Energy is the most important concept in mechanics. What is energy. There are many definitions, and here is one of them.

What is energy?

Energy is the ability of a body to do work.

Consider a body that was moving under the influence of some forces and changed its speed from v 1 → to v 2 → . In this case, the forces acting on the body have done a certain amount of work A.

The work of all forces acting on the body is equal to the work of the resultant force.

F p → = F 1 → + F 2 →

A \u003d F 1 s cos α 1 + F 2 s cos α 2 \u003d F p cos α.

Let's establish a connection between the change in the speed of the body and the work done by the forces acting on the body. For simplicity, we will assume that a single force F → acts on the body, directed along a straight line. Under the action of this force, the body moves uniformly accelerated and in a straight line. In this case, the vectors F → , v → , a → , s → coincide in direction and can be considered as algebraic quantities.

The work of the force F → is equal to A = F s . The movement of the body is expressed by the formula s = v 2 2 - v 1 2 2 a. From here:

A = F s = F v 2 2 - v 1 2 2 a = m a v 2 2 - v 1 2 2 a

A = m v 2 2 - m v 2 2 2 = m v 2 2 2 - m v 2 2 2 .

As you can see, the work done by the force is proportional to the change in the square of the speed of the body.

Definition. Kinetic energy

The kinetic energy of a body is half the product of the body's mass times the square of its speed.

Kinetic energy is the energy of the motion of a body. At zero speed it is zero.

Terem about kinetic energy

Let us turn again to the considered example and formulate a theorem on the kinetic energy of a body.

Kinetic energy theorem

The work of the force applied to the body is equal to the change in the kinetic energy of the body. This statement is also true when the body moves under the action of a force changing in magnitude and direction.

A \u003d E K 2 - E K 1.

Thus, the kinetic energy of a body of mass m, moving at a speed v → , is equal to the work that the force must do to accelerate the body to this speed.

A = m v 2 2 = E K .

To stop the body, you need to do work

A = - m v 2 2 = - E K

Kinetic energy is the energy of motion. Along with kinetic energy, there is also potential energy, that is, the energy of the interaction of bodies, which depends on their position.

For example, a body is raised above the ground. The higher it is raised, the greater the potential energy will be. When a body falls down under the influence of gravity, this force does work. Moreover, the work of gravity is determined only by the vertical displacement of the body and does not depend on the trajectory.

Important!

In general, one can talk about potential energy only in the context of those forces whose work does not depend on the shape of the body's trajectory. Such forces are called conservative (or dissipative).

Examples of dissipative forces: gravity, elastic force.

When a body moves vertically upwards, gravity does negative work.

Consider an example where the ball has moved from a point with height h 1 to a point with height h 2 .

In this case, the force of gravity has done work equal to

A \u003d - m g (h 2 - h 1) \u003d - (m g h 2 - m g h 1) .

This work is equal to the change in m g h taken with the opposite sign.

The value of E P \u003d m g h is the potential energy in the gravity field. At the zero level (on the ground) the potential energy of the body is zero.

Definition. Potential energy

Potential energy is a part of the total mechanical energy of the system located in the field of dissipative (conservative) forces. Potential energy depends on the position of the points that make up the system.

We can talk about potential energy in the gravity field, potential energy of a compressed spring, and so on.

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

A \u003d - (E P 2 - E P 1) .

It is clear that the potential energy depends on the choice of the zero level (the origin of the OY axis). We emphasize that the physical meaning is the change potential energy when moving bodies relative to each other. With any choice of the zero level, the change in potential energy will be the same.

When calculating the movement of bodies in the Earth's gravitational field, but at considerable distances from it, one must take into account the law of universal gravitation (the dependence of the force of gravity on the distance to the center of the Earth). We give a formula expressing the dependence of the potential energy of the body.

E P = - G m M r .

Here G is the gravitational constant, M is the mass of the Earth.

Potential energy of a spring

Let's imagine that in the first case we took a spring and lengthened it by x. In the second case, we first lengthened the spring by 2x and then shortened it by x. In both cases, the spring was stretched by x , but this was done in different ways.

In this case, the work of the elastic force with a change in the length of the spring by x in both cases was the same and equal to

A y p p \u003d - A \u003d - k x 2 2.

The value of E y p p \u003d k x 2 2 is called the potential energy of a compressed spring. It is equal to the work of the elastic force during the transition from a given state of the body to a state with zero deformation.

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The unit for energy in the International System of Units (SI) is the joule, while in the CGS system it is the erg.

On the physical meaning of the concept of potential energy

F → (r →) = − ∇ E p (r →) , (\displaystyle (\vec (F))((\vec (r)))=-\nabla E_(p)((\vec (r) ))))

or, in the simple one-dimensional case,

F (x) = − d E p (x) / d x , (\displaystyle F(x)=-(\rm (d))E_(p)(x)/(\rm (d))x,)

so arbitrariness of choice E p 0 (\displaystyle E_(p0)) does not affect.

Types of potential energy

In the Earth's gravitational field

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

E p = m g h , (\displaystyle \ E_(p)=mgh,)

where m(\displaystyle\m)- body mass, g(\displaystyle\g)- acceleration of gravity , h(\displaystyle\h)- the height of the position 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 qp (\displaystyle \q_(p)), in an electrostatic field with potential φ (r →) (\displaystyle \varphi ((\vec (r)))) is:

E p = q p φ (r →) . (\displaystyle \ E_(p)=q_(p)\varphi ((\vec (r))).)

For example, if the 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

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 everything is more or less obvious with the first, then often the force that arises between two stationary objects remains beyond understanding.

It is well known that the planet Earth affects all bodies located on its surface due to that is, it attracts any object with a certain force. When moving an object, changing its height, there is also a change in energy indicators. Immediately at the moment of lifting, the body has acceleration. However, at its highest point, when the object (even for a fraction of a second) is motionless, it has potential energy. The thing is that it is still drawn to itself by the field of the Earth, 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. At the same time, 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 - mass, g - acceleration h - height.

It is important to consider the height parameter in more detail, since it often causes difficulties in solving problems and understanding the value of the quantity under consideration. The fact is that any vertical movement of the body has its own start and end point. To correctly find the potential energy of the interaction of bodies, it is important to know the initial height. If it is not specified, then its value is equal to zero, that is, it coincides with the surface of the Earth. In the case where both the starting point of reference and the final height are known, it is necessary to find the difference between them. The resulting number will become the required h.

It is also important to note that the potential energy of the system can have negative meaning. Suppose we have already raised the body above the level of the Earth, therefore, it has a height, which we will call initial. When it is omitted, 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 the higher, the farther from the surface of the Earth the body is located. In order to better understand this fact, let's think: the higher you need to raise the body above the Earth, the more thorough the perfect work. The higher the value of the work of any force, the more, relatively speaking, the more energy is invested. Potential energy, in other words, is the energy of possibility.

Similarly, it is possible to measure the interaction energy of bodies when an object is stretched.

Within the framework of the topic under consideration, it is necessary to discuss separately the interaction of a charged particle and an electric field. In such a system, the potential energy of the charge will be present. Let's consider this fact in more detail. Any charge within an electric field is subject to the same force. The movement of the particle occurs 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 get 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.

denoting "action". You can call an energetic person who moves, creates a certain work, can create, act. Also, machines created by people, living and nature have energy. But that's in real life. In addition, there is a strict one that has defined and designated many types of energy - electrical, 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 there is an interaction of bodies or parts of one body, but there is no movement as such. This is the main difference from kinetic energy. For example, if you raise 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 do some work when it falls. And the possible amount of work will be equal to the potential energy of the body at a certain height h. The following formula is used to calculate this energy:

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

Two types of potential energy

There are two types of potential energy:

1. Energy in the mutual arrangement of bodies. A suspended stone possesses such energy. Interestingly, ordinary firewood or coal also have potential energy. They contain unoxidized carbon, which can be oxidized. To put it simply, burnt wood can potentially heat water.

2. Energy of elastic deformation. An example here is an elastic tourniquet, a compressed spring, or a bone-muscle-ligament system.

Potential and kinetic energy are interconnected. They can pass into each other. For example, if you throw a stone up, when moving, it first has kinetic energy. When it reaches a certain point, it freezes for a moment and gains potential energy, and then gravity pulls it down and kinetic energy reappears.

1. You met the concept of energy in the 7th grade physics course. Let's remember him. Let us suppose that some body, for example a trolley, slides down an inclined plane and moves a bar lying at its base. The cart is said to do work. Indeed, it acts on the bar with a certain elastic force, and the bar moves in this case.

Another example. The driver of a car moving at a certain speed applies the brakes, and the car stops after some time. 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. SI unit of energy - joule (1 J).

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

Potential energy is the energy of interaction of bodies or body parts, 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 does not refer to one body, but to a system of interacting bodies. In the case when we are talking 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 relationship between the work of gravity and the change in the potential energy of the body.

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

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

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

A = E p1 - E n2 = -( E p2 - E n1).

When the position of the body changes, its potential energy changes. In this way,

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

The minus sign means that when the body falls, the force of gravity does positive work, and the potential energy of the body decreases. If the body moves up, 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.

So, the potential energy of a ball flying over a volleyball net has one value relative to the net, and another value relative to the floor of the gym. It is important that the difference between the potential energies of the body at two points does not depend on the chosen 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.

Often, the surface of the Earth is taken as the zero level when determining potential energy. If a body falls from a certain height onto 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 deforms, the forces of interaction 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 the point with coordinate D l 2. Recall that the work of the elastic force in this case is equal to:

A =– .

Value = E n1 characterizes the first state of the deformed body and represents its potential energy in the first state, the 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 n1), i.e.

the work of 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 action of an external force, its potential energy increases, and the elastic force does negative work.

Questions for self-examination

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

2. What is 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. What work must be done in order to shift 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 the bag of flour lying on the shelf, and its potential energy when it is on the table, relative to the floor?

2. What work must be done to transfer a spring with a stiffness of 4 kN/m to the state 1 , stretching it by 2 cm? What extra work must be done in order to transfer the spring to the state 2 , stretching it another 1 cm? What is the change in the potential energy of the spring when it is transferred to the state 1 and from the state 1 into 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 force of gravity acting on the ball versus the height of the ball. Calculate, using the graph, the potential energy of the ball at a height of 1.5 m.

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

Work and kinetic energy