Potential energy is the law of conservation of energy. Energy
Energy is the maximum amount of work that a body can do.
There are several types of energy. For example, in mechanics:
Potential energy of gravity,
determined by height h.
- Potential energy of elastic deformation,
determined by the amount of deformation X.
- Kinetic energy - the energy of the movement of bodies,
determined by the speed of the body v.
Energy can be transferred from one body to another, and can also be transformed from one type to another.
- Total mechanical energy.
Law of energy conservation: in closed body system complete energy does not change at any interactions within this system of bodies. The law imposes restrictions on the course of processes in nature. Nature does not allow energy to appear from nowhere and disappear into nowhere. Perhaps it turns out only this way: how much one body loses energy, how much another acquires; how much one type of energy decreases, so much is added to another type.
In mechanics, to determine the types of energy, it is necessary to pay attention to three quantities: height lifting the body above the earth h, deformation x, speed body v.
body momentum
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 a 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 velocity) and is expressed by the formula:
$p↖(→)=mυ↖(→)$
The direction of the momentum vector always coincides with the direction of the velocity.
The unit of momentum in SI is the momentum of a body with a mass of $1$ kg moving at a speed of $1$ m/s, therefore, the unit of momentum is $1$ kg $·$ m/s.
If a constant force acts on a body (material point) during the time interval $∆t$, 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 momentum change over time $∆t$. Then the previous equation becomes:
$∆p↖(→)=F↖(→)∆t$
The expression $∆p↖(→)=F↖(→)∆t$ is a mathematical representation of Newton's second law.
The product of a force and its duration is called momentum of force. So 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 body motion equation. It should be noted that the same action - a change in the momentum of a point - can be obtained by a small force in a long period of time and by a large force in a small period of time.
Impulse of the system tel. Law of change of momentum
The impulse (momentum) 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.
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, the equation $∆p↖(→)=F↖(→)∆t$ can be written. Adding the left and right parts 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 is the geometric sum of the changes in the momentum of all the bodies of the system, equal to the change in the momentum of the system itself - $(∆p_(syst))↖(→)$. With this in mind, the equality $(∆p_1)↖(→)+(∆p_2) ↖(→)=((F_1)↖(→)+(F_2)↖(→))∆t$ can be written:
$(∆p_(sys))↖(→)=F↖(→)∆t$
where $F↖(→)$ is the sum of all external forces acting on the body. The result obtained means that only external forces can change the momentum of the system, 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 the momentum of a mechanical system.
Internal forces cannot change the total momentum of the system. They only change the impulses of the individual bodies of the system.
Law of conservation of momentum
From the equation $(∆p_(syst))↖(→)=F↖(→)∆t$ the momentum conservation law follows. If no external forces act on the system, then the right side of the equation $(∆p_(sys))↖(→)=F↖(→)∆t$ vanishes, which means that the total momentum of the system remains unchanged:
$(∆p_(sys))↖(→)=m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=const$
A system on which no external forces act or the resultant of external forces is equal to 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 on some direction is equal to zero, then the projection of the momentum of the system on 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), because in this direction the force of gravity does not is valid.
Jet propulsion
Consider examples that confirm the validity of the law of conservation of momentum.
Let's take a children's rubber balloon, inflate it and let it go. We will see that when the air starts to come out of it in one direction, the balloon itself will fly in the other direction. The movement of the ball is an example of jet propulsion. It is explained by the law of conservation of momentum: the total momentum of the system "ball plus air in it" before the outflow of air is zero; it must remain equal to zero during the movement; therefore, the ball moves in the direction opposite to the direction of the outflow of the jet, and with such a speed that its momentum is equal in absolute value to the momentum of the air jet.
jet propulsion called the motion of a body that occurs when a part of it separates from it at some speed. Due to the law of conservation of momentum, the direction of motion of the body is opposite to the direction of motion of the separated part.
Rocket flights are based on the principle of jet propulsion. A modern space rocket is a very complex aircraft. The mass of the rocket is the sum of the mass of the working fluid (i.e., hot gases resulting from the combustion of fuel and ejected in the form of a jet stream) and the final, or, as they say, “dry” mass of the rocket, remaining after the ejection of the working fluid from the rocket.
When a reactive gas jet is ejected from a rocket at high speed, the rocket itself rushes in the opposite direction. According to the momentum conservation law, 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 can be seen that the greater the speed of the rocket, the greater the speed of the ejected gases and the ratio of the mass of the working fluid (i.e., the mass of 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 smaller and smaller. The exact formula for the speed of a rocket was obtained in 1897 by K. E. Tsiolkovsky and bears his name.
Force work
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 done by force. The physical quantity of work is usually denoted by the letter $A$.
Force work- this is a measure of the action of a force, depending on its module and direction, as well as on the displacement of the point of application of the force. For a constant force and rectilinear movement, 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 the force is equal to the product of the modules 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 SI unit of work is joule($1$ J). $1$ J is the work done by a force of $1$ N on a path of $1$ m in the direction 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.
The work of gravity
Let us consider a body sliding along an inclined plane with an inclination angle $α$ and a height $H$.
We express $∆x$ in terms of $H$ and $α$:
$∆x=(H)/(sinα)$
Considering that 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 can be seen that the work of gravity depends on the height and does not depend on the angle of inclination of the plane.
From this 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 of gravity is zero, i.e., gravity is a conservative force (forces that have this property are called conservative).
The work of reaction forces, is zero because the reaction force ($N$) is directed perpendicular to the displacement $∆x$.
The work of the friction force
The friction force is directed opposite to the displacement $∆x$ and makes an angle $180°$ with it, so 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
$A_(tr)=μmgHctgα$
The work of the 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 termination of the force $F↖(→)$ at the point $x_0$, the spring is compressed under the action of the force $F_(control)$.
Let us determine the work of the elastic force when the coordinate of the right end of the spring changes from $х_0$ to $х$. Since the elastic force in this area changes linearly, in Hooke's law, its average value in this area can be used:
$F_(ex.av.)=(kx_0+kx)/(2)=(k)/(2)(x_0+x)$
Then the work (taking into account the fact that the directions $(F_(exp.av.))↖(→)$ and $(∆x)↖(→)$ coincide) is equal to:
$A_(exerc)=(k)/(2)(x_0+x)(x_0-x)=(kx_0^2)/(2)-(kx^2)/(2)$
It can be shown that the form of the last formula does not depend on the angle between $(F_(exp.av.))↖(→)$ and $(∆x)↖(→)$. The work of the elastic forces depends only on the deformations of the spring in the initial and final states.
Thus, the elastic force, like gravity, is a conservative force.
Power of force
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, for $1$ s).
Power is determined by the formula:
where $N$ is the power, $A$ is the work done in the time $∆t$.
Substituting $A=F|(∆r)↖(→)|cosα$ into the formula $N=(A)/(∆t)$ instead of the work $A$, we get:
$N=(F|(∆r)↖(→)|cosα)/(∆t)=Fυcosα$
The power is equal to the product of the modules 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 in $1$ 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 in order to be able to compare the performance of a steam engine and a horse: $ 1 $ hp. $= 735.5$ Tue.
In technology, larger units of power are often used - kilowatts and megawatts: $1$ 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 say that they have energy.
The word "energy" (from the Greek. energia - action, activity) is often used in everyday life. So, for example, people who can quickly do work are called energetic, with 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 of $υ$. Since kinetic energy is the energy due to motion, the zero state for it is the state in which the body is at rest. Having found the work necessary to communicate a given speed to the body, we will find its kinetic energy.
To do this, we calculate the work done on the displacement section $∆r↖(→)$ when the directions of the force vectors $F↖(→)$ and displacement $∆r↖(→)$ coincide. In this case, the work is
where $∆x=∆r$
For the movement of a point with acceleration $α=const$, the expression for movement has the form:
$∆x=υ_1t+(at^2)/(2),$
where $υ_1$ is the initial speed.
Substituting the expression for $∆x$ from $∆x=υ_1t+(at^2)/(2)$ into the equation $A=F ∆x$ and using Newton's second law $F=ma$, we get:
$A=ma(υ_1t+(at^2)/(2))=(mat)/(2)(2υ_1+at)$
Expressing the acceleration in terms of initial $υ_1$ and final $υ_2$ speeds $a=(υ_2-υ_1)/(t)$ and substituting into $A=ma(υ_1t+(at^2)/(2))=(mat)/ (2)(2υ_1+at)$ 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 velocity 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 $υ$.
From $E_K=(mυ)/(2)=(p^2)/(2m)$ it follows that the work of 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 the kinetic energy of the body(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 mutual arrangement 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 of a force that depends 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:
$A=(kx_0^2)/(2)-(kx^2)/(2)$
The potential energy of the body interacting with the Earth is called the value equal to the product of the mass $m$ of this body and the free fall acceleration $g$ and the height $h$ of the body above the Earth's surface:
The potential energy of an elastically deformed body is the value equal to half the product of the coefficient of elasticity (stiffness) $k$ of the body and the square of 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 us to give a general definition of potential energy.
The potential energy of the system is a quantity that depends on the position of the bodies, the change of 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 done by internal forces (for example, falling body to the ground under the action of gravity in the "stone-Earth" system), the energy of the system decreases. Work and change in potential energy in a system always have opposite signs.
Since work determines only the change in potential energy, only the change in energy has physical meaning in mechanics. Therefore, the choice of the zero energy level is arbitrary and is determined solely by considerations of convenience, for example, the ease of writing the corresponding equations.
The 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 the bodies (potential energy) and their speed (kinetic energy).
According to the kinetic energy theorem,
$E_k-E_(k_1)=A_p+A_(pr),$
where $А_р$ is the work of potential forces, $А_(pr)$ is the work of nonpotential 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. With this in mind, we get an expression for the law of change of 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 the total mechanical energy, and the right side is the work of nonpotential 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 nonpotential forces.
A mechanical system in which only potential forces act 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 the laws of Newtonian 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 homogeneity of time.
Uniformity of time is that, under the same initial conditions, the course of physical processes does not depend on the moment at which 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 different forms of motion of matter, different types of energy are considered: mechanical, internal (equal to the sum of the kinetic energy of the chaotic motion of molecules relative to the center of mass of the body and the potential energy of the interaction of molecules with each other), electromagnetic, chemical (which consists of the kinetic energy of the motion of electrons and electric the energy of their interaction with each other and with atomic nuclei), nuclear energy, etc. It can be seen from the foregoing that the division of energy into different types is rather arbitrary.
Natural phenomena are usually accompanied by the transformation of one type of energy into another. So, for example, the friction of parts of various mechanisms leads to the conversion of mechanical energy into heat, i.e., into 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.
Here is how the concept of energy is formulated in modern physics:
Energy is a general quantitative measure of the movement and interaction of all types of matter. Energy does not arise from nothing and does not disappear, it can only pass from one form to another. The concept of energy binds together all the phenomena of nature.
simple mechanisms. mechanism efficiency
Simple mechanisms are devices that change the magnitude or direction of the forces applied to the body.
They are used to move or lift large loads with little effort. These include the lever and its varieties - blocks (movable and fixed), a gate, an inclined plane and its varieties - a wedge, a screw, etc.
Lever arm. Lever rule
The lever is a rigid body capable of rotating around a fixed support.
The leverage rule 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 the 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 tending to turn the lever counterclockwise. Thus, $M_1=M_2$, which was to be proved.
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 have been possible. Indeed, for example, for the construction of the pyramid of Cheops, which has a height of $147$ m, more than two million stone blocks were used, the smallest of which had a mass of $2.5$ 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 leverage: $l_1=l_2=r$. 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 movable block acts similarly to a lever, whose arms 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 movable block gives a gain in strength twice.
Polyspast (block system)
An ordinary chain hoist consists of $n$ movable and $n$ fixed blocks. Applying 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 chain hoist gives a gain in strength of $2^n$ times:
$F_1=(F_2)/(2^n)$
Screw
The screw is an inclined plane wound on the axis.
The condition for the balance of forces acting on the screw has the form:
$F_1=(F_2h)/(2πr)=F_2tgα, F_1=(F_2h)/(2πR)$
where $F_1$ is an external force applied to the screw and acting at a distance $R$ from its axis; $F_2$ is the force acting in the direction of the screw axis; $h$ - screw pitch; $r$ is the average thread radius; $α$ is the angle of the thread. $R$ is the length of the lever (wrench) that rotates the screw with the force $F_1$.
Efficiency
Coefficient of performance (COP) - the ratio of useful work to all the work expended.
Efficiency is often expressed as a percentage and denoted by the Greek letter $η$ ("this"):
$η=(A_p)/(A_3) 100%$
where $A_n$ is useful work, $A_3$ is all the work expended.
Useful work is always only a part of the total work that a person expends using this or that mechanism.
Part of the work done is spent on overcoming the forces of friction. Since $А_3 > А_п$, the efficiency is always less than $1$ (or $< 100%$).
Since each of the works in this equation can be expressed as the product of the corresponding force and the distance traveled, it can be rewritten as follows: $F_1s_1≈F_2s_2$.
From this it follows that, winning with the help of the mechanism in force, we lose the same number of times on 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 to overcome friction and gravity of the parts of the devices used. Nevertheless, it can be very useful when 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 lifting force of $10$ 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: the known momenta and energies before the collision are used to determine the values of these quantities after the collision. Consider the cases of elastic and inelastic impacts.
An absolutely inelastic impact is called, 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 for a system of bodies with masses $m_1$ and $m_2$ (if we are talking about two bodies) before and after the impact:
$m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=(m_1+m_2)υ↖(→)$
Obviously, the kinetic energy of bodies is not conserved during an inelastic impact (for example, at $(υ_1)↖(→)=-(υ_2)↖(→)$ and $m_1=m_2$ it becomes equal to zero after the impact).
An absolutely elastic impact is called, in which not only the sum of impulses is preserved, but also the sum of the kinetic energies of the colliding bodies.
For an absolutely elastic impact, the equations
$m_1(υ_1)↖(→)+m_2(υ_2)↖(→)=m_1(υ"_1)↖(→)+m_2(υ"_2)↖(→);$
$(m_(1)υ_1^2)/(2)+(m_(2)υ_2^2)/(2)=(m_1(υ"_1)^2)/(2)+(m_2(υ"_2 )^2)/(2)$
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.
Kinetic energy of a mechanical system is the energy of the mechanical movement of this system.
Force F, acting on a body at rest and causing its movement, does work, and the energy of the moving body increases by the amount of work expended. Thus the work dA strength F on the path that the body has traveled during the increase in speed from 0 to v, goes to increase the kinetic energy dT body, i.e.
Using Newton's second law F=md v/dt
and multiplying both sides of the equality by the displacement d r, we get
F d r=m(d v/dt)dr=dA
Thus, a body of mass t, moving at speed v, has kinetic energy
T = tv 2 /2. (12.1)
From formula (12.1) it can be seen that the kinetic energy depends only on the mass and speed of the body, i.e., the kinetic energy of the system is a function of the state of its motion.
When deriving formula (12.1), it was assumed that the motion is considered in an inertial frame of reference, since otherwise it would be impossible to use Newton's laws. In different inertial frames of reference moving relative to each other, the speed of the body, and hence its kinetic energy, will be different. Thus, the kinetic energy depends on the choice of reference frame.
Potential energy - mechanical energy of a system of bodies, determined by their mutual arrangement and the nature of the forces of interaction between them.
Let the interaction of bodies be carried out through force fields (for example, fields of elastic forces, fields of gravitational forces), characterized by the fact that the work done by the acting forces when moving the body from one position to another does not depend on which trajectory this movement occurred, and depends only on the start and end positions. Such fields are called potential and the forces acting in them - conservative. If the work done by the force depends on the trajectory of the movement of the body from one point to another, then such a force is called dissipative; its example is the force of friction.
The body, being in a potential field of forces, has potential energy II. The work of conservative forces with an elementary (infinitely small) change in the configuration of the system is equal to the increment of potential energy, taken with a minus sign, since the work is done due to a decrease in potential energy:
Job d BUT expressed as the scalar product of the force F to move d r and expression (12.2) can be written as
F d r= -dP. (12.3)
Therefore, if the function П( r), then from formula (12.3) one can find the force F modulo and direction.
Potential energy can be determined from (12.3) as
where C is the integration constant, i.e., the potential energy is determined up to some arbitrary constant. This, however, does not affect the physical laws, since they include either the difference in potential energies in two positions of the body, or the derivative of P with respect to coordinates. Therefore, the potential energy of the body in a certain position is considered equal to zero (the zero reference level is chosen), and the body energy in other positions is counted relative to the zero level. For conservative forces
or in vector form
F=-gradП, (12.4) where
(i, j, k are the unit vectors of the coordinate axes). The vector defined by expression (12.5) is called scalar gradient P.
For it, along with the designation grad П, the designation П is also used. ("nabla") means a symbolic vector called operatorHamilton or nabla-operator:
The specific form of the P function depends on the nature of the force field. For example, the potential energy of a body of mass t, elevated to a height h above the earth's surface is
P = mgh,(12.7)
where is the height h is measured from the zero level, for which P 0 = 0. Expression (12.7) follows directly from the fact that the potential energy is equal to the work of gravity when a body falls from a height h to the surface of the earth.
Since the origin is chosen arbitrarily, the potential energy can have a negative value (kinetic energy is always positive. !} If we take as zero the potential energy of a body lying on the surface of the Earth, then the potential energy of a body located at the bottom of the mine (depth h "), P = - mgh".
Let us find the potential energy of an elastically deformed body (spring). The elastic force is proportional to the deformation:
F X ex = -kx,
where F x ex - projection of the elastic force on the axis X;k- elasticity coefficient(for spring - rigidity), and the minus sign indicates that F x ex directed in the direction opposite to the deformation X.
According to Newton's third law, the deforming force is equal in absolute value to the elastic force and is directed opposite to it, i.e.
F x =-F x ex =kx elementary work da, performed by force F x at an infinitely small deformation dx, is equal to
dA = F x dx=kxdx,
a complete job
goes to increase the potential energy of the spring. Thus, the potential energy of an elastically deformed body
P =kx 2 /2.
The potential energy of a system, like the 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.
Total mechanical energy of the system- energy of mechanical movement and interaction:
i.e., equal to the sum of the kinetic and potential energies.
The law of conservation of energy states that the energy of the body never disappears and does not reappear, it can only turn from one form to another. This law is universal. It has its own formulation in various branches of physics. Classical mechanics considers the law of conservation of mechanical energy.
The total mechanical energy of a closed system of physical bodies, between which conservative forces act, is a constant value. This is how the law of conservation of energy in Newtonian mechanics is formulated.
A closed or isolated system is considered to be a physical system that is not affected by external forces. It does not exchange energy with the surrounding space, and its own energy, which it possesses, remains unchanged, that is, it is preserved. In such a system, only internal forces act, and the bodies interact with each other. It can only convert potential energy into kinetic energy and vice versa.
The simplest example of a closed system is a sniper rifle and a bullet.
Types of mechanical forces
The forces that act inside a mechanical system are usually divided into conservative and non-conservative.
conservative forces are considered whose work does not depend on the trajectory of the body to which they are applied, but is determined only by the initial and final position of this body. The conservative forces are also called potential. The work of such forces in a closed loop is zero. Examples of conservative forces − gravity force, elastic force.
All other forces are called non-conservative. These include friction force and drag force. They are also called dissipative forces. These forces perform negative work during any motions in a closed mechanical system, and under their action the total mechanical energy of the system decreases (dissipates). It passes into other, non-mechanical types of energy, for example, into heat. Therefore, the law of conservation of energy in a closed mechanical system can be fulfilled only if there are no non-conservative forces in it.
The total energy of a mechanical system consists of kinetic and potential energy and is their sum. These types of energies can transform into each other.
Potential energy
Potential energy is called the energy of interaction of physical bodies or their parts with each other. It is determined by their mutual arrangement, that is, the distance between them, and is equal to the work that needs to be done to move the body from the reference point to another point in the field of conservative forces.
Potential energy has any motionless physical body, raised to some height, since it is affected by gravity, which is a conservative force. Such energy is possessed by water at the edge of a waterfall, a sled at the top of a mountain.
Where did this energy come from? While the physical body was being raised to a height, work was done and energy was expended. It is this energy that was stored in the raised body. And now this energy is ready to do work.
The value of the potential energy of the body is determined by the height at which the body is located relative to some initial level. We can take any point we choose as a starting point.
If we consider the position of the body relative to the Earth, then the potential energy of the body on the surface of the Earth is zero. And on top h it is calculated by the formula:
E p = mɡ h ,
where m - body mass
ɡ - acceleration of gravity
h – height of the center of mass of the body relative to the Earth
ɡ \u003d 9.8 m / s 2
When a body falls from a height h1 up to height h2 gravity does work. This work is equal to the change in potential energy and has a negative value, since the magnitude of potential energy decreases as the body falls.
A = - ( E p2 - E p1) = - ∆ E p ,
where E p1 is the potential energy of the body at height h1 ,
E p2 - potential energy of a body at a height h2 .
If the body is raised to a certain height, then work is done against the forces of gravity. In this case, it has a positive value. And the value of the potential energy of the body increases.
An elastically deformed body (compressed or stretched spring) also has potential energy. Its value depends on the stiffness of the spring and on how long it was compressed or stretched, and is determined by the formula:
E p \u003d k (∆x) 2 / 2 ,
where k - stiffness coefficient,
∆x - lengthening or contraction of the body.
The potential energy of the spring can do work.
Kinetic energy
Translated from the Greek "kinema" means "movement". The energy that a physical body receives as a result of its movement is called kinetic. Its value depends on the speed of movement.
A soccer ball rolling across the field, a sledge rolling down a mountain and continuing to move, an arrow fired from a bow - they all have kinetic energy.
If a body is at rest, its kinetic energy is zero. As soon as a force or several forces act on the body, it will begin to move. And since the body is moving, the force acting on it does work. The work of the force, under the influence of which the body from rest will go into motion and change its speed from zero to ν , is called kinetic energy body mass m .
If, at the initial moment of time, the body was already in motion, and its speed had the value v 1 , and at the end it was equal to v 2 , then the work done by the force or forces acting on the body will be equal to the increment in the kinetic energy of the body.
∆ E k = E k 2 - E k 1
If the direction of the force coincides with the direction of motion, then positive work is done, and the kinetic energy of the body increases. And if the force is directed in the direction opposite to the direction of motion, then negative work is done, and the body gives off kinetic energy.
Law of conservation of mechanical energy
Ek 1 + E p1= E k 2 + E p2
Any physical body located at some height has potential energy. But when falling, it begins to lose this energy. Where does she go? It turns out that it does not disappear anywhere, but turns into the kinetic energy of the same body.
Suppose , at some height, a load is motionlessly fixed. Its potential energy at this point is equal to the maximum value. If we let it go, it will start falling at a certain speed. Therefore, it will begin to acquire kinetic energy. But at the same time, its potential energy will begin to decrease. At the point of impact, the kinetic energy of the body will reach a maximum, and the potential energy will decrease to zero.
The potential energy of a ball thrown from a height decreases, while the kinetic energy increases. Sledges at rest on top of a mountain have potential energy. Their kinetic energy at this moment is zero. But when they start to roll down, the kinetic energy will increase, and the potential energy will decrease by the same amount. And the sum of their values will remain unchanged. The potential energy of an apple hanging on a tree is converted into its kinetic energy when it falls.
These examples clearly confirm the law of conservation of energy, which says that the total energy of a mechanical system is a constant value . The value of the total energy of the system does not change, and the potential energy is converted into kinetic energy and vice versa.
By what amount the potential energy decreases, the kinetic energy will increase by the same amount. Their amount will not change.
For a closed system of physical bodies, the equality
E k1 + E p1 = E k2 + E p2,
where E k1 , E p1
- kinetic and potential energies of the system before any interaction, E k2 , E p2
- corresponding energies after it.
The process of converting kinetic energy into potential energy and vice versa can be seen by watching a swinging pendulum.
Click on the picture
Being in the extreme right position, the pendulum seems to freeze. At this moment, its height above the reference point is maximum. Therefore, the potential energy is also maximum. And the kinetic is zero, since it does not move. But the next moment the pendulum starts moving down. Its speed increases, and, therefore, its kinetic energy increases. But as the height decreases, so does the potential energy. At the bottom point, it will become equal to zero, and the kinetic energy will reach its maximum value. The pendulum will pass this point and begin to rise up to the left. Its potential energy will begin to increase, and its kinetic energy will decrease. Etc.
To demonstrate the transformation of energy, Isaac Newton came up with a mechanical system, which is called Newton's cradle or Newton's balls .
Click on the picture
If the first ball is deflected and then released, its energy and momentum will be transferred to the last one through three intermediate balls, which will remain motionless. And the last ball will deflect with the same speed and rise to the same height as the first one. Then the last ball will transfer its energy and momentum through the intermediate balls to the first one, and so on.
A ball laid aside has the maximum potential energy. Its kinetic energy at this moment is zero. Having started moving, it loses potential energy and acquires kinetic energy, which reaches its maximum at the moment of collision with the second ball, and potential energy becomes equal to zero. Further, the kinetic energy is transferred to the second, then the third, fourth and fifth balls. The latter, having received kinetic energy, begins to move and rises to the same height at which the first ball was at the beginning of the movement. Its kinetic energy at this moment is equal to zero, and the potential energy is equal to the maximum value. Then it starts to fall and in the same way transfers energy to the balls in reverse order.
This continues for quite a long time and could continue indefinitely if there were no non-conservative forces. But in reality, dissipative forces act in the system, under the influence of which the balls lose their energy. Their speed and amplitude gradually decrease. And eventually they stop. This confirms that the law of conservation of energy is satisfied only in the absence of non-conservative forces.
If forces, friction and resistance forces do not act in a closed system, then the sum of the kinetic and potential energies of all bodies of the system remains constant.
Consider an example of the manifestation of this law. Let the body raised above the Earth have potential energy E 1 = mgh 1 and speed v 1 directed downwards. As a result of free fall, the body moved to a point with a height h 2 (E 2 = mgh 2), while its speed increased from v 1 to v 2. Therefore, its kinetic energy has increased from
Let's write the equation of kinematics:
Multiplying both sides of the equation by mg, we get:
After transformation we get:
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, there is a deviation from the law of conservation of mechanical energy. For example, when a body falls to the Earth, the kinetic energy of the body first increases as the speed increases. The resistance force also increases, which increases with increasing speed. Over time, it will compensate for gravity, and in the future, with a decrease in potential energy relative to the Earth, the kinetic energy does not increase.
This phenomenon is beyond the scope of mechanics, since the work of resistance forces leads to a change in body temperature. The heating of bodies under the action of friction is easy to detect by rubbing the palms together.
Thus, in mechanics, the law of conservation of energy has rather rigid boundaries.
The 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 transformation 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.
