The force acting on the body can change its speed both in absolute value and in direction.
An example of strength changing speed modulo- the force of the wind pressing on the sail.
Such a force causes rectilinear body movement.
An example of a force changing speed in direction − centripetal force untwisted load on a rope
This force leads to curvilinear motion.
If the body moves in a circle with a constant speed in absolute value, then its acceleration is called centripetal, it is directed to the center of the circle and is calculated by the formula:
a = v 2 / r, where v is the speed, r is the radius of the circle
a=ω 2 * r, where w is the angular velocity of the body on the circle in radians per second.
In the general case, forces act on the body, changing the speed both in direction and in absolute value. An example is shown in the figure - the gravitational force simultaneously slows down the satellite and bends its trajectory:
In such cases, the force is said to have tangential and normal components. Tangential force component- this is the one that is directed along (or against) the speed and accelerates (or slows down) the body.
Normal component of force- this is the one that acts perpendicular to the movement and changes the direction of speed.
For a curved path at any point, you can calculate the radius of curvature using the formula:
R \u003d v 2 / a n, where v is the speed of the body, and a n is the normal (perpendicular to the speed) component of acceleration.
Edit this lesson and/or add an assignment Add your own lesson and/or assignmentDepending on the shape of the trajectory, the movement can be divided into rectilinear and curvilinear. Most often, you will encounter curvilinear movements when the path is represented as a curve. An example of this type of movement is the path of a body thrown at an angle to the horizon, the movement of the Earth around the Sun, planets, and so on.
Picture 1 . Trajectory and displacement in curvilinear motion
Definition 1Curvilinear motion called the movement, the trajectory of which is a curved line. If the body moves along a curved path, then the displacement vector s → is directed along the chord, as shown in Figure 1, and l is the length of the path. The direction of the instantaneous velocity of the body is tangential at the same point of the trajectory where the moving object is currently located, as shown in Figure 2.
Figure 2. Instantaneous speed in curvilinear motion
Definition 2
Curvilinear motion of a material point called uniform when the modulus of speed is constant (motion in a circle), and uniformly accelerated with a changing direction and modulus of speed (movement of a thrown body).
Curvilinear motion is always accelerated. This is explained by the fact that even with an unchanged speed modulus, but a changed direction, there is always an acceleration.
In order to investigate the curvilinear motion of a material point, two methods are used.
The path is divided into separate sections, on each of which it can be considered straight, as shown in Figure 3.
Figure 3. Splitting curvilinear motion into translational
Now for each section, you can apply the law of rectilinear motion. This principle is accepted.
The most convenient solution method is considered to be the representation of the path as a set of several movements along arcs of circles, as shown in Figure 4. The number of partitions will be much less than in the previous method, in addition, the movement around the circle is already curvilinear.
Figure 4. Partitioning of a curvilinear motion into motions along arcs of circles
Remark 1
To record a curvilinear movement, it is necessary to be able to describe movement along a circle, to represent an arbitrary movement in the form of sets of movements along the arcs of these circles.
The study of curvilinear motion includes the compilation of a kinematic equation that describes this motion and allows you to determine all the characteristics of the motion from the available initial conditions.
Example 1
Given a material point moving along a curve, as shown in Figure 4. The centers of the circles O 1 , O 2 , O 3 are located on one straight line. Need to find a move
s → and the length of the path l during the movement from point A to B.
Solution
By condition, we have that the centers of the circle belong to one straight line, hence:
s → = R 1 + 2 R 2 + R 3 .
Since the trajectory of motion is the sum of semicircles, then:
l ~ A B \u003d π R 1 + R 2 + R 3.
Answer: s → \u003d R 1 + 2 R 2 + R 3, l ~ A B \u003d π R 1 + R 2 + R 3.
Example 2
The dependence of the path traveled by the body on time is given, represented by the equation s (t) \u003d A + B t + C t 2 + D t 3 (C \u003d 0, 1 m / s 2, D \u003d 0, 003 m / s 3) . Calculate after what period of time after the start of movement the acceleration of the body will be equal to 2 m / s 2
Solution
Answer: t = 60 s.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Curvilinear motion- this is a movement whose trajectory is a curved line (for example, a circle, an ellipse, a hyperbola, a parabola). An example of a curvilinear movement is the movement of the planets, the end of the clock hand on the dial, etc. In general curvilinear speed changes in size and direction.
Curvilinear motion of a material point it is considered uniform motion if the modulus is constant (for example, uniform motion in a circle), and uniformly accelerated if the modulus and direction change (for example, the motion of a body thrown at an angle to the horizon).
Rice. 1.19. Trajectory and displacement vector in curvilinear motion.
When moving along a curved path, it is directed along the chord (Fig. 1.19), and l is the length. The instantaneous speed of the body (that is, the speed of the body at a given point in the trajectory) is directed tangentially at that point in the trajectory where the moving body is currently located (Fig. 1.20).
Rice. 1.20. Instantaneous velocity in curvilinear motion.
Curvilinear motion is always accelerated motion. That is curvilinear acceleration is always present, even if the modulus of the speed does not change, but only the direction of the speed changes. The change in speed per unit of time is:
Where v τ , v 0 are the speeds at time t 0 + Δt and t 0, respectively.
At a given point of the trajectory, the direction coincides with the direction of the velocity of the body or is opposite to it.
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.
centripetal acceleration is the normal acceleration for uniform circular motion.
Full acceleration with equally variable curvilinear motion of the body equals:
The movement of a body along a curvilinear trajectory can be approximately represented as movement along the arcs of some circles (Fig. 1.21).
Rice. 1.21. The movement of the body during curvilinear motion.
Kinematics studies the movements of bodies without considering the causes that determine this movement.
1. Material point – a body with a mass, the dimensions of which in this problem can be neglected.
A material point is an abstraction, but its introduction facilitates the solution of practical problems (for example, planets moving around the Sun can be taken as material points in calculations).
System reference- a set of reference body, coordinate system, device for measuring time.
The simplest types of mechanical motion of bodies are translational and rotational motion. body movement is called progressive , if all its points move in the same way.
Trajectory – the line along which the body moves (material point).
Path – trajectory length, scalar value.
moving is a directed line segment connecting starting position points with its end position. Vector value.
Types of mechanical movement (rectilinear and curvilinear).
Uniform rectilinear movement called the movement in which the body for any equal intervals of time makes the same movement.
speed uniform rectilinear motion of a body is called a value equal to the ratio of the movement of the body to the period of time during which this movement occurred:.
A movement in which the speed for any equal intervals of time changes in the same way is called uniform motion .
The value characterizes the rate of change of speed. It's called acceleration. A= .
acceleration of a moving body is called a value equal to the ratio of the change in the speed of the body to the time interval during which this change occurred.
| Uniformly accelerated motion | Uniformly slow motion |
 |  |
 | |
 |  |
At rotational When a body moves, its points describe concentric circles located in parallel planes.
Curvilinear motion - the movement of a body whose trajectory is a curved line.
Examples.
LECTURE 2
Dynamics
Dynamics studies the laws of motion of bodies and the causes that cause or change this motion.
1. Newton's first law .
There are such frames of reference with respect to which moving bodies keep their speed constant if no other bodies act on them (or the influence of other bodies is compensated). Such frames of reference are called inertial (ISO).
The concept of "force" means the measure of the impact of one body on another. Force is a vector quantity; it is characterized by module (absolute value), direction and point of application. The unit of force in SI is the force that imparts an acceleration of 1 m / s 2 to a body with a mass of 1 kg. This unit is called newton, 1H \u003d 1kg ∙1 m / s 2 \u003d 1 kg ∙ m / s 2.
Newton's second law .
The acceleration received by the body is directly proportional to the resultant of all forces and inversely proportional to the mass of the body:
Let's write down the consequences of Newton's second law:
a) if equal forces act on two bodies of different masses, then the ratio of the accelerations of the bodies is inversely proportional to their masses. Let's conclude:
Since , then , or modulo ,
b) if equal forces act on two bodies of different masses, then the accelerations acquired by the bodies are directly proportional to the acting forces. Let's conclude:
; since then
Newton's third law is formulated as follows: when two bodies interact, they act on each other with forces directed along one straight line, equal in magnitude and opposite in direction:
These forces are applied to different bodies interacting with each other.
2. Law gravity was discovered by Isaac Newton; the law is formulated as follows: the force of mutual attraction of two bodies is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them:
Where G= 6.67∙10 -11 - gravitational constant.
physical meaning gravitational constant is as follows: it shows that two bodies with masses of 1 kg, located at a distance of 1 m from each other, are attracted with a force of 6.67∙10 -11 H.
Example : gravity. If only the force of gravity acts on the body, then it makes a free fall.
body weight called the force with which the body, being attracted to the Earth, acts on a horizontal support or on a suspension.
In general, in all inertial frames of reference, the weight of the body is equal in absolute value to the force of gravity. Body weight is the force applied to the support, and the force of gravity is applied to the body.
In the course of the lesson, we will look at curvilinear motion, circular motion, and some other examples. We will also discuss cases in which it is necessary to apply various models for describing the motion of a body.
Do straight lines really exist? They seem to be all around us. But let's take a closer look at the edge of the table, the case or the screen of the monitor: they always have a recess, a roughness of the material. Let's look through a microscope, and doubts about the curvature of these lines will disappear.
It turns out that a straight line is really an abstraction, something ideal and non-existent. But with the help of this abstraction, it is possible to describe many real objects, if, when considering them, their small irregularities are not important to us and we can consider them straight lines.
We have considered the simplest motion - uniform rectilinear motion. This is the same idealization as the straight line itself. Real objects move in the real world, and their trajectory cannot be perfectly straight. The car moves from city A to city B: there can be no absolutely even road between cities and it will not be possible to maintain a constant speed. Nevertheless, using the model of uniform rectilinear motion, we can describe even such a motion.
This model for describing movement is not always applicable.
1) Movement may be uneven.
2) For example, a carousel is spinning - there is movement, but not in a straight line. The same can be said about the ball that the player hits. Or about the movement of the moon around the earth. In these examples, the movement occurs along a curved path.
So, since there are such problems, we need a convenient tool to describe the movement along the curve.
Driving in a straight line and along a curve
We can consider the same trajectory of motion to be straight in one problem, but not in another. This is a convention, depends on what we are interested in in this problem.
If the problem is about a car that travels from Moscow to St. Petersburg, then the road is not straight, but at such distances we are not interested in all these turns - what happens on them is negligible. Moreover, we are talking about the average speed, which takes into account all these hitches in the corners, because of them, the average speed will simply become less. Therefore, we can move on to an equivalent problem - we can "straighten" the trajectory, while maintaining the length and speed - we get the same result. So, the model of rectilinear motion is suitable here. If the task is about the movement of the car at a particular turn or during overtaking, then the curvature of the trajectory may be important to us and we will use a different model.
Let us divide the motion along the curve into segments small enough to consider them as straight segments. Let's imagine a pedestrian who moves along a complex trajectory, avoids obstacles, but he walks and takes steps. There are no curvilinear steps, these are segments from footprint to print.
Rice. 1. Curvilinear trajectory
We have divided the movement into small segments, and we can describe the movement on each such segment as rectilinear. The shorter these straight segments are, the more accurate the approximations will be.
Rice. 2. Curvilinear motion approximation
We used such a mathematical tool as splitting into small intervals when we found the displacement during rectilinear uniformly accelerated motion: we divided the motion into sections so small that the change in speed in this section was insignificant and the movement could be considered uniform. It was easy to calculate the displacement in each such section, then it remained to add the displacement in each section and get the total.
Rice. 3. Movement with rectilinear uniformly accelerated motion
Let's start describing the curvilinear motion from the simplest model - a circle, which is described by one parameter - the radius.
Rice. 4. Circle as a model of curvilinear motion
The end of the clock hand moves at the same distance of the length of the hand from the point of its attachment. The points of the wheel rim all the time remain at the same distance from the axis - at a distance of the length of the spoke. We continue to study the motion of a material point and work within the framework of this model.
Translational and rotational motion
Translational motion is a motion in which all points of the body move in the same way: at the same speed, making the same movement. Wave your hand and follow: it is clear that the palm and shoulder moved differently. Look at the Ferris wheel: points near the axis hardly move, and the cabins move at a different speed and along different trajectories. Look at a car moving in a straight line: if you do not take into account the rotation of the wheels and the movement of parts of the motor, all points of the car move in the same way, we consider the movement of the car to be translational. Then it makes no sense to describe the movement of each point, you can describe the movement of one. The car is considered a material point. Please note that during translational movement, the line connecting any two points of the body during movement remains parallel to itself.
The second type of movement according to this classification is rotational movement. With rotational motion, all points of the body move in a circle around a single axis. This axis may cross the body, as in the case of a Ferris wheel, or it may not cross, as in the case of a car on a bend.
Rice. 5. Rotary movement
But not every movement can be attributed to any one of the two types. How to describe the movement of the pedals of a bicycle relative to the Earth - is this some kind of third type? Our model is convenient in that we can consider movement as a combination of translational and rotational motions: the pedals rotate about their axis, and the axis, together with the whole bike, moves translationally relative to the Earth.
The end of the clock hand for equal time intervals will cover the same path. That is, we can talk about the uniformity of its movement. Velocity is a vector quantity, therefore, in order for it to be constant, both its modulus and direction must not change. And if the modulus of speed does not change when moving along a circle, then the direction will change constantly.
Let's consider a uniform motion in a circle.
Why choose not to consider relocation
Consider how displacement changes when moving along a circle. The point was in one place (see Fig. 6) and passed a quarter of the circle.
Let's follow the movement during further movement - it is difficult to describe the pattern by which it changes, and such a consideration is uninformative. It makes sense to consider displacement over intervals small enough to be considered approximately equal.
Let us introduce some convenient characteristics of circular motion.
Whatever the size of the watch, in 15 minutes the end of the minute hand will always pass a quarter of the circumference of the dial. And in an hour it will make a complete revolution. In this case, the path will depend on the radius of the circle, but the angle of rotation will not. That is, the angle will also change uniformly. Therefore, in addition to the distance traveled, we will also talk about changing the angle. As we know, an angle is proportional to the arc on which it rests:
Rice. 7. Changing the angle of the arrow
Since the angle changes uniformly, it is possible, by analogy with the ground speed, showing the path that the body travels per unit time, to enter the angular velocity: the angle by which the body turns (or which the body passes) per unit time, .
That is, how many radians does a point rotate in a second. It will be measured, respectively, in rad / s.
Uniform movement in a circle is a repetitive process, or, in other words, periodic. When the point makes a full turn, it is again in its original position and the movement is repeated.
Examples of periodic phenomena in nature
Many phenomena are periodic in nature: the change of day and night, the change of seasons. Here it is clear what exactly is a period: a day and a year, respectively.
There are other periods: spatial (a pattern with periodically repeating elements, a series of trees spaced at equal intervals), periods in the recording of numbers. Periods in music, poetry.
Periodic phenomena are described by what happens in a period and the length of that period. For example, the daily cycle - sunrise-sunset and the period - the time for which everything repeats - 24 hours. Spatial pattern - a single element of the pattern and how often it repeats (or its length). In decimal representation of an ordinary fraction, this is a sequence of digits in a period (what is in brackets) and length / period is the number of digits: in 1/3 - one digit, in 1/17 - 16 digits.
Let's look at some time periods.
The period of revolution of the Earth on its axis = day + night = 24 hours.
The period of revolution of the Earth around the Sun = 365 periods of revolution day + night.
The period of revolution of the hour hand on the dial is 12 hours, the minute hand is 1 hour.
The period of oscillation of the clock pendulum is 1 s.
The period is measured in generally accepted units of time (SI second, minute, hour, etc.).
The period of the pattern is measured in units of length (m, cm), the period in decimal fraction - in the number of digits in the period.
Period is the time it takes for a point to move around a circle one complete revolution. Let's denote it capital letter.
If revolutions are made in time, then one revolution is made, obviously, in time.
To judge how often the process repeats, we introduce a value that we will call frequency.
The frequency of the appearance of the Sun in a year is 365 times. The frequency of the appearance of the full moon per year is 12, sometimes 13 times. The frequency of the arrival of spring per year is 1 time.
For uniform motion in a circle, the frequency is the number of complete revolutions that a point makes per unit of time. If revolutions are made in t seconds, then revolutions are made in every second. We denote the frequency, sometimes it is also denoted or. The frequency is measured in revolutions per second, this value was called hertz, after the name of the scientist Hertz.
Frequency and period are mutually inverse: the more often something happens, the shorter the period should be. And vice versa: the longer one period lasts, the less often the event occurs.
Mathematically, we can write the inverse proportionality: or.
So, the period is the time for which the body makes a complete revolution. It is clear that it must be related to the angular velocity: the faster the angle changes, the faster the body will return to the starting point, that is, it will make a complete revolution.
Consider one full turn. Angular velocity is the angle that a body rotates per unit of time. Through what angle should the body turn during a full rotation? 3600, or in radians. The time for a complete revolution is a period. So, by definition, the angular velocity is: .
We will also find the ground speed - it is also called linear - by considering one revolution. Point in time, one period, the body makes a complete revolution, that is, it travels a path, equal to the length circles. From here we express the speed by definition as the path divided by the time: .
If we take into account that - this is the angular velocity, then we get the relationship between the linear and angular velocity:
Task
With what frequency should the gate of the well be rotated so that the bucket rises at a speed of 1 m / s, if the radius of the section of the gate is equal to?
The task describes the rotation of the gate - we apply the model of rotational movement to it, considering the points of its surface.
Rice. 8. Gate rotation model
It is also about the movement of the bucket. The bucket is attached to the collar with a rope, and this rope is wound. This means that any part of the rope, including the one wound around the collar, moves at the same speed as the bucket. Thus, we have given the linear speed of the points on the surface of the gate.
The physical part of the solution. We are talking about the linear speed of movement in a circle, it is equal to:.
The period and frequency are mutually reciprocal quantities, we write: .
We got a system of equations, which remains only to be solved - this will be the mathematical part of the solution. Substitute the frequency in the first equation instead of: 
.
Let's express the frequency from here: .
Calculate by converting the radius to meters:
We got the answer: you need to rotate the gate with a frequency of 1.06 Hz, that is, make approximately one revolution in one second.
Imagine that we have two identical bodies moving. One is in a circle, and the other (under the same conditions and with the same characteristics), but in a regular polygon. The more sides such a polygon has, the less the movements of these two bodies will differ for us.
Rice. 9. Curvilinear motion along a circle and along a polygon
The difference is that the second body on each section (side of the polygon) moves in a straight line.
On each such segment we denote the displacement of the body . The displacement here is a two-dimensional vector , on a plane.
Rice. 10. Moving a body during curvilinear motion along a polygon
On this small section, the movement is completed in time . Divide and get the velocity vector in this section.
With an increase in the number of sides of a polygon, the length of its side will decrease: . Since the modulus of the body's velocity is constant, then the time to overcome this segment will tend to 0: .
Accordingly, the speed of the body in such a small area will be called instant speed.
The smaller the side of the polygon, the closer it will be to the tangent to the circle. Therefore, in the limiting, ideal case () we can assume that the instantaneous speed at a given point is directed tangentially to the circle.
And the sum of the displacement modules will be less and less different from the path that the point passes along the arc. Therefore, the modulo instantaneous speed will coincide with the ground speed, and all the relationships that we obtained earlier will be true for the instantaneous velocity modulo displacement. You can even designate it, meaning.
The speed is directed tangentially, we can also find its modulus. Let's find the speed at another point. Its modulus is the same, since the motion is uniform, and it is directed tangentially to the circle already at this point.
Rice. 11. Tangential body speed
This is not the same vector, they are equal in absolute value, but they have a different direction, . The speed has changed, and since it has changed, then you can calculate this change:
The change in speed per unit of time, by definition, is acceleration:
Calculate the acceleration when moving in a circle. Change of speed .
Rice. 12. Graphical subtraction of vectors
Got a vector. The acceleration is directed in the same direction (these vectors are related by the relation 
, which means they are codirectional).
The smaller the section AB, the more the velocity vectors and will coincide, and will be closer and closer to the perpendicular to both of them.
Rice. 13. Dependence of speed on the size of the area
That is, it will lie along the perpendicular to the tangent (the velocity is directed along the tangent), which means that the acceleration will be directed towards the center of the circle, along the radius. Remember from the math course: the radius drawn to the point of contact is perpendicular to the tangent.
When the body passes a small angle , the velocity vector, which is tangent to the radius, also rotates through the angle .
Proof of equality of angles
Consider the quadrilateral ACBO. The sum of the angles of a quadrilateral is 360°. (as the angles between the radii drawn at the tangent points and the tangents).
The angle between the directions of speed at points A and B () and - adjacent with a straight line AC, then 
,
Previously received, from here.
On a small section AB, the modulo movement of a point practically coincides with the path, that is, with the length of the arc: .
The ABO triangles and the triangle formed by the velocity vectors at points A and B are similar (from point A the vector has been moved parallel to itself to point B).
These triangles are isosceles (OA = OB - radii, - since the movement is uniform), they have equal angles between the sides (just proved in the branch). This means that the angles equal to each other at the base will be equal. The equality of the angles is sufficient to assert that the triangles are similar.
From the similarity of triangles, we write: side AB (and it is equal to ) refers to the radius of the circle as the modulus of change of speed refers to the modulus of speed: .
We write without vectors, because we are interested in the lengths of the sides of triangles. We all lead to acceleration, it is associated with a change in speed, or. Substitute, we get: .
The derivation of the formula turned out to be quite complicated, but you can remember the finished result and use it when solving problems.
At whatever point we find the acceleration with uniform motion around the circle, it is equal in absolute value and at any point is directed towards the center of the circle. Therefore it is also called centripetal acceleration.
Problem 2. Centripetal acceleration
Let's solve the problem.
Find the speed at which the car moves on the turn, if we consider the turn to be part of a circle with a radius of 40 m, and centripetal acceleration equals .
Condition analysis. The problem describes the movement in a circle, we are talking about centripetal acceleration. We write the formula for centripetal acceleration:
The acceleration and radius of the circle are given, it remains only to express and calculate the speed:
Or, if converted to km / h, then this is about 32 km / h.
In order to change the speed of a body, another body with some force must act on it, or, to put it more simply, a force must act. In order for a body to move in a circle with centripetal acceleration, a force must also act on it, which creates this acceleration. In the case of a car on a curve, this is the friction force, so we skid when cornering when there is ice on the roads. If we spin something on a rope, this is the tension in the rope - and we feel how it is pulled tighter. As soon as this force disappears, for example, the thread breaks, the body, in the absence of inertia forces, maintains speed - that speed directed tangentially to the circle, which was at the moment of separation. And this can be seen by following the direction of movement of this body (figure). For the same reason, we are pressed against the wall of the transport at the turn: we move by inertia in such a way as to maintain speed, we are, as it were, thrown out of the circle until we hit the wall and a force arises that imparts centripetal acceleration.
Previously, we had only one tool - the rectilinear motion model. We were able to describe another model - movements in a circle.
This is a common type of movement (turns, vehicle wheels, planets, etc.), so a separate tool was needed (it is not very convenient to approximate the trajectory with small straight segments each time).
Now we have two "bricks", which means that with their help we will be able to build buildings of a more complex shape - to solve more complex problems with combined types of movements.
These two models will be enough for us to solve most kinematic problems.
For example, such a movement can be represented as movement along the arcs of three circles. Or this example: a car drove straight down the street and accelerated, then turned and drove at a constant speed along another street.
Rice. 14. Division into sections of the trajectory of the car
We will consider three sections and apply one of the simple models to each.
Bibliography
- Sokolovich Yu.A., Bogdanova G.S. Physics: a reference book with examples of problem solving. - 2nd ed., redistribution. - X .: Vesta: publishing house "Ranok", 2005. - 464 p.
- Peryshkin A.V., Gutnik E.M. Physics. Grade 9: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300 .
- Web site " extracurricular lesson» ()
- Internet site "Class! Physics" ()
Homework
- Give examples of curvilinear motion in Everyday life. Can this movement be rectilinear in any construction of the condition?
- Determine the centripetal acceleration with which the Earth moves around the Sun.
- Two cyclists with constant speeds start simultaneously in the same direction from two diametrically opposite points of the circular track. 10 minutes after the start, one of the cyclists caught up with the other for the first time. How long after the start will the first cyclist overtake the other for the second time?