example, a carousel is spinning - there is movement, but not in a straight line. The same can be said about the ball hit by a football player. Or about the movement of the moon around the earth. In these examples, movement follows a curved path.
So, since there are such tasks, you need a convenient tool for describing movement along a curve.
Driving in a straight line and in a curve
We can consider the same trajectory of motion to be straight in one problem, but not in the other. This is a convention, it depends on what interests us in this task.
If the problem is about a car that travels from Moscow to St. Petersburg, then the road is not straight, but at such distances we are not interested in all these turns - what happens on them is negligible. Moreover, we are talking about average speed, which takes into account all these cornering hiccups, because of them, the average speed will simply become lower. Therefore, we can go to an equivalent task - we can “straighten” the trajectory, keeping the length and speed - we will get the same result. This means that the model of rectilinear motion is suitable here. If the problem is about the movement of the car at a specific turn or during overtaking, then the curvature of the trajectory may be important for us and we will use a different model.
Divide the movement along the curve into sections small enough to be considered straight line segments. Imagine a pedestrian moving along a complex trajectory, avoiding obstacles, but he walks and takes steps. There are no curved steps, these are the segments from the footprint to the footprint.
Rice. 1. Curved path
We have divided the movement into small segments, and we are able to describe the movement on each such segment as straight-line. The shorter these line segments are, the more accurate the approximations will be.
Rice. 2. Approximation of curvilinear motion
We used such a mathematical tool as splitting into small intervals when we found displacement in a straight line uniformly accelerated motion: We divided the movement into sections so small that the change in speed in this section was insignificant and the movement could be considered uniform. It was easy to calculate the displacement in each such section, then it remained to add up the displacement in each section and get the total.
Rice. 3. Moving with rectilinear uniformly accelerated motion
Let's start describing curvilinear motion from the simplest model - a circle, which is described by one parameter - the radius.
Rice. 4. Circle as a model of curvilinear motion
The end of the watch hand moves at the same distance of the length of the hand from the point of its attachment. The points of the wheel rim all the time remain at the same distance from the axle - at the distance of the spoke length. We continue to study movement material point and we work within the framework of this model.
Translational and rotational movement
Translational movement is a movement in which all points of the body move in the same way: at the same speed, making the same movement. Wave your hand and observe: it is clear that the palm and shoulder moved differently. Look at the Ferris wheel: the points near the axis hardly move, and the cabins move at a different speed and along different trajectories. Look at a car moving in a straight line: if you do not take into account the rotation of the wheels and the movement of the engine parts, all points of the car move in the same way, the movement of the car is assumed to be translational. Then it makes no sense to describe the movement of each point, you can describe the movement of one. We consider the car as a material point. Note that when moving forward, the line connecting any two points of the body during movement remains parallel to itself.
The second type of movement according to this classification is rotary motion... During rotational motion, all points of the body move in a circle around some one axis. This axis may intersect the body, as is the case with a Ferris wheel, or it may not intersect, as is the case with a car on a bend.
Rice. 5. Rotational movement
But not every movement can be attributed to one of two types. How to describe the movement of the pedals of a bicycle relative to the Earth - is this some kind of third type? Our model is convenient in that it is possible to consider movement as a combination of translational and rotational movements: the pedals rotate relative to their axis, and the axis, together with the entire bicycle, moves translationally relative to the Earth.
The end of the clock hand will follow the same path at equal time intervals. That is, we can talk about the uniformity of its movement. Velocity is a vector quantity, therefore, in order for it to be constant, both its magnitude and direction must not change. And if the speed module does not change when moving in a circle, then the direction will change constantly.
Consider a uniform motion along a circle.
Why did you choose not to consider moving
Consider how the displacement changes when moving around a circle. The point was in one place (see Fig. 6) and passed a quarter of the circle.
Let us trace the displacement during further movement - it is difficult to describe the regularity by which it changes, and such an examination is not very informative. It makes sense to consider displacement at intervals small enough to be considered approximately equal.
Let us introduce several convenient characteristics of the movement along a circle.
Regardless of the size of the watch, in 15 minutes the end of the minute hand will always cover a quarter of the dial's circumference. And it will make a full turn in an hour. In this case, the path will depend on the radius of the circle, but the angle of rotation will not. That is, the angle will also change uniformly. Therefore, in addition to the path traveled, we will also talk about changing the angle. As we know, the angle is proportional to the arc on which it rests:
Rice. 7. Changing the angle of deflection of the arrow
Since the angle changes uniformly, it is possible, by analogy with the ground speed, showing the path that the body travels per unit of time, to enter the angular velocity: the angle through which the body rotates (or that the body travels) per unit of time,.
That is, how many radians the point is rotated per second. It will be measured, respectively, in rad / s.
Uniform movement around a circle is a repetitive process, or, in other words, periodic... When the point makes a full turn, it is back in its original position and the movement is repeated.
Examples of periodic phenomena in nature
Many phenomena are of a periodic nature: change of day and night, change of seasons. Here it is clear what exactly is a period: a day and a year, respectively.
There are other periods: spatial (a pattern with periodically repeating elements, a row of trees located at equal intervals), periods in the recording of numbers. Periods in music, poetry.
Periodic events are described by what happens over a period and the length of that period. For example, the daily cycle is sunrise-sunset and the period is the time for which everything repeats itself - 24 hours. A spatial pattern is a single element of a pattern and how often it repeats (or its length). Decimal notation common fraction is a sequence of digits in a period (what is in brackets) and length / period is the number of digits: in 1/3 - one digit, in 1/17 - 16 digits.
Let's consider some time periods.
The period of the Earth's revolution around its axis = day + night = 24 hours.
The period of revolution of the Earth around the Sun = 365 periods of revolution day + night.
The period of rotation of the hour hand on the dial is 12 hours, minute - 1 hour.
The oscillation period of the clock pendulum is 1 s.
The period is measured in generally accepted units of time (second in SI, minute, hour, etc.).
The period of the pattern is measured in units of length (m, cm), the period in decimal- in the number of digits in the period.
Period- this is the time during which a point, with uniform movement around a circle, makes one complete revolution. Let's designate it with a capital letter.
If revolutions are made during the time, then one revolution is made, obviously, during the time.
To judge how often the process is repeated, we introduce a quantity that we will call so - frequency.
The frequency of the appearance of the Sun per year is 365 times. Spawn rate full moon a year - 12, sometimes 13 times. The frequency of the arrival of spring per year is 1 time.
For uniform motion around a circle, frequency is the number of complete revolutions that a point makes per unit of time. If revolutions are made in t seconds, then revolutions are made for each second. Let's denote the frequency, sometimes it is also denoted by or. The frequency is measured in revolutions per second, this value was called hertz, after the name of the scientist Hertz.
Frequency and period are reciprocal values: the more often something happens, the shorter the period should be. Conversely, the longer one period lasts, the less often the event occurs.
Mathematically, we can write inverse proportionality: or.
So, the period is the time during which the body makes a complete revolution. It is clear that it must be related to the angular velocity: the faster the angle changes, the faster the body will return to the starting point, that is, it will complete a full revolution.
Consider one complete revolution. Angular velocity is the angle by which the body rotates per unit of time. At what angle should the body turn with a full revolution? 3600, or in radians. The turnaround time is a period. Hence, by definition, the angular velocity is equal to:.
We will also find the ground speed - it is also called linear - by considering one revolution. A point in time, one period, the body makes a full revolution, that is, it travels a path equal to the length of a circle. From here we express the speed by definition as the path divided by the time:.
If we take into account that is the angular velocity, then we get the relationship between the linear and angular velocity:
Task
With what frequency should the gate of the well be rotated so that the bucket rises at a speed of 1 m / s, if the radius of the gate section is equal?
The problem describes the rotation of the gate - we apply a model of rotational motion to it, considering the points of its surface.
Rice. 8. Gate rotation model
It is also about the movement of the bucket. The bucket is attached with a rope to the collar and this rope is wound. This means that any part of the rope, including the one wound around the collar, moves at the same speed as the bucket. Thus, we have set the linear speed of the points on the surface of the gate.
The physical part of the solution... Speech about the linear speed of movement in a circle, it is equal to:.
The period and frequency are mutually reciprocal values, we write down:.
We got a system of equations, which only remains to be solved - this will be the mathematical part of the solution. Substitute the frequency in the first equation instead of: 
.
Let us express the frequency from here:.
Let's calculate by converting the radius to meters:
We got the answer: you need to rotate the gate with a frequency of 1.06 Hz, that is, make approximately one revolution in one second.
Let's imagine that we have two identical bodies moving. One - along a circle, and the other (under the same conditions and with the same characteristics), but along a regular polygon. The more sides such a polygon has, the less the movements of these two bodies will differ for us.
Rice. 9. Curvilinear movement along a circle and a polygon
The difference is that the second body on each section (side of the polygon) moves in a straight line.
On each such segment, we denote the movement of the body. Moving here is a two-dimensional vector, on a plane.
Rice. 10. Moving a body during curvilinear motion along a polygon
In this small area, the movement was completed in time. Let's divide and get the velocity vector in this section.
With an increase in the number of sides of the polygon, the length of its side will decrease:. Since the body's velocity module is constant, the time to overcome this segment will tend to 0:.
Accordingly, the speed of the body in such a small area will be called instant speed.
The smaller the side of the polygon is, the closer it will be to the tangent to the circle. Therefore, in the limiting, ideal case (), we can assume that the instantaneous velocity at a given point is directed tangentially to the circle.
And the sum of the displacement modules will differ less and less from the path that the point passes along the arc. Therefore, the modulus of the instantaneous speed will coincide with the ground speed, and all those ratios that we obtained earlier will also be true for the instantaneous speed along the displacement. You can even designate it, meaning.
The speed is directed tangentially, we can find its module too. Let's find the speed at another point. Its modulus is the same, since the movement is uniform, and it is directed tangentially to the circle already at this point.
Rice. 11. Body speed tangential
This is not the same vector, they are equal in absolute value, but they have a different direction,. The speed has changed, and since it has changed, then this change can be calculated:
The change in speed per unit of time is, by definition, an acceleration:
Let's calculate the acceleration when moving in a circle. Change in speed.
Rice. 12. Graphical subtraction of vectors
We got a vector. Acceleration is directed to the same direction (these vectors are related by the relation 
, which means they are co-directed).
The smaller the section AB, the more the velocity vectors and will coincide, and will be closer and closer to the perpendicular to both of them.
Rice. 13. Dependence of speed on the size of the section
That is, it will lie along the perpendicular to the tangent (the speed is directed tangentially), which means that the acceleration will be directed to the center of the circle, along the radius. Remember from your mathematics course: the radius drawn to the tangent point is perpendicular to the tangent.
When the body passes a small angle, the velocity vector, which is tangential to the radius, also rotates through the angle.
Proof of equality of angles
Consider a quadrilateral ASVO. The angles of a quadrilateral add up to 360 °. (like the angles between tangent radii and tangents).
The angle between the directions of the velocity at points A and B () and - adjacent at a straight line AC, then 
,
Previously received, from here.
On a small section AB, the displacement of a point in absolute value practically coincides with the path, that is, with the length of the arc:.
The ABO triangles and the triangle composed by the velocity vectors at points A and B are similar (from point A, the vector was transferred parallel to itself to point B).
These triangles are isosceles (OA = OB - radii, - since the movement is uniform), they have equal angles between the lateral sides (just proved in the branch). This means that the angles at the base that are equal to each other will be equal. Equality of angles is enough to assert that triangles are similar.
From the similarity of triangles, we write: side AB (and it is equal) refers to the radius of the circle as the modulus of speed change refers to the modulus of speed:.
We write without vectors, because we are interested in the lengths of the sides of the triangles. We all lead to acceleration, it is associated with a change in speed, or. Substituting, we get:.
The derivation of the formula turned out to be quite complicated, but you can remember the finished result and use it when solving problems.
At whatever point we find the acceleration during uniform motion along the circle, it is equal in absolute value and at any point is directed to the center of the circle. Therefore, it is also called centripetal acceleration.
Problem 2. Centripetal acceleration
Let's solve the problem.
Find the speed at which the car is moving on a bend if the bend is considered to be a part of a circle with a radius of 40 m, and the centripetal acceleration is equal to.
Analysis of the condition. The problem describes movement in a circle, we are talking about centripetal acceleration. Let's write down the formula for centripetal acceleration:
The acceleration and the radius of the circle are given, it remains only to express and calculate the speed:
Or, if translated in km / h, then this is about 32 km / h.
In order for the speed of a body to change, another body must act on it with some force, or, to put it more simply, a force must act. In order for the body to move in a circle with centripetal acceleration, the force that creates this acceleration must also act on it. In the case of a car at a bend, this is the friction force, so we are skidded at bends when the roads are icy. If we untwist something on the rope, it is the pulling force of the rope - and we feel it pulling tighter. As soon as this force disappears, for example, the thread breaks, the body, in the absence of inertial forces, maintains its speed - the speed directed tangentially to the circle that was at the moment of separation. And this can be seen by following the direction of movement of this body (figure). For the same reason, we are pressed against the wall of the vehicle at a bend: we move by inertia so as to maintain speed, we are, as it were, thrown out of the circle until we hit the wall and a force arises that will impart centripetal acceleration.
Previously, we only had one tool - the linear motion model. We were able to describe another model - movement in a circle.
This is a common type of movement (turns, wheels of vehicles, planets, etc.), so a separate tool was needed (each time it is not very convenient to zoom in on the trajectory with small straight segments).
Now we have two "bricks", which means that with their help we will be able to construct buildings of a more complex shape - to solve more complex problems with combined types of movements.
These two models will be enough for us to solve most kinematic problems.
For example, such a movement can be represented as movement along the arcs of three circles. Or such an example: a car drove straight down the street and accelerated, then turned and drove at a constant speed down another street.
Rice. 14. Dividing the vehicle trajectory into sections
We will look at three areas and apply one of the simple models to each.
Bibliography
- Sokolovich Yu.A., Bogdanova G.S. Physics: a handbook with examples of problem solving. - 2nd ed., Redistribution. - X .: Vesta: Ranok publishing house, 2005. - 464 p.
- Peryshkin A.V., Gutnik E.M. Physics. 9th grade: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., Stereotype. - M .: Bustard, 2009 .-- 300.
- Web site " Extracurricular lesson» ()
- Website "Class! Naya Physics" ()
Homework
- Give examples of curved movement in Everyday life... Can this movement be straightforward in any construction of the condition?
- Determine the centripetal acceleration with which the Earth moves around the Sun.
- Two cyclists at constant speeds start simultaneously in the same direction from two diametrically opposite points circular track... 10 minutes after the start, one of the cyclists caught up with the other for the first time. How long after the start will the first cyclist catch up with the other for the second time?
With the help of this lesson, you will be able to independently study the topic “Straight and curved motion. The movement of a body in a circle with a constant modulo velocity ”. First, we characterize rectilinear and curvilinear motion, considering how these types of motion relate the velocity vector and the force applied to the body. Consider next special case, when the body moves in a circle with a constant absolute speed.
In the previous lesson, we looked at issues related to the law universal gravitation... The topic of today's lesson is closely related to this law, we will turn to the uniform movement of the body around the circumference.
We said earlier that motion - it is a change in the position of a body in space relative to other bodies over time. The movement and direction of movement are also characterized by speed. The change in speed and the type of movement itself are associated with the action of force. If a force acts on the body, then the body changes its speed.
If the force is directed parallel to the movement of the body, then such a movement will be straightforward(fig. 1).
Rice. 1. Straight-line movement
Curvilinear there will be such a movement when the speed of the body and the force applied to this body are directed relative to each other at a certain angle (Fig. 2). In this case, the speed will change its direction.
Rice. 2. Curvilinear movement
So, at straight motion the velocity vector is directed in the same direction as the force applied to the body. A curvilinear motion is such a movement when the velocity vector and the force applied to the body are located at an angle to each other.
Consider a special case of curvilinear motion, when the body moves in a circle with a constant modulus of speed. When a body moves in a circle at a constant speed, then only the direction of the speed changes. In absolute value, it remains constant, but the direction of the velocity changes. Such a change in speed leads to the presence of an acceleration in the body, which is called centripetal.
Rice. 6. Movement along a curved path
If the trajectory of the body is a curve, then it can be represented as a set of movements along arcs of circles, as shown in Fig. 6.
In fig. 7 shows how the direction of the velocity vector changes. The speed during this movement is directed tangentially to the circle along the arc of which the body moves. Thus, its direction is constantly changing. Even if the modulus of the speed remains constant, a change in speed leads to the appearance of acceleration:
In this case acceleration will point towards the center of the circle. Therefore it is called centripetal.
Why is centripetal acceleration directed towards the center?
Recall that if a body moves along a curved trajectory, then its speed is tangential. Velocity is a vector quantity. A vector has a numerical value and a direction. Speed as the body moves continuously changes its direction. That is, the difference in speeds at different points in time will not be equal to zero (), in contrast to rectilinear uniform motion.
So, we have a change in speed over a period of time. The relation to is acceleration. We come to the conclusion that even if the speed does not change in absolute value, the body, making a uniform motion around the circle, has an acceleration.
Where is this acceleration directed? Consider fig. 3. Some body moves curvilinearly (along an arc). The speed of the body at points 1 and 2 is tangential. The body moves uniformly, that is, the modules of the velocities are equal:, but the directions of the velocities do not coincide.
Rice. 3. The movement of the body in a circle
Let's subtract the velocity from it and get the vector. To do this, you need to connect the beginnings of both vectors. Move the vector to the beginning of the vector in parallel. We finish building to the triangle. The third side of the triangle will be the vector of the speed difference (Fig. 4).
Rice. 4. Velocity difference vector
The vector is directed towards the circle.
Consider a triangle formed by the velocity vectors and the difference vector (Fig. 5).
Rice. 5. The triangle formed by the vectors of velocities
This triangle is isosceles (velocity modules are equal). This means that the angles at the base are equal. Let's write the equality for the sum of the angles of the triangle:
Let us find out where the acceleration is directed at a given point of the trajectory. To do this, we will begin to bring point 2 closer to point 1. With such unlimited diligence, the angle will tend to 0, and the angle - to. The angle between the vector of the change in speed and the vector of the speed itself is. The velocity is directed tangentially, and the velocity vector is directed to the center of the circle. This means that the acceleration is also directed towards the center of the circle. That is why this acceleration is called centripetal.
How to find centripetal acceleration?
Consider the trajectory along which the body moves. In this case, it is a circular arc (Fig. 8).
Rice. 8. The movement of the body in a circle
The figure shows two triangles: a triangle formed by velocities and a triangle formed by radii and a displacement vector. If points 1 and 2 are very close, then the displacement vector will be the same as the path vector. Both triangles are isosceles with the same apex angles. Thus, triangles are similar. This means that the corresponding sides of the triangles are related in the same way:
The movement is equal to the product of speed and time:. Substituting this formula, you can get the following expression for centripetal acceleration:
Angular velocity denoted by the Greek letter omega (ω), it tells about the angle at which the body rotates per unit of time (Fig. 9). This is the magnitude of the arc in degree measure traversed by the body in some time.
Rice. 9. Angular velocity
Note that if solid rotates, then the angular velocity for any points on this body will be constant. Closer point is located to the center of rotation or further - it does not matter, that is, it does not depend on the radius.
The unit of measurement in this case will be either degrees per second () or radians per second (). Often the word "radian" is not written, but simply written. For example, let's find what the angular velocity of the Earth is equal to. The earth makes a full turn for an hour, and in this case we can say that the angular velocity is equal to:
Also pay attention to the relationship of angular and linear velocities:
Linear speed is directly proportional to radius. The larger the radius, the greater the linear velocity. Thus, moving away from the center of rotation, we increase our linear speed.
It should be noted that movement in a circle at a constant speed is a special case of movement. However, the movement around the circle can be uneven. The speed can change not only in the direction and remain the same in magnitude, but also change in its value, i.e., in addition to changing the direction, there is also a change in the speed modulus. In this case, we are talking about the so-called accelerated motion in a circle.
What is a radian?
There are two units for measuring angles: degrees and radians. In physics, as a rule, the radian measure of the angle is the main one.
Construct a central angle that rests on an arc of length.
Depending on the shape of the trajectory, the movement can be subdivided into rectilinear and curvilinear. Most often, you can encounter curvilinear movements when the trajectory is presented in the form of a curve. An example of this type of movement is the path of a body thrown at an angle to the horizon, the movement of the Earth around the Sun, planets, and so on.
Picture 1 . Trajectory and displacement in curvilinear motion
Definition 1Curvilinear motion is called a movement whose trajectory is a curved line. If the body moves along a curved trajectory, then the displacement vector s → is directed along the chord, as shown in Figure 1, and l is the length of the trajectory. The direction of the instantaneous speed of movement of the body is tangential at the same point of the trajectory, where in this moment a moving object is located, as shown in Figure 2.
Figure 2. Instantaneous speed in curved motion
Definition 2
Curvilinear motion of a material point is called uniform when the velocity modulus is constant (movement in a circle), and uniformly accelerated with a changing direction and modulus of velocity (motion of a thrown body).
Curvilinear movement is always accelerated. This is because even with an unchanged speed modulus, and a changed direction, there is always acceleration.
In order to investigate the curvilinear motion of a material point, two methods are used.
The path is divided into separate sections, at each of which it can be considered straight-line, as shown in Figure 3.
Figure 3. Splitting curvilinear motion into translational
Now the law of rectilinear motion can be applied to each section. This principle is allowed.
The most convenient solution method is considered to represent the path as a set of several motions along circular arcs, as shown in Figure 4. The number of splits will be much less than in the previous method, in addition, the movement along the circle is already curvilinear.
Figure 4. Splitting curvilinear motion into motion along circular arcs
Remark 1
To record a curvilinear movement, it is necessary to be able to describe movement in a circle, to represent an arbitrary movement in the form of a set of movements along the arcs of these circles.
The study of curvilinear motion includes drawing up a kinematic equation that describes this motion and allows to determine all characteristics of the motion using the available initial conditions.
Example 1
A material point is given, moving along a curve, as shown in Figure 4. The centers of the circles O 1, O 2, O 3 are located on one straight line. Need to find a move
s → and the length of the path l while moving from point A to B.
Solution
By condition, we have that the centers of the circle belong to one straight line, hence:
s → = R 1 + 2 R 2 + R 3.
Since the trajectory of movement is the sum of semicircles, then:
l ~ A B = π R 1 + R 2 + R 3.
Answer: s → = R 1 + 2 R 2 + R 3, l ~ A B = π R 1 + R 2 + R 3.
Example 2
The dependence of the distance traveled by the body on time is given, represented by the equation s (t) = A + B t + C t 2 + D t 3 (C = 0.1 m / s 2, D = 0.003 m / s 3). Calculate how long after the start of movement the acceleration of the body will be equal to 2 m / s 2
Solution
Answer: t = 60 s.
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Slide captions:
Think and answer! 1. What movement is called uniform? 2. What is called the speed of uniform movement? 3. What motion is called uniformly accelerated? 4. What is body acceleration? 5. What is relocation? What is a trajectory?
Lesson topic: Straight and curved motion. The movement of the body in a circle.
Mechanical movements Rectilinear Curvilinear elliptical movement Parabolic movement Hyperbola movement Circular movement
Lesson objectives: 1. To know the main characteristics of curvilinear movement and the relationship between them. 2. Be able to apply the knowledge gained in solving experimental problems.
Study plan for the topic Study of new material Condition of rectilinear and curvilinear motion Direction of body speed during curvilinear motion Centripetal acceleration Orbital period Orbital frequency Centripetal force Fulfillment of frontal experimental tasks Independent work in the form of tests Summing up
By the type of trajectory, the movement can be: Curvilinear Rectilinear
Conditions for the rectilinear and curvilinear motion of bodies (Experiment with a ball)
page 67 Remember! Working with the tutorial
Circular motion is a special case of curved motion
Preview:
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Slide captions:
Movement characteristics - linear speed of curvilinear movement () - centripetal acceleration () - period of revolution () - frequency of revolution ()
Remember. The direction of movement of particles coincides with the tangent to the circle
In curvilinear motion, the speed of the body is directed tangentially to the circle Remember.
In curvilinear motion, the acceleration is directed towards the center of the circle. Save.
Why is the acceleration directed towards the center of the circle?
Determination of speed - speed - period of revolution r - radius of a circle
When a body moves in a circle, the modulus of the velocity vector may change or remain constant, but the direction of the velocity vector necessarily changes. Therefore, the velocity vector is a variable value. This means that movement in a circle always occurs with acceleration. Remember!
Preview:
Topic: Straight and curved motion. The movement of the body in a circle.
Goals: To study the features of curvilinear movement and, in particular, movement along a circle.
Introduce the concept of centripetal acceleration and centripetal force.
Continue work on the formation of key competencies of students: the ability to compare, analyze, draw conclusions from observations, generalize experimental data based on existing knowledge about body movement; form the ability to use basic concepts, formulas and physical laws body movements when moving on a circle.
Foster independence, teach children to cooperate, foster respect for the opinions of others, awaken curiosity and observation.
Lesson equipment:computer, multimedia projector, screen, ball on an elastic band, ball on a thread, ruler, metronome, whirligig.
Registration: "We are truly free when we have retained the ability to reason for ourselves." Ceceron.
Lesson type: a lesson in learning new material.
During the classes:
Organizing time:
Problem statement: What types of movements have we studied?
(Answer: Rectilinear uniform, rectilinear uniformly accelerated.)
Lesson plan:
- Updating basic knowledge(physical warm-up) (5 min)
- What movement is called uniform?
- What is called the speed of uniform movement?
- What motion is called uniformly accelerated?
- What is body acceleration?
- What is relocation? What is a trajectory?
- Main part. Learning new material. (11 minutes)
- Formulation of the problem:
Assignment to students:Consider the spinning of a whirligig, the spinning of a ball on a thread (demonstration of experience). How can you characterize their movements? What is common in their movement?
Teacher: This means that our task in today's lesson is to introduce the concept of rectilinear and curvilinear motion. Body movements in a circle.
(recording the topic of the lesson in notebooks).
- Lesson topic.
Slide number 2.
Teacher: To set goals, I propose to analyze the scheme mechanical movement. (types of movement, scientific nature)
Slide number 3.
- What goals will we set for our topic?
Slide number 4.
- I suggest exploring this topic as follows plan. (Highlight main)
Do you agree?
Slide number 5.
- Take a look at the picture. Consider examples of the types of trajectories found in nature and technology.
Slide number 6.
- The action of a force on a body in some cases can only lead to a change in the modulus of the velocity vector of this body, and in others - to a change in the direction of the velocity. Let us show this experimentally.
(Experiments with a ball on an elastic band)
Slide number 7
- Make a conclusion what the type of trajectory of movement depends on.
(Answer)
Now let's compare this definition with the one given in your textbook on page 67
Slide number 8.
- Consider the drawing. How can curvilinear motion be associated with circular motion?
(Answer)
That is, the curved line can be rearranged as a set of arcs of circles of different diameters.
Let's conclude: ...
(Write in a notebook)
Slide number 9.
- Consider what physical quantities characterize movement in a circle.
Slide number 10.
- Consider an example of a car moving. What is flying out from under the wheels? How does it move? How are the particles directed? How are they protected from the action of these particles?
(Answer)
Let's make a conclusion :… (About the nature of particle motion)
Slide number 11
- Let's look at how the speed is directed when the body moves in a circle. (Animation with a horse.)
Let's conclude: ... ( how the speed is directed.)
Slide number 12.
- Let us find out how the acceleration is directed during curvilinear motion, which appears here due to the fact that there is a change in speed in the direction.
(Animation with a motorcyclist.)
Let's conclude: ... ( how is the acceleration directed)
Let's write down formula in a notebook.
Slide number 13.
- Consider the drawing. Now we will find out why the acceleration is directed to the center of the circle.
(teacher's explanation)
Slide number 14.
What conclusions can be drawn about the direction of speed and acceleration?
- There are other characteristics of curvilinear motion. These include the period and frequency of rotation of the body in a circle. The speed and period are related by a ratio, which we will establish mathematically:
(Teacher writes on the chalkboard, students write in notebooks)
It is known, but the path, then.
Since then
Slide number 15.
- What is the general conclusion that one can draw about the nature of the movement in a circle?
(Answer)
Slide number 16.,
- According to Newton's II law, acceleration is always co-directed with the force, as a result of which it arises. The same is true for centripetal acceleration.
Let's make a conclusion : How is the force directed at each point of the trajectory?
(answer)
This force is called centripetal.
Let's write down formula in a notebook.
(Teacher writes on the chalkboard, students write in notebooks)
The centripetal force is created by all forces of nature.
Give examples of the action of centripetal forces by their nature:
- elastic force (stone on a rope);
- the force of gravity (planets around the sun);
- frictional force (cornering).
Slide number 17.
- For consolidation, I propose to conduct an experiment. To do this, we will create three groups.
Group I will establish the dependence of the speed on the radius of the circle.
Group II will measure the acceleration when moving in a circle.
Group III will establish the dependence of the centripetal acceleration on the number of revolutions per unit of time.
Slide number 18.
Summarizing... How does the speed and acceleration depend on the radius of the circle?
- We will conduct testing for the initial consolidation. (7 minutes)
Slide number 19.
- Assess your work in the lesson. Continue the sentences on the flyers.
(Reflection. Students voice individual answers aloud.)
Slide number 20.
- Homework: §18-19,
Exercise 18 (1, 2)
Additionally ex. 18 (5)
(Teacher comments)
Slide number 21.
Curvilinear motion Is a movement whose trajectory is a curved line (for example, a circle, ellipse, hyperbola, parabola). An example of curvilinear movement is the movement of planets, the end of the clock hand on the dial, etc. In general curvilinear speed varies in magnitude and direction.
Curvilinear motion of a material point is considered a uniform motion if the modulus is constant (for example, uniform motion around a circle), and uniformly accelerated if the modulus and direction change (for example, the movement of a body thrown at an angle to the horizon).
Rice. 1.19. Trajectory and displacement vector for curvilinear motion.
When moving along a curved trajectory, it is directed along the chord (Fig. 1.19), and l is the length. The instantaneous speed of movement of the body (that is, the speed of the body at a given point of the trajectory) is directed tangentially at that point of the trajectory where the moving body is currently located (Fig. 1.20).
Rice. 1.20. Instantaneous speed in curvilinear motion.
Curvilinear movement is always accelerated movement. That is curved acceleration is always present, even if the speed module does not change, but only the speed direction changes. The change in the speed value per unit of time is:
Where v τ, v 0 - the values of the speeds at time t 0 + Δt and t 0, respectively.
At a given point, the trajectory in the direction coincides with the direction of the body's velocity or opposite to it.
Is the change in speed in direction per unit of time:
Normal acceleration directed along the radius of curvature of the trajectory (to the axis of rotation). Normal acceleration is perpendicular to the direction of speed.
Centripetal acceleration- This is the normal acceleration when moving uniformly around the circumference.
Full acceleration with uniform curvilinear motion of the body equals:
The movement of a body along a curved trajectory can be roughly represented as movement along the arcs of some circles (Fig. 1.21).
Rice. 1.21. Body movement during curvilinear movement.
