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 above the 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,

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's consider the uniform motion of a point around 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 central angle, corresponding to the arc of a circle s, then, as is known, there is a relationship between R, α and s:

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 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.

Uniformly accelerated curvilinear motion

Curvilinear movements are movements whose trajectories are not straight, but curved lines. Planets and river waters move along curvilinear trajectories.

Curvilinear movement- this is always motion with acceleration, even if the absolute speed is constant. Curvilinear motion with constant acceleration always occurs in the plane in which the acceleration vectors and initial velocities of the point are located. In the case of curvilinear motion with constant acceleration in the xOy plane, the projections vx and vy of its velocity on the Ox and Oy axes and the x and y coordinates of the point at any time t are determined by the formulas

Uneven movement. Rough speed

No body moves at a constant speed all the time. When the car starts moving, it moves faster and faster. It can move steadily for a while, but then it slows down and stops. In this case, the car travels different distances in the same time.

Movement in which a body travels unequal lengths of path in equal intervals of time is called uneven. With such movement, the speed does not remain unchanged. In this case, we can only talk about average speed.

Average speed shows the distance a body travels per unit time. It is equal to the ratio of the displacement of the body to the time of movement. Average speed, like the speed of a body during uniform motion, is measured in meters divided by a second. In order to characterize motion more accurately, instantaneous speed is used in physics.

Body speed in this moment time or at a given point in the trajectory is called instantaneous speed. Instantaneous speed is a vector quantity and is directed in the same way as the displacement vector. You can measure instantaneous speed using a speedometer. In the International System, instantaneous speed is measured in meters divided by second.

point movement speed uneven

Movement of a body in a circle

Curvilinear motion is very common in nature and technology. It is more complex than a straight line, since there are many curved trajectories; this movement is always accelerated, even when the velocity module does not change.

But movement along any curved path can be approximately represented as movement along the arcs of a circle.

When a body moves in a circle, the direction of the velocity vector changes from point to point. Therefore, when they talk about the speed of such movement, they mean instantaneous speed. The velocity vector is directed tangentially to the circle, and the displacement vector is directed along the chords.

Uniform circular motion is a motion during which the modulus of the motion velocity does not change, only its direction changes. The acceleration of such motion is always directed towards the center of the circle and is called centripetal. In order to find the acceleration of a body moving in a circle, it is necessary to divide the square of the speed by the radius of the circle.

In addition to acceleration, the motion of a body in a circle is characterized by the following quantities:

The period of rotation of a body is the time during which the body makes one complete revolution. The rotation period is designated by the letter T and is measured in seconds.

The frequency of rotation of a body is the number of revolutions per unit time. Is the rotation speed indicated by a letter? and is measured in hertz. In order to find the frequency, you need to divide one by the period.

Linear speed is the ratio of the movement of a body to time. In order to find linear speed body along a circle, it is necessary to divide the circumference by the period (the circumference is equal to 2? multiplied by the radius).

Angular velocity - physical quantity, equal to the ratio of the angle of rotation of the radius of the circle along which the body moves to the time of movement. Angular velocity is indicated by a letter? and is measured in radians divided per second. Can you find the angular velocity by dividing 2? for a period of. Angular velocity and linear velocity among themselves. In order to find the linear speed, it is necessary to multiply the angular speed by the radius of the circle.

Figure 6. Circular motion, formulas.

Kinematics of a point. Path. Moving. Speed ​​and acceleration. Their projections onto the coordinate axes. Calculation of the distance traveled. Average values.

Kinematics of a point- a branch of kinematics that studies the mathematical description of the movement of material points. The main task of kinematics is to describe movement using a mathematical apparatus without identifying the reasons causing this movement.

Path and movement. The line along which a point on the body moves is called trajectory of movement. The path length is called the path traveled. The vector connecting the initial and end point trajectory is called moving. Speed- a vector physical quantity characterizing the speed of movement of a body, numerically equal to the ratio of movement over a short period of time to the value of this interval. A period of time is considered to be sufficiently small if the speed during uneven movement did not change during this period. The defining formula for speed is v = s/t. The unit of speed is m/s. In practice, the speed unit used is km/h (36 km/h = 10 m/s). Speed ​​is measured with a speedometer.

Acceleration- vector physical quantity characterizing the rate of change in speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred. If the speed changes equally throughout the entire movement, then the acceleration can be calculated using the formula a=Δv/Δt. Acceleration unit – m/s 2

Speed ​​and acceleration during curved motion. Tangential and normal accelerations.

Curvilinear movements– movements whose trajectories are not straight, but curved lines.

Curvilinear movement– this is always motion with acceleration, even if the absolute speed is constant. Curvilinear motion with constant acceleration always occurs in the plane in which the acceleration vectors and initial velocities of the point are located. In the case of curvilinear motion with constant acceleration in the plane xOy projections v x And v y its speed on the axis Ox And Oy and coordinates x And y points at any time t determined by formulas

v x =v 0 x +a x t, x=x 0 +v 0 x t+a x t+a x t 2 /2; v y =v 0 y +a y t, y=y 0 +v 0 y t+a y t 2 /2

A special case of curvilinear motion is circular motion. Circular motion, even uniform, is always accelerated motion: the velocity module is always directed tangentially to the trajectory, constantly changing direction, therefore circular motion always occurs with centripetal acceleration |a|=v 2 /r where r– radius of the circle.

The acceleration vector when moving in a circle is directed towards the center of the circle and perpendicular to the velocity vector.

In curvilinear motion, acceleration can be represented as the sum of the normal and tangential components: ,

Normal (centripetal) acceleration is directed towards the center of curvature of the trajectory and characterizes the change in speed in the direction:

v – instantaneous speed value, r– radius of curvature of the trajectory at a given point.

Tangential (tangential) acceleration is directed tangentially to the trajectory and characterizes the change in speed modulo.

The total acceleration with which a material point moves is equal to:

Tangential acceleration characterizes the speed of change in the speed of movement by numerical value and is directed tangentially to the trajectory.

Hence

Normal acceleration characterizes the rate of change in speed in direction. Let's calculate the vector:

4.Kinematics solid. Spinning around fixed axis. Angular velocity and acceleration. Relationship between angular and linear velocities and accelerations.

Kinematics of rotational motion.

The movement of the body can be either translational or rotational. In this case, the body is represented as a system of material points rigidly interconnected.

During translational motion, any straight line drawn in the body moves parallel to itself. According to the shape of the trajectory, the translational movement can be rectilinear or curvilinear. During translational motion, all points of a rigid body during the same period of time make movements equal in magnitude and direction. Consequently, the velocities and accelerations of all points of the body at any moment of time are also the same. To describe translational motion, it is enough to determine the movement of one point.

Rotational movement of a rigid body around a fixed axis is called such a movement in which all points of the body move in circles, the centers of which lie on the same straight line (axis of rotation).

The axis of rotation can pass through the body or lie outside it. If the axis of rotation passes through the body, then the points lying on the axis remain at rest when the body rotates. Points of a rigid body located at different distances from the axis of rotation in equal periods of time travel different distances and, therefore, have different linear velocities.

When a body rotates around a fixed axis, the points of the body undergo the same angular movement in the same period of time. The module is equal to the angle of rotation of the body around the axis in time , the direction of the angular displacement vector with the direction of rotation of the body is connected by the screw rule: if you combine the directions of rotation of the screw with the direction of rotation of the body, then the vector will coincide with the translational movement of the screw. The vector is directed along the axis of rotation.

The rate of change in angular displacement is determined by the angular velocity - ω. By analogy with linear speed, the concepts average and instantaneous angular velocity:

Angular velocity- vector quantity.

The rate of change in angular velocity is characterized by average and instantaneous

angular acceleration.

The vector and can coincide with the vector and be opposite to it

You are well aware that depending on the shape of the trajectory, movement is divided into rectilinear And curvilinear. WITH rectilinear movement we learned how to work in previous lessons, namely to solve the main problem of mechanics for this type of movement.

However, it is clear that in real world we most often deal with curvilinear movement, when the trajectory is a curved line. Examples of such movement are the trajectory of a body thrown at an angle to the horizon, the movement of the Earth around the Sun, and even the trajectory of the movement of your eyes, which are now following this note.

The question of how to solve the main task mechanics in the case of curvilinear motion, and this lesson will be devoted.

First, let's decide what fundamental differences does curvilinear movement (Fig. 1) have relative to rectilinear movement and what do these differences lead to.

Rice. 1. Trajectory of curvilinear movement

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

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 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 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 a body makes per unit time.

Frequency is indicated by a letter and measured in reciprocal seconds:

They are related by the relation:

There is a relationship between angular velocity and the frequency of rotation of the body. If we remember that 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.

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. Аyp.ru ().
2. Wikipedia ().

Homework

Having solved the problems for this lesson, you can prepare for questions 1 of the GIA and questions A1, A2 of the Unified State Exam.

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.