We know that in rectilinear motion, the direction of the velocity vector always coincides with the direction of movement. What can be said about the direction of speed and displacement in curvilinear motion? To answer this question, we will use the same technique that was used in the previous chapter when studying the instantaneous speed of rectilinear motion.

Figure 56 shows some curvilinear trajectory. Suppose 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, it cannot be assumed 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 movement of the body occurs precisely along these chords. On each of them, the body moves in a straight line and the velocity vector is directed along the chord.

Now let's make our straight sections (chords) shorter (Fig. 58). As before, on each of them the velocity vector is directed along the chord. But it can be seen that the broken line in Figure 58 already looks more like a smooth curve.

It is therefore clear that by continuing to reduce the length of the straight sections, we will, as it were, shrink them into points and the broken line will turn into a smooth curve. The speed at each point of this curve will be directed but tangent to the curve at this point (Fig. 59).

The speed of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at this point.

The fact that the speed of a point during curvilinear motion is indeed directed along a tangent is convinced, for example, by observing the work of a gochnl (Fig. 60). If you press the ends of a steel bar to a rotating grindstone, then the hot particles coming off the stone will be visible in the form of sparks. These particles travel at the same speed as

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. Spray from the wheels of a skidding car also moves tangentially to the circle (Fig. 61).

Thus, the instantaneous speed of the body at different points of the curvilinear trajectory has different directions, as shown in Figure 62. The modulus of speed can be the same at all points of the trajectory (see Figure 62) or change from point to point, from one point in time to another (Fig. 63).

With curvilinear motion, the direction of the velocity vector changes. In this case, its module, i.e., the length, can 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.

Curvilinear acceleration

Tangential acceleration characterizes the rate of change of linear velocity, and normal acceleration characterizes the rate of change in direction of motion.

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

(15)

The total acceleration modulus is:

.

Consider the uniform motion of a point along a circle. Wherein and . Let the point be in position 1 at the considered time t (Fig. 11). After time Δt, the point will be in position 2, having traveled the path Δs, equal to the arc 1-2. In this case, the speed of the point v gets an increment Δv, as a result of which the velocity vector, remaining unchanged in magnitude, will turn through an angle Δφ , coinciding in magnitude 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, we'll 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 top. 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 modulus of the vector:

.

Dividing both parts of the equation by Δt and making the limit transition, we obtain the value of centripetal acceleration:

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

Radius of curvature

The circle radius R is called radius of curvature trajectories. The reciprocal of R is called the curvature of the path:

.

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

s = Ra. (18)

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

The circle of curvature of a flat line at some 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 is dashed). 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 is the center of curvature of the curve for the same point A.

Draw at points B 1 and B 2 the tangents B 1 D and B 2 E to the circle passing through the points B 1 , A and B 2 . The normals to these tangents B 1 C and B 2 C will be the radii R of the circle and intersect at its center C. Let us introduce the angle Δα between the normals B1C and B 2 C; obviously, it is equal to the angle between the tangents B 1 D and B 2 E. Let's designate the section of the curve between the 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

On fig. 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 larger and the curvature smaller. In addition, the circle of curvature at point A 3 lies on the other side of the curve. Therefore, the magnitude 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.

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,

You are well aware that, depending on the shape of the trajectory, the 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 are most often dealing 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 your eyes, which are now following this abstract.

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 the curvilinear motion (Fig. 1) has relative to the rectilinear one and what these differences lead to.

Rice. 1. Trajectory of curvilinear motion

Let's talk about how it is convenient to describe the motion of a body during curvilinear motion.

You can break the movement into separate sections, on each of which the movement can be considered rectilinear (Fig. 2).

Rice. 2. Partitioning of curvilinear motion into segments of rectilinear motion

However, the following approach is more convenient. We will represent this movement as a set of several movements along arcs of circles (Fig. 3). 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 movement in a circle in nature are very common. From this we can conclude:

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.

Rice. 3. Partitioning of a curvilinear motion into motions along arcs of circles

So, let's start the study of curvilinear motion with the study of uniform motion in a circle. Let's see what are the fundamental differences between curvilinear and rectilinear motion. To begin with, recall that in the ninth grade we studied the fact that the speed of a body when moving along a circle is directed tangentially to the trajectory (Fig. 4). By the way, you can observe this fact in practice if you look at how sparks move when using a grindstone.

Consider the motion of a body along a circular arc (Fig. 5).

Rice. 5. The speed of the body when moving in a circle

Please note that in this case, the modulus of the body's speed at the point is equal to the modulus of the body's speed at the 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 a while. Thus, we get the familiar combination:

This is nothing more than a change in speed over a period of time, or the acceleration of a body. We can draw a very important conclusion:

Movement along a curved path is accelerated. The nature of this acceleration is a continuous change in the direction of the velocity vector.

Once again, we note that, even if it is said that the body moves uniformly in a circle, it means that the modulus of the body's velocity does not change. However, such movement is always accelerated, since the direction of the velocity changes.

In the ninth grade, you studied what this acceleration is 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 centripetal acceleration module can be calculated using the formula:

We turn to the description of the uniform motion of the 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, for definiteness, rotates clockwise. On its radius, we mark two points and (Fig. 8). Consider their movement. For some time, these points will move along the arcs of the circle and become points and . Obviously, the point has moved more than the point . 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 we carefully look 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 motion in a circle. Note that to describe the motion in a circle, we can use corner characteristics.

Let's start the consideration of motion in a circle with the simplest case - uniform motion in a circle. Recall that a uniform translational motion is a motion in which the body makes the same displacements for any equal intervals of time. By analogy, we can define uniform motion in a circle.

Uniform motion in a circle is a motion in which the body rotates through the same angles for any equal intervals of time.

Similarly to the concept of linear velocity, the concept of angular velocity is introduced.

Angular velocity of uniform motion ( called a physical quantity equal to the ratio of the angle through which the body turned to the time during which this turn occurred.

In physics, the radian measure of an angle is most commonly used. For example, angle at is equal to radians. The angular velocity is measured in radians per second:

Let's find the relationship between the angular velocity of a point and the linear velocity of this point.

Rice. 9. Relationship between angular and linear speed

During rotation, the point passes an arc of length , turning at the same time through an angle . From the definition of the radian measure of an angle, we can write:

We divide the left and right parts of the equality by the time interval , for which the movement was made, then we use the definition of angular and linear velocities:

Note that the farther the point is from the axis of rotation, the higher its linear velocity. And the points located on the very axis of rotation are fixed. 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 on the earth's surface). Thanks to such satellites, we are able to receive television signals.

Recall that earlier we introduced the concepts of period and frequency of rotation.

The period of rotation is the time of one complete revolution. The period of rotation is indicated by a letter and is measured in seconds in SI:

The frequency of rotation is a physical quantity equal to the number of revolutions that the body makes per unit of time.

The frequency is indicated by a letter and is measured in reciprocal seconds:

They are related by:

There is a relationship between the angular velocity and the frequency of rotation of the body. If we remember that a full revolution is , it is easy to see that the angular velocity is:

By substituting these expressions into the dependence between the angular and linear speed, one can obtain the dependence of the linear speed on the 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 motion in a circle.

Let's summarize. In this lesson, we started to describe curvilinear motion. We understood how to relate curvilinear motion to circular motion. Circular motion is always accelerated, and the presence of acceleration causes the fact that the speed always changes its direction. Such acceleration is called centripetal. Finally, we remembered some characteristics of motion in a circle (linear velocity, angular velocity, period and frequency of rotation) and found the relationship 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. Problems in physics. - M.: Nauka, 1988.
4. A.V. Peryshkin, V.V. Krauklis. Physics course. T. 1. - M .: State. uch.-ped. ed. min. education of the RSFSR, 1957.

1. Ayp.ru ().
2. Wikipedia ().

Homework

By solving the tasks for this lesson, you will be able to prepare for questions 1 of the GIA and questions A1, A2 of the Unified State Examination.

1. Problems 92, 94, 98, 106, 110 - Sat. tasks of A.P. Rymkevich, ed. ten
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.

Kinematics studies movement without identifying the causes that cause this movement. Kinematics is a branch of mechanics. The main task of kinematics is the mathematical determination of the position and characteristics of the movement of points or bodies in time.

Basic kinematic quantities:

- Move() - a vector connecting the start and end points.

r is the radius vector, determines the position of the MT in space.

- Speed is the ratio of path to time .

- Path is the set of points through which the body has passed.

- Acceleration - the rate of change of rate, that is, the first derivative of the rate.

2. Curvilinear acceleration: normal and tangential acceleration. Flat rotation. Angular speed, acceleration.

Curvilinear motion is a movement whose trajectory is a curved line. An example of a curvilinear movement is the movement of the planets, the end of the clock hand on the dial, etc.

Curvilinear motion It's always fast paced. That is, acceleration during curvilinear motion is always present, even if the speed modulus does not change, but only the direction of speed changes.

Change in the value of speed per unit of time - is the tangential acceleration:

Where 𝛖 τ , 𝛖 0 are the speeds at time t 0 + Δt and t 0, respectively. Tangential acceleration at a given point of the trajectory, the direction coincides with the direction of the velocity of the body or is opposite to it.

Normal acceleration is the change in speed in direction per unit of time:

Normal acceleration directed along the radius of curvature of the trajectory (toward the axis of rotation). Normal acceleration is perpendicular to the direction of velocity.

Full acceleration with an equally variable curvilinear motion of the body is equal to:

-angular velocity shows at what angle the point rotates when moving uniformly around the circle per unit of time. The SI unit is rad/s.

Flat rotation is the rotation of all velocity vectors of the points of the body in one plane.

3. Connection between velocity vectors and angular velocity of a material point. Normal, tangential and full acceleration.

Tangential (tangential) acceleration is the component of the acceleration vector directed along the tangent to the trajectory at a given point in the trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.

Normal (centripetal) acceleration is a component of the acceleration vector directed along the normal to the motion trajectory at a given point on the body motion trajectory. That is, the normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in the direction and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory.

Full acceleration in curvilinear motion, it is composed of tangential and normal accelerations according to the vector addition rule and is determined by the formula.