The concept of a material point. Trajectory. Path and movement. Reference system. Velocity and acceleration in curvilinear motion. Normal and tangential accelerations. Classification of mechanical movements.
The subject of mechanics . Mechanics is a branch of physics devoted to the study of the laws of the simplest form of motion of matter - mechanical motion.
Mechanics consists of three subsections: kinematics, dynamics and statics.
Kinematics studies the motion of bodies without taking into account the causes that cause it. It operates with such quantities as displacement, distance traveled, time, speed and acceleration.
Dynamics explores the laws and causes that cause the movement of bodies, i.e. studies the motion of material bodies under the action of forces applied to them. To the kinematic quantities are added quantities - force and mass.
ATstatic investigate the equilibrium conditions for a system of bodies.
Mechanical movement a body is the change in its position in space relative to other bodies over time.
Material point - a body, the size and shape of which can be neglected under the given conditions of motion, considering the mass of the body concentrated at a given point. The material point model is the simplest model of body motion in physics. A body can be considered a material point when its dimensions are much smaller than the characteristic distances in the problem.
To describe the mechanical movement, it is necessary to indicate the body relative to which the movement is considered. An arbitrarily chosen motionless body, in relation to which the motion of this body is considered, is called reference body .
Reference system - the reference body together with the coordinate system and clock associated with it.
Consider the motion of a material point M in a rectangular coordinate system, placing the origin at point O.
The position of the point M relative to the reference system can be set not only with the help of three Cartesian coordinates, but also with the help of one vector quantity - the radius vector of the point M drawn to this point from the origin of the coordinate system (Fig. 1.1). If are unit vectors (orts) of the axes of a rectangular Cartesian coordinate system, then
or the time dependence of the radius vector of this point
Three scalar equations (1.2) or one vector equation (1.3) equivalent to them are called kinematic equations of motion of a material point .
trajectory a material point is a line described in space by this point during its movement (the locus of the ends of the radius vector of the particle). Depending on the shape of the trajectory, rectilinear and curvilinear motions of a point are distinguished. If all parts of the trajectory of the point lie in the same plane, then the movement of the point is called flat.
Equations (1.2) and (1.3) define the trajectory of a point in the so-called parametric form. The role of the parameter is played by time t. Solving these equations jointly and excluding the time t from them, we find the trajectory equation.
long way material point is the sum of the lengths of all sections of the trajectory traversed by the point during the considered period of time.
Displacement vector material point is a vector connecting the initial and final position of the material point, i.e. increment of the radius-vector of a point for the considered time interval
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With rectilinear motion, the displacement vector coincides with the corresponding section of the trajectory. From the fact that displacement is a vector, the law of independence of motions, confirmed by experience, follows: if a material point participates in several motions, then the resulting displacement of the point is equal to the vector sum of its displacements performed by it for the same time in each of the movements separately
To characterize the movement of a material point, a vector physical quantity is introduced - speed , a quantity that determines both the speed of movement and the direction of movement at a given time.
Let a material point move along a curvilinear trajectory MN so that at time t it is at point M, and at time at point N. The radius vectors of points M and N, respectively, are equal, and the length of the arc MN is (Fig. 1.3 ).
Average speed vector points in the time interval from t before t+Δ t called the ratio of the increment of the radius-vector of a point over this period of time to its value:
The average velocity vector is directed in the same way as the displacement vector i.e. along the chord MN.
Instantaneous speed or speed at a given time . If in expression (1.5) we pass to the limit, tending to zero, then we will obtain an expression for the velocity vector of the m.t. at the time t of its passage through the t.M trajectory.
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In the process of decreasing the value, the point N approaches t.M, and the chord MN, turning around t.M, in the limit coincides in direction with the tangent to the trajectory at the point M. Therefore, the vectorand speedvmoving point directed along a tangent trajectory in the direction of motion. The velocity vector v of a material point can be decomposed into three components directed along the axes of a rectangular Cartesian coordinate system.
From a comparison of expressions (1.7) and (1.8), it follows that the projections of the velocity of a material point on the axes of a rectangular Cartesian coordinate system are equal to the first time derivatives of the corresponding coordinates of the point:
A movement in which the direction of the velocity of a material point does not change is called rectilinear. If the numerical value of the instantaneous velocity of a point remains unchanged during the movement, then such movement is called uniform.
If, in arbitrary equal time intervals, a point passes paths of different lengths, then the numerical value of its instantaneous velocity changes over time. Such movement is called uneven.
In this case, a scalar value is often used, called the average ground speed of uneven movement in a given section of the trajectory. It is equal to the numerical value of the speed of such a uniform movement, at which the same time is spent on the passage of the path, as with a given uneven movement:
Because only in the case of rectilinear motion with a constant speed in the direction, then in the general case:
The value of the path traveled by a point can be represented graphically by the area of the figure of a bounded curve v = f (t), direct t = t 1 and t = t 1 and the time axis on the speed graph.
The law of addition of speeds . If a material point simultaneously participates in several movements, then the resulting displacement, in accordance with the law of independence of motion, is equal to the vector (geometric) sum of elementary displacements due to each of these movements separately:
According to definition (1.6):
Thus, the speed of the resulting movement is equal to the geometric sum of the velocities of all movements in which the material point participates (this provision is called the law of addition of velocities).
When a point moves, the instantaneous speed can change both in magnitude and in direction. Acceleration characterizes the rate of change in the module and direction of the velocity vector, i.e. change in the magnitude of the velocity vector per unit of time.
Mean acceleration vector . The ratio of the speed increment to the time interval during which this increment occurred expresses the average acceleration:
The vector of the average acceleration coincides in direction with the vector .
Acceleration, or instantaneous acceleration is equal to the limit of the average acceleration when the time interval tends to zero:
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In projections onto the corresponding coordinates of the axis:
In rectilinear motion, the velocity and acceleration vectors coincide with the direction of the trajectory. Consider the motion of a material point along a curvilinear plane trajectory. The velocity vector at any point of the trajectory is directed tangentially to it. Let's assume that in t.M of the trajectory the speed was , and in t.M 1 it became . At the same time, we assume that the time interval during the transition of a point on the way from M to M 1 is so small that the change in acceleration in magnitude and direction can be neglected. In order to find the velocity change vector , it is necessary to determine the vector difference:
To do this, we move it parallel to itself, aligning its beginning with the point M. The difference of two vectors is equal to the vector connecting their ends is equal to the side of the AC MAC, built on the velocity vectors, as on the sides. We decompose the vector into two components AB and AD, and both, respectively, through and . Thus, the velocity change vector is equal to the vector sum of two vectors:
Thus, the acceleration of a material point can be represented as the vector sum of the normal and tangential accelerations of this point 
A-priory:
where - ground speed along the trajectory, coinciding with the absolute value of the instantaneous speed at a given moment. The vector of tangential acceleration is directed tangentially to the trajectory of the body.
Everything in the world around us is in constant motion. Movement in the general sense of the word refers to any changes that occur in nature. The simplest type of movement is mechanical movement.
From the 7th grade physics course, you know that the mechanical movement of a body is a change in its position in space relative to other bodies that occurs over time.
When solving various scientific and practical problems related to the mechanical movement of bodies, one must be able to describe this movement, i.e., determine the trajectory, speed, distance traveled, body position and some other characteristics of movement for any moment of time.
For example, when launching an aircraft from the Earth to another planet, scientists must first calculate where this planet is located relative to the Earth at the time the apparatus landed on it. And for this it is necessary to find out how the direction and module of the speed of this planet change over time and along what trajectory it moves.
From the course of mathematics, you know that the position of a point can be set using a coordinate line or a rectangular coordinate system (Fig. 1). But how to set the position of a body that has dimensions? After all, each point of this body will have its own coordinate.
Rice. 1. The position of the point can be set using the coordinate line or rectangular coordinate system
When describing the motion of a body that has dimensions, other questions arise. For example, what should be understood by the speed of a body if, while moving in space, it simultaneously rotates around its own axis? After all, the speed of different points of this body will be different both in absolute value and in direction. For example, during the daily rotation of the Earth, its diametrically opposite points move in opposite directions, and the closer the point is to the axis, the lower its speed.
How can you set the coordinate, speed and other characteristics of the movement of a body that has dimensions? It turns out that in many cases, instead of the motion of a real body, one can consider the motion of a so-called material point, i.e., a point that has the mass of this body.
For a material point, one can unambiguously determine the coordinate, speed and other physical quantities, since it has no dimensions and cannot rotate around its own axis.
Material points do not exist in nature. A material point is a concept, the use of which simplifies the solution of many problems and at the same time allows obtaining sufficiently accurate results.
- A material point is a concept introduced in mechanics to designate a body, which is considered as a point having a mass
Almost any body can be considered as a material point in cases where the distances traveled by the points of the body are very large compared to its dimensions.
For example, the Earth and other planets are considered material points when studying their movement around the Sun. In this case, the differences in the movement of different points of any planet, caused by its daily rotation, do not affect the quantities describing the annual movement.
Planets are considered material points when studying their movement around the Sun.
But when solving problems related to the daily rotation of the planets (for example, when determining the time of sunrise in different places on the surface of the globe), it makes no sense to consider a planet as a material point, since the result of the problem depends on the size of this planet and the speed of movement of points on its surface. So, for example, in the Vladimir time zone the sun will rise 1 hour later, in Irkutsk - 2 hours later, and in Moscow - 8 hours later than in Magadan.
It is legitimate to take an airplane as a material point if, for example, it is required to determine the average speed of its movement on the way from Moscow to Novosibirsk. But when calculating the air resistance force acting on a flying aircraft, it cannot be considered a material point, since the resistance force depends on the shape and speed of the aircraft.
For a material point, you can take an airplane flying from one city to another
A body moving forward 1 can be taken as a material point even if its dimensions are commensurate with the distances it travels. For example, a person standing on the step of a moving escalator is moving progressively (Fig. 2, a). At any moment in time, all points of the human body move in the same way. Therefore, if we want to describe the movement of a person (i.e., to determine how his speed, path, etc., changes over time), then it is enough to consider the movement of only one of his points. In this case, the solution of the problem is greatly simplified.
With a rectilinear motion of a body, one coordinate axis is sufficient to determine its position.
For example, the position of a trolley with a dropper (Fig. 2, b) moving along the table in a straight line and translationally can be determined at any time using a ruler located along the trajectory of movement (the trolley with a dropper is taken as a material point). In this experiment, it is convenient to take the ruler as the reference body, and its scale can serve as the coordinate axis. (Recall that the body of reference is the body, relative to which the change in the position of other bodies in space is considered.) The position of the cart with the dropper will be determined relative to the zero division of the ruler.
Rice. 2. With the translational motion of the body, all its points move in the same way
But if it is necessary to determine, for example, the path that the cart has traveled for a certain period of time, or the speed of its movement, then in addition to the ruler, you will need a device for measuring time - a clock.
In this case, the role of such a device is performed by a dropper, from which drops fall at regular intervals. By turning the faucet, drops can be made to fall at intervals of, for example, 1 s. By counting the number of gaps between traces of drops on the ruler, you can determine the corresponding time interval.
It is clear from the above examples that in order to determine the position of a moving body at any moment of time, the type of movement, the speed of the body and some other characteristics of the movement, a reference body, an associated coordinate system (or one coordinate axis if the body moves along a straight line) and a device for time measurements.
- The coordinate system, the body of reference with which it is associated, and the device for measuring time form a reference system, relative to which the movement of the body is considered
Of course, in many cases it is impossible to directly measure the coordinates of a moving body at any given time. We do not have a real opportunity, for example, to place a measuring tape and place observers with watches along a many-kilometer path of a moving car, a liner floating on the ocean, a flying aircraft, a projectile fired from an artillery gun, various celestial bodies whose movement we are observing, etc. .
Nevertheless, knowledge of the laws of physics makes it possible to determine the coordinates of bodies moving in different frames of reference, in particular, in the frame of reference associated with the Earth.
Questions
- What is a material point?
- What is the purpose of the term "material point"?
- In what cases is a moving body usually considered as a material point?
- Give an example showing that the same body in one situation can be considered a material point, but not in another.
- In what case can the position of a moving body be set using one coordinate axis?
- What is a reference system?
Exercise 1
- Is it possible to consider a car as a material point when determining the path that it traveled in 2 hours, moving at an average speed of 80 km / h; when overtaking another car?
- The plane makes a flight from Moscow to Vladivostok. Can the dispatcher, observing its movement, consider the aircraft as a material point; a passenger on this plane?
- When talking about the speed of a car, train and other vehicles, the reference body is usually not indicated. What is meant in this case by the reference body?
- The boy stood on the ground and watched his little sister ride the carousel. After the ride, the girl told her brother that he himself, and the houses, and the trees quickly rushed past her. The boy began to assert that he, along with the houses and trees, was motionless, and his sister was moving. With respect to what reference bodies did the girl and the boy consider the motion? Explain who is right in the dispute.
- Relative to what reference body is the movement considered when they say: a) the wind speed is 5 m/s; b) the log floats down the river, so its speed is zero; c) the speed of a tree floating along the river is equal to the speed of the flow of water in the river; d) any point of the wheel of a moving bicycle describes a circle; e) the sun rises in the east in the morning, moves across the sky during the day, and sets in the west in the evening?
1 Translational motion - the movement of a body in which a straight line connecting any two points of this body moves, remaining all the time parallel to its original direction. Translational motion can be both rectilinear and curvilinear. For example, the cabin of a Ferris wheel is moving forward.
Material point. Reference system.
The mechanical motion of a body is the change over time of its position relative to other bodies.
Almost all physical phenomena are accompanied by the movement of bodies. In physics, there is a special branch that studies motion - this is Mechanics.
The word "mechanics" comes from the Greek "mechane" - a machine, a device.
Under the action of various machines and mechanisms, their parts move: levers, ropes, wheels, ... Mechanics also includes finding the conditions under which the body is at rest - the conditions for the equilibrium of bodies. These issues play a huge role in the construction business. Not only material bodies can move, but also a sunbeam, a shadow, light signals, radio signals.
To study movement, one must be able to describe movement. We are not interested in how this movement arose, we are interested in the process itself. The branch of mechanics that studies motion without examining the cause that causes it is called kinematics.
The movement of each body can be considered in relation to different bodies, and relative to them, this body will make different movements: a suitcase lying in a car on the shelf of a moving train, relative to the car, it is at rest, and relative to the Earth, it moves. A balloon carried away by the wind - relative to the Earth - moves, and relative to the air - it is at rest. An aircraft flying in squadrons is at rest relative to other aircraft in formation, but moves at high speed relative to the Earth.
Therefore, any movement, as well as the rest of the body, are relative.
When answering the question whether a body is moving or at rest, we must indicate in relation to what we are considering the movement.
The body relative to which the given motion is considered is called the body of reference.
A reference body is associated with a coordinate system and a device for measuring time. This whole set forms reference system .
What does it mean to describe movement? This means that you need to define:
1. trajectory, 2. speed, 3. path, 4. body position.
The point is very simple. From the course of mathematics it is known that the position of a point can be specified using coordinates. And if we have a body that has a size? Each point will have its own coordinates. In many cases, when considering the motion of a body, the body can be taken as a material point, or a point that has the mass of this body. And for a point, you can uniquely determine the coordinates.
So, a material point is an abstract concept that is introduced to simplify the solution of problems.
The condition under which the body can be taken as a material point:
It is often possible to take a body as a material point, and provided that its dimensions are comparable to the distance traveled, when at any moment in time all points move in the same way. This type of movement is called progressive.
A sign of forward movement is the condition that a straight line, mentally drawn through any two points of the body, remains parallel to itself.
Example: a person moves on an escalator, a needle in a sewing machine, a piston in an internal combustion engine, a car body when driving on a straight road.
Different movements differ from each other in the form of the trajectory.
If the trajectory straight line- then rectilinear movement if the trajectory is curved line, then the movement is curvilinear.
Move.
Path and movement: what's the difference?
S=AB+BC+CD
A displacement is a vector (or directional line) connecting the initial position to its subsequent position.
Displacement is a vector quantity, which means it is characterized by two quantities: a numerical value or module and direction.
It is designated - S, and is measured in meters, (km, cm, mm).
If you know the displacement vector, then you can uniquely determine the position of the body.
Vectors and actions with vectors.
VECTOR DEFINITION
Vector a directed segment is called, that is, a segment for which the beginning (also called the point of application of the vector) and end are indicated.
VECTOR MODULE
The length of a directed segment representing a vector is called the length, or module, vector. The length of a vector is denoted by .
NULL VECTOR
Null vector() - a vector whose beginning and end coincide; its modulus is 0 and its direction is indefinite.
COORDINATE REPRESENTATION
Let the Cartesian coordinate system XOY be given on the plane.
Then the vector can be given by two numbers:
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These numbers https://pandia.ru/text/78/050/images/image012_18.gif" width="20" height="25 src="> in geometry are called vector coordinates, and in physics vector projections to the corresponding coordinate axes.
To find the projection of a vector, it is necessary: from the beginning and end of the vector, lower the perpendiculars on the coordinate axes.
Then the projection will be the length of the segment enclosed between the perpendiculars.
The projection can take both positive and negative values.
If the projection turned out with the “-” sign, then the vector is directed in the opposite direction of the axis on which it was projected.
With this definition of the vector, its module, a direction is given by the angle a, which is uniquely determined by the relations:
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COLLINEAR VECTORS
D) a chess piece
E) a chandelier in the room,
G) submarine,
Y) aircraft on the runway.
8. Do we pay for the journey or transportation in a taxi ride?
9. The boat passed along the lake in the north-east direction for 2 km, and then in the north direction for another 1 km. Find the geometric construction of displacement and its modulus.
Definition
A material point is a macroscopic body whose dimensions, shape, rotation and internal structure can be neglected when describing its motion.
The question of whether a given body can be considered as a material point does not depend on the size of this body, but on the conditions of the problem being solved. For example, the radius of the Earth is much less than the distance from the Earth to the Sun, and its orbital motion can be well described as the motion of a material point with a mass equal to the mass of the Earth and located in its center. However, when considering the daily motion of the Earth around its own axis, replacing it with a material point does not make sense. The applicability of the material point model to a specific body depends not so much on the size of the body itself, but on the conditions of its motion. In particular, in accordance with the theorem on the motion of the center of mass of a system during translational motion, any rigid body can be considered a material point, the position of which coincides with the center of mass of the body.
The mass, position, speed and some other physical properties of a material point at any particular moment of time completely determine its behavior.
The position of a material point in space is defined as the position of a geometric point. In classical mechanics, the mass of a material point is assumed to be constant in time and independent of any features of its motion and interaction with other bodies. In the axiomatic approach to the construction of classical mechanics, the following is accepted as one of the axioms:
Axiom
A material point is a geometric point that is associated with a scalar called mass: $(r,m)$, where $r$ is a vector in the Euclidean space referred to some Cartesian coordinate system. The mass is assumed to be constant, independent of either the position of the point in space or time.
Mechanical energy can be stored by a material point only in the form of the kinetic energy of its movement in space and (or) the potential energy of interaction with the field. This automatically means that a material point is incapable of deformation (only an absolutely rigid body can be called a material point) and rotation around its own axis and changes in the direction of this axis in space. At the same time, the model of body motion described by a material point, which consists in changing its distance from some instantaneous center of rotation and two Euler angles that set the direction of the line connecting this point with the center, is extremely widely used in many branches of mechanics.
The method of studying the laws of motion of real bodies by studying the motion of an ideal model - a material point - is the main one in mechanics. Any macroscopic body can be represented as a set of interacting material points g, with masses equal to the masses of its parts. The study of the motion of these parts is reduced to the study of the motion of material points.
The limitations of the application of the concept of a material point can be seen from this example: in a rarefied gas at high temperature, the size of each molecule is very small compared to the typical distance between molecules. It would seem that they can be neglected and the molecule can be considered a material point. However, this is not always the case: vibrations and rotations of a molecule are an important reservoir of the "internal energy" of the molecule, the "capacity" of which is determined by the size of the molecule, its structure and chemical properties. In a good approximation, a monatomic molecule (inert gases, metal vapors, etc.) can sometimes be considered as a material point, but even in such molecules at a sufficiently high temperature, excitation of electron shells due to molecular collisions is observed, followed by emission.
Exercise 1
a) a car entering the garage;
b) a car on the Voronezh - Rostov highway?
a) a car entering the garage cannot be taken as a material point, since under these conditions the dimensions of the car are significant;
b) a car on the Voronezh-Rostov highway can be taken as a material point, since the dimensions of the car are much smaller than the distance between cities.
Can it be taken as a material point:
a) a boy who walks 1 km on his way home from school;
b) a boy doing exercises.
a) When a boy, returning from school, walks a distance of 1 km to the house, then the boy in this movement can be considered as a material point, because his size is small compared to the distance he walks.
b) when the same boy does morning exercises, then he cannot be considered a material point.
When solving a whole set of problems, one can abstract from the shape and size of the body and consider it as a material point.
Definition
material point in physics, they call a body that has a mass, but whose dimensions, in comparison with the distances to other bodies, can be neglected in the problem under consideration.
The concept of "material point"
The concept of "material point" is an abstraction. In nature, material points do not exist. But the formulation of some problems of mechanics makes it possible to use this abstraction.
When we talk about a point in kinematics, it can be considered as a mathematical point. In kinematics, a point is understood as a small mark on the body or the body itself, if its dimensions are small in comparison with the distances that the body overcomes.
In such a branch of mechanics as dynamics, one must already speak of a material point, as a point that has mass. The basic laws of classical mechanics refer to a material point, a body that does not have geometric dimensions, but has mass.
In dynamics, the size and shape of the body in many cases does not affect the nature of the movement; in this case, the body can be considered as a material point. But under other conditions, the same body cannot be considered a point, since its shape and size are decisive in describing the movement of the body.
So, if a person is interested in how much time a car needs to get from Moscow to Tyumen, then it is absolutely not necessary to know how each of the wheels of the car moves. But, if a motorist is trying to squeeze his car into a narrow parking space, it is impossible to take the car for a material point, since the dimensions of the car matter. It is possible to take the Earth as a material point if we consider the movement of our planet around the Sun, but this cannot be done when studying its movement around its own axis, if we are trying to establish the reasons why day replaces night. So, one and the same body under certain conditions can be considered as a material point, under other conditions this cannot be done.
There are some types of movement in which the body can be safely taken as a material point. So, for example, during the translational motion of a rigid body, all its parts move in the same way, therefore, in such a motion, the body is usually considered as a point with a mass that is equal to the mass of the body. But if the same body rotates around its axis, then it cannot be taken as a material point.
And so, a material point is the simplest model of a body. If the body can be likened to a material point, then this greatly simplifies the solution of the problem of studying its motion.
Different types of point movement are distinguished, first of all, by the type of trajectory. In the event that the trajectory of the movement of a point is a straight line, then the movement is called rectilinear. With regard to the motion of a macroscopic body, it makes sense to speak of rectilinear or curvilinear motion of the body only when it is possible to confine ourselves to considering the displacement of one point of this body when describing the motion. In a body, in general, different points can perform different types of movement.
System of material points
If the body cannot be taken as a material point, then it can be represented as a system of material points. In this case, the body is mentally divided into infinitesimal elements, each of which can be taken as a material point.
In mechanics, each body can be represented as a system of material points. Having the laws of motion of a point, we can assume that we have a method for describing any body.
In mechanics, an essential role is played by the concept of an absolutely rigid body, which is defined as a system of material points, the distances between which are unchanged, for any interactions of this body.
Examples of problems with a solution
Example 1
The task. In what case can a body be considered a material point:
The athlete at the competition throws the core. Can the nucleus be considered a material point?
The ball rotates around its axis. A sphere is a material point?
The gymnast performs an exercise on the uneven bars.
The runner covers the distance.
Example 2
The task. Under what conditions can a moving stone be considered a material point. See fig.1 and fig.2.
Decision: On fig. 1 the size of the stone cannot be considered small in comparison with the distance to it. In this case, the stone cannot be considered a material point.
On fig. 2 the stone rotates, therefore, it cannot be considered a material point.
Answer. A stone thrown up can be considered a material point if its dimensions are small in comparison with the distance to it, and it moves forward (there will be no rotation).
