Energy is a scalar quantity. The SI unit of energy is the Joule.

Kinetic and potential energy

There are two types of energy - kinetic and potential.

DEFINITION

Kinetic energy- this is the energy that a body possesses due to its movement:

DEFINITION

Potential energy is energy that is determined by the relative position of bodies, as well as the nature of the interaction forces between these bodies.

Potential energy in the Earth's gravitational field is the energy due to the gravitational interaction of a body with the Earth. It is determined by the position of the body relative to the Earth and is equal to the work of moving the body from a given position to the zero level:

Potential energy is the energy caused by the interaction of body parts with each other. It is equal to the work of external forces in tension (compression) of an undeformed spring by the amount:

A body can simultaneously possess both kinetic and potential energy.

The total mechanical energy of a body or system of bodies is equal to the sum of the kinetic and potential energies of the body (system of bodies):

Law of energy conservation

For a closed system of bodies, the law of conservation of energy is valid:

In the case when a body (or a system of bodies) is acted upon by external forces, for example, the law of conservation of mechanical energy is not satisfied. In this case, the change in the total mechanical energy of the body (system of bodies) is equal to the external forces:

The law of conservation of energy allows us to establish a quantitative connection between various forms of motion of matter. Just like , it is valid not only for, but also for all natural phenomena. The law of conservation of energy says that energy in nature cannot be destroyed just as it cannot be created from nothing.

In its most general form, the law of conservation of energy can be formulated as follows:

• Energy in nature does not disappear and is not created again, but only transforms from one type to another.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2

Any body always has energy. In the presence of movement, this is obvious: there is speed or acceleration, which, multiplied by mass, gives the desired result. However, in the case when the body is motionless, it, paradoxically, can also be characterized as having energy.

So, it arises during movement, potential - during the interaction of several bodies. If with the first everything is more or less obvious, then often the force that arises between two motionless objects remains beyond understanding.

It is well known that planet Earth influences all bodies located on its surface due to the fact that it attracts any object with a certain force. When an object moves or its height changes, energy indicators also change. Immediately at the moment of lifting, the body has acceleration. However, at its highest point, when an object (even for a split second) is motionless, it has potential energy. The whole point is that it is still pulled towards itself by the Earth’s field, with which the desired body interacts.

In other words, potential energy always arises due to the interaction of several objects that form a system, regardless of the size of the objects themselves. Moreover, by default one of them is represented by our planet.

Potential energy is a quantity that depends on the mass of an object and the height to which it is raised. International designation - Latin letters Ep. as follows:

Where m is mass, g is acceleration h is height.

It is important to consider the height parameter in more detail, since it often becomes the cause of difficulties when solving problems and understanding the meaning of the value in question. The fact is that any vertical movement of the body has its own starting and ending point. To correctly find the potential energy of interaction between bodies, it is important to know the initial height. If it is not specified, then its value is zero, that is, it coincides with the surface of the Earth. If both the initial reference point and the final height are known, it is necessary to find the difference between them. The resulting number will become the desired h.

It is also important to note that the potential energy of a system can be negative. Suppose we have already raised the body above the level of the Earth, therefore, it has a height that we will call initial. When lowered, the formula will look like this:

Obviously, h1 is greater than h2, therefore, the value will be negative, which will give the whole formula a minus sign.

It is curious that the potential energy is higher, the further from the surface of the Earth the body is located. In order to better understand this fact, let us think: the higher the body needs to be raised above the Earth, the more thoroughly the work done. The higher the work done by any force, the more energy is invested, relatively speaking. Potential energy, in other words, is the energy of possibility.

In a similar way, you can measure the energy of interaction between bodies when an object is stretched.

Within the framework of the topic under consideration, it is necessary to separately discuss the interaction of a charged particle and an electric field. In such a system there will be potential charge energy. Let's consider this fact in more detail. Any charge located within the electric field is subject to the same force. The particle moves due to the work produced by this force. Considering that the charge itself and (more precisely, the body that created it) are a system, we also obtain the potential energy of charge movement within a given field. Since this type of energy is a special case, it was given the name electrostatic.

Kinetic energy of a mechanical system is the energy of mechanical movement of this system.

Force F, acting on a body at rest and causing it to move, does work, and the energy of a moving body increases by the amount of work expended. So the work dA strength F on the path that the body has passed during the increase in speed from 0 to v, it goes to increase kinetic energy dT bodies, 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 is clear that kinetic energy depends only on the mass and speed of the body, that is, 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 was considered in an inertial frame of reference, since otherwise it would be impossible to use Newton’s laws. In different inertial reference systems moving relative to each other, the speed of the body, and therefore its kinetic energy, will not be the same. Thus, 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 interaction forces between them.

Let the interaction of bodies be carried out through force fields (for example, a field of elastic forces, a field of gravitational forces), characterized by the fact that the work done by the acting forces when moving a body from one position to another does not depend on the trajectory along which this movement occurred, and depends only on the start and end positions. Such fields are called potential, and the forces acting in them are conservative. If the work done by a force depends on the trajectory of the body moving from one point to another, then such a force is called dissipative; an example of this is the force of friction.

A body, being in a potential field of forces, has potential energy II. The work done by conservative forces during an elementary (infinitesimal) change in the configuration of the system 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:

Work d A expressed as the dot product of force F to move d r and expression (12.2) can be written as

F d r=-dP. (12.3)

Therefore, if the function P( r), then from formula (12.3) one can find the force F by module and direction.

Potential energy can be determined based on (12.3) as

where C is the integration constant, i.e. the potential energy is determined up to some arbitrary constant. This, however, is not reflected in 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 a body in a certain position is considered equal to zero (the zero reference level is chosen), and the energy of the body in other positions is measured relative to the zero level. For conservative forces

or in vector form

F=-gradP, (12.4) where

(i, j, k- unit vectors of coordinate axes). The vector defined by expression (12.5) is called gradient of the scalar P.

For it, along with the designation grad P, the designation P is also used.  (“nabla”) means a symbolic vector called operatorHamilton or by nabla operator:

The specific form of the function P depends on the nature of the force field. For example, the potential energy of a body of mass T, raised to a height h above the Earth's surface is equal to

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 potential energy is equal to the work done by 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 the potential energy of a body lying on the surface of the Earth as zero, then the potential energy of a body located at the bottom of the shaft (depth h"), P = - mgh".

Let's find the potential energy of an elastically deformed body (spring). The elastic force is proportional to the deformation:

F X control = -kx,

Where F x control - projection of elastic force onto the axis X;k- elasticity coefficient(for a spring - rigidity), and the minus sign indicates that F x control directed in the direction opposite to the deformation X.

According to Newton’s third law, the deforming force is equal in magnitude to the elastic force and directed oppositely to it, i.e.

F x =-F x control =kx Elementary work dA, performed by force F x at an infinitesimal deformation dx, is equal to

dA = F x dx = kxdx,

a full 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 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 kinetic and potential energies.

The International System of Units (SI) unit of energy is the joule, and the GHS unit is the erg.

On the physical meaning of the concept of potential energy

or, in the simple one-dimensional case,

so the choice is arbitrary E p 0 (\displaystyle E_(p0)) has no effect.

Types of potential energy

In the Earth's gravitational field

Body potential energy E p (\displaystyle \E_(p)) in the Earth's gravitational field near the surface is approximately expressed by the formula:

Where m (\displaystyle\m)- body mass, g (\displaystyle\g)- acceleration of gravity , h (\displaystyle\h)- the height of the center of mass of the body above an arbitrarily chosen zero level.

In an electrostatic field

Potential energy of a material point carrying an electric charge q p (\displaystyle \q_(p)), in an electrostatic field with potential φ (r →) (\displaystyle \varphi ((\vec (r)))) is:

For example, if a field is created by a point charge in a vacuum, then there will be E p = q p q / 4 π ε 0 r (\displaystyle \E_(p)=q_(p)q/4\pi \varepsilon _(0)r)(recorded in the system

Potential energy is called the energy of interaction of physical bodies or their parts with each other. It is determined by their relative position, 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 action of conservative forces.

Any motionless physical body raised to some height has potential energy, since it is acted upon by gravity, which is a conservative force. Such energy is possessed by water at the edge of a waterfall, and a sled on a mountain top.

Where did this energy come from? While the physical body was raised to a height, work was done and energy was expended. It is this energy that is stored in the raised body. And now this energy is ready to do work.

The amount of potential energy of a 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 reference point.

If we consider the position of the body relative to the Earth, then the potential energy of the body on the Earth’s surface is zero. And on top h it is calculated by the formula:

E p = mɡh,

Where m - body mass

ɡ - acceleration of gravity

h– height of the body’s center of mass relative to the Earth

ɡ = 9.8 m/s 2

When a body falls from a height h 1 up to height h 2 gravity does work. This work is equal to the change in potential energy and has a negative value, since the amount of potential energy decreases when the body falls.

A = - (E p2 – E p1) = - ∆ E p ,

Where E p1 – potential energy of the body at height h 1 ,

E p2 - potential energy of the body at height h 2 .

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 amount of 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 the length to which it was compressed or stretched, and is determined by the formula:

E p = k·(∆x) 2 /2,

Where k – stiffness coefficient,

∆x– lengthening or compression of the body.

The potential energy of a spring can do work.

Kinetic energy

Translated from 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 a field, a sled rolling down a mountain and continuing to move, an arrow shot from a bow - all of them have kinetic energy.

If a body is at rest, its kinetic energy is zero. As soon as a force or several forces act on a body, it will begin to move. And since the body moves, the force acting on it does work. The work of force, under the influence of which a body from a state of rest goes into motion and changes its speed from zero to ν , called kinetic energy body mass m .

If at the initial moment of time the body was already in motion, and its speed mattered ν 1 , and at the final moment it was equal to ν 2 , then the work done by the force or forces acting on the body will be equal to the increase in the kinetic energy of the body.

∆E k = E k2 - E k1

If the direction of the force coincides with the direction of movement, 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 movement, then negative work is done, and the body gives off kinetic energy.