With the help this lesson You can independently study the topic “Rectilinear and curvilinear motion. Movement of a body in a circle with a constant absolute speed." First, we will characterize rectilinear and curvilinear motion by considering how in these types of motion the velocity vector and the force applied to the body are related. Next, we consider a special case when a body moves in a circle with a constant velocity in absolute value.
In the previous lesson we looked at issues related to the law of universal gravitation. The topic of today's lesson is closely related to this law; we will turn to the uniform motion of a body in a circle.
We said earlier that movement - This is a change in the position of a body in space relative to other bodies over time. 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 a body, then the body changes its speed.
If the force is directed parallel to the movement of the body, then such 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, when straight motion the velocity vector is directed in the same direction as the force applied to the body. A curvilinear movement is such a movement when the velocity vector and the force applied to the body are located at a certain angle to each other.
Let us consider a special case of curvilinear motion, when a body moves in a circle with a constant velocity in absolute value. When a body moves in a circle at a constant speed, only the direction of the speed changes. In absolute value it remains constant, but the direction of the velocity changes. This change in speed leads to the presence of acceleration in the body, which is called centripetal.
Rice. 6. Movement along a curved path
If the trajectory of a body’s movement is a curve, then it can be represented as a set of movements along circular arcs, as shown in Fig. 6.
In Fig. Figure 7 shows how the direction of the velocity vector changes. The speed during such a movement is directed tangentially to the circle along the arc of which the body moves. Thus, its direction is constantly changing. Even if the absolute speed remains constant, a change in speed leads to acceleration:
IN in this case acceleration will be directed towards the center of the circle. That's why it's called centripetal.
Why is centripetal acceleration directed towards the center?
Recall that if a body moves along a curved path, then its speed is directed tangentially. Velocity is a vector quantity. A vector has a numerical value and a direction. The speed continuously changes its direction as the body moves. That is, the difference in speeds at different moments of time will not be equal to zero (), in contrast to rectilinear uniform motion.
So, we have a change in speed over a certain period of time. The ratio to is acceleration. We come to the conclusion that, even if the speed does not change in absolute value, a body performing uniform motion in a circle has acceleration.
Where is this acceleration directed? Let's look at Fig. 3. Some body moves curvilinearly (along an arc). The speed of the body at points 1 and 2 is directed tangentially. The body moves uniformly, that is, the velocity modules are equal: , but the directions of the velocities do not coincide.
Rice. 3. Body movement in a circle
Subtract the speed from it and get the vector. To do this, you need to connect the beginnings of both vectors. In parallel, move the vector to the beginning of the vector. We build up to a triangle. The third side of the triangle will be the velocity difference vector (Fig. 4).
Rice. 4. Velocity difference vector
The vector is directed towards the circle.
Let's consider a triangle formed by the velocity vectors and the difference vector (Fig. 5).
Rice. 5. Triangle formed by velocity vectors
This triangle is isosceles (the velocity modules are equal). This means that the angles at the base are equal. Let us write down the equality for the sum of the angles of a triangle:
Let's 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 will tend to . The angle between the velocity change vector and the velocity vector itself is . The speed is directed tangentially, and the vector of speed change is directed towards 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?
Let's consider the trajectory along which the body moves. In this case it is a circular arc (Fig. 8).
Rice. 8. Body movement in a circle
The figure shows two triangles: a triangle formed by velocities, and a triangle formed by radii and displacement vector. If points 1 and 2 are very close, then the displacement vector will coincide with the path vector. Both triangles are isosceles with the same vertex angles. Thus, the triangles are similar. This means that the corresponding sides of the triangles are equally related:
The displacement is equal to the product of speed and time: . Substituting this formula, we can obtain the following expression for centripetal acceleration:
Angular velocity denoted by the Greek letter omega (ω), it indicates the angle through which the body rotates per unit time (Fig. 9). This is the magnitude of the arc in degree measure traversed by the body over some time.
Rice. 9. Angular velocity
Please note that if solid rotates, then the angular velocity for any points on this body will be a constant value. Whether the point is located closer to the center of rotation or further away is not important, i.e. 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. The Earth makes a complete rotation in one hour, and in this case we can say that the angular velocity is equal to:
Also pay attention to the relationship between angular and linear speeds:
Linear speed is directly proportional to the radius. The larger the radius, the greater the linear speed. Thus, moving away from the center of rotation, we increase our linear speed.
It should be noted that circular motion at a constant speed is a special case of motion. However, the movement around the circle may be uneven. Speed can change not only in direction and remain the same in magnitude, but also change in value, i.e., in addition to a change in direction, there is also a change in the magnitude of velocity. 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 angle is the main one.
Let's build central angle, which rests on an arc of length .
Today we will continue to study movement. We considered cases when bodies moved only rectilinearly, that is, in a straight line. But how often do we encounter such a movement in life? Of course not. Bodies usually move along curved trajectories. The movement of planets, trains, animals - all this will be an example of curvilinear movement. It is more difficult to describe such a movement. The coordinates will change along at least two axes, for example OX and OY. Let's compare how the velocity and displacement vectors are directed during rectilinear and curvilinear motion. When a body moves in a straight line, the direction of the velocity vector and the displacement vector always coincide. In order to answer the same question in the case of curvilinear motion, consider the figure. Suppose that a body moves from point M1 to point M2 along an arc. The path is the length of the arc, the displacement is the vector M1M2. In geometry, such a segment is called a chord. We see that the direction of velocity and displacement do not coincide. For curvilinear motion, we will talk about instantaneous speed. The instantaneous speed of the body at each point of the curvilinear trajectory is directed tangent to the trajectory at this point. You can verify this by observing the splashes from under the wheels of the car; they also fly out tangentially to the circumference of the wheel. Please note that the speed at each point of the curvilinear trajectory is different direction, therefore, even if the velocity module remains the same, if the direction of movement has changed, then a new vector must be considered. Since the speed is constantly changing, it follows that the acceleration will also change. Therefore, curvilinear motion is motion with acceleration. Suppose a body moves along some curvilinear trajectory. There can be countless such trajectories; is it really true that each of them will have to describe its own laws of motion? It turns out that individual parts of the trajectory can be approximately represented as circular arcs. And the curvilinear movement itself, in most cases, can be represented as a set of movements along circular arcs of different radii. By studying circular motion, we will be able to describe more complex cases movements. Let us remember that if the speed of a body and the force acting on it are directed along one straight line, then the body moves rectilinearly, and if they are directed along intersecting straight lines, then the body moves curvilinearly. Determine what trajectory a stone rotating on a thread will fly if the thread suddenly breaks? The instantaneous speed of the stone is directed along a tangent to the curved line, therefore, at the moment of breaking, according to the law of inertia, the body will move while maintaining the same speed, that is, along the same tangent. The truck moves along a curved path. The speed of movement modulo is constant. Can we say that the acceleration of the truck is zero? It is impossible to say that the acceleration of the truck is zero, since the speed has a different direction at each point of the curvilinear trajectory, therefore, even if the velocity module remains the same, a new vector must be considered. Since the speed is constantly changing, it follows that the acceleration will also change. We already know that the cause of acceleration is force. Indicate in which areas of the curvilinear motion the force acted?
Justify your answer. Body position marks are made on the trajectory at regular intervals. The force acted in the area 0-3. The body moved in a straight line, but the speed of the body changed (the body moved accelerated), that is, under the influence of force. The force operated in area 7-8. The magnitude of the speed did not change, but the direction changed (the body moved accelerated), that is, under the influence of force.
We know that all bodies attract each other. In particular, the Moon, for example, is attracted to the Earth. But the question arises: if the Moon is attracted to the Earth, why does it revolve around it instead of falling towards the Earth?
In order to answer this question, it is necessary to consider the types of motion of bodies. We already know that movement can be uniform and uneven, but there are other characteristics of movement. In particular, depending on the direction, rectilinear and curvilinear movement are distinguished.
Straight-line movement
It is known that a body moves under the influence of a force applied to it. You can do a simple experiment showing how the direction of movement of a body will depend on the direction of the force applied to it. To do this, you will need an arbitrary small object, a rubber cord and a horizontal or vertical support.
Ties the cord at one end to the support. At the other end of the cord we attach our object. Now, if we pull our object a certain distance and then release it, we will see how it begins to move in the direction of the support. Its movement is caused by the elastic force of the cord. This is how the Earth attracts all bodies on its surface, as well as meteorites flying from space.
Only instead of the elastic force, the force of attraction acts. Now let’s take our object with an elastic band and push it not in the direction towards/away from the support, but along it. If the object were not secured, it would simply fly away. But since it is held by a cord, the ball, moving to the side, slightly stretches the cord, which pulls it back, and the ball slightly changes its direction towards the support.
Curvilinear movement in a circle
This happens at every moment of time; as a result, the ball does not move along the original trajectory, but also not straight to the support. The ball will move around the support in a circle. The trajectory of its movement will be curvilinear. This is how the Moon moves around the Earth without falling on it.
This is how the Earth's gravity captures meteorites that fly close to the Earth, but not directly at it. These meteorites become satellites of the Earth. Moreover, how long they will stay in orbit depends on what their initial angle of motion was relative to the Earth. If their movement was perpendicular to the Earth, then they can remain in orbit indefinitely. If the angle was less than 90˚, then they will move in a descending spiral, and will gradually fall to the ground.
Circular motion with a constant modulus speed
Another point to note is that the speed of curvilinear motion around a circle varies in direction, but is the same in value. And this means that movement in a circle with a constant absolute speed occurs uniformly accelerated.
Since the direction of movement changes, it means that the movement occurs with acceleration. And since it changes equally at each moment of time, therefore, the movement will be uniformly accelerated. And the force of gravity is the force that causes constant acceleration.
The Moon moves around the Earth precisely because of this, but if suddenly the Moon’s movement ever changes, for example, a very large meteorite crashes into it, then it may well leave its orbit and fall to the Earth. We can only hope that this moment never comes. Such things.
Municipal budgetary educational institution "Chubaevskaya secondary school" of the Urmara district of the Chechen Republic
PHYSICS LESSON in 9TH GRADE
“Rectilinear and curvilinear movement.
Movement of a body in a circle."
Teacher: Stepanova E.A.
Chubaevo – 2013
Subject: Rectilinear and curvilinear movement. Movement of a body in a circle with a constant absolute speed.
Objectives of the lesson: to give students an idea of rectilinear and curvilinear motion, frequency, period. Introduce formulas for finding these quantities and units of measurement.
Educational objectives: to form the concept of rectilinear and curvilinear motion, the quantities that characterize it, the units of measurement of these quantities and formulas for calculation.
Developmental tasks: continue to develop the skills to apply theoretical knowledge to solve practical problems, develop interest in the subject and logical thinking.
Educational objectives: continue to develop students’ horizons; the ability to keep notes in notebooks, observe, notice patterns in phenomena, and justify their conclusions.
Equipment: Presentation. Computer. Multimedia projector Ball, ball on a string, inclined chute, ball, toy car, spinning top, clock model with hands, stopwatches
Lesson progress
I. Organizational moment. Introductory word from the teacher. Hello, my young friends! Let me start our lesson today with these lines: “Terrible mysteries of nature hang everywhere in the air” (N. Zabolotsky, poem “Mad Wolf”) (slide 1)
2. Updating knowledge
- What types of movement do you know?- What is the difference between rectilinear and curvilinear movements?- Compare trajectory and path for straight and curved movements. Teacher: We know that all bodies attract each other. In particular, the Moon, for example, is attracted to the Earth. But the question arises: if the Moon is attracted to the Earth, why does it revolve around it instead of falling towards the Earth? (sl-)In order to answer this question, it is necessary to consider the types of motion of bodies. We already know that movement can be uniform and uneven, but there are other characteristics of movement (slide)
3. Problem situation: How are the following movements different?
Demonstrations: falling a ball in a straight line, rolling a ball along a straight chute. And along a circular path, the rotation of a ball on a string, the movement of a toy car on the table, the movement of a ball thrown at an angle to the horizon...( by type of trajectory)
Teacher: Based on the type of trajectory, these movements can be divide for movement in a straight line and along a curved line .(slide)
Let's try to give definitions curvilinear and rectilinear movements. ( Writing in a notebook) rectilinear movement– movement along a straight path. Curvilinear movement is movement along an indirect (curved) trajectory.
4. So, the topic of the lesson
Rectilinear and curvilinear movement. Circular movement(slide)
Teacher: Let's consider two examples of curvilinear movement: along a broken line and along a curve (draw). How are these trajectories different?
Students: In the first case, the trajectory can be divided into straight sections and each section can be considered separately. In the second case, you can divide the curve into circular arcs and straight sections. T.ob. this movement can be considered as a sequence of movements occurring along circular arcs of different radii. Therefore, to study curvilinear motion, you need to study movement in a circle.(slide 15)
Message 1 Movement of a body in a circle
In nature and technology very often there are movements whose trajectories are not straight, but curved lines. This is a curvilinear movement. Planets and artificial satellites of the Earth move along curvilinear trajectories in outer space, and on Earth all kinds of means of transport, parts of machines and mechanisms, river waters, atmospheric air, etc.
If you press the end of a steel rod against a rotating grindstone, the hot particles coming off the stone will be visible in the form of sparks. These particles fly at the speed they had at the moment they left the stone. It is clearly seen that the direction of movement of the sparks coincides with the tangent to the circle at the point where the rod touches the stone. On a tangent Splashes from the wheels of a skidding car are moving. (Sketch.)
Direction and velocity module
Teacher: Thus, the instantaneous speed of the body in different points curvilinear trajectory has a different direction. In absolute terms, the speed can be the same everywhere or vary from point to point. (slide)
But even if the speed module does not change, it cannot be considered constant. Speed is a vector quantity. For vector quantity module and direction are equally important. And once speed changes, which means there is acceleration. Therefore, curvilinear movement is always accelerating movement, even if the absolute value of the speed is constant .(slide)(video1)
Acceleration body moving uniformly in a circle at any point centripetal, i.e. directed along the radius of the circle towards its center. At any point, the acceleration vector is perpendicular to the velocity vector. (Draw)
Modulus of centripetal acceleration: a c =V 2 /R ( write the formula), where V is the linear velocity of the body, and R is the radius of the circle. (slide)
Centripetal force is a force acting on a body during curvilinear motion at any time, always directed along the radius of the circle towards the center (as is centripetal acceleration). And the force acting on a body is proportional to acceleration. F=ma, then
Characteristics of body motion in a circle
Circular motion is often characterized not by the speed of movement, but by the period of time during which the body makes one full revolution. This quantity is called circulation period and is designated by the letter T. ( Write period definition). When moving in a circle, a body will return to its original point in a certain period of time. Therefore, the circular motion is periodic.
A period is the time of one complete revolution.
If a body makes N revolutions in time t, then how to find the period? (formula)
Let's find the connection between the period of revolution T and the magnitude of velocity for uniform motion in a circle of radius R. Because V=S/t = 2πR/T. ( Write the formula in your notebook)
Message2 A period is a quantity that occurs quite often in nature and technology. Yes, we know. That the Earth rotates around its axis and the average rotation period is 24 hours. A complete revolution of the Earth around the Sun occurs in approximately 365.26 days. The impellers of hydraulic turbines make one full revolution in a time of 1 second. A helicopter rotor has a rotation period of 0.15 to 0.3 seconds. The period of blood circulation in humans is approximately 21-22 seconds.
Teacher: The movement of a body in a circle can be characterized by another quantity - the number of revolutions per unit time. They call her frequency circulation: ν= 1/T. Frequency unit: s -1 =Hz. ( Write definition, unit and formula)(slide)
How to find the frequency if a body makes N revolutions in time t (formula)
Teacher: What conclusion can be drawn about the relationship between these quantities? (period and frequency are reciprocal quantities)
Message3 The crankshafts of tractor engines have a rotation speed of 60 to 100 revolutions per second. The gas turbine rotor rotates at a frequency of 200 to 300 rps. Bullet. Flying out of a Kalashnikov assault rifle, it rotates at a frequency of 3000 rps. To measure frequency, there are instruments, so-called frequency measuring circles, based on optical illusions. On such a circle there are black stripes and frequencies. When such a circle rotates, the black stripes form a circle at a frequency corresponding to this circle. Tachometers are also used to measure frequency. (slide)
Connection Rotation speed and rotation period
ℓ - circumference
ℓ=2πr V=2πr/T
Additional characteristics of circular motion. (slide)
Teacher: Let us remember what quantities characterize rectilinear motion?
Movement, speed, acceleration.
Teacher: by analogy, movement in a circle - the same quantities - angular displacement, angular velocity and angular acceleration.
Angular displacement: (slide) This is the angle between two radii. Designated – Measured in rad or deg.
Teacher: Let's remember from the algebra course how the radian is related to the degree?
2pi rad = 360 deg. Pi = 3.14, then 1 rad = 360/6.28 = 57 degrees.
Angular velocity w=
Unit of measurement of angular velocity - rad/s
Teacher:. Think about what the angular velocity will be equal to if the body has made one full revolution?
Student. Since the body has completed a full revolution, the time of its movement is equal to the period, and the angular displacement is 360° or 2. Therefore, the angular velocity is equal to.
Teacher: So what did we talk about today? (about curvilinear motion)
5. Questions for consolidation.
What kind of movement is called curvilinear?
Which motion is a special case of curvilinear motion?
What is the direction of instantaneous velocity during curvilinear motion?
Why is acceleration called centripetal?
What are period and frequency called? What units are they measured in?
How are these quantities interrelated?
How can we describe curvilinear motion?
What is the direction of acceleration of a body moving in a circle with a constant velocity?
6. Experimental work
Measure the period and frequency of a body suspended on a thread and rotating in a horizontal plane.
(on your desks you have bodies suspended by threads, a stopwatch. Rotate the body in a horizontal plane evenly and measure the time of 10 complete rotations. Calculate the period and frequency)
7. Consolidation. Problem solving. (slide)
A.S. Pushkin. "Ruslan and Lyudmila"
There is a green oak near the Lukomorye,
Golden chain on the oak tree
Day and night the cat is a scientist
Everything goes around and around in a chain.
Q: What is this cat movement called? Determine the frequency and period and angular velocity if in 2 minutes. He does 12 laps. (answer: 0.1 1/s, T=10s, w=0.628rad/s)
P.P. Ershov “The Little Humpbacked Horse”
Well, this is how our Ivan goes
Behind the ring on the okiyan
The little hunchback flies like the wind,
And the start for the first evening
I covered a hundred thousand versts
And I didn’t rest anywhere.
Q: How many times did the Little Humpbacked Horse circle the Earth during the first evening? The earth has the shape of a ball, and one mile is approximately 1066 m. (answer: 2.5 times)
8.Test Checking the assimilation of new material(tests on paper)
Test 1.
1. An example of curvilinear movement is...
a) falling of a stone;
b) turn the car to the right;
c) sprinter running 100 meters.
2. The minute hand of a clock makes one full revolution. What is the period of circulation?
a) 60 s; b) 1/3600 s; c) 3600 s.
3. A bicycle wheel makes one revolution in 4 s. Determine the rotation speed.
a) 0.25 1/s; b) 4 1/s; c) 2 1/s.
4. The propeller of a motor boat makes 25 revolutions in 1 s. What is the angular velocity of the propeller?
a) 25 rad/s; b) /25 rad/s; c) 50 rad/s.
5. Determine the rotation speed of the electric drill drill if its angular speed is 400 .
a) 800 1/s; b) 400 1/s; c) 200 1/s.
Answers: b; V; A; V; V.
Test 2.
1. An example of curvilinear movement is...
a) movement of the elevator;
b) a ski jump from a springboard;
c) a cone falling from the lower branch of a spruce tree in calm weather.
The second hand of a watch makes one full revolution. What is its circulation frequency?
a) 1/60 s; b) 60 s; c) 1 s.
3. The car wheel makes 20 revolutions in 10 s. Determine the period of revolution of the wheel?
a) 5 s; b) 10 s; c) 0.5 s.
4. The rotor of a powerful steam turbine makes 50 revolutions in 1 s. Calculate the angular velocity.
a) 50 rad/s; b)/50 rad/s; c) 10 rad/s.
5. Determine the rotation period of the bicycle sprocket if the angular velocity is equal.
a) 1 s; b) 2 s; c)0.5 s.
Answers: b; A; V; V; b.
Self-test
9. Reflection.
Let's fill it out together ZUH mechanism (I know, I found out, I want to know)
10.Summing up, grades for the lesson
11. Homework paragraphs 18,19,
home study: calculate, if possible, all the characteristics of any rotating body (bicycle wheel, minute hand of a clock)
Ya. I. Perelman. Entertaining physics. Book 1 and 2 - M.: Nauka, 1979.
S. A. Tikhomirova. Didactic material in physics. Physics in fiction. 7 – 11 grades. – M.: Enlightenment. 1996.
Slide 2
Lesson topic: Rectilinear and curvilinear motion. Movement of a body in a circle.
Slide 3
Mechanical movements Rectilinear Curvilinear Motion along an ellipse Motion along a parabola Motion along a hyperbola Motion along a circle
Slide 4
Lesson objectives: 1. Know the basic characteristics of curvilinear motion and the relationship between them. 2. Be able to apply the acquired knowledge when solving experimental problems.
Slide 5
Topic study plan
Studying new material Conditions for rectilinear and curvilinear motion Direction of body speed during curvilinear motion Centripetal acceleration Period of revolution Frequency of revolution Centripetal force Performing frontal experimental tasks Independent work in the form of tests Summing up
Slide 6
According to the type of trajectory, the movement can be: Curvilinear Rectilinear
Slide 7
Conditions for rectilinear and curvilinear motion of bodies (Experiment with a ball)
Slide 8
p.67 Remember! Working with the textbook
Slide 9
Circular motion is a special case of curvilinear motion
Slide 10
Characteristics of motion – linear speed of curvilinear motion () – centripetal acceleration () – period of revolution () – frequency of revolution ()
Slide 11
Remember. The direction of particle movement coincides with the tangent to the circle
Slide 12
In curvilinear motion, the speed of the body is directed tangentially to the circle. Remember.
Slide 13
During curvilinear motion, acceleration is directed towards the center of the circle. Remember.
Slide 14
Why is acceleration directed towards the center of the circle?
Slide 15
Determination of speed - speed - period of revolution r - radius of a circle
Slide 16
When a body moves in a circle, the magnitude of the velocity vector can change or remain constant, but the direction of the velocity vector necessarily changes. Therefore, the velocity vector is a variable quantity. This means that motion in a circle always occurs with acceleration.
Remember!
Slide 17
Centripetal force elastic force friction force gravitational force Model of the hydrogen atom
Slide 18
1. Establish the dependence of speed on radius2. Measure the acceleration when moving in a circle3. Establish the dependence of centripetal acceleration on the number of revolutions per unit time.
Experiment
Slide 19
Option 1Option 2 1. The body moves uniformly in a circle in a clockwise direction counterclockwise What is the direction of the acceleration vector during such movement? a) 1; b) 2; c) 3; d) 4. 2. The car moves with a constant absolute speed along the trajectory of the figure. At which of the indicated points on the trajectory is the centripetal acceleration minimum and maximum? 3. How many times will the centripetal acceleration change if the speed material point increase decrease by 3 times? a) will increase 9 times; b) will decrease by 9 times; c) will increase 3 times; d) will decrease by 3 times. Independent work
Slide 20
Continue the sentence Today in the lesson I realized that... I liked something in the lesson that... What made me happy in the lesson... I am satisfied with my work because... I would like to recommend...
Slide 21
Homework: §18-19, ex. 18 (1, 2) Additionally ex. 18 (5) Thank you for your attention. Thanks for the lesson!
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