If the elementary displacement d is written in the form:
According to Newton's II law:
The quantity is called kinetic energy
The work of the resultant of all forces acting on a particle is equal to the change in the kinetic energy of the particle.
or another entry
kinetic dissipative scalar physical
If A > 0, then WC increases (falls)
If A > 0, then WC decreases (throwing).
Moving bodies have the ability to do work even if no forces from other bodies act on them. If a body moves at a constant speed, then the sum of all forces acting on the body is equal to 0 and no work is done. If a body acts with some force in the direction of motion on another body, then it is able to do work. In accordance with Newton's third law, a force of the same magnitude will be applied to a moving body, but directed in the opposite direction. Thanks to the action of this force, the speed of the body will decrease until it comes to a complete stop. The energy WC caused by the motion of a body is called kinetic. A completely stopped body cannot do any work. WC depends on speed and body weight. Changing the direction of speed does not affect kinetic energy.
Newton's first law postulates the presence of such a phenomenon as the inertia of bodies. Therefore it is also known as the Law of Inertia. Inertia is the phenomenon of a body maintaining its speed of movement (both in magnitude and direction) when no forces act on the body. To change the speed of movement, a certain force must be applied to the body. Naturally, the result of the action of forces of equal magnitude on different bodies will be different. Thus, bodies are said to have inertia. Inertia is the property of bodies to resist changes in their current state. The amount of inertia is characterized by body weight. There are such reference systems, called inertial, relative to which the material point in the absence external influences maintains the magnitude and direction of its speed indefinitely.
Newton's second law is a differential law of motion that describes the relationship between a force applied to a material point and the resulting acceleration of that point. In fact, Newton's second law introduces mass as a measure of the manifestation of inertia of a material point in the selected inertial reference frame (IFR). In an inertial reference frame, the acceleration that a material point receives is directly proportional to the resultant of all forces applied to it and inversely proportional to its mass.
Law of Thirds. This law explains what happens to two interacting bodies. Let us take for example a closed system consisting of two bodies. The first body can act on the second with some force, and the second can act on the first with force. How do the forces compare? Newton's third law states: the action force is equal in magnitude and opposite in direction to the reaction force. We emphasize that these forces are applied to different bodies, and therefore are not compensated at all. An action always has an equal and opposite reaction, otherwise the interactions of two bodies on each other are equal and directed in opposite directions.
4 ) The principle of relativity- fundamental physical principle, according to which all physical processes in inertial reference systems proceed in the same way, regardless of whether the system is stationary or in a state of uniform and rectilinear motion.
It follows that all laws of nature are the same in all inertial frames of reference.
There is a distinction between Einstein's principle of relativity (which is given above) and Galileo's principle of relativity, which states the same thing, but not for all laws of nature, but only for the laws of classical mechanics, implying the applicability of Galileo's transformations, leaving open the question of the applicability of the principle of relativity to optics and electrodynamics .
In modern literature, the principle of relativity in its application to inertial frames of reference (most often in the absence of gravity or when it is neglected) usually appears terminologically as Lorentz covariance (or Lorentz invariance).
5)Forces in nature.
Despite the variety of forces, there are only four types of interactions: gravitational, electromagnetic, strong and weak.
Gravitational forces are noticeably manifested on a cosmic scale. One of the manifestations of gravitational forces is the free fall of bodies. The earth imparts to all bodies the same acceleration, which is called the acceleration of gravity g. It varies slightly depending on geographic latitude. At the latitude of Moscow it is 9.8 m/s2.
Electromagnetic forces act between particles having electric charges. Strong and weak interactions manifest themselves inside atomic nuclei and in nuclear transformations.
Gravitational interaction exists between all bodies with masses. The law of universal gravitation, discovered by Newton, states:
The force of mutual attraction between two bodies, which can be taken as material points, is directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
The proportionality coefficient y is called the gravitational constant. It is equal to 6.67 10-11 N m2/kg2.
If only the gravitational force from the Earth acts on the body, then it is equal to mg. This is the force of gravity G (without taking into account the rotation of the Earth). The force of gravity acts on all bodies on Earth, regardless of their movement.
When a body moves with the acceleration of gravity (or even with a lower acceleration directed downward), the phenomenon of complete or partial weightlessness is observed.
Complete weightlessness - no pressure on the stand or gimbal. Weight is the pressure force of a body on a horizontal support or the tensile force of a thread from a body suspended from it, which arises due to gravitational attraction given body to Earth.
The forces of attraction between bodies are indestructible, while the weight of the body can disappear. Thus, in a satellite that moves at escape velocity around the Earth, there is no weight, just like in an elevator falling with acceleration g.
Examples of electromagnetic forces are the forces of friction and elasticity. There are sliding friction forces and rolling friction forces. The sliding friction force is much greater than the rolling friction force.
The friction force depends in a certain interval on the applied force, which tends to move one body relative to another. By applying a force of varying magnitude, we will see that small forces cannot move the body. In this case, a compensating force of static friction arises.
In the absence of forces shifting the body, the static friction force is zero. The static friction force acquires its greatest significance at the moment when one body begins to move relative to another. In this case, the static friction force becomes equal to the sliding friction force:
where n is the coefficient of friction, N is the force of normal (perpendicular) pressure. The friction coefficient depends on the substance of the rubbing surfaces and their roughness.
6) Law of conservation of momentum ( The law of conservation of momentum) states that the vector sum of the impulses of all bodies (or particles) of a closed system is a constant quantity.
In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. From Newton's laws it can be shown that when moving in empty space, momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces.
Like any of the fundamental conservation laws, the law of conservation of momentum describes one of the fundamental symmetries - the homogeneity of space.
Center of mass in mechanics- this is a geometric point that characterizes the movement of a body or a system of particles as a whole. The concept of center of mass is widely used in physics.
The motion of a rigid body can be considered as a superposition of the motion of the center of mass and the rotational motion of the body around its center of mass. In this case, the center of mass moves in the same way as a body with the same mass, but infinitely small dimensions (material point) would move. The latter means, in particular, that all Newton's laws are applicable to describe this movement. In many cases, you can completely ignore the size and shape of a body and consider only the movement of its center of mass. It is often convenient to consider the movement of a closed system in a reference system associated with the center of mass. Such a reference system is called the center of mass system (C-system), or the center of inertia system. In it, the total momentum of a closed system always remains equal to zero, which makes it possible to simplify the equations of its motion.
Energy- a scalar physical quantity, which is a unified measure of various forms of motion of matter and a measure of the transition of the motion of matter from one form to another. Mechanical work is a physical quantity that is a scalar quantitative measure of the action of a force or forces on a body or system, depending on the numerical magnitude and direction of the force (forces) and on the movement of the point (points) of the body or system. Energy is a measure of the ability of a physical system to perform work, Therefore, quantitatively, energy and work are expressed in the same units.
Mechanical work and mechanical energy are identified.
Power- a physical quantity equal to the ratio of work performed over a certain period of time to this period of time.
Kinetic energy- the energy of a mechanical system, depending on the speed of movement of its points. The kinetic energy of translational and rotational motion is often released. The SI unit of measurement is the Joule. More strictly, kinetic energy is the difference between the total energy of the system and its rest energy; Thus, kinetic energy is the part of the total energy due to motion.
Potential energy- a scalar physical quantity that characterizes the ability of a certain body (or material point) to do work due to its location in the field of action of forces. A correct definition of potential energy can only be given in a field of forces, the work of which depends only on the initial and final position of the body, but not on the trajectory of its movement. Such forces are called conservative. Also potential energy is a characteristic of the interaction of several bodies or a body and a field. Any physical system tends to a state with the lowest potential energy. The potential energy in the Earth's gravitational field near the surface is approximately expressed by the formula:
where Ep is the potential energy of the body, m is the mass of the body, g is the acceleration of gravity, h is the height of the center of mass of the body above an arbitrarily chosen zero level.
On the physical meaning of the concept of potential energy
If kinetic energy can be determined for one separate body, then potential energy always characterizes at least two bodies or the position of a body in an external field.
Kinetic energy is characterized by speed; potential - by the relative position of the bodies.
The main physical meaning is not the value of potential energy itself, but its change.
8) In physics, mechanical energy describes the sum of potential and kinetic energy available in the components of a mechanical system. Mechanical energy is the energy associated with the movement of an object or its position. Law of conservation of mechanical energy states that if a body or system is subjected to only conservative forces, then the total mechanical energy of that body or system remains constant. In an isolated system, where only conservative forces act, the total mechanical energy is conserved.
Related information.
Kinetic energy- the energy of a mechanical system, depending on the speed of movement of its points. The kinetic energy of translational and rotational motion is often released. The SI unit of measurement is Joule. More strictly, kinetic energy is the difference between the total energy of a system and its rest energy; Thus, kinetic energy is the part of the total energy due to motion.
Let us consider the case when a body of mass m there is a constant force (it can be the resultant of several forces) and force vectors and the movements are directed along one straight line in one direction. In this case, the work done by the force can be defined as A = F∙s. The modulus of force according to Newton's second law is equal to F = m∙a, and the displacement module s with uniformly accelerated rectilinear motion is associated with the modules of the initial υ 1 and final υ 2 speed and acceleration A expression
From here we get to work
A physical quantity equal to half the product of a body’s mass and the square of its speed is calledkinetic energy of the body .
Kinetic energy is represented by the letter E k .
Then equality (1) can be written as follows:
A = E k 2 – E k 1 . (3)
Kinetic energy theorem:
the work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body.
Since the change in kinetic energy is equal to the work of force (3), the kinetic energy of a body is expressed in the same units as work, i.e., in joules.
If the initial speed of movement of a body of mass T is zero and the body increases its speed to the value υ , then the work done by the force is equal to the final value of the kinetic energy of the body:
(4)
Physical meaning kinetic energy:
The kinetic energy of a body moving with a speed v shows how much work must be done by a force acting on a body at rest in order to impart this speed to it.
Potential energy- the minimum work that must be done to move a body from a certain reference point to a given point in the field of conservative forces. Second definition: potential energy is a function of coordinates, which is a term in the Lagrangian of the system and describes the interaction of the elements of the system. Third definition: potential energy is the energy of interaction. Units [J]
Potential energy is assumed to be zero for a certain point in space, the choice of which is determined by the convenience of further calculations. The process of selecting a given point is called potential energy normalization. It is also clear that the correct definition of potential energy can only be given in the field of forces, the work of which depends only on the initial and final position of the body, but not on the trajectory of its movement. Such forces are called conservative.
The potential energy of a body raised above the Earth is the energy of interaction between the body and the Earth by gravitational forces. The potential energy of an elastically deformed body is the energy of interaction of individual parts of the body with each other by elastic forces.
Potential are calledstrength , the work of which depends only on the initial and final position of a moving material point or body and does not depend on the shape of the trajectory.
In a closed trajectory, the work done by the potential force is always zero. Potential forces include gravitational forces, elastic forces, electrostatic forces and some others.
Powers , the work of which depends on the shape of the trajectory, are callednon-potential . When a material point or body moves along a closed trajectory, the work done by the nonpotential force is not equal to zero.
Potential energy of interaction of a body with the Earth.
Let's find the work done by gravity F t when moving a body of mass T vertically down from a height h 1 above the Earth's surface to a height h 2 (Fig. 1).
If the difference h 1 –h 2 is negligible compared to the distance to the center of the Earth, then the force of gravity F T during the movement of the body can be considered constant and equal mg.
Since the displacement coincides in direction with the gravity vector, the work done by gravity is equal to
A = F∙s = m∙g∙(h l –h 2). (5)
Let us now consider the movement of a body along an inclined plane. When moving a body down an inclined plane (Fig. 2), the force of gravity F T = m∙g does work
A = m∙g∙s∙cos a = m∙g∙h, (6)
Where h– height of the inclined plane, s– displacement module equal to the length of the inclined plane.
Movement of a body from a point IN exactly WITH along any trajectory (Fig. 3) can be mentally imagined as consisting of movements along sections of inclined planes with different heights h", h" etc. Work A gravity all the way from IN V WITH equal to the sum of work on individual sections of the route:
Where h 1 and h 2 – heights from the Earth’s surface at which the points are located, respectively IN And WITH.
Equality (7) shows that the work of gravity does not depend on the trajectory of the body and is always equal to the product of the gravity modulus and the difference in heights in the initial and final positions.
When moving downward, the work of gravity is positive, when moving up it is negative. The work done by gravity on a closed trajectory is zero .
Equality (7) can be represented as follows:
A = – (m∙g∙h 2 – m∙g∙h l). (8)
A physical quantity equal to the product of the mass of a body by the acceleration modulus of free fall and the height to which the body is raised above the surface of the Earth is calledpotential energy interaction between the body and the Earth.
Work done by gravity when moving a body of mass T from a point located at a height h 2 , to a point located at a height h 1 from the Earth's surface, along any trajectory, is equal to the change in the potential energy of interaction between the body and the Earth, taken with the opposite sign.
A= – (ER 2 – ER 1). (9)
Potential energy is indicated by the letter ER.
The value of the potential energy of a body raised above the Earth depends on the choice of the zero level, i.e., the height at which the potential energy is assumed to be zero. It is usually assumed that the potential energy of a body on the Earth's surface is zero.
With this choice of the zero level, the potential energy ER body at height h above the Earth's surface is equal to the product of mass m bodies to the free fall acceleration module g and distance h it from the surface of the Earth:
Ep = m∙g∙h. (10)
Physical meaning potential energy of interaction of a body with the Earth:
the potential energy of a body on which gravity acts is equal to the work done by gravity when moving the body to the zero level.
Unlike the kinetic energy of translational motion, which can only have positive values, the potential energy of a body can be both positive and negative. Body mass m, located at a height h, Where h 0 ( h 0 – zero height), has negative potential energy:
Ep = –m∙gh
Potential energy of gravitational interaction
Potential energy of gravitational interaction of a system of two material points with masses T And M, located at a distance r one from the other is equal
(11)
Where G is the gravitational constant, and the zero of the potential energy reference ( Ep= 0) accepted at r = ∞. Potential energy of gravitational interaction of a body with mass T with the Earth, where h– height of the body above the Earth’s surface, M 3 – mass of the Earth, R 3 is the radius of the Earth, and the zero of the potential energy reading is chosen at h= 0.
(12)
Under the same condition of choosing zero reference, the potential energy of gravitational interaction of a body with mass T with Earth for low altitudes h(h« R 3) equal to
Ep = m∙g∙h,
where is the magnitude of the acceleration due to gravity near the Earth's surface.
Potential energy of an elastically deformed body
Let us calculate the work done by the elastic force when the deformation (elongation) of the spring changes from a certain initial value x 1 to final value x 2 (Fig. 4, b, c).
The elastic force changes as the spring deforms. To find the work of the elastic force, you can take the average value of the force modulus (since the elastic force depends linearly on x) and multiply by the displacement module:
(13)
Where 
From here
(14)
A physical quantity equal to half the product of the rigidity of a body by the square of its deformation is calledpotential energy elastically deformed body:
From formulas (14) and (15) it follows that the work of the elastic force is equal to the change in the potential energy of an elastically deformed body, taken with the opposite sign:
A = –(ER 2 – ER 1). (16)
If x 2 = 0 and x 1 = x, then, as can be seen from formulas (14) and (15),
ER = A.
Then physical meaning potential energy of a deformed body
the potential energy of an elastically deformed body is equal to the work done by the elastic force when the body transitions to a state in which the deformation is zero.
Energy is a scalar quantity. The SI unit of energy is the Joule.
Kinetic and potential energy
There are two types of energy - kinetic and potential.
DEFINITION
Kinetic energy- this is the energy that a body possesses due to its movement:
DEFINITION
Potential energy is energy that is determined by the relative position of bodies, as well as the nature of the interaction forces between these bodies.
Potential energy in the Earth's gravitational field is the energy due to the gravitational interaction of a body with the Earth. It is determined by the position of the body relative to the Earth and is equal to the work of moving the body from a given position to the zero level:
Potential energy is the energy caused by the interaction of body parts with each other. It is equal to the work of external forces in tension (compression) of an undeformed spring by the amount:
A body can simultaneously possess both kinetic and potential energy.
The total mechanical energy of a body or system of bodies is equal to the sum of the kinetic and potential energies of the body (system of bodies):
Law of energy conservation
For a closed system of bodies, the law of conservation of energy is valid:
In the case when a body (or a system of bodies) is acted upon by external forces, for example, the law of conservation of mechanical energy is not satisfied. In this case, the change in the total mechanical energy of the body (system of bodies) is equal to the external forces:
The law of conservation of energy allows us to establish a quantitative connection between various forms of motion of matter. Just like , it is valid not only for, but also for all natural phenomena. The law of conservation of energy says that energy in nature cannot be destroyed just as it cannot be created from nothing.
In the most general view The law of conservation of energy can be formulated as follows:
- Energy in nature does not disappear and is not created again, but only transforms from one type to another.
Examples of problem solving
EXAMPLE 1
| Exercise | A bullet flying at a speed of 400 m/s hits an earthen shaft and travels 0.5 m to a stop. Determine the resistance of the shaft to the movement of the bullet if its mass is 24 g. |
| Solution | The drag force of the shaft is an external force, so the work done by this force is equal to the change in the kinetic energy of the bullet: Since the resistance force of the shaft is opposite to the direction of movement of the bullet, the work done by this force is: Change in bullet kinetic energy: Thus, we can write: where does the resistance force of the earthen rampart come from: Let's convert the units to the SI system: g kg. Let's calculate the resistance force: |
| Answer | The shaft resistance force is 3.8 kN. |
EXAMPLE 2
| Exercise | A load weighing 0.5 kg falls from a certain height onto a plate weighing 1 kg, mounted on a spring with a stiffness coefficient of 980 N/m. Determine the magnitude of the greatest compression of the spring if at the moment of impact the load had a speed of 5 m/s. The impact is inelastic. |
| Solution | Let us write down a load + plate for a closed system. Since the impact is inelastic, we have: where does the velocity of the plate with the load after impact come from:
According to the law of conservation of energy, the total mechanical energy of the load together with the plate after impact is equal to the potential energy of the compressed spring: |
This paragraph does not provide any new information, but it does highlight and clarify some important features of potential energy that should be noted.
Potential energy - the energy of interaction between bodies
It is important to clearly understand that kinetic energy is a quantity related to one body, and potential energy is always the energy of interaction of at least two bodies (or parts of one body) with each other. The concept of potential energy refers to a system of bodies, and not to one body. If there are several bodies in a system, then the total potential energy of the system is equal to the sum of the potential energies of all pairs of interacting bodies (any body interacts with each of the others).
Rice. 6.15
Potential energy characterizes the interaction of bodies precisely because the very concept of force always refers to two bodies: the body on which the force acts, and the body from which it acts.
When obtaining an expression for kinetic energy, we did not use this feature of the force, immediately replacing it in the formula for work with the product of mass and acceleration according to Newton’s second law. That is why the concept of kinetic energy refers to one body.
We obtained the expression for potential energy using the known dependence of forces on the location of interacting bodies, without using the equations of motion. The equality A = -AEp determines the potential energy regardless of the equations of motion. Therefore, potential energy is simply another characteristic (along with force) of the mutual action of bodies on each other.
Often, when deriving a formula connecting the change in potential energy with the work of forces, one of the bodies of the system is taken as stationary. Thus, when considering the fall of a body to the Earth under the influence of gravity, the displacement of the Earth is neglected. Therefore, the work of the interaction forces between the Earth and the body is reduced to the work of only one force acting on the body.
Or another example. A compressed or extended spring acting on a body is usually fixed at one end, and that end of the spring does not move (in fact, it is attached to the globe). In this case, the work is performed only by the elastic force of the deformed spring applied to the body.
Because of this, the potential energy of a system of two bodies gets used to being considered as the energy of one body. This can lead to confusion.
In fact, in all cases the following statement is true: the change in the potential energy of two bodies interacting with forces that depend only on the distance between the bodies is equal to the work of these forces taken with a minus sign:
A = F12- Ar, + F21 ¦ Ar2 = ~ = -AEp. (6.7.1)
Here F12 is the force acting on body 1 from body 2, and F21 is the force acting on body 2 from body 1 (Fig. 6.15).
Zero potential energy level
According to equation (6.7.1), the work of interaction forces determines not the potential energy itself, but its change.
Since work determines only the change in potential energy, then only the change in energy in mechanics has physical meaning. Therefore, you can arbitrarily choose the state of the system in which its potential energy is considered equal to zero. This state corresponds to a zero level of potential energy. Not a single phenomenon in nature or technology is determined by the value of potential energy itself. What is important is the difference between the potential energy values in the final and initial states of the system of bodies.
The choice of the zero level is made in different ways and is dictated solely by considerations of convenience, i.e., the simplicity of writing the equation expressing the law of conservation of energy. Typically, the state of the system with minimum energy is chosen as the state with zero potential energy. Then potential energy is always positive.
A spring has a minimum potential energy in the absence of deformation, while a stone has a minimum potential energy when it lies on the surface
2
Earth. Therefore, in the first case, Ep = ^i^L (Fig. 6.16), and in the second case, Ep = mgh (Fig. 6.17). But you can add any constant value C to these expressions, and that’s okay
2
won't change. We can assume that E = ^^ + C and E = mgh + C.
g D R
If in the second case we set C = -mgh0, then this will mean that the zero energy level is taken to be the energy at a height hQ above the Earth’s surface.
ABOUT
h
m
oh oh
Sometimes it is not possible to choose the zero potential energy level so that the minimum energy is zero. So, for example, the potential energy of two bodies interacting through the forces of universal gravity can be written as follows:
m-i t.* -G-
+ C. Fig. 6.18
As r -» 0, the first term tends to -°o. Therefore, the minimum energy value can be considered equal to zero only at C = °o. But, of course, you cannot use equations that include an infinite quantity. Therefore, here it is more convenient to put C = O and thereby take the potential energy in a state in which the bodies are infinitely distant from each other (r = °o) as the zero level. Then the zero level will correspond not to the minimum energy, but to the maximum. For any finite value of g, the potential energy is negative (Fig. 6.18).
Independence of potential energy from the choice of reference frame
Let us note once again that the concept of potential energy makes sense for systems in which the interaction forces are conservative, that is, they depend only on the distance between the bodies or their parts. Accordingly, potential energy depends on the distance between bodies or their parts: on the height of the stone above the Earth’s surface, on the length of the spring, on the distance between point bodies. Potential energy does not directly depend on the coordinates of bodies. (Only insofar as distances are functions of coordinates can we speak of dependence on coordinates.) This implies a very important conclusion, which is usually not paid attention to. Since the distances in all reference systems, moving and stationary, are the same, the potential energy does not depend on the choice of reference system.
But how can this be? After all, AEp = -A, and the work depends on the choice of the reference system. This is where the fact clearly manifests itself that potential energy is the energy of interaction between two bodies, and its change is determined by the work of forces acting on both bodies. When moving from a stationary system to a moving one, the work done by both forces changes, but the total work remains unchanged. In fact, if in some reference frame work is done in time At
A1 = $12 " + ^21 " A?2"
then in another system moving relative to the first, the work is equal to
A2 = F12 (Dgi + Ar0) + F21 (Ar2 + Ar0),
where Ar0 is the movement of the reference systems relative to each other during the time At.
Since, according to Newton’s third law, F12 = ~F21, then
F12 ¦ Ar0 + F2j Ar0 = 0. Therefore, At = A2.
Differences between potential and kinetic energy
Kinetic energy depends only on the speeds of bodies, and potential energy depends only on the distances between them.
Further, positive work of internal forces always leads to an increase in kinetic energy, but necessarily reduces potential energy:
AEk=A, but AEp = -A.
Kinetic energy is always positive, but potential energy can be either positive or negative.
The change in kinetic energy is always equal to the work of the forces acting on the body, and the change in potential energy is equal (with a minus sign) to the work of only conservative forces (but not friction forces that depend on speed).
Both potential and kinetic energy are functions of the state of the system, i.e. they are precisely determined if the coordinates and velocities of all bodies of the system are known.
