Mechanical energy and its types. Mechanical energy of the body What determines the mechanical energy of the body
The purpose of this article is to reveal the essence of the concept of “mechanical energy”. Physics widely uses this concept both practically and theoretically.
Work and Energy
Mechanical work can be determined if the force acting on a body and the displacement of the body are known. There is another way to calculate mechanical work. Let's look at an example:
The figure shows a body that can be in different mechanical states (I and II). The process of transition of a body from state I to state II is characterized by mechanical work, that is, during the transition from state I to state II, the body can perform work. When performing work, the mechanical state of the body changes, and the mechanical state can be characterized by one physical quantity - energy.
Energy is a scalar physical quantity of all forms of motion of matter and options for their interaction.
What is mechanical energy equal to?
Mechanical energy is a scalar physical quantity that determines the ability of a body to do work.
A = ∆E
Since energy is a characteristic of the state of a system at a certain point in time, work is a characteristic of the process of changing the state of the system.
Energy and work have the same units of measurement: [A] = [E] = 1 J.
Types of mechanical energy
Mechanical free energy is divided into two types: kinetic and potential.
Kinetic energy is the mechanical energy of a body, which is determined by the speed of its movement.
E k = 1/2mv 2
Kinetic energy is inherent in moving bodies. When they stop, they perform mechanical work.
In different reference systems, the velocities of the same body at an arbitrary moment in time can be different. Therefore, kinetic energy is a relative quantity; it is determined by the choice of the reference system.
If a force (or several forces at the same time) acts on a body during movement, the kinetic energy of the body changes: the body accelerates or stops. In this case, the work of the force or the work of the resultant of all forces that are applied to the body will be equal to the difference in kinetic energies:
A = E k1 - E k 2 = ∆E k
This statement and formula was given a name - kinetic energy theorem.
Potential energy name the energy caused by the interaction between bodies.
When a body weighs m from high h the force of gravity does the work. Since work and energy change are related by an equation, we can write a formula for the potential energy of a body in a gravitational field:
Ep = mgh
Unlike kinetic energy E k potential E p may have a negative value when h<0 (for example, a body lying at the bottom of a well).
Another type of mechanical potential energy is strain energy. Compressed to distance x spring with stiffness k has potential energy (strain energy):
E p = 1/2 kx 2
Deformation energy has found wide application in practice (toys), in technology - automatic machines, relays and others.
E = E p + E k
Total mechanical energy bodies call the sum of energies: kinetic and potential.
Law of conservation of mechanical energy
Some of the most accurate experiments carried out in the mid-19th century by the English physicist Joule and the German physicist Mayer showed that the amount of energy in closed systems remains unchanged. It only passes from one body to another. These studies helped discover law of energy conservation:
The total mechanical energy of an isolated system of bodies remains constant during any interactions of the bodies with each other.
Unlike impulse, which does not have an equivalent form, energy has many forms: mechanical, thermal, energy of molecular motion, electrical energy with charge interaction forces, and others. One form of energy can be converted into another, for example, kinetic energy is converted into thermal energy during the braking process of a car. If there are no friction forces and no heat is generated, then the total mechanical energy is not lost, but remains constant in the process of movement or interaction of bodies:
E = E p + E k = const
When the friction force between bodies acts, then a decrease in mechanical energy occurs, however, even in this case it is not lost without a trace, but turns into thermal (internal). If an external force performs work on a closed system, then the mechanical energy increases by the amount of work performed by this force. If a closed system performs work on external bodies, then the mechanical energy of the system is reduced by the amount of work performed by it.
Each type of energy can be completely transformed into any other type of energy.
In mechanics, there are two types of energy: kinetic and potential. Kinetic energy call the mechanical energy of any freely moving body and measure it by the work that the body could do when it slows down to a complete stop.
Let the body IN, moving at speed v, begins to interact with another body WITH and at the same time it slows down. Therefore the body IN affects the body WITH with some force F and on the elementary section of the path ds does work
According to Newton's third law, body B is simultaneously acted upon by a force -F, the tangent component of which -F τ causes a change in the numerical value of the body's speed. According to Newton's second law
Hence,
The work done by the body until it comes to a complete stop is:
So, the kinetic energy of a translationally moving body is equal to half the product of the mass of this body by the square of its speed:
(3.7)
From formula (3.7) it is clear that the kinetic energy of a body cannot be negative ( Ek ≥ 0).
If the system consists of n progressively moving bodies, then to stop it it is necessary to brake each of these bodies. Therefore, the total kinetic energy of a mechanical system is equal to the sum of the kinetic energies of all bodies included in it:
(3.8)
From formula (3.8) it is clear that E k depends only on the magnitude of the masses and speeds of movement of the bodies included in it. In this case, it does not matter how the body mass m i gained speed ν i. In other words, the kinetic energy of a system is a function of its state of motion.
Speeds ν i depend significantly on the choice of reference system. When deriving formulas (3.7) and (3.8), it was assumed that the motion is considered in an inertial reference frame, since otherwise Newton's laws could not be used. However, in different inertial reference systems moving relative to each other, the speed ν i i th body of the system, and, consequently, its Eki and the kinetic energy of the entire system will not be the same. Thus, the kinetic energy of the system depends on the choice of the reference frame, i.e. is the quantity relative.
Potential energy- this is the mechanical energy of a system of bodies, determined by their relative position and the nature of the interaction forces between them.
Numerically, the potential energy of a system in its given position is equal to the work that will be done by the forces acting on the system when moving the system from this position to the one where the potential energy is conventionally assumed to be zero ( E n= 0). The concept of “potential energy” applies only to conservative systems, i.e. systems in which the work of the acting forces depends only on the initial and final positions of the system. So, for a load weighing P, raised to a height h, the potential energy will be equal En = Ph (E n= 0 at h= 0); for a load attached to a spring, E n = kΔl 2 / 2, Where Δl- elongation (compression) of the spring, k– its stiffness coefficient ( E n= 0 at l= 0); for two particles with masses m 1 And m 2, attracted by the law of universal gravitation, 
, Where γ
– gravitational constant, r– distance between particles ( E n= 0 at r → ∞).
Let's consider the potential energy of the Earth system - a body of mass m, raised to a height h above the surface of the Earth. The decrease in the potential energy of such a system is measured by the work of gravitational forces performed during the free fall of a body to the Earth. If a body falls vertically, then
Where E no– potential energy of the system at h= 0 (the “-” sign indicates that the work is done due to the loss of potential energy).
If the same body falls down an inclined plane of length l and with an angle of inclination α to the vertical ( lcosα = h), then the work done by the gravitational forces is equal to the previous value:
If, finally, the body moves along an arbitrary curvilinear trajectory, then we can imagine this curve consisting of n small straight sections Δl i. The work done by the gravitational force on each of these sections is equal to
Along the entire curvilinear path, the work done by the gravitational forces is obviously equal to:
So, the work of gravitational forces depends only on the difference in heights of the starting and ending points of the path.
Thus, a body in a potential (conservative) field of forces has potential energy. With an infinitesimal change in the configuration of the system, the work of conservative forces is equal to the increase in potential energy taken with a minus sign, since the work is done due to the decrease in potential energy:
In turn, work dA expressed as the dot product of force F to move dr, so the last expression can be written as follows:
(3.9)
Therefore, if the function is known E n(r), then from expression (3.9) one can find the force F by module and direction.
For conservative forces
Or in vector form
Where
(3.10)
The vector defined by expression (3.10) is called gradient of the scalar function P; i, j, k- unit vectors of coordinate axes (orts).
Specific type of function P(in our case E n) depends on the nature of the force field (gravitational, electrostatic, etc.), as was shown above.
Total mechanical energy W system is equal to the sum of its kinetic and potential energies:
From the definition of the potential energy of a system and the examples considered, it is clear that this energy, like kinetic energy, is a function of the state of the system: it depends only on the configuration of the system and its position in relation to external bodies. Consequently, the total mechanical energy of the system is also a function of the state of the system, i.e. depends only on the position and velocities of all bodies in the system.
Total mechanical energy characterizes the movement and interaction of bodies, therefore, it depends on the speeds and relative positions of the bodies.
The total mechanical energy of a closed mechanical system is equal to the sum of the kinetic and potential energy of the bodies of this system:
Law of energy conservation
The law of conservation of energy is a fundamental law of nature.
In Newtonian mechanics, the law of conservation of energy is formulated as follows:
The total mechanical energy of an isolated (closed) system of bodies remains constant.
In other words:
Energy does not arise from nothing and does not disappear anywhere, it can only move from one form to another.
Classic examples of this statement are: a spring pendulum and a pendulum on a string (with negligible damping). In the case of a spring pendulum, during the process of oscillation, the potential energy of the deformed spring (which has a maximum in the extreme positions of the load) transforms into the kinetic energy of the load (reaching a maximum at the moment the load passes the equilibrium position) and vice versa. In the case of a pendulum on a string, the potential energy of the load is converted into kinetic energy and vice versa.
2 Equipment
2.1 Dynamometer.
2.2 Laboratory tripod.
2.3 Weight weighing 100 g – 2 pcs.
2.4 Measuring ruler.
2.5 A piece of soft cloth or felt.
3 Theoretical background
The experimental setup diagram is shown in Figure 1.
The dynamometer is mounted vertically in the tripod leg. A piece of soft cloth or felt is placed on the tripod. When hanging weights from the dynamometer, the tension of the dynamometer spring is determined by the position of the pointer. In this case, the maximum elongation (or static displacement) of the spring X 0 occurs when the elastic force of a spring with stiffness k balances the force of gravity of the load with mass T:
kx 0 =mg, (1)
Where g = 9.81 - free fall acceleration.
Hence,
Static displacement characterizes the new equilibrium position O" of the lower end of the spring (Fig. 2).
If the load is pulled down a distance A from point O" and release at point 1, then periodic oscillations of the load occur. At points 1 and 2, called turning points, the load stops, reversing its direction of movement. Therefore, at these points the speed of the load is v = 0.
Maximum speed v m ax the load will be at the midpoint O. Two forces act on the oscillating load: the constant force of gravity mg and variable elastic force kx. Potential energy of a body in a gravitational field at an arbitrary point with coordinate X equal to mgx. The potential energy of a deformed body is correspondingly equal to .
In this case, the point X = 0, corresponding to the pointer position for an unstretched spring.
The total mechanical energy of a load at an arbitrary point is the sum of its potential and kinetic energy. Neglecting friction forces, we use the law of conservation of total mechanical energy.
Let us equate the total mechanical energy of the load at point 2 with the coordinate -(X 0 -A) and at point O" with coordinate -X 0 :
Opening the brackets and carrying out simple transformations, we reduce formula (3) to the form
Then the maximum load speed module
The spring constant can be found by measuring the static displacement X 0 . As follows from formula (1),
Let's summarize some results. In the previous paragraphs it was clarified that:
1) if individual bodies of the system move at certain speeds, then work can be obtained from them by reducing the kinetic energy of these bodies:
where is equal to the sum of changes in the kinetic energy of all bodies of the system;
2) if any conservative forces act in a system of bodies, then work can also be obtained by reducing
potential energy of this system:
Therefore, we can say that the total work that such a system can produce will always be equal to
The sum of the potential and kinetic energies of a system of bodies is called the total energy of the system:
The total energy of a system determines the work that can be obtained from a given system of bodies when it interacts with any other bodies not included in this system.
Let us first determine what can happen to the energy of an isolated system if the bodies are given the opportunity to move freely under the influence of internal forces.
Let a body of mass be at a height above the Earth’s surface and have a speed (Fig. 5.33). In this position the body will have kinetic energy and potential energy. The total energy of the system will be equal to
Let us assume that the body has moved to a height and its speed has become equal. During this movement, the force of gravity will do work
All this work will be spent on increasing the kinetic energy of the body:
(There is no friction or external forces.) Let’s substitute the value of work into this expression and rearrange the terms of the equation:
The left side of the found expression determines the total energy of the system for the initial moment of time:
The right side determines the total energy of the system for a final moment of time:
As a result, we can write:
It turned out that when the bodies of an isolated system move only under the influence of internal forces, the total energy of the system does not change. When bodies moved, only part of the potential energy was converted into kinetic energy. This is the law of conservation of energy, which can be formulated as follows: in an isolated system of bodies, the total energy remains constant throughout the movement of the bodies; in the system only transformations of energy from one type to another occur.
It also follows that if any external forces act on the system, then changes in the total energy of the system are equal to the work of these external forces.
If friction forces act in a system, then the total energy of the system decreases as bodies move. It is spent working against these forces. At the same time, the work of friction forces produces heating. As mentioned earlier, when friction forces work, mechanical motion is converted into thermal motion. The amount of heat released is exactly equal to the decrease in the total mechanical energy of the system.
System particles could be any body, gas, mechanism, solar system, etc.
The kinetic energy of a system of particles, as mentioned above, is determined by the sum of the kinetic energies of the particles included in this system.
The potential energy of the system is the sum of own potential energy particles of the system, and the potential energy of the system in the external field of potential forces.
The intrinsic potential energy is determined by the relative arrangement of particles belonging to a given system (i.e., its configuration), between which potential forces act, as well as the interaction between individual parts of the system. It can be shown that the work of all internal potential forces when changing the configuration of the system is equal to the decrease in the system’s own potential energy:
. (3.23)
Examples of intrinsic potential energy are the energy of intermolecular interaction in gases and liquids, and the energy of electrostatic interaction of stationary point charges. An example of external potential energy is the energy of a body raised above the surface of the Earth, since it is caused by the action of a constant external potential force on the body - the force of gravity.
Let us divide the forces acting on a system of particles into internal and external, and internal into potential and non-potential. Let us represent (3.10) in the form
Let us rewrite (3.24) taking into account (3.23):
The quantity, the sum of the kinetic and self-potential energy of the system, is total mechanical energy of the system. Let us rewrite (3.25) as:
i.e., the increase in the mechanical energy of the system is equal to the algebraic sum of the work of all internal non-potential forces and all external forces.
If we put in (3.26) A external=0 (this equality means that the system is closed) and (which is equivalent to the absence of internal non-potential forces), we obtain:
Both equalities (3.27) are expressions law of conservation of mechanical energy: the mechanical energy of a closed system of particles, in which there are no non-potential forces, is conserved during movement, Such a system is called conservative. With a sufficient degree of accuracy, the Solar System can be considered a closed conservative system. When a closed conservative system moves, the total mechanical energy is conserved, while the kinetic and potential energy changes. However, these changes are such that the increase in one of them is exactly equal to the decrease in the other.
If a closed system is not conservative, i.e., non-potential forces act in it, for example, friction forces, then the mechanical energy of such a system decreases, as it is spent on work against these forces. The law of conservation of mechanical energy is only a separate manifestation of the universal law of conservation and transformation of energy existing in nature: energy is never created or destroyed, it can only pass from one form to another or be exchanged between individual parts of matter. At the same time, the concept of energy is expanded by the introduction of concepts about new forms of it besides mechanical - energy of the electromagnetic field, chemical energy, nuclear energy, etc. The universal law of conservation and transformation of energy covers those physical phenomena to which Newton's laws do not apply. This law has independent significance, since it was obtained on the basis of generalizations of experimental facts.
Example 3.1. Find the work done by an elastic force acting on a material point along a certain x axis. Force obeys law, where x is the displacement of the point from the initial position (in which x=x 1), - unit vector in the x-axis direction.
Let us find the elementary work of elastic force when moving a point by an amount dx. In formula (3.1) for elementary work we substitute the expression for force:
.
Then we find the work of force, perform integration along the axis x ranging from x 1 before x:
. (3.28)
Formula (3.28) can be used to determine the potential energy of a compressed or stretched spring, which is initially in a free state, i.e. x 1 =0(coefficient k called the spring constant). The potential energy of a spring under compression or tension is equal to the work against elastic forces, taken with the opposite sign:
.
Example 3.2 Application of the theorem on the change in kinetic energy.
Find the minimum speed u, which must be communicated to the projectile, so that it rises to a height H above the Earth's surface(neglect air resistance).
Let's direct the coordinate axis from the center of the Earth in the direction of flight of the projectile. The initial kinetic energy of the projectile will be spent working against the potential forces of the Earth's gravitational attraction. Formula (3.10) taking into account formula (3.3) can be represented as:
.
Here A– work against the force of gravitational attraction of the Earth (
, g gravitational constant, r– distance measured from the center of the Earth). The minus sign appears due to the fact that the projection of the force of gravitational attraction on the direction of movement of the projectile is negative. Integrating the last expression and taking into account that T(R+H)=0, T(R) = mυ 2 /2, we get:
Having solved the resulting equation for υ, we find:
where is the acceleration of gravity on the Earth's surface.
