Dynamics studies the movement of bodies taking into account the reasons that cause this movement.
Dynamics is based on Newton's laws.
I law. There are inertial reference systems (IRS), in which a material point (body) maintains a state of rest or uniform rectilinear motion until the influence of other bodies takes it out of this state.
The property of a body to maintain a state of rest or uniform rectilinear motion in the absence of influence of other bodies on it is called inertia.
ISO is a reference system in which a body, free from external influences, is at rest or moves uniformly in a straight line.
An inertial reference system is one that is at rest or moves uniformly in a straight line relative to any ISO.
A reference system moving with acceleration relative to the ISO is non-inertial.
Newton's First Law, also called the law of inertia, was first formulated by Galileo. Its content boils down to 2 statements:
1) all bodies have the property of inertia;
2) there are ISOs.
Galileo's principle of relativity: all mechanical phenomena occur in the same way in all ISOs, i.e. It is impossible to establish by any mechanical experiments inside an ISO whether a given ISO is at rest or moves uniformly in a straight line.
In most practical problems, a reference system rigidly connected to the Earth can be considered an ISO.
It is known from experience that under the same influences, different bodies change their speed differently, i.e. acquire different accelerations, the acceleration of bodies depends on their mass.
Weight- a measure of the inertial and gravitational properties of a body. With the help of precise experiments it has been established that the inertial and gravitational masses are proportional to each other. Choosing units in such a way that the proportionality coefficient becomes equal to one, we obtain that m and = m g, so we simply talk about the mass of the body.
[m]=1kg is the mass of a platinum-iridium cylinder, the diameter and height of which are h=d=39mm.
To characterize the action of one body on another, the concept of force is introduced.
Force- a measure of the interaction of bodies, as a result of which the bodies change their speed or are deformed.
Force is characterized by its numerical value, direction, and point of application. The straight line along which a force acts is called line of action of force. The simultaneous action of several forces on a body is equivalent to the action of one force, called resultant or the resulting force and equal to their geometric sum:
Newton's second law - the fundamental law of the dynamics of translational motion - answers the question of how the motion of a body changes under the influence of forces applied to it.
II law. The acceleration of a material point is directly proportional to the force acting on it, inversely proportional to its mass and coincides in direction with the acting force.
Where is the resultant force.
The force can be expressed by the formula
, 
1N is a force under the influence of which a body weighing 1 kg receives an acceleration of 1 m/s 2 in the direction of the force.
Newton's second law can be written in another form by introducing the concept of momentum:
.
Pulse- a vector quantity, numerically equal to the product of the body’s mass and its speed and co-directed with the speed vector.
Date: __________ Deputy Director for HR:___________
Subject; Newton's second law for rotational motion
Target:
Educational: identify and write down Newton’s second law in mathematical form; explain the relationship between the quantities included in the formulas of this law;
Developmental: develop logical thinking, the ability to explain the manifestations of Newton’s second law in nature;
Educational : to develop interest in studying physics, to cultivate hard work and responsibility.
Lesson type: learning new material.
Demonstrations: the dependence of the acceleration of a body on the force acting on it.
Equipment: cart with light wheels, rotating disk, set of weights, spring, block, block.
DURING THE CLASSES
Organizing time
Updating students' basic knowledge
Chain of formulas (reproduce formulas):
II. Motivation for students' learning activities
Teacher. Using Newton's laws, one can not only explain observed mechanical phenomena, but also predict their course. Let us recall that the direct main task of mechanics is to find the position and velocity of a body at any moment of time, if its position and velocity at the initial moment of time and the forces that act on it are known. This problem is solved using Newton's second law, which we will study today.
III. Learning new material
1. Dependence of body acceleration on the force acting on it
A more inert body has a larger mass, a less inert body has a smaller one:
2. Newton's second law
Newton's second law of dynamics establishes a connection between kinematic and dynamic quantities. Most often it is formulated as follows: the acceleration that a body receives is directly proportional to the mass of the body and has the same direction as the force:
where is acceleration, is the resultant of the forces acting on the body, N; m - body weight, kg.
If we determine force from this expression, we obtain the second law of dynamics in the following formulation: the force acting on a body is equal to the product of the body’s mass and the acceleration provided by this force.
Newton formulated the second law of dynamics somewhat differently, using the concept of momentum (momentum of a body). Impulse is the product of a body's mass and its speed (the same as the amount of motion) - one of the measures of mechanical motion: Impulse (amount of motion) is a vector quantity. Since the acceleration is
Newton formulated his law as follows: the change in the momentum of a body is proportional to the acting force and occurs in the direction of the straight line along which this force acts.
It is worth considering another formulation of the second law of dynamics. In physics, a vector quantity is widely used, which is called the impulse of a force - this is the product of a force and the time of its action: Using this, we get 
. The change in the momentum of a body is equal to the impulse of the force that acts on it.
Newton's second law of dynamics generalized an extremely important fact: the action of forces does not cause motion itself, but only changes it; force causes a change in speed, i.e. acceleration, not speed itself. The direction of force coincides with the direction of velocity only in the partial case of rectilinear evenly accelerated (Δ 0) motion. For example, during the movement of a body thrown horizontally, the force of gravity is directed downward, and the speed forms a certain angle with the force, which changes during the flight of the body. And in the case of uniform motion of a body in a circle, the force is always directed perpendicular to the speed of the body.
The SI unit of force is determined based on Newton's second law. The unit of force is called [H] and is defined as follows: a force of 1 newton imparts an acceleration of 1 m/s2 to a body weighing 1 kg. Thus,
Examples of application of Newton's second law
As an example of the application of Newton's second law, we can consider, in particular, measuring body weight using weighing. An example of the manifestation of Newton's second law in nature can be the force that acts on our planet from the Sun, etc.
Limits of application of Newton's second law:
1) the reference system must be inertial;
2) the speed of the body must be much less than the speed of light (for speeds close to the speed of light, Newton’s second law is used in impulse form: ).
IV. Fixing the material
Problem solving
1. A body weighing 500 g is simultaneously acted upon by two forces of 12 N and 4 N, directed in the opposite direction along one straight line. Determine the magnitude and direction of acceleration.
Given: m = 500 g = 0.5 kg, F1 = 12 N, F2 = 4 N.
Find: a - ?
According to Newton's second law: , where Let's draw the Ox axis, then the projection F = F1 - F2. Thus,
Answer: 16 m/s2, acceleration is directed in the direction of the greater force.
2. The coordinate of the body changes according to the law x = 20 + 5t + 0.5t2 under the action of a force of 100 N. Find the mass of the body.
Given: x = 20 + 5t + 0.5t2, F = 100H
Find: m - ?
Under the influence of a force, the body moves at a uniform acceleration. Consequently, its coordinate changes according to the law:
According to Newton's second law:
Answer: 100 kg.
3. A body weighing 1.2 kg acquired a speed of 12 m/s at a distance of 2.4 m under the influence of a force of 16 N. Find the initial speed of the body.
Given: = 12 m/s, s = 2.4m, F = 16H, m = 1.2 kg
Find: 0 - ?
Under the influence of a force, a body acquires acceleration according to Newton's second law:
For uniformly accelerated motion:
From (2) we express time t:
and substitute for t in (1):
Let's substitute the expression for acceleration:
Answer: 8.9 m/s.
V. Lesson summary
Frontal conversation with questions
1. How are such physical quantities as acceleration, force and body mass related to each other?
2. Or can we use the formula to say that the force acting on a body depends on its mass and acceleration?
3. What is the momentum of a body (quantity of motion)?
4. What is a force impulse?
5. What formulations of Newton’s second law do you know?
6. What important conclusion can be drawn from Newton’s second law?
VI. Homework
Work through the relevant section of the textbook.
Solve problems:
1. Find the acceleration modulus of a body weighing 5 kg under the action of four forces applied to it, if:
a) F1 = F3 = F4 = 20 H, F2 = 16 H;
b) F1 = F4 = 20 H, F2 = 16 H, F3 = 17 H.
2. A body weighing 2 kg, moving in a straight line, changed its speed from 1 m/s to 2 m/s in 4 s.
a) With what acceleration was the body moving?
b) What force acted on the body in the direction of its movement?
c) How did the body’s momentum (amount of motion) change during the time under consideration?
d) What is the impulse of the force acting on the body?
e) What distance did the body travel during the considered time of movement?
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.
The branch of mechanics that studies the movement of material bodies together with the physical causes that cause this movement is called dynamics. The basic ideas and quantitative laws of dynamics arose and are developing on the basis of centuries of human experience: observations of the movement of earthly and celestial bodies, industrial practice and specially designed experiments.
The great Italian physicist Galileo Galilei experimentally established that a material point (body) sufficiently distant from all other bodies (i.e., not interacting with them) will maintain its state of rest or uniform rectilinear motion. This position of Galileo was confirmed by all subsequent experiments and constitutes the content of the first fundamental law of dynamics, the so-called law of inertia. In this case, rest should be considered as a special case of uniform and rectilinear motion, when .
This law is equally valid both for the movement of giant celestial bodies and for the movement of the smallest particles. The property of material bodies to maintain a state of uniform and rectilinear motion is called inertia.
Uniform and linear motion of a body in the absence of external influences is called motion by inertia.
The reference system in relation to which the law of inertia holds is called an inertial reference system. The inertial frame of reference is almost exactly the heliocentric frame. In view of the enormous distance to the stars, their movement can be neglected and then the coordinate axes directed from the Sun to three stars that do not lie in the same plane will be motionless. Obviously, any other reference frame moving uniformly and rectilinearly relative to the heliocentric frame will also be inertial.
The physical quantity characterizing the inertia of a material body is its mass. Newton defined mass as the amount of matter contained in a body. This definition cannot be considered exhaustive. Mass characterizes not only the inertia of a material body, but also its gravitational properties: the force of attraction experienced by a given body from another body is proportional to their masses. Mass determines the total energy reserve of a material body.
The concept of mass allows us to clarify the definition of a material point. A material point is a body, when studying the movement of which one can abstract from all its properties except mass. Each material point, therefore, is characterized by the magnitude of its mass. In Newtonian mechanics, which is based on Newton's laws, the mass of a body does not depend on the position of the body in space, its speed, the action of other bodies on the body, etc. Mass is an additive quantity, i.e. The mass of a body is equal to the sum of the masses of all its parts. However, the property of additivity is lost at speeds close to the speed of light in vacuum, i.e. in relativistic mechanics.
Einstein showed that the mass of a moving body depends on speed
,
(2.1)
where m 0 - mass of a body at rest, - speed of body movement, s – speed of light in vacuum.
From (2.1) it follows that when bodies move at low speeds c, the mass of the body is equal to the rest mass, i.e. m=m 0 ; at c mass m.
Summarizing the results of Galileo's experiments on the fall of heavy bodies, Kepler's astronomical laws on the motion of planets, and the data of his own research, Newton formulated the second fundamental law of dynamics, which quantitatively linked the change in the motion of a material body with the forces causing this change in motion. Let us dwell on the analysis of this most important concept.
In general, the force 
- is a physical quantity that characterizes the action exerted by one body on another. This vector quantity is determined by a numerical quantity or modulus 
, direction in space and point of application.
If two forces act on a material point 
And 
, then their action is equivalent to the action of one force
,
obtained from the known triangle of forces (Fig. 2.1). If n-forces act on a body, the total action is equivalent to the action of one resultant, which is the geometric sum of forces:
.
(2.2)
The dynamic manifestation of force is that under the influence of force a material body experiences acceleration. The static action of force leads to the fact that elastic bodies (springs) are deformed under the influence of forces, and gases are compressed.
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Under the influence of forces, the movement ceases to be uniform and rectilinear and acceleration appears ( |
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), its direction coincides with the direction of the force. Experience shows that the acceleration received by a body under the influence of a force is inversely proportional to the magnitude
its masses:
or 
.
(2.3)
Equation (2.3) represents the mathematical notation of the second fundamental law of dynamics:
the vector of force acting on a material point is numerically equal to the product of the mass of the point and the acceleration vector arising under the action of this force.
Because acceleration
,
Where 
- unit vectors, 
- projections of acceleration onto the coordinate axes, then
.
(2.4)
If we denote , then expression (2.4) can be rewritten in terms of projections of forces onto the coordinate axes:
In the SI system, the unit of force is the newton.
According to (2.3), a newton is a force that imparts an acceleration of 1 m/s 2 to a mass of 1 kg. It's easy to see that
.
Newton's second law can be written differently if we introduce the concept of body momentum (m) and force impulse (Fdt). Let's substitute in
(2.3) expression for acceleration
,
we get 
or 
.
(2.5)
Thus, the elementary impulse of force acting on a material point during the time interval dt is equal to the change in the momentum of the body over the same period of time.
Indicating the impulse of the body
,
we obtain the following expression for Newton's second law:
.
In relativistic mechanics, at c, the basic law of dynamics and the momentum of the body, taking into account the dependence of mass on speed (2.1.), will be written in the following form
,
.
Until now, we have considered only one side of the interaction between bodies: the influence of other bodies on the nature of the movement of a given selected body (material point). Such influence cannot be one-sided; interaction must be mutual. This fact is reflected by the third law of dynamics, formulated for the case of interaction of two material points: if the material point m 2
experiences from the side of the material point m 1
a force equal to 
, then m 1
experiences from the outside
m 2
force 
, equal in magnitude and opposite in direction 
:
.
These forces always act along a straight line passing through the points m 1 and m 2 , as shown in Figure 2.2. Figure 2.2 A applies
|
to the case when the interaction forces between points are repulsive forces. In Figure 2.2, b the case of attraction is depicted. |
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MATERIAL POINT AND SOLID BODY
Brief theory
As a measure of the mechanical action of one body on another, a vector quantity called by force. Within the framework of classical mechanics, they deal with gravitational forces, as well as elastic and frictional forces.
The force of gravitational attraction, acting between two material points, in accordance with law of universal gravitation, is proportional to the product of the masses of the points and , is inversely proportional to the square of the distance between them and is directed along the straight line connecting these points:
, (3.1)
Where G=6.67∙10 -11 m 3 /(kg∙s 2) - gravitational constant.
Gravity is the force of attraction in the gravitational field of a celestial body:
| , | (3.2) |
where is body weight; - acceleration of free fall, - mass of the celestial body, - distance from the center of mass of the celestial body to the point at which the acceleration of free fall is determined (Fig. 3.1).
Weight - this is the force with which a body acts on a support or suspension that is motionless relative to a given body. For example, if a body with a support (suspension) is motionless relative to the Earth, then the weight is equal to the force of gravity acting on the body from the Earth. Otherwise the weight 
, where is the acceleration of the body (with support) relative to the Earth.
Elastic forces
Any real body, under the influence of forces applied to it, is deformed, that is, it changes its size and shape. If, after the cessation of forces, the body returns to its original size and shape, the deformation is called elastic. The force acting on the body (spring) is counteracted by an elastic force. Taking into account the direction of action for the elastic force, the formula holds:
| , | (3.3) |
Where k- coefficient of elasticity (stiffness in the case of a spring), - absolute deformation. The statement about the proportionality between elastic force and deformation is called Hooke's law. This law is valid only for elastic deformations.
As a quantity characterizing the deformation of the rod, it is natural to take the relative change in its length:
Where l 0 - length of the rod in the undeformed state, Δ l– absolute elongation of the rod. Experience shows that for rods made of this material, the relative elongation ε during elastic deformation, it is proportional to the force per unit cross-sectional area of the rod:
, (3.5)
Where E- Young's modulus (a value characterizing the elastic properties of a material). This value is measured in pascals (1Pa=1N/m2). Attitude F/S represents normal voltage σ , since strength F directed normal to the surface.
Friction forces
When a body moves along the surface of another body or in a medium (water, oil, air, etc.), it encounters resistance. This is the force of resistance to movement. It is the resultant of the resistance forces of the body shape and friction: 
. The friction force is always directed along the contact surface in the direction opposite to the movement. If there is liquid lubricant, this will already be viscous friction between layers of liquid. The situation is similar with the movement of a body completely immersed in the medium. In all these cases, the friction force depends on the speed in a complex way. For dry friction this force depends relatively little on speed (at low speeds). But static friction cannot be determined unambiguously. If the body is at rest and there is no force tending to move the body, it is equal to zero. If there is such a force, the body will not move until this force becomes equal to a certain value called the maximum static friction. The static friction force can have values from 0 to , which is reflected in the graph (Fig. 3.2, curve 1) by a vertical segment. According to Fig. 3.2 (curve 1), the sliding friction force with increasing speed first decreases somewhat and then begins to increase. Laws dry friction boil down to the following: the maximum static friction force, as well as the sliding friction force, do not depend on the area of contact of the rubbing bodies and turn out to be approximately proportional to the magnitude of the normal pressure force pressing the rubbing surfaces to each other:
| , | (3.6) |
where is a dimensionless coefficient of proportionality, called the coefficient of friction (rest or sliding, respectively). It depends on the nature and condition of the rubbing surfaces, in particular on their roughness. In the case of sliding, the coefficient of friction is a function of speed.
Rolling friction formally obeys the same laws as sliding friction, but the friction coefficient in this case turns out to be much smaller.
Force viscous friction goes to zero along with the speed. At low speeds it is proportional to the speed:
where is a positive coefficient characteristic of a given body and a given environment. The value of the coefficient depends on the shape and size of the body, the state of its surface and on the property of the medium called viscosity. This coefficient also depends on the speed, but at low speeds in many cases it can be practically considered constant. At high speeds, the linear law becomes quadratic, that is, the force begins to increase in proportion to the square of the speed (Fig. 3.2, curve 2).
Newton's first law: Every body is in a state of rest or uniform and rectilinear motion until the influence of other bodies forces it to change this state.
Newton's first law states that a state of rest or uniform linear motion does not require any external influences to maintain it. This reveals a special dynamic property of bodies called inertia. Accordingly, Newton's first law is also called law of inertia, and the movement of a body free from external influences is coasting.
Experience shows that every body “exhibits resistance” to any attempts to change its speed - both in magnitude and in direction. This property, expressing the degree of intractability of a body to changes in its speed, is called inertia. It manifests itself to different degrees in different bodies. The measure of inertia is a quantity called mass. A body with greater mass is more inert, and vice versa. Within the framework of Newtonian mechanics, mass has the following two most important properties:
1) mass is an additive quantity, that is, the mass of a composite body is equal to the sum of the masses of its parts;
2) the mass of the body as such is a constant quantity that does not change during its movement.
Newton's second law: under the action of the resultant force the body acquires acceleration
Forces and are applied to different bodies. These forces are of the same nature.
Impulse – vector quantity equal to the product of a body’s mass and its speed:
| , | (3.10) |
where is the momentum of the body, is the mass of the body, is the speed of the body.
For a point included in the system of points:
, (3.11)
where is the rate of change of momentum i-th point of the system; - the sum of internal forces acting on i-th point from the side of all points of the system; - the resultant external force acting on i-th point of the system; N- number of points in the system.
Basic equation of translational motion dynamics for a system of points:
, (3.12)
Where 
- rate of change of the system impulse; - the resulting external force acting on the system.
Basic equation of translational motion dynamics solid:
 ,
| (3.13) |
where is the resultant force acting on the body; - speed of the center of mass of the body, 
rate of change of momentum of the body's center of mass.
Questions for self-study
1. Name the groups of forces in mechanics and give them a definition.
2. Define resultant force.
3. Formulate the law of universal gravitation.
4. Define gravity and acceleration of gravity. What parameters do these physical quantities depend on?
5. Obtain an expression for the first escape velocity.
6. Tell us about body weight and the conditions under which it changes. What is the nature of this force?
7. Formulate Hooke’s law and indicate the limits of its applicability.
8. Explain dry and viscous friction. Explain how the force of dry and viscous friction depends on the speed of the body.
9. Formulate Newton's first, second and third laws.
10. Give examples of the implementation of Newton's laws.
11. Why is Newton’s first law called the law of inertia?
12. Define and give examples of inertial and non-inertial reference systems.
13. Tell us about the mass of a body as a measure of inertia, list the properties of mass in classical mechanics.
14. Give the definition of body impulse and force impulse, indicate the units of measurement of these physical quantities.
15. Formulate and write down the basic law of the dynamics of translational motion for an isolated material point, a point of a system, a system of points and a rigid body.
16. A material point begins to move under the influence of force Fx, the time dependence of which is shown in the figure. Draw a graph reflecting the dependence of the magnitude of the projection of the impulse p x from time.
Examples of problem solving
3
.1
. A cyclist rides on a circular horizontal platform, the radius of which is , and the coefficient of friction depends only on the distance to the center of the platform according to the law 
where is the constant. Find the radius of a circle with a center at point , along which a cyclist can ride at maximum speed. What is this speed?
Given: Find:
R, r(vmax), v max.
The problem considers the movement of a cyclist in a circle. Since the cyclist’s speed is constant in absolute value, he moves with centripetal acceleration under the influence of several forces: gravity, ground reaction force and friction force (Fig. 3.4).
Applying Newton's second law, we get:
++ + =m.(1)
Having chosen the coordinate axes (Fig. 1.3), we write equation (1) in projections onto these axes:
Considering that F tr =μF N = 
mg, we obtain an expression for speed:
. (2)
To find the radius r, at which the cyclist’s speed is maximum, it is necessary to investigate the function v(r) to the extremum, that is, find the derivative and equate it to zero:
= 
=0. (3)
The denominator of the fraction (3) cannot be equal to zero, then from the equality of the numerator to zero we obtain an expression for the radius of the circle at which the speed is maximum:
Substituting expression (4) into (2), we obtain the required maximum speed:
.
Answer: 
.
On a smooth horizontal plane lies a board of mass m1 and on it a block of mass m2. A horizontal force is applied to the block, increasing with time according to the law where c is a constant. Find the dependence on the acceleration of the board and the block if the coefficient of friction between the board and the block is equal. Draw approximate graphs of these dependencies.
Given: Find:
m 1, 1.
m2, 2.
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The problem considers the translational motion of two contacting bodies (a board and a block), between which a friction force acts. There is no friction force between the board and the plane. Force F, applied to the block, increases with time, therefore, up to a certain point in time, the block and the board move together with the same acceleration, and when the block begins to overtake the board, it will slide along it. The friction force is always directed in the direction opposite to the relative speed. Therefore, the friction forces acting on the board and the block are directed as shown in Figure 3.5, and . Let the starting point of time be t= 0 coincides with the beginning of the movement of the bodies, then the friction force will be equal to the maximum static friction force (where the normal reaction force of the board is balanced by the gravity force of the block). The acceleration of the board occurs under the influence of one friction force, directed in the same way as the force.
The dependence of the acceleration of the board and the acceleration of the block on time can be found from the equation of Newton's second law, written for each body. Since the vertical forces acting on each of the bodies are compensated, the equations of motion for each of the bodies can be written in scalar form (for projections onto the OX axis):
Considering that , = , we can get:
. (1)
From the system of equations (1) one can find the moment of time , taking into account that when 
:
.
By solving the system of equations (1) for , we can obtain:
(at ). (2)
When accelerating and are different, but the friction force has a certain value 
, Then:
(3)
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A graph of acceleration versus time for bodies and can be constructed based on expressions (2) and (3). When the graph is a straight line coming from the origin. When the graph is straight, parallel to the x-axis, the graph is straight, going up more steeply (Fig. 3.6).
Answer: when accelerating 
at 
. Here .
3.3. In the installation (Figure 3.7) the angle is known φ inclined plane with the horizon and the coefficient of friction between the body and the inclined plane. The masses of the block and thread are negligible, there is no friction in the block. Assuming that at the initial moment both bodies are motionless, find the mass ratio at which the body:
1) will begin to descend;
2) will begin to rise;
3) will remain at rest.
Given: Find:
Solution:
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The problem considers two bodies connected by a thread and performing translational motion. The body of mass is acted upon by the force of gravity, the normal reaction force of the inclined plane, the tension force of the thread and the friction force. Only gravity and the tension of the thread act on the body (Fig. 3.7). Under equilibrium conditions, the accelerations of the first and second bodies are zero, and the friction force is the static friction force, and its direction is opposite to the direction of possible motion of the body. Applying Newton's second law for the first and second bodies, we obtain the system of equations:
(1)
Due to the weightlessness of the thread and block. Selecting the coordinate axes (Fig. 3.7 A, 3.7 b), we write down for each body the equation of motion in projections onto these axes. The body will begin to descend (Fig. 3.7 A) given that:
(2)
When solving system (2) jointly, one can obtain
(3)
Taking into account that expression (3) can be written as:
(4)
