Kinematics – studies the movement of bodies without considering the reasons that determine this movement.
Mat.point – has no dimensions, but the mass of the entire body is concentrated at the matte point.
Progressive – a movement in which the straight line connected to the body remains || to myself.
Kinetic levels of movement of the matte point:
Trajectory – a line described by a mating point in space.
Moving – increment of the radius vector of a point over the considered period of time.
Speed – Speed of movement of the matte point.
Vector average speed<> is called the ratio of the increment of the radius vector of a point to a period of time.
Instantaneous speed – a value equal to the first derivative of the radius vector of a moving point with respect to time.
Instant Speed Module equal to the first derivative of the path with respect to time.
The components are equal to the time derivatives of the coordinates.
Uniform - a movement in which a body travels identical paths in equal periods of time.
Uneven – movement in which the speed changes both in magnitude and direction.
Acceleration and its components.
Acceleration – a physical quantity that determines the rate of change in speed, both in magnitude and in direction.
Medium acceleration uneven movement in the time interval from t to t+t is called a vector quantity equal to the ratio of the change in speed to the time interval t: . Instant acceleration mat.points at time t will be the limit of the average acceleration. ..
determines modulo.
determines by direction. i.e. is equal to the first derivative with respect to time of the velocity modulus, thereby determining the rate of change in velocity modulo.
The normal component of acceleration is directed along the normal to the trajectory to the center of its curvature (therefore it is also called centripetal acceleration).
Complete the acceleration of a body is the geometric sum of the tangential and normal components.
If a n =?,a T =?
1,2,3 Newton's laws.
Based on the Dynamics of the mat.point Newton's three laws lie.
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.
Inertia – the desire of the body to maintain a state of rest or uniform rectilinear motion.
Newton's laws are valid only in inertial reference frame .
Inertial reference frame - a system that is either at rest or moving uniformly and rectilinearly relative to some other inertial system.
Body weight – physical quantity, which is one of the main characteristics of matter, determining its inertial (inertial mass) and gravitational (gravitational mass) properties.
Strength – a vector quantity, which 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 – 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.
Impulse (number of movement) – a vector quantity, numerically equal to the product of the mass of a material point and its speed and having the direction of speed.
A more general formulation of the 2nd law of N. (equation of motion MT): the rate of change of momentum of a material point is equal to the force acting on it.
Corollary from 2zN: the principle of independence of the action of forces: if several forces act on a machine at the same time, then each of these forces imparts acceleration to the machine according to 23N, as if there were no other forces.
Newton's third law. Any action of mt (bodies) on each other is in the nature of interaction; the forces with which the mt act on each other are always equal in magnitude, oppositely directed and act along the straight line connecting these points.
Body impulse, force. Law of conservation of momentum.
Inner forces – interaction forces between mechanical system elements.
External forces – forces with which external bodies act on the body of the system.
In a mechanical system of bodies, according to Newton’s 3rd law, the forces acting between these bodies will be equal and oppositely directed, i.e. the geometric sum of internal forces is 0.
Let's write 2зН, for each ofnmechanical system bodies(ms):
…………………
Let's add these equations:
Because the geometric sum of internal forces ms over 3zN is equal to 0, then:
where is the momentum of the system.
In the absence of external forces (closed system):
, i.e.
This is itlaw of conservation of momentum : The momentum of the closed system is conserved, i.e. does not change over time.
Center of mass, movement of the center of mass.
Center of mass (center of inertia) MT system is called an imaginary point WITH, the position of which characterizes the mass distribution of this system.
Radius vector this point is equal to:
Speed center of mass (cm):
; , i.e. impulse of the system equal to the product the mass of the system by the speed of its center of mass.
Because then:, i.e.:
Law of motion of the center of mass: the center of mass of the system moves as a mt 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 acting on the system.
Kinematics of rotational motion of a material point.
Angular velocity – vector quantity equal to the first derivative of the angle of rotation of the body with respect to time.
The vector is directed along the axis of rotation according to the rule of the right screw.
Linear speed of point:
In vector form: , and the module is equal to:.
If =const, then the rotation is uniform.
Rotation period (T) – the time during which the point makes one full revolution. ().
Rotational speed ( n ) – the number of complete revolutions made by a body during its uniform motion in a circle, per unit time. ;.
Angular acceleration – vector quantity equal to the first derivative of the angular velocity with respect to time: . When accelerated, when slowed down.
Tangential acceleration component:
Normal component: .
Formulas for the relationship between linear and angular quantities:
At :
Moment of power.
moment of force F relative to a fixed point O is a physical quantity determined by the vector product of the radius vector r, drawn from point O to point A of application of force, to force F.
Here is a pseudo-vector, its direction coincides with the direction of translational movement of the right propeller when it rotates open.
Module moment of force is equal to .
Moment of force about a fixed axis z is a scalar quantity equal to the projection onto this axis of the vector moment of force, defined relative to an arbitrary point O of this axis z. The value of the moment does not depend on the choice of the position of point O on a given axis.
Moment of inertia of a rigid body. Steiner's theorem.
Moment of inertia system (body) relative to the axis of rotation is a physical quantity equal to the sum of the products of the masses n mt of the system by the square of their distances to the axis in question.
At continuous distribution wt.
Steiner's theorem: the moment of inertia of a body J relative to any axis of rotation is equal to the moment of its inertia J C relative to a parallel axis passing through the center of mass C of the body, added to the product of the mass m of the body by the square of the distance A between axes:
Basic equation for the dynamics of rotational motion.
Let the force F be applied to point B, located at a distance r from the axis of rotation, - the angle between the direction of the force and the radius vector r. When the body rotates through an infinitesimal angle , the point of application B travels the path, and the work is equal to the product of the projection of the force on the direction of the displacement by the magnitude of the displacement:
Considering that , we write:
Where is the moment of force relative to the axis.
Work with body rotation is equal to the product of the moment of the acting force and the angle of rotation.
When a body rotates, work goes towards increasing its kinetic energy:
But,, therefore
Considering that we get:
This one is relative to a fixed axis.
If the axis of rotation coincides with the main axis of inertia passing through the center of mass, then: .
Moment of impulse. Law of conservation of angular momentum.
Momentum (momentum) mt A relative to a fixed point O – physical quantity determined by the vector product:
where r is the radius vector drawn from point O to point A; - impulse mt.-pseudovector, its direction coincides with the direction of translational movement of the right propeller when it rotates open.
Module angular momentum vector:
Moment of impulse relative to a fixed axis z is a scalar quantity L z equal to the projection onto this axis of the angular momentum vector defined relative to an arbitrary point O of this axis.
Because , then the angular momentum of an individual particle:
Momentum of a rigid body relative to the axis is the sum of the angular momentum of individual particles, and since , That:
That. the angular momentum of a rigid body relative to an axis is equal to the product of the moment of inertia of the body relative to the same axis and the angular velocity.
Let's differentiate the last equation: , i.e.:
this is it equation of dynamics of rotational motion of a rigid body relative to a fixed axis: The derivative of the angular momentum of a rigid body relative to an axis is equal to the moment of force relative to the same axis.
It can be shown that there is a vector equality:
In a closed system, the moment of external forces and, from where: L = const, this expression is law of conservation of angular momentum: the angular momentum of the closed system is conserved, i.e. does not change over time.
Work of force. Power.
Energy – a universal measure of various forms of movement and interaction.
Work of force – a quantity characterizing the process of energy exchange between interacting bodies in mechanics.
If the body moves straight forward and it affects him constant force that makes a certain angle with the direction of movement, then the work of this force is equal to the product of the projection of the force F s on the direction of movement, multiplied by the displacement of the point of application of the force:
Elementary work force on displacement is called a scalar quantity equal to:, where,,.
The work of force on the trajectory section from 1 to 2 is equal to the algebraic sum of elementary work on individual infinitesimal sections of the path:
If the graph shows the dependence of F s on S, then Job determined on the graph by the area of the shaded figure.
When , then A>0
When , then A<0,
When , then A=0.
Power – speed of work.
Those. power is equal to the scalar product of the force vector and the speed vector with which the point of application of the force moves.
Kinetic and potential energy of translational and rotational motion.
Kinetic energy of a mechanical system – the energy of mechanical movement of this system. dA=dT. By 2зН, multiply by and we get:;
From here:.
Kinetic energy of the system – is a function of the state of its movement, it is always , and depends on the choice of the reference system.
Potential energy – mechanical energy of a system of bodies, determined by their relative position and the nature of the interaction forces between them.
If a force field is characterized by the fact that the work done by the acting forces when moving a body from one position to another does not depend on the trajectory along which this movement occurred, but depends only on the initial and final positions, then such a field is called potential, and the forces acting in it are conservative, if the work depends on the trajectory, then such a force is dissipative .
Because the work is done due to the loss of potential energy, then: ;;, where C is the integration constant, i.e. energy is determined up to some arbitrary constant.
If the forces are conservative, then:
- Gradient of scalar P. (also denoted ).
Because Since the reference point is chosen arbitrarily, the potential energy can have a negative value. (at P=-mgh’).
Let's find the potential energy of the spring.
Elastic force: , according to 3зН:F x = -F x control =kx;
dA=F x dx=kxdx;.
The potential energy of a system is a function of the state of the system; it depends only on the configuration of the system and on its position in relation to external bodies.
Kinetic energy of rotation
Mechanical energy. Law of conservation of mechanical energy.
Total mechanical energy of the system – energy of mechanical movement and interaction: E=T+P, i.e. equal to the sum of kinetic and potential energies.
Let F 1 ’…F n ’ be the resultant internal conservative forces. F 1 …F n - resultants of external conservative forces. f 1 …f n . Let us write the equations 2зН for these points:
Let's multiply each equation by , taking into account that.
Let's add up the equations:
First term on the left side:
Where dT is the increment in the kinetic energy of the system.
The second term is equal to the elementary work of internal and external forces, taken with a minus sign, i.e. equal to the elementary increment of potential energy dP of the system.
The right side of the equality specifies the work of external non-conservative forces acting on the system. That.:
If there are no external non-conservative forces, then:
d(T+P)=0;T+P=E=const
Those. the total mechanical energy of the system remains constant. Law of conservation of mechanical energy : in a system of bodies between which only conservative forces act, the total mechanical energy is conserved, i.e. does not change over time.
Absolutely elastic impact.
Impact (impact)
Recovery rate
absolutely inelastic , if =1 then absolutely elastic.
Strike line
Central strike
Absolutely elastic impact - a collision of 2 bodies, as a result of which no deformations remain in both interacting bodies and all the kinetic energy that the bodies possessed before the impact is converted back into kinetic energy after the impact.
For an absolutely elastic impact, the law of conservation of momentum and the law of conservation of energy are satisfied.
Conservation laws:
m 1 v 1 +m 2 v 2 =m 1 v’ 1 +m 2 v’ 2
after transformations:
from where:v 1 +v 1 ’=v 2 +v 2 ’
solving the last level and the next to the last one we find:
Absolutely inelastic impact.
Impact (impact) – a collision of 2 or more bodies, in which the interaction lasts a very short time. During an impact, external forces can be neglected.
Recovery rate – the ratio of the normal component of the relative speed of bodies after and before the impact.
If =0 for colliding bodies, then such bodies are called absolutely inelastic , if =1 then absolutely elastic.
Strike line – a straight line passing through the point of contact of the bodies and normal to the surface of their contact.
Central strike - such an impact in which the bodies before the impact move along a straight line passing through their center of mass.
Absolutely inelastic impact – a collision of 2 bodies, as a result of which the bodies unite, moving further as a single whole.
Law of conservation of momentum:
If the balls moved towards each other, then with a completely inelastic impact the balls move in the direction of greater momentum.
Gravitational field, tension, potential.
Law of universal gravitation: between any two points there is a force of mutual attraction, directly proportional to the product of the masses of these points and inversely proportional to the square of the distance between them:
G – Gravitational constant (G=6.67*10 -11 Hm 2 /(kg) 2)
Gravitational interaction between two bodies is carried out using gravitational fields , or gravitational field. This field is generated by bodies and is a form of existence of matter. The main property of the field is that any body brought into this field is affected by the force of gravity:
The vector does not depend on the mass and is called the gravitational field strength.
Gravitational field strength is determined by the force acting from the field per mt of unit mass, and coincides in direction with the acting force, tension is the force characteristic of the gravitational field.
Gravitational field homogeneous if the tension at all its points is the same, and central , if at all points of the field the intensity vectors are directed along straight lines that intersect at one point.
The gravitational field of gravity is a carrier of energy.
At a distance R the force acts on the body:
When moving this body a distance dR, work is expended:
The minus sign appears because force and displacement in this case are opposite in direction.
The work expended in the gravitational field does not depend on the trajectory of movement, i.e. The gravitational forces are conservative, and the gravitational field is potential.
If then P 2 =0, then we write:
Gravitational field potential – a scalar quantity determined by the potential energy of a body of unit mass at a given point in the field or the work of moving a unit mass from a given point in the field to infinity. That.:
Equipotential – surfaces for which the potential is constant.
The relationship between potential and tension.
The min sign indicates that the tension vector is directed towards decreasing potential.
If the body is at height h, then
Non-inertial reference system. Inertia forces during accelerated translational motion of the reference system.
Non-inertial – a reference system moving relative to an inertial reference frame with acceleration.
The laws of H can be applied in a non-inertial frame of reference, if we take into account the forces of inertia. In this case, the inertial forces must be such that, together with the forces caused by the influence of bodies on each other, they impart to the body the acceleration that it has in non-inertial reference systems, i.e.:
Inertia forces during accelerated translational motion of the reference system.
Those. The angle of deviation of the thread from the vertical is equal to:
Relative to the frame of reference associated with the cart, the ball is at rest, which is possible if the force F is balanced by an equal and opposite force F in directed to it, i.e.:
Inertial forces acting on a body at rest in a rotating frame of reference.
Let the disk rotate uniformly with angular velocity around a vertical axis passing through its center. Pendulums are installed on the disk at different distances from the axis of rotation (balls are suspended on threads). When the pendulums rotate together with the disk, the balls deviate from the vertical by a certain angle.
In the inertial reference frame associated with the room, a force equal to and directed perpendicular to the axis of rotation of the disk acts on the ball. It is the resultant of gravity and the tension force of the thread:
When the motion of the ball is established, then:
those. The greater the distance R from the ball to the axis of rotation of the disk and the greater the angular velocity of rotation, the greater the angles of deflection of the pendulum threads.
Relative to the frame of reference associated with the rotating disk, the ball is at rest, which is possible if the force is balanced by an equal and opposite force directed to it.
The force called centrifugal force of inertia , is directed horizontally from the axis of rotation of the disk and is equal to:.
Hydrostatic pressure, Archimedes' law, law of jet continuity.
Hydroaeromechanics - a branch of mechanics that studies the equilibrium and movement of liquids and gases, their interaction with each other and the solid bodies flowing around them.
Incompressible fluid - a liquid whose density is the same everywhere and does not change with time.
Pressure – physical quantity determined by the normal force acting on the sides of the liquid per unit area:
Pascal's law – the pressure in any place of a fluid at rest is the same in all directions, and the pressure is equally transmitted throughout the entire volume occupied by the fluid at rest.
If the liquid is not compressible, then with the cross section S of the liquid column, its height h and density, the weight is:
And the pressure on the lower base:, i.e. pressure varies linearly with altitude. Pressure is called hydrostatic pressure .
It follows from this that the pressure on the lower layers of the liquid will be greater than on the upper ones, which means that a buoyant force determined by Archimedes' law: a body immersed in a liquid (gas) is acted upon by an upward buoyant force from this liquid, equal to the weight of the liquid displaced by the body:
Flow – fluid movement. Flow – a collection of particles of a moving fluid. Current lines – graphic representation of fluid movement.
Fluid flow steady (stationary) , if the shape of the arrangement of streamlines, as well as the values of velocities at each point do not change over time.
In 1 s, a volume of liquid equal to will pass through the section S 1, and through S 2 -, here it is assumed that the speed of the liquid in the section is constant. If the fluid is not compressible, then an equal volume will pass through both sections:
This is it jet continuity equation for an incompressible fluid.
Bernoulli's law.
The fluid is ideal, the motion is stationary.
In a short period of time, the liquid moves from sections S 1 and S 2 to sections S’ 1 and S’ 2.
According to the law of conservation of energy, the change in the total energy of an ideal incompressible fluid is equal to the work of external forces to move the mass of the fluid:
where E 1 and E 2 are the total energies of the liquid of mass m at the cross sections S 1 and S 2, respectively.
On the other hand, A is the work done when moving the entire fluid contained between sections S 1 and S 2 during the considered period of time. To transfer mass from S 1 to S’ 1, the liquid must move a distance and from S 2 to S’ 2 a distance ., where F 1 = p 1 S 1 and F 2 = -p 2 S 2.
The total energies E 1 and E 2 will be the sum of the kinetic and potential energies of the liquid mass:
Considering that
divide the equation by:
because sections were chosen arbitrarily, then:
This expression is Bernoulli's equation – expression of the law of conservation of energy, as applied to the steady flow of an ideal fluid.
p- This static (excessive) pressure ,
- dynamic pressure.
- hydrostatic pressure.
From the Bernoulli equation and the continuity equation it follows that when a liquid flows through a horizontal pipe having different sections, the fluid speed is greater in places of narrowing, and the static pressure is greater in wider places.
Torricelli formula.
Let's consider two sections (at the level h 1 and h 2), write the Bernoulli equation for them:
Because p 1 =p 2 =Atm., then:
from the level of continuity it follows that,
If S 1 >>S 2, then and the term can be neglected:
this expression is Torricelli's formula .
Internal friction (viscosity). Flow regimes.
Viscosity – the ability of real liquids to resist the movement of one part of the liquid relative to another.
Speed gradient – the value shows how quickly the speed changes when moving from layer to layer, in the direction perpendicular to the movement of the layers, i.e. friction force:
Where viscosity is a proportionality coefficient depending on the nature of the liquid.
Flow modes:
Laminar – a flow in which each selected thin layer slides relative to its neighbors without mixing with them.
This flow is observed at low speeds of its movement.
Turbulent – a flow in which intense vortex formation and mixing of the liquid occur along the flow.
Liquid particles acquire velocity components perpendicular to the flow, so they can move from one layer to another. Due to the large velocity gradient at the pipe surface, vortices are formed.
Liquid viscosity is the transfer of momentum between contacting layers.- kinematic viscosity.
R e – Reynolds number , the nature of the movement depends on it:
R e<=1000, то ламинарное
1000<=R e <=2000, переход от ламинарного к турбулентному.
R e =2300, then turbulent
Stokes method.
It is based on measuring the speed of small spherical bodies slowly moving in a liquid.
A ball falling vertically downward in a liquid is acted upon by 3 forces:
Gravity: (ball density)
Archimedes' force: (fluid density)
Resistance force (Stokes): .
With uniform ball movement:
projections:
Poiseuille method.
Based on laminar flow of liquid in a thin capillary.
In a liquid, let us mentally select a cylindrical layer of radius r and thickness dr, the internal friction force acting on the lateral surface of this layer is equal to:
where dS is the lateral surface, there is (-), because As the radius increases, the speed decreases.
The viscous force is balanced by the pressure force acting on the base:
During time t, a volume of liquid will flow out of the pipe:
Surface tension.
A liquid is characterized by a short-range order in the arrangement of particles, i.e. their ordered arrangement, repeating at distances comparable to interatomic ones.
Radius of molecular action ( r =10 -9 m) – From a distance greater than this radius, the forces of intermolecular interaction can be neglected.
The resulting forces of all molecules of the surface layer exert a pressure on the liquid, called molecular or internal.
Molecules on the surface have additional energy called surface energy .,
where sigma is surface tension.
where is the surface tension force acting per unit length of the liquid surface contour.,
this work is done due to a decrease in surface energy, i.e.:
those. surface tension is equal to the force of surface tension acting per unit length of the contour of the liquid surface.
Surface active – substances that affect the surface tension of a liquid.
(soap - , salt/sugar -)
Wetting and non-wetting.
Contact angle – the angle between the tangents to the surface of the liquid and solid.
The equilibrium condition for a drop is the equality to zero of the sum of the projections of surface tension forces on the direction of the tangent to the surface of the solid body:;
From this condition it follows that:
wetting
no wetting
Fluid equilibrium condition:
Full wetting:
Complete non-wetting:
Pressure under a curved liquid surface. Laplace's formula.
If the surface of the liquid is not flat, but curved, then it exerts excess (additional) pressure on the liquid, because surface tension forces act, for a convex surface it is positive, and for a concave surface it is negative. Each infinitesimal element of the contour length is acted upon by a surface tension force: 
tangent to the surface of the sphere.
Decomposing it into two components, we see that the geometric sum is equal to zero, i.e. the resultant of the surface tension forces acting on the cut segment is directed perpendicular to the section plane. And is equal to:
This is the formula for excess (additional) pressure for a convex surface.
For concave:
These two formulas are special cases of Laplace's formula, which determines the excess pressure for an arbitrary liquid surface of double curvature:
Capillary phenomena.
Capillarity – the phenomenon of changes in the height of liquid in capillaries.
the liquid in the capillary rises or is released to a height h at which the pressure of the liquid column (hydrostatic pressure) is balanced by excess pressure, i.e.
As the first of three laws. Therefore this law is called Newton's first law.
First Law mechanics, or law of inertia was formulated by Newton as follows:
Any body is kept in a state of rest or uniform rectilinear motion until it changes this state under the influence of applied forces.
Surrounding any body, whether it is at rest or moving, there are other bodies, some or all of which somehow act on the body and influence the state of its motion. To find out the influence of surrounding bodies, it is necessary to study each individual case.
Let us consider any body at rest that has no acceleration, and the speed is constant and equal to zero. Let's say it will be a ball suspended on a rubber cord. It is at rest relative to the Earth. There are many different bodies around the ball: the cord on which it hangs, many objects in the room and other rooms, and, of course, the Earth. However, the action of all these bodies on the ball is not the same. If, for example, you remove furniture from a room, this will not have any effect on the ball. But if you cut the cord, the ball, under the influence of the Earth, will begin to fall down with acceleration. But until the cord was cut, the ball was at rest. This simple experiment shows that of all the bodies surrounding the ball, only two noticeably influence it: the rubber cord and the Earth. Their combined influence ensures the ball's state of rest. As soon as one of these bodies, the cord, was removed, the state of peace was disrupted. If it were possible to remove the Earth, this would also disturb the peace of the ball: it would begin to move in the opposite direction.
From here we come to the conclusion that the actions of two bodies on the ball - the cord and the Earth - compensate (balance) each other. When they say that the actions of two or more bodies compensate each other, this means that the result of their joint action is the same as if these bodies did not exist at all.
The considered example, as well as other similar examples, allow us to draw the following conclusion: if the actions of bodies compensate each other, then the body under the influence of these bodies is at rest.
Thus we have come to one of basic laws of mechanics which is called Newton's first law:
There are such reference systems relative to which moving bodies maintain their speed constant if they are not acted upon by other bodies or the action of other bodies is compensated.
However, as it turned out over time, Newton's first law is satisfied only in inertial reference systems. Therefore, from the point of view of modern concepts, Newton’s law is formulated as follows:
Reference systems relative to which a free body, when compensating for external influences, moves uniformly and rectilinearly are called inertial reference systems.
Free body in this case, a body is called that is not affected by other bodies.
It must be remembered that Newton's first law deals with bodies that can be represented as material points.
DEFINITION
Formulation of Newton's first law. There are such reference systems relative to which a body maintains a state of rest or a state of uniform rectilinear motion if other bodies do not act on it or the action of other bodies is compensated.
Description of Newton's first law
For example, the ball on the thread hangs at rest because the force of gravity is compensated by the tension of the thread.
Newton's first law is true only in . For example, bodies that are at rest in the cabin of an airplane that is moving uniformly can begin to move without any influence on them from other bodies if the airplane begins to maneuver. In transport, during sudden braking, passengers fall, although no one is pushing them.
Newton's first law shows that a state of rest and state do not require external influences for their maintenance. The property of a free body to maintain its speed unchanged is called inertia. Therefore, Newton's first law is also called law of inertia. Uniform rectilinear motion of a free body is called motion by inertia.
Newton's first law contains two important statements:
- all bodies have the property of inertia;
- inertial frames of reference exist.
It should be remembered that Newton's first law deals with bodies that can be taken as .
The law of inertia is by no means obvious, as it might seem at first glance. His discovery put an end to one long-standing misconception. Before this, for centuries it was believed that in the absence of external influences on the body, it can only be in a state of rest, that rest is, as it were, the natural state of the body. For a body to move at a constant speed, it is necessary for another body to act on it. Everyday experience seemed to confirm this: in order for a cart to move at a constant speed, it must be pulled at all times by a horse; in order for the table to move on the floor, it must be continuously pulled or pushed, etc. Galileo Galilei was the first to point out that this is not true, that in the absence of external influence a body can not only be at rest, but also move rectilinearly and uniformly. Rectilinear and uniform motion is, therefore, the same “natural” state of bodies as rest. In fact, Newton's first law says that there is no difference between a body at rest and uniform motion in a straight line.
It is impossible to test the law of inertia experimentally, because it is impossible to create conditions under which the body would be free from external influences. However, the opposite can always be traced. Anyway. when a body changes the speed or direction of its movement, you can always find a reason - the force that caused this change.
Examples of problem solving
EXAMPLE 1
EXAMPLE 2
| Exercise | A light toy car stands on a table in a uniformly and rectilinearly moving train. When the train braked, the car rolled forward without any external influence. Is the law of inertia fulfilled: a) in the reference frame associated with the train during its rectilinear uniform motion? while braking? b) in the reference frame associated with the Earth? |
| Answer | a) the law of inertia is satisfied in the reference frame associated with the train during its linear motion: the toy car is at rest relative to the train, since the action from the Earth is compensated by the action from the table (reaction of the support). When braking, the law of inertia is not satisfied, since braking is a movement with and the train in this case is not an inertial frame of reference. b) in the reference frame associated with the Earth, the law of inertia is satisfied in both cases - with uniform motion of the train, the toy car moves relative to the Earth at a constant speed (train speed); When the train brakes, the car tries to keep its speed relative to the Earth unchanged, and therefore rolls forward. |
Isaac Newton's Second Law of Motion describes what happens when an external force acts on a massive body at rest or in uniform linear motion. What happens to the body from which this external force is applied? This situation is described 3 Newton's law of motion. It says: The force of action is equal to the force of reaction.
Movements in 1687 in his seminal work Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), in which he formalized the description of how massive bodies move under the influence of external forces.
Newton uncovered the earlier work of Galileo Galilei, which developed the first precise laws of mass motion, according to Greg Botun, a professor of physics at Oregon State University. Galileo's experiments showed that all bodies accelerate at the same rate, regardless of size or mass. Newton also criticized and expanded on the work of René Descartes, who also published a set of laws of nature in 1644, two years after Newton's birth. Descartes' laws are very similar to Newton's first law of motion.
Forces always occur in pairs; when one body pushes towards another, the second body reacts just as strongly. For example, when you push a cart, the cart pushes back towards you; when you pull the rope, the rope swings back towards you; and when gravity pulls you toward the ground, the ground resists your feet. A simplified version of this phenomenon has been expressed as "You cannot touch without being touched."
If body A exerts a force F on body B, then body B exerts an equal and opposite force -F back on body A. The mathematical expression for this is FAB = -FBA
The subscript AB indicates that A exerts a force on B, and BA indicates that B exerts a force on A. The minus sign indicates that the forces are in opposite directions. Often FAB and FBA are referred to as action force and reaction force; however, the choice of which is completely arbitrary.
If one object is much, much more massive than another, especially if the first object is grounded, virtually all of the acceleration is transferred to the second object, and the acceleration of the first object can be safely ignored. For example, if you were to throw a ball to the west, you don't have to assume that you actually caused the Earth's rotation to speed up a little while the ball was in the air. However, if you are standing on roller skates and you throw a bowling ball, you will start to move backwards at a noticeable speed.
One might ask, “If two forces are equal and opposite, why don’t they cancel each other out?” In fact, in some cases they do. Consider a book resting on a table. The weight of the book pushes onto the table with a force mg, and the table resists the book with an equal and opposite force. In this case, the forces cancel each other. The reason for this is that both forces act on the same body, and Newton's third law describes two different bodies acting on each other.
Consider a horse and cart. The horse pulls the cart and the cart leans back onto the horse. The two forces are equal and opposite, so why is the cart moving at all? The reason is that the horse also exerts a force on the ground, which is external to the horse's system, and the ground exerts a reaction on the horse, causing it to accelerate.
Newton's 3rd law in action
Rockets traveling through space embrace all three of Newton's laws of motion.
When the engines fire and propel the rocket forward, it is the result of a reaction. The engine burns fuel, which accelerates towards the rear of the ship. This causes the force in the opposite direction to propel the rocket forward. Motors can also be used on the sides of the rocket to change the direction of travel, or on the front to create a reaction force to slow the rocket down.
And if, while working outside the rocket, the astronaut's rope breaks and they walk away from the rocket, they can use one of their tools, for example, to change direction and return to the rocket. An astronaut can throw a hammer in the exact opposite direction of where they want to go. The hammer will fly away from the rocket very quickly, and the astronaut will return very slowly to the rocket. This is why Newton's third law is considered a fundamental principle of rocket science.
