With the help of this lesson you can independently study the topic “Rectilinear and curvilinear movement. 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 will consider special case when a body moves in a circle with a constant absolute speed.
In the previous lesson we looked at issues related to the law universal gravity. 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 with constant speed, then only the direction of velocity 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 by curvilinear trajectory
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 centripetal acceleration towards the center?
Recall that if a body moves along a curved path, then its speed is directed tangentially. Speed is vector quantity. A vector has a numerical value and a direction. The speed continuously changes its direction as the body moves. That is, the speed difference in various moments time will not be equal to zero (), unlike 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.
Consider a triangle, formed by vectors speeds and 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 on 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: triangle, formed by speeds, and a triangle formed by the radii and the 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 Greek letter omega (ω), it talks about the angle at 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
Let us note that if a rigid body 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. How larger 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 .
We know that during rectilinear motion, the direction of the velocity vector always coincides with the direction of movement. What can be said about the direction of velocity and displacement during curved motion? To answer this question, we will use the same technique that we used in previous chapter when studying instantaneous speed rectilinear motion.
Figure 56 shows a certain curved trajectory. Let us assume that a body moves along it from point A to point B.
In this case, the path traveled by the body is an arc A B, and its displacement is a vector. Of course, we cannot assume that the speed of the body during movement is directed along the displacement vector. Let us draw a series of chords between points A and B (Fig. 57) and imagine that the body’s movement occurs precisely along these chords. On each of them the body moves rectilinearly and the velocity vector is directed along the chord.
Let us now make our straight sections (chords) shorter (Fig. 58). As before, on each of them the velocity vector is directed along the chord. But it is clear that broken line in Figure 58 it looks more like a smooth curve.
It is clear, therefore, that by continuing to reduce the length of the straight sections, we will, as it were, pull them into points and the broken line will turn into a smooth curve. The speed at each point of this curve will be directed tangentially to the curve at this point (Fig. 59).
The speed of movement of a body at any point on a curvilinear trajectory is directed tangentially to the trajectory at that point.
The fact that the speed of a point during curvilinear movement is really directed along a tangent is convinced by, for example, observation of the operation of the gochnla (Fig. 60). If you press the ends 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 at which
they possessed at the moment of separation from the stone. It is clearly seen that the direction of the sparks always coincides with the tangent to the circle at the point where the rod touches the stone. The splashes from the wheels of a skidding car also move tangentially to the circle (Fig. 61).
Thus, the instantaneous speed of the body in different points curvilinear trajectory has various directions, as shown in Figure 62. The velocity module can be the same at all points of the trajectory (see Figure 62) or vary from point to point, from one moment in time to another (Figure 63).
Kinematics of a point. Path. Moving. Speed and acceleration. Their projections on coordinate axes. Calculation of the distance traveled. Average values.
Kinematics of a point- section of kinematics that studies mathematical description movement of material points. The main task of kinematics is to describe movement using a mathematical apparatus without identifying the reasons causing this movement.
Path and movement. The line along which a point on the body moves is called trajectory of movement. The path length is called the path traveled. The vector connecting the initial and end point trajectory is called moving. Speed- vector physical quantity, characterizing the speed of movement of the body, numerically equal to the ratio movements over a short period of time to the value of this interval. The time period is considered sufficiently small if the speed at uneven movement did not change during this period. The defining formula for speed is v = s/t. The unit of speed is m/s. In practice, the speed unit used is km/h (36 km/h = 10 m/s). Speed is measured with a speedometer.
Acceleration- vector physical quantity characterizing the rate of change in speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred. If the speed changes equally throughout the entire movement, then the acceleration can be calculated using the formula a=Δv/Δt. Acceleration unit – m/s 2
Speed and acceleration during curved motion. Tangential and normal accelerations.
Curvilinear movements– movements whose trajectories are not straight, but curved lines.
Curvilinear movement– this is always motion with acceleration, even if the absolute speed is constant. Curvilinear movement with constant acceleration always occurs in the plane in which the acceleration vectors and initial speeds points. In the case of curvilinear motion with constant acceleration in the plane xOy projections v x And v y its speed on the axis Ox And Oy and coordinates x And y points at any time t determined by formulas
v x =v 0 x +a x t, x=x 0 +v 0 x t+a x t+a x t 2 /2; v y =v 0 y +a y t, y=y 0 +v 0 y t+a y t 2 /2
A special case of curvilinear motion is circular motion. Circular motion, even uniform, is always accelerated motion: the velocity module is always directed tangentially to the trajectory, constantly changing direction, therefore circular motion always occurs with centripetal acceleration |a|=v 2 /r where r– radius of the circle.
The acceleration vector when moving in a circle is directed towards the center of the circle and perpendicular to the velocity vector.
In curvilinear motion, acceleration can be represented as the sum of the normal and tangential components: ,
Normal (centripetal) acceleration is directed towards the center of curvature of the trajectory and characterizes the change in speed in the direction:
v – instantaneous speed value, r– radius of curvature of the trajectory at a given point.
Tangential (tangential) acceleration is directed tangentially to the trajectory and characterizes the change in speed modulo.
Full acceleration, with which it moves material point, equals:
Tangential acceleration characterizes the speed of change in the speed of movement along numerical value and is directed tangentially to the trajectory.
Hence
Normal acceleration characterizes the rate of change in speed in direction. Let's calculate the vector:
4.Kinematics solid. Spinning around fixed axis. Angular velocity and acceleration. Relationship between angular and linear velocities and accelerations.
Kinematics of rotational motion.
The movement of the body can be either translational or rotational. In this case, the body is represented as a system of material points rigidly interconnected.
During translational motion, any straight line drawn in the body moves parallel to itself. According to the shape of the trajectory, the translational movement can be rectilinear or curvilinear. During translational motion, all points of a rigid body during the same period of time make movements equal in magnitude and direction. Consequently, the velocities and accelerations of all points of the body at any moment of time are also the same. To describe translational motion, it is enough to determine the movement of one point.
Rotational movement rigid body around a fixed axis is called such a movement in which all points of the body move in circles, the centers of which lie on the same straight line (axis of rotation).
The axis of rotation can pass through the body or lie outside it. If the axis of rotation passes through the body, then the points lying on the axis remain at rest when the body rotates. Points of a rigid body located at different distances from the axis of rotation in equal periods of time travel different distances and, therefore, have different linear velocities.
When a body rotates around a fixed axis, the points of the body undergo the same angular movement in the same period of time. Module equal to angle rotation of a body around an axis in time , the direction of the angular displacement vector with the direction of rotation of the body is connected by the screw rule: if you combine the directions of rotation of the screw with the direction of rotation of the body, then the vector will coincide with forward movement screw The vector is directed along the axis of rotation.
The rate of change in angular displacement is determined by the angular velocity - ω. By analogy with linear speed, the concepts average and instantaneous angular velocity:
Angular velocity- vector quantity.
The rate of change in angular velocity is characterized by average and instantaneous
angular acceleration.
The vector and can coincide with the vector and be opposite to it
You are well aware that depending on the shape of the trajectory, movement is divided into rectilinear And curvilinear. We learned how to work with rectilinear motion in previous lessons, namely, to solve the main problem of mechanics for this type of motion.
However, it is clear that in real world we most often deal with curvilinear movement, when the trajectory is a curved line. Examples of such movement are the trajectory of a body thrown at an angle to the horizon, the movement of the Earth around the Sun, and even the trajectory of the movement of your eyes, which are now following this note.
The question of how to solve the main task mechanics in the case of curvilinear motion, and this lesson will be devoted.
First, let's decide what fundamental differences does curvilinear movement (Fig. 1) have relative to rectilinear movement and what do these differences lead to.
Rice. 1. Trajectory of curvilinear movement
Let's talk about how it is convenient to describe the movement of a body during curvilinear motion.
You can divide the movement into separate sections, in each of which the movement can be considered rectilinear (Fig. 2).
Rice. 2. Dividing curvilinear motion into sections of rectilinear motion
However, the following approach is more convenient. We will imagine this movement as a combination of several movements along circular arcs (Fig. 3). Please note that there are fewer such partitions than in the previous case, in addition, the movement along the circle is curvilinear. In addition, examples of motion in a circle are very common in nature. From this we can conclude:
In order to describe curvilinear movement, you need to learn to describe movement in a circle, and then represent arbitrary movement in the form of sets of movements along circular arcs.
Rice. 3. Partitioning curvilinear motion into motion along circular arcs
So, let's begin the study of curvilinear motion by studying uniform motion in a circle. Let's figure out what are the fundamental differences between curvilinear movement and rectilinear movement. To begin with, let us remember that in ninth grade we studied the fact that the speed of a body when moving in a circle is directed tangent to the trajectory (Fig. 4). By the way, you can observe this fact experimentally if you watch how sparks move when using a sharpening stone.
Let's consider the movement of a body along a circular arc (Fig. 5).
Rice. 5. Body speed when moving in a circle
Please note that in this case the modulus of the body’s velocity at the point equal to modulus body velocity at point:
However, the vector is not equal to the vector. So, we have a velocity difference vector (Fig. 6):
Rice. 6. Velocity difference vector
Moreover, the change in speed occurred after some time. So we get the familiar combination:
This is nothing more than a change in speed over a period of time, or acceleration of a body. A very important conclusion can be drawn:
Movement along a curved path is accelerated. The nature of this acceleration is a continuous change in the direction of the velocity vector.
Let us note once again that, even if it is said that the body moves uniformly in a circle, it is meant that the modulus of the body’s velocity does not change. However, such movement is always accelerated, since the direction of speed changes.
In ninth grade, you studied what this acceleration is equal to and how it is directed (Fig. 7). Centripetal acceleration is always directed towards the center of the circle along which the body is moving.
Rice. 7. Centripetal acceleration
The module of centripetal acceleration can be calculated by the formula:
Let us move on to the description of the uniform motion of a body in a circle. Let's agree that the speed that you used while describing the translational motion will now be called linear speed. And by linear speed we will understand the instantaneous speed at the point of the trajectory of a rotating body.
Rice. 8. Movement of disk points
Consider a disk that rotates clockwise for definiteness. On its radius we mark two points and (Fig. 8). Let's consider their movement. Over time, these points will move along the arcs of the circle and become points and. It is obvious that the point has moved more than the point . From this we can conclude that the farther a point is from the axis of rotation, the greater the linear speed it moves
However, if you look closely at the points and , we can say that the angle by which they turned relative to the axis of rotation remained unchanged. It is the angular characteristics that we will use to describe the movement in a circle. Note that to describe circular motion we can use corner characteristics.
Let's start considering the motion in a circle from the very simple case– uniform movement around a circle. Let us recall that uniform translational motion is a movement in which the body makes equal movements over any equal periods of time. By analogy, we can give the definition of uniform motion in a circle.
Uniform circular motion is a motion in which the body rotates through equal angles over any equal intervals of time.
Similar to the concept of linear velocity, the concept of angular velocity is introduced.
Angular velocity of uniform motion ( is a physical quantity equal to the ratio of the angle through which the body turned to the time during which this rotation occurred.
In physics, the radian measure of angle is most often used. For example, the angle at equal to radians. Angular velocity is measured in radians per second:
Let's find the connection between the angular speed of rotation of a point and the linear speed of this point.
Rice. 9. Relationship between angular and linear speed
When rotating, a point passes an arc of length , turning at an angle . From the definition of the radian measure of an angle we can write:
Let's divide the left and right sides of the equality by the period of time during which the movement was made, then use the definition of angular and linear velocities:
Please note that the further a point is from the axis of rotation, the higher its linear speed. And the points located on the axis of rotation itself are motionless. An example of this is a carousel: the closer you are to the center of the carousel, the easier it is for you to stay on it.
This dependence of linear and angular velocities is used in geostationary satellites (satellites that are always above the same point earth's surface). Thanks to such satellites, we are able to receive television signals.
Let us remember that earlier we introduced the concepts of period and frequency of rotation.
The rotation period is the time of one full revolution. The rotation period is indicated by a letter and measured in SI seconds:
Rotation frequency is a physical quantity equal to the number of revolutions a body makes per unit time.
Frequency is indicated by a letter and measured in reciprocal seconds:
They are related by the relation:
There is a relationship between angular velocity and the frequency of rotation of the body. If we remember that full turn is equal to , it is easy to see that the angular velocity is:
Substituting these expressions into the relationship between angular and linear speed, we can obtain the dependence of linear speed on period or frequency:
Let us also write down the relationship between centripetal acceleration and these quantities:
Thus, we know the relationship between all the characteristics of uniform circular motion.
Let's summarize. In this lesson we began to describe curvilinear motion. We understood how we can connect curvilinear motion with circular motion. Circular motion is always accelerated, and the presence of acceleration determines the fact that the speed always changes its direction. This acceleration is called centripetal. Finally, we remembered some characteristics of circular motion (linear speed, angular velocity, period and rotation frequency) and found the relationships between them.
Bibliography
- G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10. - M.: Education, 2008.
- A.P. Rymkevich. Physics. Problem book 10-11. - M.: Bustard, 2006.
- O.Ya. Savchenko. Physics problems. - M.: Nauka, 1988.
- A.V. Peryshkin, V.V. Krauklis. Physics course. T. 1. - M.: State. teacher ed. min. education of the RSFSR, 1957.
- Аyp.ru ().
- Wikipedia ().
Homework
Having solved the problems for this lesson, you can prepare for questions 1 of the GIA and questions A1, A2 of the Unified State Exam.
- Problems 92, 94, 98, 106, 110 - Sat. problems A.P. Rymkevich, ed. 10
- Calculate the angular velocity of the minute, second and hour hands of the clock. Calculate the centripetal acceleration acting on the tips of these arrows if the radius of each is one meter.
Depending on the shape of the trajectory, movement can be divided into rectilinear and curvilinear. Most often you encounter curvilinear movements when the trajectory is represented as a curve. An example of this type of motion 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 movement in curved motion
Definition 1Curvilinear movement called a movement whose trajectory is a curved line. If a body moves along a curved path, then the displacement vector s → is directed along the chord, as shown in Figure 1, and l is the length of the path. The direction of the instantaneous speed of movement of the body goes tangentially at the same point of the trajectory where at this moment the moving object is located, as shown in Figure 2.
Figure 2. Instantaneous speed during curvilinear movement
Definition 2
Curvilinear motion of a material point called uniform when the velocity module is constant (circular motion), and uniformly accelerated when the direction and velocity module are changing (movement of an thrown body).
Curvilinear motion is always accelerated. This is explained by the fact that even with an unchanged velocity module and a changed direction, acceleration is always present.
In order to study 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, as shown in Figure 3.
Figure 3. Partitioning curvilinear motion into translational ones
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 movements along circular arcs, as shown in Figure 4. The number of partitions will be much less than in the previous method, in addition, the movement along the circle is already curvilinear.
Figure 4. Partitioning curvilinear motion into motion along circular arcs
Note 1
To record curvilinear movement, you must be able to describe movement in a circle, and represent arbitrary movement in the form of sets of movements along the arcs of these circles.
The study of curvilinear motion includes the compilation of a kinematic equation that describes this motion and allows, based on available data, initial conditions determine all movement characteristics.
Example 1
Given a material point moving along a curve, as shown in Figure 4. The centers of circles O 1, O 2, O 3 are located on the same straight line. Need to find displacement
s → and path length l while moving from point A to B.
Solution
By condition, we have that the centers of the circle belong to the same 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 after what period of time 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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