What is called the kinetic energy of the body. Kinetic and potential energy
Opening law of conservation of momentum, which asserts that the vector sum of the impulses of all bodies (or particles) of a closed system is a constant value, showed that the mechanical motion of bodies has a quantitative measure that is preserved during any interactions of bodies. This measure is momentum. However, only with the help of this law it will not be possible to give a complete explanation of all the laws of motion and interaction of bodies.
Consider an example. A 9 gram bullet at rest is absolutely harmless. But during the shot, when in contact with an obstacle, the bullet deforms it. It is obvious that such a destructive effect is obtained as a result of the fact that the bullet has a special energy.
Let's consider another example. Two identical plasticine balls move towards each other with the same speed. When they collide, they stop and merge into one body.
The sum of the momenta of the balls before the collision and after the collision is the same and equal to zero, the momentum conservation law is satisfied. What happens to plasticine balls when they collide, except for a change in the speed of movement? The balls deform and heat up.
An increase in the temperature of bodies during a collision can be observed, for example, when a hammer strikes a lead or copper rod. A change in body temperature indicates changes in the speed of the chaotic thermal motion of atoms that make up the body. Consequently, mechanical motion did not disappear without a trace, it turned into another form of motion of matter.
Let's go back to the question we posed above. Is there a measure of the motion of matter in nature that is preserved during any transformations of one form of motion into another? Experiments and observations have shown that such a measure of motion exists in nature. They called it energy.
energy called a physical quantity, which is a quantitative measure of various forms of motion of matter.
To accurately define energy as a physical quantity, it is necessary to find its relationship with other quantities, choose a unit of measurement and find ways to measure it.
mechanical energy called a physical quantity, which is a quantitative measure of mechanical movement.
In physics, as such a quantitative measure of translational mechanical motion, when it arises from other forms of motion or is transformed into other forms of motion, a value equal to half the product of the mass of the body and the square of the speed of its motion is accepted. This physical quantity is called kinetic energy of the body and is marked with the letter E with index To:
E k \u003d mv 2 / 2
Since speed is a quantity that depends on the choice of frame of reference, the value of the kinetic energy of a body depends on the choice of frame of reference.
There is a theorem about kinetic energy. "The work of the resultant force applied to the body is equal to the change in its kinetic energy":
A \u003d E k2 -E k1
This theorem will be valid both when the body moves under the action of a constant force, and when the body moves under the action of a changing force, the direction of which does not coincide with the direction of movement. Kinetic energy is the energy of motion. It turns out, kinetic energy of the body mass m, moving at a speed v is equal to the work that the force applied to a body at rest must do to give it this speed:
A \u003d mv 2 / 2 \u003d E to
If the body moves with a speed v, then to stop it completely, work must be done:
A \u003d -mv 2 / 2 \u003d -E to
The unit of work in the international system is the work done by force 1 Newton on a way 1 meter when moving in the direction of the force vector. This unit of work is called Joule.
1 J \u003d 1 kg m 2 / s 2
Since work equals change in energy, energy is measured in the same unit as work. Unit of energy in SI - 1J.
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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 took place, 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 the 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 A 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, 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 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 infinitesimal 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.
One of the characteristics of any system is its kinetic and potential energy. If any force F exerts an action on a body at rest in such a way that the latter begins to move, then work dA is performed. In this case, the value of the kinetic energy dT becomes the higher, the more work is done. In other words, we can write the equality:
Considering the path dR traveled by the body and the developed speed dV, we will use the second one for the force:
An important point: this law can be used if an inertial frame of reference is taken. The choice of system affects the energy value. Internationally, energy is measured in joules (J).
It follows that a particle or body, characterized by the speed of movement V and mass m, will be:
T = ((V * V)*m) / 2
It can be concluded that kinetic energy is determined by speed and mass, in fact, representing a function of motion.
Kinetic and potential energy allow you to describe the state of the body. If the first, as already mentioned, is directly related to the movement, then the second is applied to a system of interacting bodies. Kinetic and are usually considered for examples where the force binding the bodies does not depend on In this case, only the initial and final positions are important. The most famous example is the gravitational interaction. But if the trajectory is also important, then the force is dissipative (friction).
In simple terms, potential energy is the ability to do work. Accordingly, this energy can be considered as the work that must be done to move the body from one point to another. That is:
If the potential energy is denoted as dP, then we get:
A negative value indicates that work is being done by decreasing dP. For the known function dP, it is possible to determine not only the modulus of force F, but also its direction vector.
A change in kinetic energy is always associated with potential energy. This is easy to understand if you remember the systems. The total value of T + dP when moving the body always remains unchanged. Thus, the change in T always occurs in parallel with the change in dP, they seem to flow into each other, transforming.
Since kinetic and potential energy are interrelated, their sum is the total energy of the system under consideration. In relation to molecules, it is and is always present, as long as there is at least thermal motion and interaction.
When performing calculations, a reference system and any arbitrary moment taken as the initial one are selected. It is possible to accurately determine the value of potential energy only in the zone of action of such forces, which, when doing work, do not depend on the trajectory of movement of any particle or body. In physics, such forces are called conservative. They are always interconnected with the law of conservation of total energy.
An interesting point: in a situation where external influences are minimal or leveled, any system under study always tends to its state when its potential energy tends to zero. For example, a tossed ball reaches the limit of its potential energy at the top of the trajectory, but at the same moment it starts moving down, converting the accumulated energy into movement, into work performed. It is worth noting once again that for potential energy there is always an interaction of at least two bodies: for example, in the example with the ball, it is influenced by the gravity of the planet. Kinetic energy can be calculated individually for each moving body.
Kinetic energy - a scalar physical quantity equal to half the product of the mass of the body and the square of its speed.
To understand what the kinetic energy of a body is, consider the case when a body of mass m under the action of a constant force (F=const) moves in a straight line with uniform acceleration (a=const). Let us determine the work of the force applied to the body when changing the modulus of the velocity of this body from v1 to v2.
As we know, the work of a constant force is calculated by the formula. Since in the case we are considering, the direction of the force F and the displacement s coincide, then , and then we get that the work of the force is equal to A = Fs. According to Newton's second law, we find the force F=ma. For rectilinear uniformly accelerated motion, the formula is valid:
From this formula we express the displacement of the body:
We substitute the found values of F and S into the work formula, and we get:
From the last formula it can be seen that the work of the force applied to the body, when the speed of this body changes, is equal to the difference between two values of a certain quantity. And mechanical work is a measure of energy change. Therefore, on the right side of the formula is the difference between the two values of the energy of a given body. This means that the quantity is the energy due to the movement of the body. This energy is called kinetic. It is designated Wk.
If we take the work formula we have derived, then we get
The work done by a force when the speed of a body changes is equal to the change in the kinetic energy of this body
There is also:
Potential energy:
In the formula we used:
Kinetic energy
Body mass
body speed
In § 88, the expression was called the kinetic energy of the body. Let us consider in more detail the content of this concept.
Let us assume that the body of mass was initially motionless (Fig. 5.8). A force acted on it under the influence of which the body traveled a distance, acquiring speed. At the same time, the force did the work and the relation will take place
If we take another body of mass and perform the same work with the same force, then for the resulting movement the relation will again be true
where is the final velocity of the body of mass
The same work of force informs bodies with different masses always the same stock of motion, and this is expressed by the equality
Thus, the kinetic energy of a body can be considered as a measure of the stock of motion of a given body. With the help of this measure, it is possible to compare with each other those stocks of motion that different bodies or systems of bodies have. The remarkable thing is that this measure takes into account any movement, regardless of their direction.
Therefore, it can be used to calculate not only ordered motions of bodies, but also disordered, chaotic motions occurring in complex systems of many bodies. Using, for example, the concept of kinetic energy, one can quantify the reserve of motion that a certain mass of gas has. Gas molecules perform continuous chaotic motions. The sum of the kinetic energies of these molecules will determine the energy of the entire mass of the gas, i.e., it will give a quantitative characteristic of the intensity of thermal motion stored in this gas. It will also give a quantitative idea of the state of motion of the system of bodies as a whole.
Note that it is impossible to get an idea of the state of internal motions in a system of bodies using the momentum vector. Let's take, for example, two bodies of the same mass that move in opposite directions with equal velocities in absolute value. The amount of movement of each of the bodies will be equal. The amount of motion of the entire system as a whole, equal to the vector sum of the quantities of motion of individual bodies, will be equal to zero.
Knowing only this result (the momentum of the system is zero), we cannot even say whether the bodies of the system are moving at all. The kinetic energy of such a system will be equal Knowing this, firstly, we can conclude that there is movement in a given system of bodies, and secondly, we can judge how large the stock of this movement is.
Consider the case when a body of mass moving at a speed (Fig. 5.9) meets another body (for example, a spring). When interacting, forces arise that slow down the movement of the body and cause deformation or movement of another body. Thus, it turns out that a moving body, when meeting with others
bodies can do some work to deform or set these bodies in motion. Let's find this job.
According to Newton's third law, at any time, the force of the body on the spring is equal to the force developed by the spring: Therefore, the work of the body during its braking is equal to the work of the spring with the opposite sign:
Substituting we get
This gives us the right to assert that the kinetic energy of any body determines the work that a moving body can do during a stop when interacting with other bodies. Kinetic energy acts as a measure of the efficiency of a moving body. This is also evidenced by the origin of the word "energy". In Greek, the word "energy" means activity, efficiency.
So, each moving body is capable of producing a certain amount of work. This work is determined by the mass and speed of the body. If the body performs this work during the interaction, then the movement of the body begins to disappear. When work is done, the movement of a body is transformed into the movement of other bodies or their parts. In this case, the transformation of mechanical motion into other forms of motion of matter, for example, the transformation of mechanical motion into thermal motion, can also occur.
Final conclusion: kinetic energy is a measure of the body's stock of motion and at the same time determines the work that the body is able to do when interacting with other bodies.
Kinetic energy is equal to half the product of the mass of the body and the square of its speed:
It is clear from the equation that the units of kinetic energy are the same as the units of work: (§ 89).
