Curvilinear movement. Movement of a body along a curved path

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 means 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 centripetal acceleration module 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 moved more 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.

References:

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 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.

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 . This figure also shows 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).

It follows from this:

(1.9).

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

ANGULAR VALUES.

1 . Rotation angle φ . During curvilinear motion, the body not only goes some way and makes some movement, but also turns 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.

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 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 motion (Fig. 1) relative to rectilinear motion 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. Dividing curvilinear movement into sections of rectilinear movement

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 velocity of the body at a point is equal to the modulus of the velocity of the body at the point:

However, a vector is not equal to a 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 some 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 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. 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, angle b is 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 located above the same point on the 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 that a body makes per unit of 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:

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

References

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. Аyp.ru ().
2. 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 - Sat. 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 is one meter.

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 velocity, 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. At the same time 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, will rotate 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 Δφ from (16) here, we obtain an expression for the modulus of the vector:

.

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 take a closer look at this concept.

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 infinitely approach point A (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 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.