Due to its location in the field of action of forces. Another definition: potential energy is a function of coordinates, which is a term in the Lagrangian of the system and describes the interaction of elements of the system. The term "potential energy" was coined in the 19th century by Scottish engineer and physicist William Rankine.
The SI unit of energy is the Joule.
Potential energy is assumed to be zero for a certain configuration of bodies in space, the choice of which is determined by the convenience of further calculations. The process of choosing this configuration is called normalization potential energy.
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.
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.
Kinetic energy
Let's consider a system consisting of one particle and write the equation of motion:
There is a resultant of all forces acting on a body.
Let us scalarly multiply the equation by the displacement of the particle. Considering that , we get:
- moment of inertia of the body
- angular velocity of the body.
Law of energy conservation.
From a fundamental point of view, according to Noether’s theorem, the law of conservation of energy is a consequence of the homogeneity of time and in this sense is universal, that is, inherent in systems of very different physical natures. In other words, for each specific closed system, regardless of its nature, it is possible to determine a certain quantity called energy, which will be conserved over time. Moreover, the fulfillment of this conservation law in each specific system is justified by the subordination of this system to its specific laws of dynamics, which generally differ for different systems.
However, in different branches of physics, for historical reasons, the law of conservation of energy is formulated differently, and therefore speaks of the conservation of various types of energy. For example, in thermodynamics, the law of conservation of energy is expressed as the first law of thermodynamics.
Since the law of conservation of energy does not apply to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it is more correct to call it not a law, but the principle of conservation of energy.
From a mathematical point of view, the law of conservation of energy is equivalent to the statement that a system of differential equations describing the dynamics of a given physical system has a first integral of motion associated with
Body impulse
The momentum of a body is a quantity equal to the product of the mass of the body and its speed.
It should be remembered that we are talking about a body that can be represented as a material point. The momentum of the body ($p$) is also called the momentum. The concept of momentum was introduced into physics by René Descartes (1596–1650). The term “impulse” appeared later (impulsus in Latin means “push”). Momentum is a vector quantity (like speed) and is expressed by the formula:
$p↖(→)=mυ↖(→)$
The direction of the momentum vector always coincides with the direction of the velocity.
The SI unit of impulse is the impulse of a body with a mass of $1$ kg moving at a speed of $1$ m/s; therefore, the unit of impulse is $1$ kg $·$ m/s.
If a constant force acts on a body (material point) during a period of time $∆t$, then the acceleration will also be constant:
$a↖(→)=((υ_2)↖(→)-(υ_1)↖(→))/(∆t)$
where $(υ_1)↖(→)$ and $(υ_2)↖(→)$ are the initial and final velocities of the body. Substituting this value into the expression of Newton's second law, we get:
$(m((υ_2)↖(→)-(υ_1)↖(→)))/(∆t)=F↖(→)$
Opening the brackets and using the expression for the momentum of the body, we have:
$(p_2)↖(→)-(p_1)↖(→)=F↖(→)∆t$
Here $(p_2)↖(→)-(p_1)↖(→)=∆p↖(→)$ is the change in momentum over time $∆t$. Then the previous equation will take the form:
$∆p↖(→)=F↖(→)∆t$
The expression $∆p↖(→)=F↖(→)∆t$ is a mathematical representation of Newton's second law.
The product of a force and the duration of its action is called impulse of force. That's why the change in the momentum of a point is equal to the change in the momentum of the force acting on it.
The expression $∆p↖(→)=F↖(→)∆t$ is called equation of body motion. It should be noted that the same action - a change in the momentum of a point - can be achieved by a small force over a long period of time and by a large force over a short period of time.
Impulse of the system tel. Law of Momentum Change
The impulse (amount of motion) of a mechanical system is a vector equal to the sum of the impulses of all material points of this system:
$(p_(syst))↖(→)=(p_1)↖(→)+(p_2)↖(→)+...$
The laws of change and conservation of momentum are a consequence of Newton's second and third laws.
Let us consider a system consisting of two bodies. The forces ($F_(12)$ and $F_(21)$ in the figure with which the bodies of the system interact with each other are called internal.
Let, in addition to internal forces, external forces $(F_1)↖(→)$ and $(F_2)↖(→)$ act on the system. For each body, we can write the equation $∆p↖(→)=F↖(→)∆t$. Adding the left and right sides of these equations, we get:
$(∆p_1)↖(→)+(∆p_2)↖(→)=((F_(12))↖(→)+(F_(21))↖(→)+(F_1)↖(→)+ (F_2)↖(→))∆t$
According to Newton's third law, $(F_(12))↖(→)=-(F_(21))↖(→)$.
Hence,
$(∆p_1)↖(→)+(∆p_2)↖(→)=((F_1)↖(→)+(F_2)↖(→))∆t$
On the left side there is a geometric sum of changes in the impulses of all bodies of the system, equal to the change in the impulse of the system itself - $(∆p_(syst))↖(→)$. Taking this into account, the equality $(∆p_1)↖(→)+(∆p_2) ↖(→)=((F_1)↖(→)+(F_2)↖(→))∆t$ can be written:
$(∆p_(syst))↖(→)=F↖(→)∆t$
where $F↖(→)$ is the sum of all external forces, acting on the body. The result obtained means that the momentum of the system can only be changed by external forces, and the change in the momentum of the system is directed in the same way as the total external force.
This is the essence of the law of change in momentum of a mechanical system.
Internal forces cannot change the total momentum of the system. They only change the impulses of individual bodies of the system.
Law of conservation of momentum
The law of conservation of momentum follows from the equation $(∆p_(syst))↖(→)=F↖(→)∆t$. If no external forces act on the system, then the right side of the equation $(∆p_(syst))↖(→)=F↖(→)∆t$ becomes zero, which means the total momentum of the system remains unchanged:
A system on which no external forces act or the resultant of external forces is zero is called closed.
The law of conservation of momentum states:
The total momentum of a closed system of bodies remains constant for any interaction of the bodies of the system with each other.
The result obtained is valid for a system containing an arbitrary number of bodies. If the sum of external forces is not equal to zero, but the sum of their projections to some direction is equal to zero, then the projection of the system’s momentum to this direction does not change. So, for example, a system of bodies on the surface of the Earth cannot be considered closed due to the force of gravity acting on all bodies, however, the sum of the projections of impulses on the horizontal direction can remain unchanged (in the absence of friction), since in this direction the force of gravity does not works.
Jet propulsion
Let us consider examples that confirm the validity of the law of conservation of momentum.
Let's take a children's rubber ball, inflate it and release it. We will see that when the air begins to leave it in one direction, the ball itself will fly in the other. The motion of a ball is an example of jet motion. It is explained by the law of conservation of momentum: the total momentum of the “ball plus air in it” system before the air flows out is zero; it must remain equal to zero during movement; therefore, the ball moves in the direction opposite to the direction of flow of the jet, and at such a speed that its momentum is equal in magnitude to the momentum of the air jet.
Jet motion call the movement of a body that occurs when some part of it is separated from it at any speed. Due to the law of conservation of momentum, the direction of movement of the body is opposite to the direction of movement of the separated part.
Rocket flights are based on the principle of jet propulsion. Modern space rocket is a very complex aircraft. The mass of the rocket consists of the mass of the working fluid (i.e., hot gases formed as a result of fuel combustion and emitted in the form of a jet stream) and the final, or, as they say, “dry” mass of the rocket remaining after the working fluid is ejected from the rocket.
When a jet of gas is ejected from a rocket at high speed, the rocket itself rushes in the opposite direction. According to the law of conservation of momentum, the momentum $m_(p)υ_p$ acquired by the rocket must be equal to the momentum $m_(gas)·υ_(gas)$ of the ejected gases:
$m_(p)υ_p=m_(gas)·υ_(gas)$
It follows that the speed of the rocket
$υ_p=((m_(gas))/(m_p))·υ_(gas)$
From this formula it is clear that the greater the speed of the rocket, the greater the speed of the emitted gases and the ratio of the mass of the working fluid (i.e., the mass of the fuel) to the final (“dry”) mass of the rocket.
The formula $υ_p=((m_(gas))/(m_p))·υ_(gas)$ is approximate. It does not take into account that as the fuel burns, the mass of the flying rocket becomes less and less. The exact formula for rocket speed was obtained in 1897 by K. E. Tsiolkovsky and bears his name.
Work of force
The term “work” was introduced into physics in 1826 by the French scientist J. Poncelet. If in everyday life only human labor is called work, then in physics and, in particular, in mechanics it is generally accepted that work is performed by force. The physical quantity of work is usually denoted by the letter $A$.
Work of force is a measure of the action of a force, depending on its magnitude and direction, as well as on the movement of the point of application of the force. For a constant force and linear displacement, the work is determined by the equality:
$A=F|∆r↖(→)|cosα$
where $F$ is the force acting on the body, $∆r↖(→)$ is the displacement, $α$ is the angle between the force and the displacement.
The work of force is equal to the product of the moduli of force and displacement and the cosine of the angle between them, i.e., the scalar product of the vectors $F↖(→)$ and $∆r↖(→)$.
Work is a scalar quantity. If $α 0$, and if $90°
When several forces act on a body, the total work (the sum of the work of all forces) is equal to the work of the resulting force.
The unit of work in SI is joule($1$ J). $1$ J is the work done by a force of $1$ N along a path of $1$ m in the direction of action of this force. This unit is named after the English scientist J. Joule (1818-1889): $1$ J = $1$ N $·$ m. Kilojoules and millijoules are also often used: $1$ kJ $= 1,000$ J, $1$ mJ $= $0.001 J.
Work of gravity
Let us consider a body sliding along an inclined plane with an angle of inclination $α$ and a height $H$.
Let us express $∆x$ in terms of $H$ and $α$:
$∆x=(H)/(sinα)$
Considering that the force of gravity $F_т=mg$ makes an angle ($90° - α$) with the direction of movement, using the formula $∆x=(H)/(sin)α$, we obtain an expression for the work of gravity $A_g$:
$A_g=mg cos(90°-α) (H)/(sinα)=mgH$
From this formula it is clear that the work done by gravity depends on the height and does not depend on the angle of inclination of the plane.
It follows that:
- the work of gravity does not depend on the shape of the trajectory along which the body moves, but only on the initial and final position of the body;
- when a body moves along a closed trajectory, the work done by gravity is zero, i.e., gravity is a conservative force (forces that have this property are called conservative).
Work of reaction forces, is equal to zero, since the reaction force ($N$) is directed perpendicular to the displacement $∆x$.
Work of friction force
The friction force is directed opposite to the displacement $∆x$ and makes an angle of $180°$ with it, therefore the work of the friction force is negative:
$A_(tr)=F_(tr)∆x·cos180°=-F_(tr)·∆x$
Since $F_(tr)=μN, N=mg cosα, ∆x=l=(H)/(sinα),$ then
$A_(tr)=μmgHctgα$
Work of elastic force
Let an external force $F↖(→)$ act on an unstretched spring of length $l_0$, stretching it by $∆l_0=x_0$. In position $x=x_0F_(control)=kx_0$. After the force $F↖(→)$ ceases to act at point $x_0$, the spring is compressed under the action of force $F_(control)$.
Let us determine the work of the elastic force when the coordinate of the right end of the spring changes from $x_0$ to $x$. Since the elastic force in this area changes linearly, Hooke’s law can use its average value in this area:
$F_(control av.)=(kx_0+kx)/(2)=(k)/(2)(x_0+x)$
Then the work (taking into account the fact that the directions $(F_(control av.))↖(→)$ and $(∆x)↖(→)$ coincide) is equal to:
$A_(control)=(k)/(2)(x_0+x)(x_0-x)=(kx_0^2)/(2)-(kx^2)/(2)$
It can be shown that the form of the last formula does not depend on the angle between $(F_(control av.))↖(→)$ and $(∆x)↖(→)$. The work of elastic forces depends only on the deformations of the spring in the initial and final states.
Thus, the elastic force, like the force of gravity, is a conservative force.
Power power
Power is a physical quantity measured by the ratio of work to the period of time during which it is produced.
In other words, power shows how much work is done per unit of time (in SI - per $1$ s).
Power is determined by the formula:
where $N$ is power, $A$ is work done during time $∆t$.
Substituting into the formula $N=(A)/(∆t)$ instead of the work $A$ its expression $A=F|(∆r)↖(→)|cosα$, we obtain:
$N=(F|(∆r)↖(→)|cosα)/(∆t)=Fυcosα$
Power is equal to the product of the magnitudes of the force and velocity vectors and the cosine of the angle between these vectors.
Power in the SI system is measured in watts (W). One watt ($1$ W) is the power at which $1$ J of work is done for $1$ s: $1$ W $= 1$ J/s.
This unit is named after the English inventor J. Watt (Watt), who built the first steam engine. J. Watt himself (1736-1819) used a different unit of power - horsepower(hp), which he introduced so that the performance of a steam engine and a horse could be compared: $1$ hp. $= 735.5$ W.
In technology, larger power units are often used - kilowatt and megawatt: $1$ kW $= 1000$ W, $1$ MW $= 1000000$ W.
Kinetic energy. Law of change of kinetic energy
If a body or several interacting bodies (a system of bodies) can do work, then they are said to have energy.
The word “energy” (from the Greek energia - action, activity) is often used in everyday life. For example, people who can do work quickly are called energetic, having great energy.
The energy possessed by a body due to motion is called kinetic energy.
As in the case of the definition of energy in general, we can say about kinetic energy that kinetic energy is the ability of a moving body to do work.
Let us find the kinetic energy of a body of mass $m$ moving with a speed $υ$. Since kinetic energy is energy due to motion, its zero state is the state in which the body is at rest. Having found the work necessary to impart a given speed to a body, we will find its kinetic energy.
To do this, let’s calculate the work in the area of displacement $∆r↖(→)$ when the directions of the force vectors $F↖(→)$ and displacement $∆r↖(→)$ coincide. In this case the work is equal
where $∆x=∆r$
For the motion of a point with acceleration $α=const$, the expression for displacement has the form:
$∆x=υ_1t+(at^2)/(2),$
where $υ_1$ is the initial speed.
Substituting into the equation $A=F·∆x$ the expression for $∆x$ from $∆x=υ_1t+(at^2)/(2)$ and using Newton’s second law $F=ma$, we obtain:
$A=ma(υ_1t+(at^2)/(2))=(mat)/(2)(2υ_1+at)$
Expressing the acceleration through the initial $υ_1$ and final $υ_2$ velocities $a=(υ_2-υ_1)/(t)$ and substituting in $A=ma(υ_1t+(at^2)/(2))=(mat)/ (2)(2υ_1+at)$ we have:
$A=(m(υ_2-υ_1))/(2)·(2υ_1+υ_2-υ_1)$
$A=(mυ_2^2)/(2)-(mυ_1^2)/(2)$
Now equating the initial speed to zero: $υ_1=0$, we obtain an expression for kinetic energy:
$E_K=(mυ)/(2)=(p^2)/(2m)$
Thus, a moving body has kinetic energy. This energy is equal to the work that must be done to increase the speed of the body from zero to the value $υ$.
From $E_K=(mυ)/(2)=(p^2)/(2m)$ it follows that the work done by a force to move a body from one position to another is equal to the change in kinetic energy:
$A=E_(K_2)-E_(K_1)=∆E_K$
The equality $A=E_(K_2)-E_(K_1)=∆E_K$ expresses theorem on the change in kinetic energy.
Change in body kinetic energy(material point) for a certain period of time is equal to the work done during this time by the force acting on the body.
Potential energy
Potential energy is the energy determined by the relative position of interacting bodies or parts of the same body.
Since energy is defined as the ability of a body to do work, potential energy is naturally defined as the work done by a force, depending only on the relative position of the bodies. This is the work of gravity $A=mgh_1-mgh_2=mgH$ and the work of elasticity:
$A=(kx_0^2)/(2)-(kx^2)/(2)$
Potential energy of the body interacting with the Earth, they call a quantity equal to the product of the mass $m$ of this body by the acceleration of free fall $g$ and the height $h$ of the body above the Earth’s surface:
The potential energy of an elastically deformed body is a value equal to half the product of the elasticity (stiffness) coefficient $k$ of the body and the squared deformation $∆l$:
$E_p=(1)/(2)k∆l^2$
The work of conservative forces (gravity and elasticity), taking into account $E_p=mgh$ and $E_p=(1)/(2)k∆l^2$, is expressed as follows:
$A=E_(p_1)-E_(p_2)=-(E_(p_2)-E_(p_1))=-∆E_p$
This formula allows you to give general definition potential energy.
The potential energy of a system is a quantity that depends on the position of the bodies, the change in which during the transition of the system from the initial state to the final state is equal to the work of the internal conservative forces of the system, taken with the opposite sign.
The minus sign on the right side of the equation $A=E_(p_1)-E_(p_2)=-(E_(p_2)-E_(p_1))=-∆E_p$ means that when work is performed by internal forces (for example, a fall bodies on the ground under the influence of gravity in the “rock-Earth” system), the energy of the system decreases. Work and changes in potential energy in a system always have opposite signs.
Since work determines only a change in potential energy, then only a change in energy has a physical meaning in mechanics. Therefore, the choice of the zero energy level is arbitrary and determined solely by considerations of convenience, for example, the ease of writing the corresponding equations.
Law of change and conservation of mechanical energy
Total mechanical energy of the system the sum of its kinetic and potential energies is called:
It is determined by the position of bodies (potential energy) and their speed (kinetic energy).
According to the kinetic energy theorem,
$E_k-E_(k_1)=A_p+A_(pr),$
where $A_p$ is the work of potential forces, $A_(pr)$ is the work of non-potential forces.
In turn, the work of potential forces is equal to the difference in the potential energy of the body in the initial $E_(p_1)$ and final $E_p$ states. Taking this into account, we obtain an expression for law of change mechanical energy:
$(E_k+E_p)-(E_(k_1)+E_(p_1))=A_(pr)$
where the left side of the equality is the change in total mechanical energy, and the right side is the work of non-potential forces.
So, law of change of mechanical energy reads:
The change in the mechanical energy of the system is equal to the work of all non-potential forces.
A mechanical system in which only potential forces, is called conservative.
In a conservative system $A_(pr) = 0$. this implies law of conservation of mechanical energy:
In a closed conservative system, the total mechanical energy is conserved (does not change with time):
$E_k+E_p=E_(k_1)+E_(p_1)$
The law of conservation of mechanical energy is derived from Newton's laws of mechanics, which are applicable to a system of material points (or macroparticles).
However, the law of conservation of mechanical energy is also valid for a system of microparticles, where Newton’s laws themselves no longer apply.
The law of conservation of mechanical energy is a consequence of the uniformity of time.
Uniformity of time is that for the same initial conditions the course of physical processes does not depend on at what point in time these conditions are created.
The law of conservation of total mechanical energy means that when the kinetic energy in a conservative system changes, its potential energy must also change, so that their sum remains constant. This means the possibility of converting one type of energy into another.
In accordance with various forms the movements of matter are considered different kinds energy: mechanical, internal (equal to the sum of the kinetic energy of the chaotic movement of molecules relative to the center of mass of the body and the potential energy of interaction of molecules with each other), electromagnetic, chemical (which consists of the kinetic energy of the movement of electrons and the electrical energy of their interaction with each other and with atomic nuclei ), nuclear, etc. From the above it is clear that the division of energy into different types quite conditional.
Natural phenomena are usually accompanied by the transformation of one type of energy into another. For example, friction of parts of various mechanisms leads to the conversion of mechanical energy into heat, i.e. internal energy. In heat engines, on the contrary, internal energy is converted into mechanical energy; in galvanic cells, chemical energy is converted into electrical energy, etc.
Currently, the concept of energy is one of the basic concepts of physics. This concept is inextricably linked with the idea of the transformation of one form of movement into another.
This is how the concept of energy is formulated in modern physics:
Energy is a general quantitative measure of movement and interaction of all types of matter. Energy does not appear from nothing and does not disappear, it can only move from one form to another. The concept of energy links together all natural phenomena.
Simple mechanisms. Efficiency of mechanisms
Simple mechanisms are devices that change the magnitude or direction of forces applied to a body.
They are used to move or lift large loads with little effort. These include the lever and its varieties - blocks (movable and fixed), gates, inclined plane and its varieties - wedge, screw, etc.
Lever arm. Leverage rule
The lever is solid, capable of rotating around a fixed support.
The rule of leverage says:
A lever is in equilibrium if the forces applied to it are inversely proportional to their arms:
$(F_2)/(F_1)=(l_1)/(l_2)$
From the formula $(F_2)/(F_1)=(l_1)/(l_2)$, applying the property of proportion to it (the product of the extreme terms of a proportion is equal to the product of its middle terms), we can obtain the following formula:
But $F_1l_1=M_1$ is the moment of force tending to turn the lever clockwise, and $F_2l_2=M_2$ is the moment of force trying to turn the lever counterclockwise. Thus, $M_1=M_2$, which is what needed to be proven.
The lever began to be used by people in ancient times. With its help it was possible to lift heavy stone slabs during the construction of the pyramids in Ancient Egypt. Without leverage this would not be possible. After all, for example, for the construction of the Cheops pyramid, which has a height of $147$ m, more than two million stone blocks were used, the smallest of which weighed $2.5$ tons!
Nowadays, levers are widely used both in production (for example, cranes) and in everyday life (scissors, wire cutters, scales).
Fixed block
The action of a fixed block is similar to the action of a lever with equal arms: $l_1=l_2=r$. The applied force $F_1$ is equal to the load $F_2$, and the equilibrium condition is:
Fixed block used when you need to change the direction of a force without changing its magnitude.
Movable block
The moving block acts similarly to a lever, the arms of which are: $l_2=(l_1)/(2)=r$. In this case, the equilibrium condition has the form:
where $F_1$ is the applied force, $F_2$ is the load. The use of a moving block gives a double gain in strength.
Pulley hoist (block system)
An ordinary chain hoist consists of $n$ moving and $n$ fixed blocks. Using it gives a gain in strength of $2n$ times:
$F_1=(F_2)/(2n)$
Power chain hoist consists of n movable and one fixed block. The use of a power pulley gives a gain in strength of $2^n$ times:
$F_1=(F_2)/(2^n)$
Screw
A screw is an inclined plane wound around an axis.
The equilibrium condition for the forces acting on the propeller has the form:
$F_1=(F_2h)/(2πr)=F_2tgα, F_1=(F_2h)/(2πR)$
where $F_1$ is the external force applied to the propeller and acting at a distance $R$ from its axis; $F_2$ is the force acting in the direction of the propeller axis; $h$ — propeller pitch; $r$ is the average thread radius; $α$ is the angle of inclination of the thread. $R$ — lever length ( wrench), rotating the screw with a force $F_1$.
Efficiency
Coefficient useful action(efficiency) - the ratio of useful work to all expended work.
Efficiency is often expressed as a percentage and is denoted by the Greek letter $η$ (“this”):
$η=(A_п)/(A_3)·100%$
where $A_n$ is useful work, $A_3$ is all expended work.
Useful work always constitutes only a part of the total work that a person expends using one or another mechanism.
Part of the work done is spent on overcoming frictional forces. Since $A_3 > A_n$, the efficiency is always less than $1$ (or $< 100%$).
Since each of the works in this equality can be expressed as a product of the corresponding force and the distance traveled, it can be rewritten as follows: $F_1s_1≈F_2s_2$.
It follows that, winning with the help of a mechanism in force, we lose the same number of times along the way, and vice versa. This law is called the golden rule of mechanics.
The golden rule of mechanics is an approximate law, since it does not take into account the work of overcoming friction and gravity of the parts of the devices used. Nevertheless, it can be very useful in analyzing the operation of any simple mechanism.
So, for example, thanks to this rule, we can immediately say that the worker shown in the figure, with a double gain in the force of lifting the load by $10$ cm, will have to lower the opposite end of the lever by $20$ cm.
Collision of bodies. Elastic and inelastic impacts
The laws of conservation of momentum and mechanical energy are used to solve the problem of the motion of bodies after a collision: from the known impulses and energies before the collision, the values of these quantities after the collision are determined. Let us consider the cases of elastic and inelastic impacts.
An impact is called absolutely inelastic, after which the bodies form a single body moving at a certain speed. The problem of the speed of the latter is solved using the law of conservation of momentum of a system of bodies with masses $m_1$ and $m_2$ (if we are talking about two bodies) before and after the impact:
$m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=(m_1+m_2)υ↖(→)$
It is obvious that the kinetic energy of bodies during an inelastic impact is not conserved (for example, for $(υ_1)↖(→)=-(υ_2)↖(→)$ and $m_1=m_2$ it becomes equal to zero after the impact).
An impact in which not only the sum of impulses is conserved, but also the sum of the kinetic energies of the impacting bodies is called absolutely elastic.
For an absolutely elastic impact, the following equations are valid:
$m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=m_1(υ"_1)↖(→)+m_2(υ"_2)↖(→);$
$(m_(1)υ_1^2)/(2)+(m_(2)υ_2^2)/(2)=(m_1(υ"_1)^2)/(2)+(m_2(υ"_2 )^2)/(2)$
where $m_1, m_2$ are the masses of the balls, $υ_1, υ_2$ are the velocities of the balls before the impact, $υ"_1, υ"_2$ are the velocities of the balls after the impact.
If forces, friction and resistance forces do not act in a closed system, then the sum of the kinetic and potential energy of all bodies of the system remains a constant value.
Let's consider an example of the manifestation of this law. Let a body raised above the Earth have potential energy E 1 = mgh 1 and velocity v 1 directed downward. As a result of free fall, the body moved to a point with height h 2 (E 2 = mgh 2), while its speed increased from v 1 to v 2. Consequently, its kinetic energy increased from
Let's write the kinematics equation:
Multiplying both sides of the equality by mg, we get:
After transformation we get:
Let us consider the restrictions that were formulated in the law of conservation of total mechanical energy.
What happens to mechanical energy if a friction force acts in the system?
In real processes where friction forces act, a deviation from the law of conservation of mechanical energy is observed.
For example, when a body falls to Earth, the kinetic energy of the body initially increases as the speed increases.
The resistance force also increases, which increases with increasing speed. Over time, it will compensate for the force of gravity, and in the future, as the potential energy decreases relative to the Earth, the kinetic energy does not increase.
A change in thermal (or internal) energy occurs as a result of the work of friction or resistance forces. It is equal to the change in mechanical energy. Thus, the sum of the total energy of bodies during interaction is a constant value (taking into account the conversion of mechanical energy into internal energy).
Energy is measured in the same units as work. As a result, we note that there is only one way to change mechanical energy - to do work.
The muscles that move the parts of the body perform mechanical work.
Work in a certain direction is the product of the force (F) acting in the direction of movement of the body along the path it has traversed (S): A = F * S.
Doing work requires energy. Therefore, as work is performed, the energy in the system decreases. Since in order for work to be done, a supply of energy is necessary, the latter can be defined as follows: Energy is the ability to do work, it is a certain measure of the “resource” available in a mechanical system to perform it. In addition, energy is a measure of the transition from one type of motion to another.
In biomechanics, the following main types of energy are considered:
- * potential, depending on the relative position of the elements of the mechanical system of the human body;
- * kinetic translational motion;
- * kinetic rotational movement;
- * potential deformation of system elements;
- * thermal;
- * metabolic processes.
The total energy of a biomechanical system is equal to the sum of all listed types of energy.
By lifting a body, compressing a spring, you can accumulate energy in potential form for later use. Potential energy is always associated with one force or another acting from one body on another. For example, the Earth acts by gravity on a falling object, a compressed spring acts on a ball, and a drawn bowstring acts on an arrow.
Potential energy is the energy that a body possesses due to its position in relation to other bodies, or due to the relative position of parts of one body.
Therefore, gravitational force and elastic force are potential.
Gravitational potential energy: Ep = m * g * h
Potential energy of elastic bodies:
where k is the spring stiffness; x is its deformation.
From the above examples it is clear that energy can be stored in the form of potential energy (lifting a body, compressing a spring) for later use.
In biomechanics, two types of potential energy are considered and taken into account: due to the relative position of the body’s links to the Earth’s surface (gravitational potential energy); related to elastic deformation elements of the biomechanical system (bones, muscles, ligaments) or any external objects (sports equipment, equipment).
Kinetic energy is stored in the body during movement. A moving body does work due to its loss. Since the parts of the body and the human body perform translational and rotational movements, the total kinetic energy (Ek) will be equal to:
where m is mass, V is linear velocity, J is the moment of inertia of the system, u is angular velocity.
Energy enters the biomechanical system due to metabolic metabolic processes occurring in the muscles. The change in energy that results in work being done is not a highly efficient process in a biomechanical system, that is, not all the energy goes into useful work. Part of the energy is lost irreversibly, turning into heat: only 25% is used to perform work, the remaining 75% is converted and dissipated in the body.
For a biomechanical system, the law of conservation of energy of mechanical motion is applied in the form:
Epol = Ek + Epot + U,
where Epol is the total mechanical energy of the system; Ek is the kinetic energy of the system; Epot - potential energy of the system; U- internal energy systems that represent primarily thermal energy.
The total energy of mechanical movement of a biomechanical system is based on the following two energy sources: metabolic reactions in the human body and mechanical energy of the external environment (deformable elements of sports equipment, equipment, supporting surfaces; opponents during contact interactions). This energy is transmitted through external forces.
A feature of energy production in a biomechanical system is that one part of the energy during movement is spent on performing the necessary motor action, the other goes to the irreversible dissipation of stored energy, the third is saved and used during subsequent movement. When calculating the energy expended during movements and the mechanical work performed during this process, the human body is represented in the form of a model of a multi-link biomechanical system, similar to the anatomical structure. The movements of an individual link and the movements of the body as a whole are considered in the form of two simpler types of movement: translational and rotational.
The total mechanical energy of some i-th link (Epol) can be calculated as the sum of potential (Epot) and kinetic energy (Ek). In turn, Ek can be represented as the sum of the kinetic energy of the center of mass of the link (Ec.c.m.), in which the entire mass of the link is concentrated, and the kinetic energy of rotation of the link relative to the center of mass (Ec.Vr.).
If the kinematics of movement of the link is known, this general expression for the total energy of the link will have the form:
Newton kinetic impulse
where mi is the mass of the i-th link; g - free fall acceleration; hi is the height of the center of mass above some zero level (for example, above the Earth’s surface at a given location); - speed of translational motion of the center of mass; Ji is the moment of inertia of the ith link relative to the instantaneous axis of rotation passing through the center of mass; u - instantaneous angular velocity of rotation relative to the instantaneous axis.
The work to change the total mechanical energy of the link (Ai) during operation from moment t1 to moment t2 is equal to the difference in energy values at the final (Ep(t2)) and initial (Ep(t1)) moments of movement:
Naturally, in in this case work is spent on changing the potential and kinetic energy of the link.
If the amount of work Ai > 0, that is, the energy has increased, then they say that positive work has been done on the link. If AI< 0, то есть энергия звена уменьшилась, - отрицательная работа.
The mode of work to change the energy of a given link is called overcoming if the muscles perform positive work on the link; inferior if the muscles perform negative work on the link.
Positive work is done when the muscle contracts against an external load, goes to accelerate the parts of the body, the body as a whole, sports equipment, etc. Negative work is done if the muscles resist stretching due to the action of external forces. This occurs when lowering a load, going down stairs, or resisting a force that exceeds the strength of the muscles (for example, in arm wrestling).
Spotted Interesting Facts ratio of positive and negative muscle work: negative muscle work is more economical than positive; preliminary execution of negative work increases the magnitude and efficiency of the positive work that follows it.
The greater the speed of movement of the human body (during track and field running, skating, skiing, etc.), the greater the part of the work spent not on the useful result - moving the body in space, but on moving the links relative to the GCM. Therefore, at high speeds, the main work is spent on accelerating and braking the body parts, since as the speed increases, the acceleration of the movement of the body parts increases sharply.
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One of the most important laws, according to which the physical quantity - energy is conserved in an isolated system. All known processes in nature, without exception, obey this law. In an isolated system, energy can only be converted from one form to another, but its quantity remains constant.
In order to understand what the law is and where it comes from, let’s take a body of mass m, which we drop to the Earth. At point 1, our body is at height h and is at rest (velocity is 0). At point 2 the body has a certain speed v and is at a distance h-h1. At point 3 the body has maximum speed and it almost lies on our Earth, that is, h = 0
At point 1 the body has only potential energy, since the speed of the body is 0, so the total mechanical energy is equal.
After we released the body, it began to fall. When falling, the potential energy of a body decreases, as the height of the body above the Earth decreases, and its kinetic energy increases, as the speed of the body increases. In section 1-2 equal to h1, the potential energy will be equal to
And the kinetic energy will be equal at that moment ( - the speed of the body at point 2):
The closer a body becomes to the Earth, the less its potential energy, but at the same moment the speed of the body increases, and because of this, kinetic energy. That is, at point 2 the law of conservation of energy works: potential energy decreases, kinetic energy increases.
At point 3 (on the surface of the Earth), the potential energy is zero (since h = 0), and the kinetic energy is maximum (where v3 is the speed of the body at the moment of falling to the Earth). Since , the kinetic energy at point 3 will be equal to Wk=mgh. Consequently, at point 3 the total energy of the body is W3=mgh and is equal to the potential energy at height h. The final formula for the law of conservation of mechanical energy will be:
The formula expresses the law of conservation of energy in a closed system in which only conservative forces act: the total mechanical energy of a closed system of bodies interacting with each other only by conservative forces does not change with any movements of these bodies. Only mutual transformations of the potential energy of bodies into their kinetic energy and vice versa occur.
In Formula we used.
