How is kinetic energy different from potential energy? Kinetic energy vs potential energy
Kinetic energy is the energy of the motion of a body. Accordingly, if we have some object that has at least some mass and at least some speed, then it also has kinetic energy. However, with respect to different reference systems, this kinetic energy for the same object can be different.
Example. There is a grandmother who, relative to the earth of our planet, is at rest, that is, she does not move and, say, sits at a bus stop waiting for her bus. Then, relative to our planet, its kinetic energy is zero. But if you look at the same grandmother from the Moon or from the Sun, relative to which you can observe the movement of the planet and, accordingly, this grandmother, who is on our planet, then the grandmother will already have kinetic energy relative to the mentioned celestial bodies. And then the bus comes. This same grandmother quickly gets up and runs to take her place. Now, relative to the planet, it is no longer at rest, but is moving quite to itself. This means that it has kinetic energy. And the fatter the grandmother and faster, the greater her kinetic energy.
There are several fundamental types of energy - the main ones. Let me tell you, for example, about mechanical. These include kinetic energy, which depends on the speed and mass of the object, potential energy, which depends on where you take the zero level of potential energy, and on the position where this object is relative to the zero level of potential energy. That is, potential energy is energy that depends on the position of the object. This energy characterizes the work done by the field in which the object is located, as it moves.
Example. You carry a huge box in your hands and fall. The box is on the floor. It turns out that you will have a zero level of potential energy, respectively, at the floor level. Then the upper part of the box will have more potential energy, since it is above the floor and above the zero level of potential energy.
It is foolish to talk about energy without mentioning the law of its conservation. Thus, according to the law of conservation of energy, these two types of energy, which describe the state of an object, do not come from anywhere and do not disappear anywhere, but only pass into each other.
And here is an example. I am falling from the height of the house, initially having potential energy relative to the ground at the moment before the jump, and my kinetic energy is negligible, so we can equate it to zero. So I tear off the legs from the cornice and my potential energy begins to decrease, as the height at which I am is getting smaller and smaller. At the same moment, when falling down, I gradually acquire kinetic energy, as I fall down with increasing speed. At the time of the fall, I already have the maximum kinetic energy, but the potential energy is zero, such things.
Another fundamental physical concept, the concept of energy, is closely related to the concept of work. Since mechanics studies, firstly, the movement of bodies, and secondly, the interaction of bodies with each other, it is customary to distinguish between two types of mechanical energy: kinetic energy, due to the movement of the body, and potential energy due to the interaction of the body with other bodies.
Kinetic energy mechanical system called energy,depending on the velocities of the points of this system.
The expression for kinetic energy can be found by determining the work of the resultant force applied to a material point. Based on (2.24), we write the formula for the elementary work of the resultant force:
Because 
, then dА = mυdυ. (2.25)
To find the work of the resultant force when the body speed changes from υ 1 to υ 2, we integrate the expression (2.29):
(2.26)
Since work is a measure of the transfer of energy from one body to another, then on
Based on (2.30), we write that the quantity 
is the kinetic energy
bodies: 
whence instead of (1.44) we get
(2.27)
The theorem expressed by formula (2.30) is usually called kinetic energy theorem . In accordance with it, the work of forces acting on a body (or system of bodies) is equal to the change in the kinetic energy of this body (or system of bodies).
From the kinetic energy theorem it follows physical meaning of kinetic energy : The kinetic energy of a body is equal to the work that it is capable of doing in the process of reducing its speed to zero. The more "reserve" of kinetic energy a body has, the more work it can do.
The kinetic energy of the system is equal to the sum of the kinetic energies of the material points of which this system consists:
(2.28)
If the work of all forces acting on the body is positive, then the kinetic energy of the body increases, if the work is negative, then the kinetic energy decreases.
Obviously, the elementary work of the resultant of all forces applied to the body will be equal to the elementary change in the kinetic energy of the body:
dA = dE k. (2.29)
In conclusion, we note that kinetic energy, like the speed of movement, has a relative character. For example, the kinetic energy of a passenger sitting on a train will be different if we consider the movement relative to the roadbed or relative to the car.
§2.7 Potential energy
The second type of mechanical energy is potential energy is the energy due to the interaction of bodies.
Potential energy does not characterize any interaction of bodies, but only one that is described by forces that do not depend on speed. Most of the forces (gravity, elasticity, gravitational forces, etc.) are just that; the only exception is the force of friction. The work of the forces under consideration does not depend on the shape of the trajectory, but is determined only by its initial and final positions. The work of such forces on a closed trajectory is zero.
Forces whose work does not depend on the shape of the trajectory, but depends only on the initial and final position of a material point (body) are called potential or conservative forces .
If a body interacts with its environment through potential forces, then the concept of potential energy can be introduced to characterize this interaction.
Potential called the energy due to the interaction of bodies and depending on their relative position.
Find the potential energy of a body raised above the ground. Let a body of mass m move uniformly in a gravitational field from position 1 to position 2 along the surface, the section of which by the drawing plane is shown in Fig. 2.8. This section is the trajectory of a material point (body). If there is no friction, then three forces act on the point:
1) force N from the side of the surface is normal to the surface, the work of this force is zero;
2) gravity mg, the work of this force A 12;
3) thrust force F from some driving body (internal combustion engine, electric motor, person, etc.); the work of this force will be denoted as A T .
Consider the work of gravity when moving a body along an inclined plane of length ℓ (Fig. 2.9). As you can see from this figure, the work is equal to
A" = mgℓ cosα = mgℓ cos(90° + α) = - mgℓ sinα
From the triangle BCD we have ℓ sinα = h, so the last formula implies:
The trajectory of the body (see Fig. 2.8) can be schematically represented by small sections of an inclined plane, therefore, for the work of gravity on the entire trajectory 1 -2, the expression is true
A 12 \u003d mg (h 1 -h 2) \u003d- (mg h 2 - mg h 1) (2.30)
So, the work of gravity does not depend on the trajectory of the body, but depends on the difference in the heights of the location of the initial and final points of the trajectory.
the value
e p = mg h (2.31)
called potential energy material point (body) with mass m raised above the ground to a height h. Therefore, formula (2.30) can be rewritten as follows:
A 12 \u003d \u003d - (En 2 - En 1) or A 12 \u003d \u003d -ΔEn (2.32)
The work of gravity is equal to the change in the potential energy of bodies, taken with the opposite sign, i.e., the difference between its final and initialvalues (potential energy theorem ).
Similar reasoning can be given for an elastically deformed body.
(2.33)
Note that the difference in potential energies has a physical meaning as a quantity that determines the work of conservative forces. In this regard, it makes no difference to what position, configuration, zero potential energy should be attributed.
One very important consequence can be obtained from the potential energy theorem: conservative forces are always directed in the direction of decreasing potential energy. The established pattern is manifested in the fact that any system, left to itself, always tends to move into a state in which its potential energy has the smallest value. This is principle of minimum potential energy .
If the system in a given state does not have a minimum potential energy, then this state is called energetically unfavorable.
If the ball is at the bottom of a concave bowl (Fig. 2.10, a), where its potential energy is minimal (compared to its values in neighboring positions), then its state is more favorable. The equilibrium of the ball in this case is sustainable: if you move the ball to the side and release it, it will return to its original position again.
Energetically unfavorable, for example, is the position of the ball on top of a convex surface (Fig. 2.10, b). The sum of the forces acting in this case on the ball is equal to zero, and therefore, this ball will be in equilibrium. However, this balance is unstable: the slightest impact is enough for it to slide down and thereby move into a state of energetically more favorable, i.e. less
P 
potential energy.
At indifferent equilibrium (Fig. 2.10, c) the potential energy of the body is equal to the potential energy of all its possible nearest states.
In figure 2.11, you can indicate some limited area of \u200b\u200bspace (for example, cd), in which the potential energy is less than outside it. This area was named potential hole .
Issues under consideration:
General theorems of the dynamics of a mechanical system. Kinetic energy: a material point, a system of material points, an absolutely rigid body (with translational, rotational and plane motion). Koenig's theorem. Work of a force: the elementary work of forces applied to a rigid body; on final displacement, gravity, sliding friction forces, elastic forces. Elementary work of the moment of force. Power of force and pair of forces. Theorem on the change in the kinetic energy of a material point. The theorem on the change in the kinetic energy of variable and unchanging mechanical systems (differential and integral form). Potential force field and its properties. equipotential surfaces. Potential feature. Potential energy. Law of conservation of total mechanical energy.
5.1 Kinetic energy
a) material point:
Definition: The kinetic energy of a material point is half the product of the mass of this point and the square of its speed:
Kinetic energy is a scalar positive quantity.
In the SI system, the unit of energy is the joule:
1 j \u003d 1 N?m.
b) systems of material points:
The kinetic energy of a system of material points is the sum of the kinetic energies of all points in the system:
(127)
c) absolutely rigid body:
1) in translational motion.
The speeds of all points are the same and equal to the speed of the center of mass, i.e. , Then:
Where M- body mass.
The kinetic energy of a rigid body moving forward is equal to half the product of the mass of the body M to the square of its speed.
2) during rotation.
The velocities of the points are determined by the Euler formula:
(130)
Speed module:
(131)
Kinetic energy of a body during rotational motion:
(133)
Where: z- axis of rotation.
The kinetic energy of a rigid body rotating around a fixed axis is equal to half the product of the moment of inertia of this body about the axis of rotation and the square of the angular velocity of the body.
3) with flat motion.
The speed of any point is determined through the pole:
(134)
Planar motion consists of translational motion at the speed of the pole and rotational motion around this pole, then the kinetic energy is the sum of the energy of translational motion and the energy of rotational motion.
Kinetic energy through the "A" pole in plane motion:
(135)
It is best to take the center of mass for the pole, then:
(136)
This is convenient because the moments of inertia about the center of mass are always known.
The kinetic energy of a rigid body during plane-parallel motion is the sum of the kinetic energy of translational motion together with the center of mass and the kinetic energy from rotation around a fixed axis passing through the center of mass and perpendicular to the plane of motion.
It is often convenient to take the instantaneous center of velocities as the pole. Then:
(137)
Given that, by definition of the instantaneous center of velocities, its velocity is zero, then .
Kinetic energy relative to the instantaneous center of velocities:
(138)
It must be remembered that to determine the moment of inertia relative to the instantaneous center of velocities, it is necessary to apply the Huygens-Steiner formula:
(139)
This formula is preferable in cases where the instantaneous center of velocities is at the end of the rod.
4) Koenig's theorem.
Let us assume that the mechanical system, together with the coordinate system passing through the center of mass of the system, moves translationally relative to the fixed coordinate system. Then, on the basis of the theorem on the addition of velocities in the case of a complex movement of a point, the absolute speed of an arbitrary point of the system can be written as the vector sum of the translational and relative velocities:
(140)
where: - the speed of the beginning of the moving coordinate system (transfer speed, i.e. the speed of the center of mass of the system);
The speed of a point relative to a moving coordinate system (relative speed). Omitting intermediate calculations, we get:
(141)
This equality defines Koenig's theorem.
Formulation: The kinetic energy of the system is equal to the sum of the kinetic energy that a material point located at the center of mass of the system and having a mass equal to the mass of the system and the kinetic energy of the system's motion relative to the center of mass would have.
5.2Force work.
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
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 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 the stone is 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.
Kinetic energy is, by definition, a quantity equal to half the mass of a moving body multiplied by the square of the speed of this body. This is one of the most important terms in modern mechanics. In short, it is the energy of motion, or the difference between the total energy and the rest energy. Yet its essence is not fully considered in modern science.
Kinetic energy (from Gr. Kinema - movement) of a body in the state
The immobility is zero. Often this value is associated not only with mass and speed. So, according to one definition, kinetic energy is the work done at a certain speed. Measured in joules.
The kinetic energy of a system is a quantity that is closely related to the speed of each of its points.
It is considered both in translational and rotational motion. The first case has already been explained in detail above, this is half the mass of any object multiplied by its speed squared. And the kinetic energy of rotation of the body is represented as the sum of the kinetic energies of each of the elementary volumes of the given body. Or as the value of the moment of inertia multiplied by the square of the angular velocity, divided by two.
Suppose there is any rigid body that moves around the axis
motionless, passing through it. This object can be divided into small elementary volumes, each of which has its own elementary mass. The body in question moves around a fixed axis. In this case, each of the elementary volumes describes a circle of the corresponding radius. Their rotation is the same. And therefore the kinetic energy of a given body is the sum of the kinetic energies of all its elementary volumes moving around the axis. A simplified version of this formula is half the product of the square of the angular velocity and the moment of inertia.
In some cases, kinetic energy is the sum of both translational and rotational energy. For example, a cylinder rolling without slipping along an inclined line. Moving forward, he performs however, while he also moves around his axis.
One of the components of the kinetic energy of rotation is what was mentioned above. It depends on the total body mass, as well as on its distribution with respect to the axis of rotation. What is it? This is a measure of the inertia of motion around an axis, just as in translational motion the measure of inertia is mass. This is a very important value. The larger the moment of inertia, the more difficult it is to bring the body into a state of rotation. Angular velocity characterizes the speed with which a rigid body moves around its axis. The unit of measure is rad/s. Angular velocity is the ratio of the angle of rotation to the time interval during which this angle passes a rotating object.
The kinetic energy theorem can be formulated approximately as follows: the work of the resultant force applied to a certain body is equivalent to a change in the kinetic energy of this body.
