Effective potential energy. Kinetic and potential energy

Denoting "action". You can call an energetic person who moves, creates certain work, can create, act. Machines created by people, living things and nature also have energy. But this is in ordinary life. In addition, there is a strict one that has defined and designated many types of energy - electric, magnetic, atomic, etc. However, now we will talk about potential energy, which cannot be considered in isolation from kinetic energy.

Kinetic energy

This energy, according to the concepts of mechanics, is possessed by all bodies that interact with each other. And in this case we are talking about the movement of bodies.

Potential energy

This type of energy is created when the interaction of bodies or parts of one body occurs, but there is no movement as such. This is the main difference from kinetic energy. For example, if you lift a stone above the ground and hold it in this position, it will have potential energy, which can turn into kinetic energy if the stone is released.

Energy is usually associated with work. That is, in this example, the released stone can produce some work as it falls. And the possible amount of work will be equal to the potential energy of the body at a certain height h. To calculate this energy, the following formula is used:

A=Fs=Ft*h=mgh, or Ep=mgh, where:
Ep - potential energy of the body,
m - body weight,
h is the height of the body above the ground,
g is the acceleration of free fall.

Two types of potential energy

Potential energy has two types:

1. Energy in the relative position of bodies. A suspended stone has such energy. Interestingly, ordinary wood or coal also has potential energy. They contain unoxidized carbon that can oxidize. To put it simply, burnt wood can potentially heat up the water.

2. Energy of elastic deformation. Examples here include an elastic band, a compressed spring, or a “bone-muscle-ligament” system.

Potential and kinetic energy are interrelated. They can transform into each other. For example, if you throw a stone up, it initially has kinetic energy as it moves. When it reaches a certain point, it will freeze for a moment and gain potential energy, and then gravity will pull it down and kinetic energy will arise again.

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

1. You were introduced to the concept of energy in the 7th grade physics course. Let's remember him. Suppose that some body, for example a cart, slides down an inclined plane and moves a block lying at its base. They say that the cart does work. Indeed, it acts on the block with a certain elastic force and the block moves.

Another example. The driver of a car moving at a certain speed presses the brake, and after some time the car stops. In this case, the car also does work against the friction force.

They say that if a body can do work, then it has energy.

Energy is denoted by the letter E. The SI unit of energy is joule (1 J).

2. There are two types of mechanical energy - potential and kinetic.

Potential energy is the energy of interaction between bodies or parts of a body, depending on their relative position.

All interacting bodies have potential energy. So, any body interacts with the Earth, therefore, the body and the Earth have potential energy. The particles that make up bodies also interact with each other, and they also have potential energy.

Since potential energy is the energy of interaction, it refers not to one body, but to a system of interacting bodies. In the case when we talk about the potential energy of a body raised above the Earth, the system consists of the Earth and the body raised above it.

3. Let's find out what the potential energy of a body raised above the Earth is. To do this, we will find the connection between the work of gravity and the change in the potential energy of the body.

Let the body have mass m falls from a height h 1 to height h 2 (Fig. 72). In this case, the displacement of the body is equal to h = h 1 – h 2. The work done by gravity in this area will be equal to:

A = F cord h = mgh = mg(h 1 – h 2), or
A = mgh 1 – mgh 2 .

Magnitude mgh 1 = E n1 characterizes the initial position of the body and represents its potential energy in the initial position, mgh 2 = E n2 is the potential energy of the body in its final position. The formula can be rewritten as follows:

A = E p1 – E n2 = –( E p2 – E p1).

When the position of a body changes, its potential energy changes. Thus,

the work done by gravity is equal to the change in the potential energy of the body, taken with the opposite sign.

The minus sign means that when a body falls, gravity does positive work, and the potential energy of the body decreases. If a body moves upward, then the force of gravity does negative work, and the potential energy of the body increases.

4. When determining the potential energy of a body, it is necessary to indicate the level relative to which it is measured, called zero level.

Thus, the potential energy of a ball flying over a volleyball net has one value relative to the net, but another value relative to the gym floor. It is important that the difference in the potential energies of the body at two points does not depend on the selected zero level. This means that the work done due to the potential energy of the body does not depend on the choice of the zero level.

When determining potential energy, the Earth's surface is often taken as the zero level. If a body falls from a certain height to the surface of the Earth, then the work done by gravity is equal to the potential energy: A = mgh.

Hence, the potential energy of a body raised to a certain height above the zero level is equal to the work done by gravity when the body falls from this height to the zero level.

5. Any deformed body has potential energy. When a body is compressed or stretched, it is deformed, the interaction forces between its particles change and an elastic force arises.

Let the right end of the spring (see Fig. 68) move from the point with coordinate D l 1 to point with coordinate D l 2. Recall that the work done by the elastic force is equal to:

A =– .

Value = E n1 characterizes the first state of the deformed body and represents its potential energy in the first state, value = E n2 characterizes the second state of the deformed body and represents its potential energy in the second state. You can write:

A = –(E p2 – E p1), i.e.

the work done by the elastic force is equal to the change in the potential energy of the spring, taken with the opposite sign.

The minus sign shows that as a result of the positive work done by the elastic force, the potential energy of the body decreases. When a body is compressed or stretched under the influence of an external force, its potential energy increases, and the elastic force does negative work.

Self-test questions

1. When can we say that a body has energy? What is the unit of energy?

2. What is called potential energy?

3. How to calculate the potential energy of a body raised above the Earth?

4. Does the potential energy of a body raised above the Earth depend on the zero level?

5. How to calculate the potential energy of an elastically deformed body?

Task 19

1. How much work must be done to transfer a bag of flour weighing 2 kg from a shelf located at a height of 0.5 m relative to the floor to a table located at a height of 0.75 m relative to the floor? What is the potential energy of a bag of flour lying on the shelf relative to the floor and its potential energy when it is on the table?

2. What work must be done to transform a spring with a stiffness of 4 kN/m into the state 1 , stretching it by 2 cm? What additional work must be done to bring the spring to the state 2 , stretching it another 1 cm? What is the change in potential energy of the spring when it is transferred to the state 1 and from the state 1 in a state 2 ? What is the potential energy of the spring in the state 1 and able 2 ?

3. Figure 73 shows a graph of the dependence of the force of gravity acting on the ball on the height of the ball. Using the graph, calculate the potential energy of the ball at a height of 1.5 m.

4. Figure 74 shows a graph of the elongation of the spring as a function of the force acting on it. What is the potential energy of the spring when it extends 4 cm?

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.

25.12.2014

Lesson 32 (10th grade)

Subject. Potential energy

1. Work of gravity

Let's calculate the work, using this time not Newton's second law, but an explicit expression for the forces of interaction between bodies depending on the distances between them. This will allow us to introduce the concept of potential energy - energy that depends not on the velocities of bodies, but on the distances between bodies (or on the distances between parts of the same body).
Let's first calculate the work gravity when a body (for example, a stone) falls vertically down. At the initial moment of time the body was at a height h 1 above the Earth's surface, and at the final moment of time - at a height h 2 (Fig.6.5). Body movement module.

The directions of the gravity and displacement vectors coincide. According to the definition of work (see formula (6.2)) we have

Let now the body be thrown vertically upward from a point located at a height h 1, above the surface of the Earth, and it reached a height h 2 (Fig.6.6). Vectors and are directed in opposite directions, and the displacement module . We write the work of gravity as follows:

If a body moves in a straight line so that the direction of movement makes an angle with the direction of gravity ( Fig.6.7), then the work done by gravity is:

From a right triangle BCD it's clear that . Hence,

Formulas (6.12), (6.13), (6.14) make it possible to notice an important regularity. When a body moves in a straight line, the work done by gravity in each case is equal to the difference between two values ​​of a quantity that depends on the positions of the body at the initial and final moments of time. These positions are determined by the heights h 1 And h 2 bodies above the Earth's surface.
Moreover, the work done by gravity when moving a body of mass m from one position to another does not depend on the shape of the trajectory along which the body moves. Indeed, if a body moves along a curve Sun (Fig.6.8), then, presenting this curve in the form of a stepped line consisting of vertical and horizontal sections of short length, we see that in the horizontal sections the work done by gravity is zero, since the force is perpendicular to the displacement, and the sum of the work in the vertical sections is equal to the work done would be the force of gravity when moving a body along a vertical segment of length h 1 -h 2.

Thus, the work done when moving along a curve is Sun is equal to:

When a body moves along a closed trajectory, the work done by gravity is zero. In fact, let the body move along a closed contour VSDMV (Fig.6.9). At the sites Sun And DM the force of gravity performs work that is equal in absolute value, but opposite in sign. The sum of these works is zero. Consequently, the work done by gravity on the entire closed loop is also zero.

Forces with such properties are called conservative.
So, the work of gravity does not depend on the shape of the body's trajectory; it is determined only by the initial and final positions of the body. When a body moves along a closed path, the work done by gravity is zero.

2. Work of elastic force

Like gravity, elastic force is also conservative. To verify this, let's calculate the work done by the spring when moving the load.
Figure 6.10a shows a spring in which one end is fixed and a ball is attached to the other end. If the spring is stretched, then it acts on the ball with a force ( Fig. 6.10, b), directed towards the equilibrium position of the ball, in which the spring is not deformed. The initial elongation of the spring is . Let us calculate the work done by the elastic force when moving a ball from a point with coordinate x 1 to the point with coordinate x 2. From Figure 6.10, c it is clear that the displacement module is equal to:

where is the final elongation of the spring.

It is impossible to calculate the work of the elastic force using formula (6.2), since this formula is valid only for a constant force, and the elastic force does not remain constant when the spring deformation changes. To calculate the work of the elastic force, we will use a graph of the dependence of the modulus of the elastic force on the coordinates of the ball ( Fig.6.11).

At a constant value of the projection of force on the displacement of the point of application of force, its work can be determined from the dependence graph F x from x and that this work is numerically equal to the area of ​​the rectangle. With arbitrary dependence F x from x, dividing the displacement into small segments, within each of which the force can be considered constant, we will see that the work will be numerically equal to the area of ​​the trapezoid.
In our example, the work of the elastic force on moving the point of its application numerically equal to the area of ​​the trapezoid BCDM. Hence,

According to Hooke's law and . Substituting these expressions for the forces into equation (6.17) and taking into account that , we get

Or finally

We considered the case when the directions of the elastic force and the displacement of the body coincided: . But it would be possible to find the work of the elastic force when its direction is opposite to the movement of the body or makes an arbitrary angle with it, as well as when the body moves along a curve of arbitrary shape.
In all these cases, body movements under the influence elastic forces we would arrive at the same formula for work (6.18). The work of elastic forces depends only on the deformation of the spring in both the initial and final states.
Thus, the work of the elastic force does not depend on the shape of the trajectory and, like gravity, the elastic force is conservative.

3. Potential energy

Using Newton's second law, that in the case of a moving body, the work of forces of any nature can be represented as the difference between two values ​​of a certain quantity depending on the speed of the body - the difference between the values ​​of the kinetic energy of the body at the final and initial moments of time:

If the interaction forces between the bodies are conservative, then, using explicit expressions for the forces, we have shown that the work of such forces can also be represented as the difference between two values ​​of a certain quantity, depending on the relative position of the bodies (or parts of one body):

Here are the heights h 1 And h 2 determine the relative position of the body and the Earth, and the elongations and determine the relative position of the turns of the deformed spring (or the values ​​of the deformations of another elastic body).
A value equal to the product of body mass m to the acceleration of free fall g and to the height h bodies above the Earth's surface are called potential energy of interaction between the body and the Earth(from the Latin word “potency” - position, opportunity).
Let us agree to denote potential energy by the letter E p:

A value equal to half the product of the elasticity coefficient k body per square of deformation is called potential energy of an elastically deformed body:

In both cases, potential energy is determined by the location of the bodies of the system or parts of one body relative to each other.
By introducing the concept of potential energy, we are able to express the work of any conservative forces through a change in potential energy. A change in a quantity is understood as the difference between its final and initial values, therefore .
Therefore, both equations (6.20) can be written as follows:

where .
The change in the potential energy of the body is equal to the work done by the conservative force, taken with the opposite sign.
This formula allows us to give a general definition of potential energy.
Potential energy system is a quantity dependent 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 “-” sign in formula (6.23) does not mean that the work of conservative forces is always negative. It only means that the change in potential energy and the work of forces in the system always have opposite signs.
For example, when a stone falls to Earth, its potential energy decreases, but gravity does positive work ( A>0). Hence, A and have opposite signs in accordance with formula (6.23).
Zero level of potential energy. According to equation (6.23), the work of conservative interaction forces determines not the potential energy itself, but its change.
Since work determines only the change in potential energy, then only the change in energy in mechanics has physical meaning. Therefore, you can arbitrarily choose state of a system in which its potential energy counts equal to zero. This state corresponds to a zero level of potential energy. Not a single phenomenon in nature or technology is determined by the value of potential energy itself. What is important is the difference between the potential energy values ​​in the final and initial states of the system of bodies.
The choice of the zero level is made in different ways and is dictated solely by considerations of convenience, i.e., the simplicity of writing the equation expressing the law of conservation of energy.
Typically, the state of the system with minimum energy is chosen as the state with zero potential energy. Then the potential energy is always positive or equal to zero.
So, the potential energy of the “body - Earth” system is a quantity that depends on the position of the body relative to the Earth, equal to the work of a conservative force when moving a body from the point where it is located to the point corresponding to the zero level of potential energy of the system.
For a spring, the potential energy is minimal in the absence of deformation, and for a “stone-Earth” system - when the stone lies on the surface of the Earth. Therefore, in the first case , and in the second case . But you can add any constant value to these expressions C, and it won't change anything. It can be assumed that .
If in the second case we put , then this will mean that the zero energy level of the “stone-Earth” system is taken to be the energy corresponding to the position of the stone at a height h 0 above the surface of the Earth.
An isolated system of bodies tends to a state in which its potential energy is minimal.
If you don't hold the body, it falls to the ground ( h=0); If you release a stretched or compressed spring, it will return to its undeformed state.
If the forces depend only on the distances between the bodies of the system, then the work of these forces does not depend on the shape of the trajectory. Therefore, work can be represented as the difference between the values ​​of a certain function, called potential energy, in the final and initial states of the system. The value of the potential energy of the system depends on the nature of the acting forces, and to determine it it is necessary to indicate the zero reference level.