Uniform curvilinear movement of a point. Kinematics of curvilinear motion

During curvilinear motion, the direction of the velocity vector changes. At the same time, its module, i.e. length, may also change. In this case, the acceleration vector is decomposed into two components: tangent to the trajectory and perpendicular to the trajectory (Fig. 10). The component is called tangential(tangential) acceleration, component – normal(centripetal) acceleration.

Acceleration during curved motion

Tangential acceleration characterizes the rate of change in linear speed, and normal acceleration characterizes the rate of change in direction of movement.

The total acceleration is equal to the vector sum of the tangential and normal accelerations:

(15)

The total acceleration module is equal to:

.

Let us consider the uniform motion of a point along a circle. Wherein And . Let at the considered moment of time t the point is in position 1 (Fig. 11). After time Δt, the point will be in position 2, having passed the path Δs, equal to arc 1-2. In this case, the speed of point v increases Δv, as a result of which the velocity vector, remaining unchanged in magnitude, rotates through an angle Δφ , coinciding in size with the central angle based on an arc of length Δs:

(16)

where R is the radius of the circle along which the point moves. Let's find the increment of the velocity vector. To do this, let's move the vector so that its beginning coincides with the beginning of the vector. Then the vector will be represented by a segment drawn from the end of the vector to the end of the vector . This segment serves as the base of an isosceles triangle with sides and and angle Δφ at the apex. If the angle Δφ is small (which is true for small Δt), for the sides of this triangle we can approximately write:

.

Substituting here Δφ from (16), we obtain an expression for the vector modulus:

.

Dividing both sides of the equation by Δt and passing to the limit, we obtain the value of centripetal acceleration:

Here the quantities v And R are constant, so they can be taken beyond the limit sign. The ratio limit is the speed modulus It is also called linear speed.

Radius of curvature

The radius of the circle R is called radius of curvature trajectories. The inverse of R is called the curvature of the trajectory:

.

where R is the radius of the circle in question. If α is the central angle corresponding to the arc of a circle s, then, as is known, the relationship between R, α and s holds:

s = Rα. (18)

The concept of radius of curvature applies not only to a circle, but also to any curved line. The radius of curvature (or its inverse value - curvature) characterizes the degree of curvature of the line. The smaller the radius of curvature (respectively, the greater the curvature), the more strongly the line is curved. Let's consider this concept in more detail.

The circle of curvature of a flat line at a certain point A is the limiting position of a circle passing through point A and two other points B 1 and B 2 as they approach point A infinitely (in Fig. 12 the curve is drawn by a solid line, and the circle of curvature by a dotted line). The radius of the circle of curvature gives the radius of curvature of the curve in question at point A, and the center of this circle gives the center of curvature of the curve for the same point A.

At points B 1 and B 2, draw tangents B 1 D and B 2 E to a circle passing through points B 1, A and B 2. The normals to these tangents B 1 C and B 2 C will represent the radii R of the circle and will intersect at its center C. Let us introduce the angle Δα between the normals B1 C and B 2 C; obviously, it is equal to the angle between the tangents B 1 D and B 2 E. Let us denote the section of the curve between points B 1 and B 2 as Δs. Then according to formula (18):

.

Circle of curvature of a flat curved line

Determining the curvature of a plane curve at different points

In Fig. Figure 13 shows circles of curvature of a flat line at different points. At point A 1, where the curve is flatter, the radius of curvature is greater than at point A 2, respectively, the curvature of the line at point A 1 will be less than at point A 2. At point A 3 the curve is even flatter than at points A 1 and A 2, so the radius of curvature at this point will be greater and the curvature less. In addition, the circle of curvature at point A 3 lies on the other side of the curve. Therefore, the value of curvature at this point is assigned a sign opposite to the sign of curvature at points A 1 and A 2: if the curvature at points A 1 and A 2 is considered positive, then the curvature at point A 3 will be negative.

This topic will be devoted to a more complex type of movement - CURVILINEAR. As you might guess, curvilinear is a movement whose trajectory is a curved line. And, since this movement is more complex than a rectilinear one, those physical quantities that were listed in the previous chapter are no longer enough to describe it.

For the mathematical description of curvilinear motion, there are 2 groups of quantities: linear and angular.

LINEAR QUANTITIES.

1. Moving. In section 1.1 we did not clarify the difference between the concept

Fig. 1.3 path (distance) and the concept of movement,

since in rectilinear motion these

differences do not play a fundamental role, and

These quantities are designated by the same letter -

howl S. But when dealing with curvilinear motion,

this issue needs to be clarified. So what is the path

(or distance)? – This is the length of the trajectory

movements. That is, if you track the trajectory

movement of the body and measure it (in meters, kilometers, etc.), you will get a value called path (or distance) S(see Fig. 1.3). Thus, the path is a scalar quantity that is characterized only by a number.

Fig.1.4 And movement is the shortest distance between

the starting point of the path and the end point of the path. And, since

the movement has a strict direction from the beginning

path to its end, then it is a vector quantity

and is characterized not only by numerical value, but also

direction (Fig. 1.3). It's not hard to guess what if

the body moves along a closed trajectory, then to

the moment it returns to the initial position, the displacement will be zero (see Fig. 1.4).

2 . Linear speed. In section 1.1 we gave a definition of this quantity, and it remains valid, although then we did not specify that this speed is linear. What is the direction of the linear velocity vector? Let's turn to Fig. 1.5. A fragment is shown here

curvilinear trajectory of the body. Any curved line is a connection between arcs of different circles. Figure 1.5 shows only two of them: circle (O 1, r 1) and circle (O 2, r 2). At the moment the body passes along the arc of a given circle, its center becomes a temporary center of rotation with a radius equal to the radius of this circle.

The vector drawn from the center of rotation to the point where the body is currently located is called the radius vector. In Fig. 1.5, radius vectors are represented by vectors and . Also shown in this figure are the linear velocity vectors: the linear velocity vector is always directed tangentially to the trajectory in the direction of movement. Consequently, the angle between the vector and the radius vector drawn to a given point on the trajectory is always equal to 90°. If a body moves with a constant linear speed, then the magnitude of the vector will not change, while its direction changes all the time depending on the shape of the trajectory. In the case shown in Fig. 1.5, the movement is carried out with a variable linear speed, so the modulus of the vector changes. But, since during curvilinear movement the direction of the vector always changes, a very important conclusion follows from this:

in curvilinear motion there is always acceleration! (Even if the movement is carried out at a constant linear speed.) Moreover, the acceleration in question in this case will be called linear acceleration in the future.

3 . Linear acceleration. Let me remind you that acceleration occurs when speed changes. Accordingly, linear acceleration appears when the linear speed changes. And the linear speed during curvilinear movement can change both in magnitude and in direction. Thus, the total linear acceleration is decomposed into two components, one of which affects the direction of the vector, and the second affects its magnitude. Let's consider these accelerations (Fig. 1.6). In this picture

rice. 1.6

ABOUT

shows a body moving along a circular path with the center of rotation at point O.

An acceleration that changes the direction of a vector is called normal and is designated . It is called normal because it is directed perpendicular (normal) to the tangent, i.e. along the radius to the center of the turn . It is also called centripetal acceleration.

The acceleration that changes the magnitude of the vector is called tangential and is designated . It lies on the tangent and can be directed either towards the direction of the vector or opposite to it :

If linear speed increases, then > 0 and their vectors are codirectional;

If linear speed decreases, then< 0 и их вектора противоположно

directed.

Thus, these two accelerations always form a right angle (90º) with each other and are components of the total linear acceleration, i.e. The total linear acceleration is the vector sum of the normal and tangential acceleration:

Let me note that in this case we are talking specifically about a vector sum, but in no case about a scalar sum. To find the numerical value of , knowing and , you need to use the Pythagorean theorem (the square of the hypotenuse of a triangle is numerically equal to the sum of the squares of the legs of this triangle):

(1.8).

This implies:

(1.9).

We will consider what formulas to calculate using a little later.

ANGULAR VALUES.

1 . Angle of rotation φ . During curvilinear motion, the body not only goes some way and makes some movement, but also rotates through a certain angle (see Fig. 1.7(a)). Therefore, to describe such a movement, a quantity is introduced that is called the angle of rotation, denoted by the Greek letter φ (read “fi”) In the SI system, the angle of rotation is measured in radians (symbol "rad"). Let me remind you that one full revolution is equal to 2π radians, and the number π is a constant: π ≈ 3.14. in Fig. 1.7(a) shows the trajectory of a body along a circle of radius r with the center at point O. The angle of rotation itself is the angle between the radius vectors of the body at some instants of time.

2 . Angular velocity ω – this is a quantity that shows how the angle of rotation changes per unit time. (ω - Greek letter, read “omega”.) In Fig. 1.7(b) shows the position of a material point moving along a circular path with the center at point O, at intervals of time Δt . If the angles through which the body rotates during these intervals are the same, then the angular velocity is constant, and this movement can be considered uniform. And if the angles of rotation are different, then the movement is uneven. And, since angular velocity shows how many radians

the body rotated in one second, then its unit of measurement is radians per second

(denoted by " rad/s »).

rice. 1.7

A). b). Δt

Δt

Δt

ABOUT φ ABOUT Δt

3 . Angular acceleration ε is a quantity that shows how it changes per unit time. And since the angular acceleration ε appears when the angular velocity changes ω , then we can conclude that angular acceleration occurs only in the case of non-uniform curvilinear motion. The unit of measurement for angular acceleration is “ rad/s 2 "(radians per second squared).

Thus, table 1.1 can be supplemented with three more values:

Table 1.2

Now we can proceed directly to the consideration of all types of curvilinear movement, and there are only three of them.

We more or less 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 the real world we most often deal with curvilinear motion, 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.

This lesson will be devoted to the question of how the main problem of mechanics is solved in the case of curvilinear motion.

To begin with, let’s determine what fundamental differences exist in curvilinear movement (Fig. 1) relative to rectilinear movement, and what 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.

The movement can be divided into separate sections, in each of which the movement can be considered rectilinear (Fig. 2).

Rice. 2. Partitioning curvilinear motion into translational motions

However, the following approach is more convenient. We will imagine this movement as a combination of several movements along circular arcs (see 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 circular motion 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. 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 in a circle (Fig. 4).

Rice. 4. Body speed when moving in a circle

Please note that in this case, the modulus of the velocity of the body at point A is equal to the modulus of the velocity of the body at point B.

However, a vector is not equal to a vector. So, we have a velocity difference vector (see Fig. 5).

Rice. 5. Speed ​​difference at points A and B.

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 a body moves uniformly in a circle, it is meant that the modulus of the body’s velocity does not change, but such motion is always accelerated, since the direction of the speed changes.

In ninth grade, you studied what this acceleration is and how it is directed (see Fig. 6). Centripetal acceleration is always directed towards the center of the circle along which the body is moving.

Rice. 6.Centripetal acceleration

The modulus of centripetal acceleration can be calculated using 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. 7. Movement of disk points

Consider a disk that rotates clockwise for definiteness. On its radius we mark two points A and B. And consider their movement. Over time, these points will move along circular arcs and become points A’ and B’. It is obvious that point A has made a greater movement than point B. From this we can conclude that the farther the point is from the axis of rotation, the greater the linear speed it moves.

However, if you look closely at points A and B, you can say that the angle θ by which they turned relative to the axis of rotation O remained unchanged. It is the angular characteristics that we will use to describe the movement in a circle. Note that to describe motion in a circle, you can use corner characteristics. First of all, let us recall the concept of the radian measure of angles.

An angle of 1 radian is a central angle whose arc length is equal to the radius of the circle.

Thus, it is easy to notice that, for example, the angle in is equal to radians. And, accordingly, you can convert any angle given in degrees into radians by multiplying it by and dividing by . The angle of rotation during rotational motion is similar to the movement during translational motion. Note that radian is a dimensionless quantity:

therefore the designation "rad" is often omitted.

Let's start considering motion in a circle with the simplest case - uniform motion in 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. Likewise,

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 is a physical quantity equal to the ratio of the angle through which the body turned to the time during which this rotation occurred.

Angular velocity is measured in radians per second, or simply in reciprocal seconds.

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

Point A rotates through an arc of length S, turning through an angle φ. From the definition of the radian measure of an angle we can write that

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 angular and 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.

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 designated by a letter and measured in seconds in the SI system:

Rotation frequency is the number of revolutions 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 a full revolution is equal to , it is easy to see that the angular velocity is:

In addition, if we remember how we defined the concept of radian, it will become clear how to connect the linear speed of a body with the angular speed:

.

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 speed, period and frequency of rotation), and found the relationships between them.

Bibliography:

1. G. Ya. Myakishev, B. B. Bukhovtsev, N. N. Sotsky. Physics 10. – M.: Education, 2008.
2. A. P. Rymkevich. Physics. Problem book 10-11. – M.: Bustard, 2006.
3. O. Ya. Savchenko. Physics problems. – M.: Nauka, 1988.
4. A. V. Peryshkin, V. V. Krauklis. Physics course. T. 1. – M.: State. teacher ed. min. education of the RSFSR, 1957.

1. Encyclopedia ().
2. Аyp.ru ().
3. Wikipedia ().

Homework:

Having solved the problems for this lesson, you will be able to prepare for questions 1 of the State Examination and questions A1, A2 of the Unified State Exam.

1. Problems 92, 94, 98, 106, 110 sb. problems A. P. Rymkevich ed. 10 ()
2. 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 of them is one meter.
3. Consider the following questions and their answers:

Question: Are there points on the Earth's surface at which the angular velocity associated with the Earth's daily rotation is zero?

Answer: Eat. These points are the geographic poles of the Earth. The speed at these points is zero because at these points you will be on the axis of rotation.

With the help of 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 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 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: 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 degrees passed 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. 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 construct a central angle that rests on an arc of length .

The concepts of speed and acceleration are naturally generalized to the case of a material point moving along curvilinear trajectory. The position of the moving point on the trajectory is specified by the radius vector r drawn to this point from some fixed point ABOUT, for example, the origin of coordinates (Fig. 1.2). Let at a moment in time t the material point is in position M with radius vector r = r (t). After a short time D t, it will move to position M 1 with radius - vector r 1 = r (t+ D t). Radius - the vector of the material point will receive an increment determined by the geometric difference D r = r 1 - r . Average speed over time D t is called the quantity

Average speed direction V Wed matches with vector direction D r .

Average speed limit at D t® 0, i.e. derivative of the radius - vector r by time

(1.9)

called true or instant speed of a material point. Vector V directed tangentially to the trajectory of a moving point.

Acceleration A is called a vector equal to the first derivative of the velocity vector V or the second derivative of the radius - vector r by time:

(1.10)

(1.11)

Let us note the following formal analogy between speed and acceleration. From an arbitrary fixed point O 1 we will plot the velocity vector V moving point at all possible times (Fig. 1.3).

End of vector V called speed point. The geometric locus of the velocity points is a curve called speed hodograph. When a material point describes a trajectory, the corresponding velocity point moves along the hodograph.

Rice. 1.2 differs from Fig. 1.3 by notation only. Radius – vector r replaced by velocity vector V , the material point - to the velocity point, the trajectory - to the hodograph. Mathematical operations on a vector r when finding the speed and above the vector V when found, the accelerations are completely identical.

Speed V directed along a tangential trajectory. That's why accelerationa will be directed tangentially to the speed hodograph. It can be said that acceleration is the speed of movement of the speed point along the hodograph. Hence,