Curvilinear motion definition and formulas. Lesson summary "Rectilinear and curvilinear motion

This topic will focus on a more complex type of movement − CURVILINEAR. How easy it is to guess curvilinear is a movement whose trajectory is a curved line. And, since this motion is more complicated than a rectilinear one, the 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 VALUES.

1. moving. In Section 1.1, we did not specify the difference between the concept

Fig. 1.3 paths (distances) and the concept of displacement,

because in rectilinear motion these

differences do not play a fundamental role, and

These values are denoted by the same letter

howl S. But when dealing with curvilinear motion,

this issue needs to be clarified. So what's the path

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

movement. That is, if you trace the trajectory

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

Fig.1.4 And displacement is the shortest distance between

the start point of the path and the end point of the path. And because

movement has a strict direction from the beginning

Way to its end, then it is a vector quantity

and is characterized not only by a numerical value, but also

direction (fig.1.3). It is easy to guess that if

the body moves along a closed path, then to

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

2 . Line speed. In section 1.1, we gave the definition of this quantity, and it remains valid, although at that time we did not specify that this speed is linear. What is the direction of the linear velocity vector? Let's turn to Figure 1.5. Here is a fragment

curvilinear trajectory of the body. Any curved line is a connection between the arcs of different circles. Figure 1.5 shows only two of them: a circle (O 1, r 1) and a circle (O 2, r 2). At the moment the body passes along the arc of the 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 the linear velocity vectors: the linear velocity vector is always directed tangentially to the trajectory in the direction of motion. Therefore, the angle between the vector and the radius vector drawn to a given point of the trajectory is always 90°. If the body moves at a constant linear speed, then the module 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 module of the vector changes. But, since the direction of the vector always changes during curvilinear motion, a very important conclusion follows from this:

Curvilinear motion always has acceleration! (Even if the movement is carried out at a constant linear speed.) Moreover, the acceleration in question in this case, in what follows we will call linear acceleration.

3 . Linear acceleration. Let me remind you that acceleration occurs when the speed changes. Accordingly, linear acceleration appears in the case of a change in linear speed. And the linear speed during curvilinear motion can change both modulo and 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 modulus. Consider these accelerations (Fig. 1.6). In this picture

rice. 1.6

a body is shown 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 denoted. It is called normal because it is directed perpendicular (normally) 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 modulus of the vector is called tangential and is denoted. It lies on the tangent and can be directed both towards the direction of the vector and opposite to it. :

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

If the line speed decreases, then< 0 и их вектора противоположно

directed.

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

I note that in this case we are talking about the vector sum, but in no case about the scalar sum. To find the numerical value, knowing and , it is necessary 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).

By what formulas to calculate and consider a little later.

ANGULAR VALUES.

1 . Angle of rotation φ . In curvilinear motion, the body not only travels some path 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, which 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 (denoted "rad"). Let me remind you that one full turn is equal to 2π radians, and the number π is a constant: π ≈ 3.14. in fig. 1.7 (a) shows the trajectory of the 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 value showing how the angle of rotation changes per unit of time. (ω - Greek letter, read "omega".) In fig. 1.7 (b) shows the position of a material point moving along a circular path with a 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 the angular velocity indicates how many radians

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

(denoted by " rad/s »).

rice. 1.7

a). b). Δt

Δt

O φ O Δt

3 . Angular acceleration ε is a value that shows how it changes per unit of 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 angular acceleration is " rad/s 2 ” (radian per second squared).

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

Table 1.2

№	physical quantity	determination of quantity	quantity designation	unit
1.	path	is the distance traveled by a body in the course of its motion	S	m (meter)
2.	speed	is the distance traveled by a body in a unit of time (e.g. 1 second)	υ	m/s (meter per second)
3.	acceleration	is the amount by which the speed of a body changes per unit of time	a	m/s 2 (meter per second squared)
4.	time		t	s (second)
5.	angle of rotation	is the angle through which the body rotates in the process of curvilinear motion	φ	rad (radian)
6.	angular velocity	is the angle that the body rotates per unit of time (for example, in 1 sec.)	ω	rad/s (radians per second)
7.	angular acceleration	is the amount by which the angular velocity changes per unit of time	ε	rad/s 2 (radian per second squared)

Now you can go directly to the consideration of all types of curvilinear motion, and there are only three of them.

With the help of this lesson, you will be able to independently study the topic “Rectilinear and curvilinear motion. The motion of a body in a circle with a constant modulo velocity. First, we characterize rectilinear and curvilinear motion by considering how the velocity vector and the force applied to the body are related in these types of motion. Next, we consider a special case when the body moves along a circle with a constant modulo speed.

In the previous lesson, we considered 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.

Earlier we said that traffic - this is a change in the position of a body in space relative to other bodies over time. Movement and direction of movement are characterized, among other things, by speed. The change in speed and the type of movement itself are associated with the action of a 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 a movement will be straightforward(Fig. 1).

Rice. 1. Rectilinear motion

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 motion

So, at rectilinear motion the velocity vector is directed in the same direction as the force applied to the body. BUT curvilinear movement is such a movement when the velocity vector and the force applied to the body are located at some angle to each other.

Consider a special case of curvilinear motion, when the body moves in a circle with a constant speed in absolute value. When a body moves in a circle at a constant speed, only the direction of the speed changes. Modulo it remains constant, but the direction of the velocity changes. Such a change in speed leads to the presence of an acceleration in the body, which is called centripetal.

Rice. 6. Movement along a curved path

If the body's trajectory is a curve, then it can be represented as a set of movements along arcs of circles, as shown in Fig. 6.

On fig. 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 modulo speed remains constant, a change in speed leads to an acceleration:

In this case acceleration will be directed towards the center of the circle. That is why it is called centripetal.

Why is centripetal acceleration directed towards the center?

Recall that if a body moves along a curved path, then its velocity is tangential. Velocity is a vector quantity. A vector has a numerical value and a direction. The speed as the body moves continuously changes its direction. That is, the difference in speeds at different times 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. Relation to is acceleration. We come to the conclusion that, even if the speed does not change in absolute value, a body that performs uniform motion in a circle has an acceleration.

Where is this acceleration directed? Consider Fig. 3. Some body moves curvilinearly (in an arc). The speed of the body at points 1 and 2 is tangential. The body moves uniformly, that is, the modules of the velocities are equal: , but the directions of the velocities do not coincide.

Rice. 3. Movement of the body in a circle

Subtract the speed from and get the vector . To do this, you need to connect the beginnings of both vectors. In parallel, we 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 the velocity vectors and the difference vector (Fig. 5).

Rice. 5. Triangle formed by velocity vectors

This triangle is isosceles (velocity modules are equal). So the angles at the base are equal. Let's write the equation for the sum of the angles of a triangle:

Find out where the acceleration is directed at a given point of the trajectory. To do this, we begin to bring point 2 closer to point 1. With such an unlimited diligence, the angle will tend to 0, and the angle - to. The angle between the velocity change vector and the velocity vector itself is . The speed is directed tangentially, and the velocity change vector 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?

Consider the trajectory along which the body moves. In this case, this is an arc of a circle (Fig. 8).

Rice. 8. Movement of the body in a circle

The figure shows two triangles: a triangle formed by the velocities, and a triangle formed by the radii and the displacement vector. If points 1 and 2 are very close, then the displacement vector will be the same as the path vector. Both triangles are isosceles with the same vertex angles. So the triangles are similar. This means that the corresponding sides of the triangles are in the same ratio:

The displacement is equal to the product of speed and time: . Substituting this formula, you can get the following expression for centripetal acceleration:

Angular velocity denoted by the Greek letter omega (ω), it indicates at what angle the body rotates per unit of time (Fig. 9). This is the magnitude of the arc, in degrees, traversed by the body in some time.

Rice. 9. Angular speed

Note that if a rigid body rotates, then the angular velocity for any points on this body will be a constant value. The point is closer to the center of rotation or farther - it does not matter, that is, it does not depend on the radius.

The unit of measurement in this case will be either degrees per second (), or radians per second (). Often the word "radian" is not written, but simply written. For example, let's find what the angular velocity of the Earth is. The earth makes a full 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 velocities:

The 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 motion in a circle at a constant speed is a special case of motion. However, circular motion can also be uneven. The speed can change not only in direction and remain the same in absolute value, but also change in its value, i.e., in addition to changing direction, there is also a change in the speed module. In this case, we are talking about the so-called accelerated circular motion.

What is a radian?

There are two units for measuring angles: degrees and radians. In physics, as a rule, the radian measure of an angle is the main one.

Let's construct a central angle , which relies on an arc of length .

The concepts of speed and acceleration are naturally generalized to the case of motion of a material point along curvilinear trajectory. The position of the moving point on the trajectory is given by the radius vector r drawn to this point from some fixed point O, for example, the origin (Fig. 1.2). Let at the moment t material point is in position M with radius vector r = r (t). After a short time D t, it will move to the position M 1 with radius - vector r 1 = r (t+ D t). Radius - the vector of a 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 the direction of the vector D r .

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

(1.9)

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

Note the following formal analogy between velocity 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 locus of velocity points is a curve called speed hodograph. When a material point describes a trajectory, the speed point corresponding to it moves along the hodograph.

Rice. 1.2 differs from fig. 1.3 only by designations. 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 over the vector V when finding the acceleration are completely identical.

Speed V directed along a tangent path. That's why accelerationa will be directed tangentially to the velocity hodograph. It can be said that acceleration is the speed of movement of the high-speed point along the hodograph. Consequently,

Depending on the shape of the trajectory, the movement is divided into rectilinear and curvilinear. 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, the movement of the planets, the end of the clock hand on the dial, etc.

Figure 1. Trajectory and displacement in curvilinear motion

Definition

Curvilinear motion is a motion whose trajectory is a curved line (for example, a circle, an ellipse, a hyperbola, a parabola). When moving along a curvilinear trajectory, the displacement vector $\overrightarrow(s)$ is directed along the chord (Fig. 1), and l is the length of the trajectory. The instantaneous speed of the body (that is, the speed of the body at a given point of the trajectory) is directed tangentially at that point of the trajectory where the moving body is currently located (Fig. 2).

Figure 2. Instantaneous velocity during curvilinear motion

However, the following approach is more convenient. You can imagine this movement as a combination of several movements along the arcs of circles (see Fig. 4.). There will be fewer such partitions than in the previous case, in addition, the movement along the circle is itself curvilinear.

Figure 4. Partitioning a curvilinear motion into motions along arcs of circles

Conclusion

In order to describe curvilinear motion, one must learn to describe motion along a circle, and then represent an arbitrary motion as a set of motions along arcs of circles.

The task of studying the curvilinear motion of a material point is to compile a kinematic equation that describes this motion and allows, according to given initial conditions, to determine all the characteristics of this motion.