EntranceDynamics of translational motion of a material point. Newton's laws
Dynamics of a material point and translational motion of a rigid body
Newton's first law. Weight. Force
Newton's first law: every material point (body) maintains a state of rest or uniform rectilinear motion until the influence of other bodies forces it to change this state. The desire of a body to maintain a state of rest or uniform rectilinear motion is called inertia. Therefore, Newton's first law is also called law of inertia.
Newton's first law is not satisfied in every frame of reference, and those systems in relation to which it is satisfied are called inertial reference systems.
Weight body - a physical quantity that is one of the main characteristics of matter, determining its inertial ( inert mass) and gravitational ( gravitational mass) properties. At present, it can be considered proven that the inertial and gravitational masses are equal to each other (with an accuracy of at least 10–12 of their values).
So, force is a vector quantity that is a measure of the mechanical impact on a body from other bodies or fields, as a result of which the body acquires acceleration or changes its shape and size.
Newton's second law
Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the mechanical motion of a material point (body) changes under the influence of forces applied to it.
a~ F (T = const) . (6.1)
a~ 1 /t (F = const). (6.2)
a =kF/ m. (6.3)
In SI proportionality coefficient k= 1. Then
(6.4)
(6.5)
Vector quantity
(6.6)
numerically equal to the product of the mass of a material point and its speed and having the direction of speed is called impulse (amount of movement) this material point.
Substituting (6.6) into (6.5), we get
(6.7)
Expression (6.7) is called equation of motion of a material point.
The SI unit of force is newton(N): 1 N is a force that imparts an acceleration of 1 m/s 2 to a mass of 1 kg in the direction of the force:
1 N = 1 kg m/s 2 .
Newton's second law is valid only in inertial frames of reference. Newton's first law can be derived from the second.
In mechanics, it is of great importance principle of independent action of forces: if several forces act simultaneously on a material point, then each of these forces imparts acceleration to the material point according to Newton’s second law, as if there were no other forces.
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Newton's third law
The interaction between material points (bodies) is determined Newton's third law.
F 12 = – F 21 , (7.1)
Newton's third law allows for the transition from dynamics separate material point to dynamics systems material points.
Friction forces
In mechanics we will consider various forces: friction, elasticity, gravity.
Friction forces, which prevent sliding of contacting bodies relative to each other.
External friction is called friction that occurs in the plane of contact of two contacting bodies during their relative movement.
Depending on the nature of their relative movement, they speak of sliding friction, rolling or spinning.
Internal friction called friction between parts of the same body, for example between different layers of liquid or gas. If bodies slide relative to each other and are separated by a layer of viscous liquid (lubricant), then friction occurs in the lubricant layer. In this case they talk about hydrodynamic friction(the lubricant layer is quite thick) and boundary friction (the thickness of the lubricant layer is 0.1 µm or less). 
Sliding friction force F tr is proportional to force N normal pressure with which one body acts on another:
F tr = f N ,
Where f - sliding friction coefficient, depending on the properties of the contacting surfaces.
In the limiting case (beginning of body sliding) F=F tr. or P sin 0 = f N = f P cos 0, where
f = tg 0 .
For smooth surfaces, intermolecular attraction begins to play a certain role. For them it is applied sliding friction law
F tr = f ist (N + Sp 0 ) ,
Where R 0 - additional pressure caused by intermolecular attractive forces, which quickly decrease with increasing distance between particles; S - contact area between bodies; f ist - true coefficient of sliding friction.
A radical way to reduce friction is to replace sliding friction with rolling friction (ball and roller bearings, etc.). The rolling friction force is determined according to the law established by Coulomb:
F tr = f To N / r , (8.1)
Where r- radius of the rolling body; f k - rolling friction coefficient, having the dimension dim f k =L. From (8.1) it follows that the rolling friction force is inversely proportional to the radius of the rolling body.
Law of conservation of momentum. Center of mass
A set of material points (bodies) considered as a single whole is called mechanical system. The forces of interaction between material points of a mechanical system are called - internal. The forces with which external bodies act on material points of the system are called external. A mechanical system of bodies that is not acted upon by external forces is called closed(or isolated). If we have a mechanical system consisting of many bodies, then, according to Newton’s third law, the forces acting between these bodies will be equal and oppositely directed, i.e. the geometric sum of internal forces is equal to zero.
Let's write down Newton's second law for each of n mechanical system bodies:
Adding these equations term by term, we get
But since the geometric sum of the internal forces of a mechanical system according to Newton’s third law is equal to zero, then
(9.1)
Where 
- impulse of the system. Thus, the time derivative of the momentum of a mechanical system is equal to the geometric sum of the external forces acting on the system.
In the absence of external forces (we consider a closed system)
The last expression is law of conservation of momentum: The momentum of a closed-loop system is conserved, that is, it does not change over time.
Experiments prove that it is also true for closed systems of microparticles (they obey the laws of quantum mechanics). This law is universal, i.e. the law of conservation of momentum - fundamental law of nature.
The law of conservation of momentum is a consequence of a certain property of the symmetry of space - its homogeneity. Homogeneity of space lies in the fact that during parallel transfer in space of a closed system of bodies as a whole, its physical properties and laws of motion do not change, in other words, they do not depend on the choice of the position of the origin of the inertial reference system.
Center of mass(or center of inertia) of a system of material points is called an imaginary point WITH, the position of which characterizes the mass distribution of this system. Its radius vector is equal to
Where m i And r i- mass and radius vector, respectively i th material point; n- number of material points in the system; 
– mass of the system. Center of mass speed
Considering that pi
= m i v i,a 
there is momentum R systems, you can write
(9.2)
that is, the momentum of the system is equal to the product of the mass of the system and the speed of its center of mass.
Substituting expression (9.2) into equation (9.1), we obtain
(9.3)
that is, the center of mass of the system moves as a material point in which the mass of the entire system is concentrated and on which a force acts equal to the geometric sum of all external forces applied to the system. Expression (9.3) is law of motion of the center of mass.
Translational motion is a mechanical movement of a system of points (body), in which any straight line segment associated with a moving body, the shape and dimensions of which do not change during movement, remains parallel to its position at any previous moment in time. If a body moves translationally, then to describe its movement it is enough to describe the movement of an arbitrary point (for example, the movement of the center of mass of the body).
One of the most important characteristics of the movement of a point is its trajectory, which in general is a spatial curve that can be represented as conjugate arcs of different radii, each emanating from its own center, the position of which can change over time. In the limit, a straight line can be considered as an arc whose radius is equal to infinity.
In this case, it turns out that during translational motion, at each given moment in time, any point of the body rotates around its instantaneous center of rotation, and the length of the radius at a given moment is the same for all points of the body. The velocity vectors of the points of the body, as well as the accelerations they experience, are identical in magnitude and direction.
For example, an elevator car moves forward. Also, to a first approximation, the Ferris wheel cabin makes translational motion. However, strictly speaking, the movement of the Ferris wheel cabin cannot be considered progressive.
The basic equation for the dynamics of translational motion of an arbitrary system of bodies
The rate of change of the system's momentum is equal to the main vector of all external forces acting on this system.
Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the mechanical motion of a material point (body) changes under the influence of forces applied to it. Considering the action of various forces on a given material point (body), the acceleration acquired by the body is always directly proportional to the resultant of these applied forces:
When the same force acts on bodies with different masses, the accelerations of the bodies turn out to be different, namely
Taking into account (1) and (2) and the fact that force and acceleration are vector quantities, we can write
Relationship (3) is Newton's second law: the acceleration acquired by a material point (body), proportional to the force causing it, coincides with it in direction and is inversely proportional to the mass of the material point (body). In the SI measurement system, the proportionality coefficient is k= 1. Then
Considering that the mass of a material point (body) in classical mechanics is constant, in expression (4) the mass can be entered under the derivative sign:
Vector quantity
numerically equal to the product of the mass of a material point by its speed and having the direction of speed, is called the impulse (amount of motion) of this material point. Substituting (6) into (5), we obtain
This expression is a more general formulation of Newton's second law: the rate of change of momentum of a material point is equal to the force acting on it.
Main characteristics of forward movement:
1.path - any movement along the trajectory
2.moving is the shortest path.
As well as force, impulse, mass, speed, acceleration, etc.
The number of degrees of freedom is the minimum number of coordinates (parameters), the specification of which completely determines the position of the physical system in space.
In translational motion, all points of the body at each moment of time have the same speed and acceleration.
The law of conservation of angular momentum (the law of conservation of angular momentum) is one of the fundamental conservation laws. It is expressed mathematically through the vector sum of all angular momentum relative to the selected axis for a closed system of bodies and remains constant until the system is acted upon by external forces. In accordance with this, the angular momentum of a closed system in any coordinate system does not change with time.
The law of conservation of angular momentum is a manifestation of the isotropy of space with respect to rotation. It is a consequence of Newton's second and third laws.
Experimental studies of the interactions of various bodies - from planets and stars to atoms and elementary particles - have shown that in any system of bodies interacting with each other, in the absence of the action of forces from other bodies not included in the system, or the sum of the acting forces is equal to zero, the geometric sum of the momenta of the bodies remains unchanged.
A system of bodies that do not interact with other bodies not included in this system is called a closed system.
P-Pulse
(with vectors)
14. Differences between rotational and translational motion. Kinematics of rotational motion. Rotational motion is a type of mechanical motion. During the rotational motion of an absolutely rigid body, its points describe circles located in parallel planes. Translational motion is a mechanical movement of a system of points (body), in which any straight line segment associated with a moving body, the shape and dimensions of which do not change during movement, remains parallel to its position at any previous moment in time .[ There is a close and far-reaching analogy between the motion of a rigid body around a fixed axis and the motion of an individual material point (or the translational motion of a body). Each linear quantity from the kinematics of a point corresponds to a similar quantity from the kinematics of rotation of a rigid body. Coordinate s corresponds to angle φ, linear velocity v - angular velocity w, linear (tangential) acceleration a - angular acceleration ε. Comparative movement parameters:
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Forward movement |
Rotational movement |
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Move S |
Angular displacement φ |
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Linear speed |
Angular velocity |
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Acceleration |
Angular acceleration |
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Moment of inertia I |
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Momentum |
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Moment of force M |
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Job: |
Job: |
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Kinetic energy |
Kinetic energy |
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Law of Conservation of Momentum (LCM) |
Law of Conservation of Angular Momentum (LACM) |
When describing the rotational motion of a rigid body relative to a stationary body in a given reference system, it is customary to use vector quantities of a special kind. In contrast to the polar vectors r (radius vector), v (velocity), a (acceleration) discussed above, the direction of which naturally follows from the nature of the quantities themselves, the direction of the vectors characterizing rotational motion coincides with the axis of rotation, therefore they are called axial (Latin axis – axis).
Elementary rotation dφ is an axial vector, the magnitude of which is equal to the angle of rotation dφ, and the direction along the axis of rotation OO" (see Fig. 1.4) is determined by the rule of the right screw (rotation angle of a rigid body).
Fig.1.4. To determine the direction of the axial vector
Linear displacement dr of an arbitrary point A of a rigid body is related to the radius vector r and rotation dφ by the relation dr=rsinα dφ or in vector form through the vector product:
dr= (1.9)
Relation (1.9) is valid precisely for an infinitesimal rotation dφ.
Angular velocity ω is an axial vector determined by the derivative of the rotation vector with respect to time:
Vector ω, like vector dφ, is directed along the axis of rotation according to the rule of the right screw (Fig. 1.5).
Fig.1.5. To determine the direction of the vector
Angular acceleration β is an axial vector determined by the derivative of the angular velocity vector with respect to time:
β=dω/dt=d2φ/dt2=ω"=φ""
With accelerated motion, the direction of vector β coincides with ω (Fig. 1.6, a), and with slow motion, vectors β and ω are directed opposite to each other (Fig. 1.6, b).
Fig.1.6. Relationship between the directions of vectors ω and β
Important note: the solution to all problems involving the rotation of a rigid body around a fixed axis is similar in form to problems involving the rectilinear motion of a point. It is enough to replace the linear quantities x, vx, ax with the corresponding angular quantities φ, ω and β, and we obtain equations similar to (1.6) - (1.8).
Treatment period-
(The time it takes a body to complete one revolution)
Frequency (number of revolutions per unit time) -
DYNAMICS OF FORWARD MOTION
Newton's first law
Kinematics deals with the description of the simplest types of mechanical movements. In this case, the reasons causing changes in the position of the body relative to other bodies are not affected, and the reference system is chosen for reasons of convenience when solving a particular problem. In principle, one can take any of an infinite number of reference systems.
However, the laws of mechanics in different reference systems have, strictly speaking, different forms. The problem arises of choosing a reference system in which the laws of mechanics would be as simple as possible. Such a reference system is obviously most convenient for describing mechanical phenomena.
Let us find out what the acceleration of a particle in some arbitrary frame of reference depends on. What is the reason for this acceleration? It has been established experimentally that this reason can be both the action of certain bodies on a given particle and the properties of the reference system itself (see. §1.8).
Newton suggested that there is a reference system in which the acceleration of a material point is due only to its interaction with other bodies and does not depend on the choice of the reference system. A material point, not subject to the action of any other bodies, moves relative to such a frame of reference rectilinearly and uniformly, or, as they say, by inertia. Such a reference system is called inertial,
The statement that inertial frames of reference exist constitutes the content of the first law of classical mechanics - Galileo's - Newton's law of inertia - is this: There are reference systems called inertial, in which, in the absence of the influence of other bodies, the particle maintains a stationary state of motion: it moves uniformly and rectilinearly (in a particular case, it is at rest).
The inertial frame of reference is heliocentric reference frame, whose origin is connected with the Sun. Reference systems moving uniformly in a straight line relative to an inertial frame are also inertial. Frames of reference moving with acceleration relative to an inertial frame are non-inertial.
For these reasons, the Earth's surface is, strictly speaking, a non-inertial frame of reference. However, in many problems, the reference frame associated with the Earth can be considered inertial to a first approximation.
Questions for self-control
What reference systems are called inertial? Why are these systems very useful for describing mechanical movements?
What factors determine the value of acceleration in inertial reference systems?
Can the reference frame associated with the Earth be considered inertial?
State Newton's first law.
The ability of a body to maintain a state of uniform rectilinear motion or rest in inertial frames of reference is called inertia of the body. The measure of body inertia is weight. Mass is a scalar quantity, measured in kilograms (kg) in the SI system.
The measure of interaction is a quantity called by force. Force is a vector quantity, measured in Newtons (N) in the SI system.
Newton's second law. In inertial systems, a material point moves with acceleration if the sum of all forces acting on it is not equal to zero, and the product of the point’s mass and its acceleration is equal to the sum of these forces, i.e.:
Since the mass of a point is a positive quantity, its acceleration vector is always directed along the sum of all forces acting on it, i.e. 
.
When solving problems using Newton's second law, it is important to remember the following:
if a point moves in a straight line, then its acceleration vector is directed along the movement with an accelerated nature of the movement, for a slow nature of the movement - against the movement;
if a point moves in a circle at an accelerated rate, then the tangential acceleration vector is directed along the linear velocity vector; if the motion is slow, the opposite is true. The normal acceleration vector is directed towards the center of rotation.
.
It should be remembered that forces, as measures of interaction, are always born in pairs.
If a body makes translational motion 1, then the vectors of forces acting on it are transferred to the center of mass of this body. This allows us to reduce the problem to the movement of one material point of a rigid body.
To successfully solve most problems using Newton's laws, it is necessary to adhere to a certain sequence of actions (a kind of algorithm).
Main points of the algorithm.
1. Analyze the condition of the problem and find out with which bodies the material point in question interacts. Based on this, determine the amount of forces acting on it. (Suppose the number of forces acting on the body is equal to 
.) Then make a schematically correct drawing on which to plot all the forces acting on the point.
2. Using the condition of the problem, determine the direction of acceleration of the point under consideration, and depict the acceleration vector in the figure.
3. Write Newton’s second law in vector form, i.e.:
Where 
forces acting on a point.
4. Select an inertial reference system. Draw in the figure a rectangular Cartesian coordinate system, the OX axis of which is usually directed along the acceleration vector, the OY and OZ axes are directed perpendicular to the OX axis.
5. Using the basic property of vector equalities, write down Newton’s second law for projections of vectors onto the coordinate axes, i.e.:
(2.3)
6. If in a problem, in addition to forces and accelerations, it is necessary to determine coordinates and speed, then in addition to Newton’s second law, it is also necessary to use kinematic equations of motion. Having written down a system of equations, it is necessary to pay attention to the fact that the number of equations is equal to the number of unknowns in this problem.
Questions for self-control
Define strength. In what SI units is force measured?
What is the property of inertia of a body? What physical quantity is a measure of the inertia of a body? In what SI units is the mass of bodies measured?
Give the formulation of Newton's second law for inertial frames of reference.
Give the formulation of Newton's third law.
Example 1.
In an elevator cabin, a load of mass hangs on a dynamometer 
. Dynamometer shows strength 
. Determine the acceleration of the load. Is it possible to answer the question in which direction the load is moving?
R 
decision. On a body moving with acceleration 
, two bodies act: the Earth with gravity 
and spring with force 
. Let's depict the forces in the figure. Let's assume that the elevator's acceleration vector is directed upward. Let's depict the vector in the figure. We write Newton's second law in vector form:
.
We select the OX axis in the direction of acceleration. We write Newton's second law for projections of vectors onto this axis:
From this equality we find the projection of acceleration onto the OX axis:
.
Since the projection of acceleration onto the OX axis is positive, the assumption that the elevator acceleration vector is directed vertically upward is true. It is not possible to determine the direction of movement of the elevator, since the indicated direction of the acceleration vector corresponds to two types of movement: a) uniformly accelerated movement vertically upward; b) uniformly slow motion vertically downwards.
Newton's second law in non-inertial frames of reference. Inertia forces.
2 Consider a non-inertial reference frame 
, rotating at a constant angular speed 
around an axis moving translationally at speed 
relative to inertial 
systems.
In this case, the acceleration of a point in the inertial frame () is related to the acceleration in the non-inertial frame ( 
) ratio (see §1.8):
Where 
– acceleration of the non-inertial system relative to the inertial system 
, 
linear velocity of a point in a non-inertial frame.
From the last relation, instead of acceleration, we substitute in equality (1), we obtain the expression:
This ratio is Newton's second law for a non-inertial frame of reference.
Inertia forces. Let us introduce some conventions:
1. 
– forward inertial force;
2. 
– Coriolis force;
3
– centrifugal force of inertia.
In problems, the translational force of inertia is depicted against the acceleration vector of translational motion of a non-inertial reference frame ( 
), centrifugal force of inertia –– from the center of rotation along the radius ( 
); the direction of the Coriolis force is determined by the rule gimlet for the cross product of vectors 
.
Strictly speaking, inertial forces are not forces in the full sense, because Newton's third law does not hold for them, i.e. they are not paired and arise only during the transition from inertial frames of reference to non-inertial frames.
Questions for self-control
§2.4. Forces in mechanics
In mechanics, one non-contact long-range force is considered - force of universal gravity, which can act on the body in question at a great distance (for example, the Earth attracts the Moon), and five contact forces: elastic force, reaction force, body weight, elastic force, friction force and resistance force.
§2.5. The force of universal gravity. Gravity.
Acceleration of gravity.
The force of universal gravitation arises in the process of interaction between bodies with masses and is calculated from the relation:
.
.
(2.6)
got the name gravitational constant. Its value in the SI system is equal to 
.
WITH 
The forces of mutual attraction are directed along one straight line connecting these material points. The law of universal gravitation is valid for bodies whose sizes are small compared to the distance between them. If the sizes of the bodies are comparable to the distance between them, then to calculate the force of interaction between them, proceed as follows.
Each of the bodies is divided into infinitesimal parts, the sizes of which can be neglected in comparison with the distance between them. Next, the forces of interaction between each part of one body and each part of another body are calculated. The total force of mutual attraction is equal to the sum of the forces acting from all elements of one body on all elements of another body.
Having carried out such reasoning for homogeneous balls, it can be shown that the resulting force of attraction is calculated according to the formula given earlier. In this case, the mass of the balls is taken, and the distance between the centers of the balls is taken as the distance.
For a body interacting with a planet, the distance from the center of the planet to the center of mass of the body is taken as the distance. Let us give the formula for the force of attraction between bodies and planets:
.
(2.7)
Usually, the force of attraction of a body to a planet is called gravity, the value of which is usually calculated using the formula 
,
Where 
body mass, 
magnitude of the free fall acceleration vector .
The force of gravity is directed towards the center of the Earth, applied to the center of gravity of the body.
Relation (2.7) allows us to establish a connection between the magnitude of the acceleration of gravity and the mass of the planet, its radius and height from the point in question to the surface of the planet:
. (2.8)
On the surface of the planet, i.e. When 
, for the acceleration of free fall the formula is valid
.
(2.9)
Questions for self-control
By what ratio is the magnitude of the force of universal gravity calculated?
Define gravity.
What determines the acceleration of free falling bodies?
Reaction forces arise when a body interacts with various structures that limit its position in space. For example, a body suspended on a thread is acted upon by a reaction force, usually called the force tension. The tension force of the thread is always directed along the thread. There is no formula for calculating its value. Usually its value is found either from Newton's first or second law.
Reaction forces also include forces acting on a particle on a smooth surface. They call her normal reaction force, denote 
. The reaction force is always directed perpendicular to the surface under consideration. A force acting on a smooth surface from the side of the body is called normal pressure force (
). According to Newton's third law, the reaction force is equal in magnitude to the force of normal pressure, but the vectors of these forces are opposite in direction.
Body weight- this is the force with which a body, due to the gravity of the Earth, presses on a horizontal support or stretches a vertical suspension.
If the scales move with acceleration, then the weight can be either greater or less than the force of gravity.
Questions for self-control
What forces are commonly called reaction forces?
Define body weight.
In what cases are body weight and gravity the same?
P 
example5
.
Determine the weight of the boy's mass 
in an elevator moving vertically upward with acceleration 
. How many times does the boy's weight differ from gravity?
Solution. The boy in the elevator is acted upon by two bodies: a) the Earth with gravity; b) elevator floor with reaction force 
. Let's depict these forces in the figure. Let us show in this figure the direction of the elevator acceleration vector. Let's write Newton's second law in vector form:
.
We choose the Earth's surface as the inertial reference system, and direct the OX axis along the acceleration vector of the elevator. Let's write Newton's second law in projection onto this axis:
From this equation we find the magnitude of the reaction force:
.
Substituting digital data in the SI system, we find the reaction force:
By definition, weight is numerically equal to the reaction force, i.e. 
.
Let's find how many times the boy's weight differs from the force of gravity:
.
WITH 
silt elasticity.
Elastic forces arise in bodies if the bodies are deformed, i.e. if the shape of the body or its volume is changed. When the deformation stops, the elastic forces disappear. It should be noted that, although elastic forces arise during deformation of bodies, deformation does not always lead to the emergence of elastic forces.
Elastic forces arise in bodies that are capable of restoring their shape after the cessation of external influence. Such bodies and the corresponding deformations are called elastic. At plastic deformation changes do not completely disappear after the cessation of external influence.
A striking example of the manifestation of elastic forces can be the forces arising in springs subject to deformation. For elastic deformations that occur in deformed bodies, the elastic force is always proportional to the magnitude of the deformation, i.e.:
, (5)
Where 
coefficient of elasticity (or stiffness) of the spring, 
spring deformation vector.
This statement is called Hooke's law.
The greater the rigidity of a body, the less it deforms under a given force. Magnitude 
determined by the geometric dimensions of the body and the material from which it is made. If the shape of a body (rod, spring or rubber band) begins to change significantly, then the proportionality between 
And 
is violated (see Fig. 2.2).
The elastic force is directed along the thread, rod or spring. The force is applied at the point of contact.
A thread– a model of a body with zero mass and a dedicated axis, which is capable of bending under an infinitesimal load. Therefore, it can be thrown over the block, and the tension force will be the same everywhere.
Spring– a model of a body (usually with zero mass) that acts on the body in question not only in an extended, but also in a compressed state. Moreover, Hooke's law holds true for a spring not only in tension, but also in compression.
Questions for self-control
What forces are commonly called elastic forces?
Which deformations are called elastic and which plastic?
Formulate Hooke's law and indicate the limits of applicability of Hooke's law.
Example 6
.
A thread is thrown through a light, frictionless rotating block. At one end of the thread there is a body of mass 
, on the other - a body of mass 
. Determine the magnitude of the tension force of the thread and the magnitude of the acceleration of the bodies.
Solution. Let us depict all the forces acting on the bodies and on the block. Let us consider the process of movement of bodies connected by a thread thrown over a block. The thread is weightless and inextensible, therefore, the magnitude of the tension force on any section of the thread will be the same, i.e. 
And 
.
P 
the displacements of bodies over any period of time will be the same, and, therefore, at any moment of time the values of the velocities and accelerations of these bodies will be the same.
Since the block rotates without friction and is weightless, it follows that the tension force of the thread on both sides of the block will be the same, i.e.: 
.
This implies the equality of the tension forces of the thread acting on the first and second bodies, i.e. 
.
Let us depict in the figure the acceleration vectors of the first and second bodies. Let us depict two OX axes. Let's direct the first axis along the acceleration vector of the first body, the second - along the acceleration vector of the second body.
Let's write Newton's second law for each body in projection onto these coordinate axes:
Considering that 
, and expressing from the first equation 
, substitute it into the second equation, we get
From the last equality we find the acceleration value:
.
From equality (1) we find the magnitude of the tension force:
Friction force. Law of dry friction.
When bodies come into contact, interaction is observed between them. The force characterizing this interaction is called the surface reaction force, denoted 
, and are represented as the sum of the forces that make it up: 
, Where 
– normal surface reaction force, directed perpendicular to this surface, 
– friction force, directed along this surface.
Upon contact of smooth surfaces 
And 
. The simplest relationship between the moduli of forces that make up the surface reaction force is formulated in the form of the law of dry friction:
When sliding, the modulus of the friction force is directly proportional to the modulus of the normal reaction force:
.
Proportionality factor 
– sliding friction coefficient does not depend either on the area of the contacting surfaces or on the speed of their relative movement.
If sliding does not occur, then maximum possible value The static friction force is equal to the sliding friction force:
.
Z 
The value and direction of the static friction force is determined from the condition of the body being immobile relative to the support.
With a gradual increase (over time) of strength 
applied along the rubbing surfaces, a similar increase in the static friction force occurs (Fig. 2.3). The forces acting along the surface are compensated, so the body is at rest.
When the force module reaches the value 
, the modulus of the static friction force reaches its maximum value, and then the friction force no longer balances the external force, and the body begins to slide, accelerating (Fig. 2.3).
Questions for self-control
Examples of problem solving
Example 9
.
On an inclined plane with an angle of inclination 
there is a body of mass 
. The coefficient of friction between the body and the inclined plane is equal to 
. A force directed upward along an inclined plane is applied to a body. What must be the magnitude of this force for the body to move up the inclined plane with acceleration?
R 
decision. A body moving upward along an inclined plane is acted upon by external bodies: a) Earth with gravity directed vertically downwards; b) an inclined plane with a reaction force directed perpendicular to the inclined plane; c) an inclined plane with friction force 
, directed against the movement of the body; d) external body with force 
, directed upward along an inclined plane.
Under the influence of these forces, the body moves uniformly accelerated up the inclined plane, and, therefore, the acceleration vector is directed along the movement of the body.
Let's depict the acceleration vector in the figure. Let's write Newton's second law in vector form:
Let us choose a rectangular Cartesian coordinate system, the OX axis of which is directed along the acceleration of the body, and the OY axis is directed perpendicular to the inclined plane.
Let's write Newton's second law in projections onto these coordinate axes and obtain the following equations:
The sliding friction force is related to the reaction force by the following relationship:
. (3)
From equality (2) we find the magnitude of the reaction force 
and substitute into equality (3), we have the following expression for the friction force:
. (4)
Substituting the right side of equality (4) into equality (1) instead of the friction force, we obtain the following equation for calculating the magnitude of the required force:
Let's calculate the magnitude of the force 
:
The power of resistance.
When bodies move in liquids and gases, friction forces also arise, but they differ significantly from the forces of dry friction. These forces are called viscous friction forces, or resistance forces. Viscous friction forces arise only during the relative motion of bodies. Resistance forces depend on many factors, namely: on the size and shape of bodies, on the properties of the medium (density, viscosity), on the speed of relative motion. At low speeds, the drag force is directly proportional to the speed of the body relative to the medium, i.e.:
, (2.11)
Where 
– vector of the speed of movement of the body relative to the medium.
At high speeds, the drag force is proportional to the square of the speed of the body relative to the medium, i.e.:
, (2.12)
Where 
some proportionality coefficients, called resistance coefficients.
Questions for self-control
Under what conditions does resistance force arise?
What formula is used to calculate the friction force for low speeds?
What formula is used to calculate the friction force at high speed?
The basic equation of the dynamics of a material point is nothing more than a mathematical expression of Newton’s second law:
. (2.13)
In a rectangular Cartesian coordinate system, the basic equation of dynamics in projections on the coordinate axes has the form:
(2.14)
In an inertial frame of reference, the sum of all forces includes only forces that are measures of interactions; in non-inertial frames, the sum of forces includes inertial forces.
From a mathematical point of view, relation (9) is a differential equation of motion of a point in vector form. Its solution is the main problem of the dynamics of a material point.
Questions for self-control
What relation is the basic equation of dynamics?
What do the equations of dynamics look like in a rectangular Cartesian coordinate system?
Example 1. , we obtain the desired dependence of speed on time: 
1 Translational motion of a rigid body is such a motion in which every straight line invariably connected with the body moves parallel to itself.
2 Material for additional study
*Task of increased complexity
Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the mechanical motion of a material object (point, body) changes under the influence of forces applied to it.
In dynamics, two types of problems are considered, the solutions of which are found on the basis of Newton's second law. Problems of the first type are to, knowing the motion of a body, determine the forces acting on it. A classic example of solving such a problem is Newton’s discovery of the law of universal gravitation: knowing the laws of planetary motion established by Kepler based on observational results, Newton proved that this motion occurs under the influence of a force inversely proportional to the square of the distance between the planet and the Sun.
Problems of the second type are fundamental in dynamics and consist in determining the law of its motion (equation of motion) based on the forces acting on the body. To solve these problems it is necessary to know the initial conditions, i.e. the position and speed of a body at the moment it begins to move under the influence of given forces. Examples of such problems are the following: a) using the magnitude and direction of the velocity of the projectile at the moment of its departure from the barrel and the force of gravity and air resistance acting on the projectile during its movement, find the law of motion of the projectile, in particular its trajectory, horizontal flight range, time of movement to the goal; b) using the known speed of the car at the moment of braking and the braking force, find the time of movement and the distance to the stop.
Newton's second law is formulated as follows: the acceleration acquired by a material point (body) is directly proportional to the acting force, coincides with it in direction and inversely proportional to the mass of the material point (body):
Where k- proportionality coefficient, depending on the choice of system of units. In the international system (SI) k=1, therefore
(2.4)
Newton's second law is usually written in the following form:
or
(2.5)
Vector mv=p called impulse or amount of movement. Unlike acceleration and speed, impulse is a characteristic of a moving body, reflecting not only the kinematic measure of movement (speed), but also its most important dynamic property - mass.
Thus, we can write:
(2.6)
Expression (2.6) is a more general formulation of Newton’s second law: the rate of change of momentum of a material point is equal to the force acting on it.
This equation is called equation of motion of a material point.
The SI unit of force is newton (N):
1 N is the force that imparts an acceleration of 1 m/s 2 to a body weighing 1 kg in the direction of the force:
1 N = 1 kg*1 m/s 2.
When several forces act on a material point, principle of independent action of forces: if several forces act simultaneously on a material point, then each of these forces imparts to the material point an acceleration determined by Newton’s second law as if there were no other forces:
where is the power 
called resultant forces or resultant force.
Thus, if several forces act on a body simultaneously, then, according to the principle of independence of action of forces, under the force F Newton's second law refers to the resultant force.
Newton's second law is valid only in inertial frames of reference. Newton's first law can be obtained from the second law: if the resultant force is equal to zero, the acceleration is also zero, i.e. the body is at rest or moving uniformly.
Course work
topic: “Dynamics of forward motion”
Moscow 2013
Introduction
Newton's first law
Newton's second law
Newton's third law
Law of Gravity
Non-inertial frames of reference
Basic formulas for the dynamics of translational motion
Introduction
Dynamics is the branch of mechanics that studies the movement of material bodies together with the causes that cause this movement. Dynamics can be divided into classical, relativistic and quantum. This chapter examines classical dynamics. It is assumed that the speed of movement of bodies is significantly less than the speed of light (v<
The beginning of classical mechanics was laid by the work of Galileo, and classical mechanics itself as a science was formed after the work of I. Newton. Classical dynamics is based on Newton’s three laws, formulated by him in 1687. These laws are a generalization of human experience and Newton’s merit is that from a huge number of experimental facts he was able to identify the main ones, which became the cornerstones of classical physics.
The mechanical motion of a body can be decomposed into translational and rotational and, accordingly, the dynamics of translational and rotational motions can be considered separately. To describe the dynamics of translational motion, in addition to kinematic characteristics, it is necessary to introduce a number of new concepts, the most important of which are the concepts of mass and force.
1. Newton's first law
Newton's first law: Every material point maintains a state of rest or uniform rectilinear motion until the influence of other bodies forces it to change this state.
Mathematically, this law can be written in the form = const or v = 0 at F = 0,
where F is the force acting on the point. Both equalities can be replaced by one: a = 0 for F = 0.
Before Galileo's work, it was believed that in order to maintain motion at a constant speed, some force must be applied to a body. Everyday experience spoke about this; the position about the presence of force was embedded in the physical teachings of Aristotle. Galileo took into account the presence of friction forces and, through logical reasoning, came to the conclusion formulated by Newton’s first law. Inertia is the tendency of a body to maintain a state of rest or uniform linear motion. Experience shows that all bodies have inertia. The concept of inertia is discussed in more detail below. A reference system is called inertial if Newton's first law is satisfied in it. Therefore, Newton's first law is sometimes called the law of inertia. In addition to inertial ones, there are also non-inertial reference systems, i.e. such systems in which Newton’s first law is not satisfied (an accelerating car, a centrifuge, etc.). Non-inertial frames of reference are discussed below.
If you remember Newton's second law
then it turns out that the first law follows from the second at. This causes some confusion. Why proclaim as a law an elementary consequence of another law?
If the forces are known, then it follows. On the other hand, how do you know that there is no force acting on the body? You can say that, if, then and. It turns out to be a vicious circle.
Example: A falling elevator is an inertial system, although it is accelerating relative to the ground. Here the body moves at a constant speed if no external forces act on it.
The meaning of the first law is that if no external forces act on a body, then there will be a frame of reference in which this body is at rest or moving at a constant speed. There are an infinite number of such systems.
In the “astronomical reference frame,” the center of the coordinate system is connected to the Sun, and the axes are directed to the fixed stars. With very high accuracy, such a system is inertial.
mechanics mass inertial
2. Newton's second law
To formulate Newton's second law, it is necessary to introduce the concepts of mass and force. It is known that every body resists attempts to change its state of motion. This property of bodies is called inertia. The main characteristic of the inert properties of a body is mass. There are various definitions of mass.
Mass is a physical quantity that determines the inertial properties of a body. In order to use this definition, it is necessary to indicate the method for measuring inertial properties. You can, for example, consider the change in the motion of different bodies under the influence of the same force. By comparing the accelerations acquired by different bodies, it is possible to obtain comparative estimates for masses. In this case, bodies with greater mass receive less acceleration.
Mass is the amount of substance contained in a body. Newton gave this definition of mass. This is a fairly general, but not entirely strict definition (within the framework of the theory of relativity, mass can change during movement).
There is also the concept of gravitational mass, which can be defined using the gravitational interaction between two masses, described by Newton's law
where G = 6.67·10 - 11 m3/kg·s2 is the gravitational constant, m1 and m2 are the masses of the bodies, r is the distance between the bodies.
The unit of mass is 1 kg - the mass of the standard stored at the International Bureau of Weights and Measures (Paris). Force is a vector quantity that is a measure of the mechanical impact on a body from other bodies or fields, as a result of which the body acquires acceleration or changes its shape and size. Within the framework of classical mechanics, several most frequently encountered types of forces can be distinguished. Among the fundamental forces that cannot be reduced to simpler ones are gravitational and electromagnetic forces. A special case of gravitational force is gravity. Often we have to deal with elastic forces and friction forces. Let's look at these forces in more detail. Gravitational forces are described by Newton's formula given above. If we take the mass of the Earth M as the mass, and the radius of the Earth R as r, we obtain an expression for the force of gravity
The value P determines the force with which all bodies with mass m are attracted to the ground. The weight of a body is the force with which the body acts on a horizontal support. If we do not take into account the rotation of the Earth and consider a frame of reference that is motionless relative to the Earth, then the weight of a body coincides with its gravity. In more complex cases, inertial forces must be taken into account (see below).
Elastic forces arise when bodies are deformed (tension or compression, bending, torsion) and are caused by intermolecular interaction. When a spring is stretched from its equilibrium position by an amount x, an elastic force arises
Here k is the spring stiffness, a constant characterizing the elastic properties of the spring. The minus sign indicates that the force is directed in the direction opposite to the displacement of the spring and tends to return the spring to its equilibrium position. Friction forces appear when bodies in contact move relative to each other. Friction between the surfaces of two solid bodies in the absence of any layer between them is called dry friction. There are static friction, sliding friction and rolling friction. If a force F acts on a body lying on a flat rough surface, but the body does not move, then the force F is balanced by the friction force.
This force is called the static friction force. It acts on the body from the surface at the contact boundary and is determined by the formula
The sliding friction force is determined by the formula
where k is the friction coefficient, N is the support reaction force. It determines the force with which bodies are pressed against each other (normal pressure force). The above formula is sometimes called the Coulomb-Amonton law.
The forces of static friction and sliding friction are often combined into one, which is determined by the formula
The graph of this force looks like
The rolling friction force is small compared to the sliding friction forces, and we do not consider it here.
Electrical and magnetic forces will be discussed in the relevant sections of electromagnetism. At the atomic and nuclear levels, instead of forces, interactions are usually considered, which are described in terms of energy.
Newton's second law: The acceleration acquired by a material point is directly proportional to the force acting on it and inversely proportional to the mass of the point:
This law is usually written in the form
Here force and acceleration are considered as vectors.
The SI unit of force is 1N (newton) - this is the force under the influence of which a body weighing 1 kg acquires an acceleration of 1 m/s2
Note that mass and force are additive quantities, i.e. the mass of a system of material points is determined by the expression
and the action of several forces can be replaced by the action of one
If F = 0, then Newton’s second law implies a = 0. It follows that in the absence of external forces v = const, i.e. statement contained in Newton's first law. In fact, the value of the first law is that it states the existence of inertial frames of reference. The momentum of a material point is the quantity
Newton's second law is the fundamental law of translational motion dynamics.
Newton's third law
We considered the action of other bodies on the selected body. In fact, there is interaction between different bodies, i.e. the selected body also affects other bodies.
Newton's third law: The forces with which interacting bodies act on each other are equal in magnitude and opposite in direction.
If the body is at rest on a horizontal plane, then the diagram of the acting forces has the form
normal pressure N is related to gravity by the relation
For a body moving along a rough horizontal plane under the action of force F, the following basic forces can be introduced, shown in the figure:
As noted above, the friction force is described by the expression
where k is the friction coefficient.
Law of Gravity
Of the many forces capable of acting on a material body, the forces of universal gravity should be singled out. They constitute the law discovered by Newton and which made it possible to explain the movement of celestial bodies and the origin of gravity. Newton's three laws, together with the law of gravity, allowed Newton to create celestial mechanics and explain Kepler's laws, the movement of planets, comets, satellites and other celestial bodies.
Newton's law of gravity. Two material points with masses and, located at a distance r from each other, are attracted with a force directly proportional to the masses of these points and inversely proportional to the square of the distance between them:
Here G = 6.67·10 - 11 m3/kg·s2 is the gravitational constant. In this case, the force is directed along the line connecting the points.
This formula is valid for material points, i.e. when the sizes of bodies can be neglected compared to the distance between them. If the sizes of the bodies are comparable to the distance between the bodies, it is necessary to use the integration operation.
As already noted, from the law of gravitation it is easy to obtain an expression for the acceleration of gravity
where M and are the mass and radius of the Earth.
Example 1. Determine the change in the acceleration of gravity when the height of the rise above the Earth's surface changes.
Solution. The acceleration due to gravity is given by the formula
where is the radius of the Earth, h is the height of the rise. When we get
acceleration of gravity on the Earth's surface.
The resulting formula shows that a noticeable change in g can be expected at altitudes comparable to the Earth’s radius km.
Question. Why do astronauts experience a feeling of weightlessness at an altitude of km?
Example 2. Determine the first and second escape velocities, i.e. the speed at which the rocket will orbit the Earth or leave the Earth.
Solution. Let's make a drawing
The first escape velocity is determined from the condition
From here we get
To determine the second escape velocity, we will find the work that must be done to remove the rocket from the Earth
From the law of conservation of energy
Similarly, you can find the third escape velocity at which the rocket will leave the solar system.
Non-inertial frames of reference
Newton's laws are valid only in an inertial frame of reference. In particular, in an accelerated elevator, in the absence of external forces, the trajectory of a material point will differ from a straight line. If you measure the weight of a body in an accelerating elevator using a spring scale, then in an ascending and descending elevator the readings of the scales will be different and different from the readings in a stationary elevator.
A reference system is called non-inertial if it moves with acceleration relative to the inertial frame. If and are the acceleration of a material point in inertial and non-inertial systems, and is the acceleration of the reference frame, then
Geometrically it looks like
Newton's laws can be written in non-inertial systems if inertial forces are added to the action of external forces:
where is the acceleration of a material point relative to a non-inertial reference frame. The value of the inertial force depends on the choice of a non-inertial reference system and the nature of the motion of the material point in this system. According to the two motions of the body - translational and rotational - both translationally moving and rotating non-inertial reference frames are used. Note that the inertial force differs from other forces in that it exists only in a non-inertial reference frame and for it it is impossible to indicate those specific forces with the parties on which it operates. In particular, inertial forces do not obey Newton's third law - there is no counterforce for them. Accordingly, in non-inertial systems the laws of conservation of energy, momentum and angular momentum may not be satisfied. Note that the connection between the forces of inertia and the forces of gravity underlies Einstein's general theory of relativity.
Let us consider the simplest cases of manifestation of inertial forces.
) Accelerated translational motion of the reference system. If in the inertial frame of reference Newton's equation has the form
then in the non-inertial system we get
If in a non-inertial system the material point is at rest (), then
This formula gives an expression for the force of inertia in translationally moving non-inertial systems.
) Centrifugal force of inertia. Let us consider a material point fixed on a rotating disk.
The force of inertia acts on the point
which is called the centrifugal force of inertia. It is directed radially from the center of rotation. Using vector notation, we write this force in vector form
It is easy to verify the validity of this formula by constructing the corresponding figure and indicating the directions of the vectors.
) Coriolis forces. In a rotating reference frame, centrifugal force acts on both a stationary and a moving body. In addition, a material point moving in a rotating reference frame is subject to an additional force associated with the movement of this point.
The Coriolis force is the force associated with the movement of a material point in a rotating coordinate system. A more complete name for this force is the Coriolis inertial force. The action of this force is shown in the figure.
If the disk does not rotate, the material point, in the absence of external forces, moves along straight line OA. In a rotating disk, the trajectory of a material point relative to the disk will be represented by the OB curve. Consequently, with respect to the rotating reference frame, the material point is acted upon by a force FK directed perpendicular to the speed v (the speed is given relative to the disk, i.e. in a non-inertial coordinate system). It can be shown that the Coriolis force is given by
This formula remains valid in any direction of speed (not necessarily along the radius).
So, in an arbitrary non-inertial frame of reference, the fundamental law of dynamics has the form
Here the force F is caused by the interaction between the bodies, and the forces Fi, Fc and FK are associated with the accelerated movement of the reference frame.
Note that in a non-inertial reference frame, when using the laws of conservation of energy and momentum, it is necessary to take into account the action of inertial forces.
Basic formulas for the dynamics of translational motion
Pulse
Newton's second law
Newton's third law
The force of gravitational interaction
Dry friction force
Center of mass coordinates
Equation of motion in a non-inertial reference frame
Inertia force
Centrifugal force of inertia
Coriolis force
List of used literature and sources
1. Trofimova T.I. Physics course, M.: Higher school, 1998, 478 p.
Trofimova T.I. Collection of problems for the course of physics, M.: Higher school, 1996, 304s
Volkenshtein V.S. Collection of problems for the general course of physics, St. Petersburg: “Special Literature”, 1999, 328 p.
Trofimova T.I., Pavlova Z.G. Collection of problems for a physics course with solutions, M.: Higher School, 1999, 592 p.
All solutions to the “Collection of problems for the general course of physics” by V.S. Volkenstein, M.: Ast, 1999, book 1, 430 pp., book 2, 588 pp.
Krasilnikov O.M. Physics. Methodological guidelines for processing observation results. M.: MISIS, 2002, 29 p.
Suprun I.T., Abramova S.S. Physics. Guidelines for performing laboratory work, Elektrostal: EPI MISiS, 2004, 54 p.