Potential energy of bodies interacting through gravitational forces. referat

If only conservative forces act on the system, then we can introduce for it the concept potential energy. Any arbitrary position of the system, characterized by setting the coordinates of its material points, we will conditionally take as zero. The work done by conservative forces during the transition of the system from the considered position to zero is called potential energy of the system in first position

The work of conservative forces does not depend on the transition path, and therefore the potential energy of the system at a fixed zero position depends only on the coordinates of the material points of the system in the considered position. In other words, the potential energy of the system U is a function of only its coordinates.

The potential energy of the system is not uniquely defined, but up to an arbitrary constant. This arbitrariness cannot affect physical conclusions, since the course of physical phenomena may depend not on the absolute values ​​of the potential energy itself, but only on its difference in various states. The same differences do not depend on the choice of an arbitrary constant.

Let the system move from position 1 to position 2 along some path 12 (Fig. 3.3). work BUT 12 performed by conservative forces during such a transition can be expressed in terms of potential energies U 1 and U 2 in states 1 And 2 . For this purpose, let us imagine that the transition is made through position O, i.e., along the path 1O2. Since the forces are conservative, then BUT 12 = BUT 1O2 = BUT 1O + BUT O2 = BUT 1O - BUT 2O. By definition of potential energy U 1 = A 1 O , U 2 = A 2O. In this way,

A 12 = U 1 – U 2 , (3.10)

i.e., the work of conservative forces is equal to the decrease in the potential energy of the system.

Same job BUT 12 , as shown earlier in (3.7), can be expressed in terms of the kinetic energy increment by the formula

BUT 12 = TO 2 – TO 1 .

Equating their right-hand sides, we get TO 2 – TO 1 = U 1 – U 2 , whence

TO 1 + U 1 = TO 2 + U 2 .

The sum of the kinetic and potential energies of a system is called its total energy E. In this way, E 1 = E 2 , or

Eº K+U= const. (3.11)

In a system with only conservative forces, the total energy remains unchanged. Only transformations of potential energy into kinetic energy and vice versa can occur, but the total energy supply of the system cannot change. This position is called the law of conservation of energy in mechanics.

Let us calculate the potential energy in some simplest cases.

a) Potential energy of a body in a uniform gravitational field. If a material point located at a height h, will fall to the zero level (i.e., the level for which h= 0), then gravity will do work A=mgh. Therefore, on top h material point has potential energy U=mgh+C, where FROM is an additive constant. An arbitrary level can be taken as zero, for example, floor level (if the experiment is carried out in a laboratory), sea level, etc. Constant FROM is equal to potential energy at zero level. Setting it equal to zero, we get

U=mgh. (3.12)

b) Potential energy of a stretched spring. The elastic forces that occur when a spring is stretched or compressed are central forces. Therefore, they are conservative, and it makes sense to talk about the potential energy of a deformed spring. They call her elastic energy. Denote by x spring extension,T. e. difference x = l – l 0 lengths of the spring in the deformed and undeformed states. Elastic force F depends on stretch. If stretching x not very large, then it is proportional to it: F = – kx(Hooke's law). When the spring returns from the deformed to the undeformed state, the force F does the job

If the elastic energy of the spring in the undeformed state is assumed to be equal to zero, then

c) Potential energy of gravitational attraction of two material points. According to Newton's law of universal gravitation, the gravitational force of attraction of two point bodies is proportional to the product of their masses mm and is inversely proportional to the square of the distance between them:

where G is gravitational constant.

The force of gravitational attraction, as a central force, is conservative. It makes sense for her to talk about potential energy. When calculating this energy, one of the masses, for example M, can be considered as stationary, and the other as moving in its gravitational field. When moving mass m from infinity, gravitational forces do work

where r- distance between masses M And m in final state.

This work is equal to the loss of potential energy:

Usually potential energy at infinity U¥ is taken equal to zero. With such an agreement

The quantity (3.15) is negative. This has a simple explanation. Attractive masses have maximum energy at an infinite distance between them. In this position, the potential energy is considered to be zero. In every other position it is smaller, i.e. negative.

Let us now assume that, along with conservative forces, dissipative forces also act in the system. The work of all forces BUT 12 during the transition of the system from position 1 to position 2 is still equal to the increment of its kinetic energy TO 2 – TO one . But in the case under consideration, this work can be represented as the sum of the work of conservative forces and the work of dissipative forces. The first work can be expressed in terms of the loss of potential energy of the system: Therefore

Equating this expression to the increment of kinetic energy, we obtain

where E=K+U is the total energy of the system. Thus, in the case under consideration, the mechanical energy E system does not remain constant, but decreases, since the work of dissipative forces is negative.

> Gravitational potential energy

What's happened gravitational energy: potential energy of gravitational interaction, formula for gravitational energy and Newton's law of universal gravitation.

Gravitational energy is the potential energy associated with the gravitational force.

Learning task

• Calculate the gravitational potential energy for two masses.

Key Points

Terms

• Potential energy is the energy of an object in its position or chemical state.
• Newton's gravitational backwater - each point universal mass attracts another with the help of a force that is directly proportional to their masses and inversely proportional to the square of their distance.
• Gravity is the net force on the ground that pulls objects toward the center. Created by rotation.

Example

What will be the gravitational potential energy of a 1 kg book at a height of 1 m? Since the position is set close to the earth's surface, the gravitational acceleration will be constant (g = 9.8 m/s 2), and the energy of the gravitational potential (mgh) reaches 1 kg ⋅ 1 m ⋅ 9.8 m/s 2 . This can also be seen in the formula:

If you add the mass and the earth's radius.

Gravitational energy reflects the potential associated with the force of gravity, because it is necessary to overcome the earth's gravity in order to do work on lifting objects. If an object falls from one point to another inside a gravitational field, then the force of gravity will do positive work, and the gravitational potential energy will decrease by the same amount.

Let's say we have a book left on the table. When we move it from the floor to the top of the table, a certain external intervention works against the gravitational force. If it falls, then this is the work of gravity. Therefore, the process of falling reflects the potential energy accelerating the mass of the book and transforming into kinetic energy. As soon as the book touches the floor, the kinetic energy becomes heat and sound.

The gravitational potential energy is affected by the height relative to a specific point, the mass and strength of the gravitational field. So the book on the table is inferior in gravitational potential energy to the heavier book below. Remember that height cannot be used in calculating gravitational potential energy unless gravity is constant.

local approximation

The strength of the gravitational field is affected by location. If the distance change is insignificant, then it can be neglected, and the force of gravity can be made constant (g = 9.8 m/s 2). Then for the calculation we use a simple formula: W = Fd. The upward force is equated to weight, so work is related to mgh, resulting in the formula: U = mgh (U is potential energy, m is the mass of the object, g is the acceleration of gravity, h is the height of the object). The value is expressed in joules. The change in potential energy is conveyed as

General formula

However, if we encounter major changes in distance, then g cannot remain constant and calculus and the mathematical definition of work must be applied. To calculate the potential energy, one can integrate the gravitational force with respect to the distance between the bodies. Then we get the formula for gravitational energy:

U = -G + K, where K is the constant of integration and is equal to zero. Here the potential energy goes to zero when r is infinite.

energy is called a scalar physical quantity, which is a single measure of various forms of the motion of matter and a measure of the transition of the motion of matter from one form to another.

To characterize various forms of motion of matter, the corresponding types of energy are introduced, for example: mechanical, internal, energy of electrostatic, intranuclear interactions, etc.

Energy obeys the law of conservation, which is one of the most important laws of nature.

Mechanical energy E characterizes the movement and interaction of bodies and is a function of the speeds and relative positions of the bodies. It is equal to the sum of kinetic and potential energies.

Kinetic energy

Let us consider the case when a body of mass m a constant force \(~\vec F\) acts (it can be the resultant of several forces) and the vectors of force \(~\vec F\) and displacement \(~\vec s\) are directed along one straight line in one direction. In this case, the work done by the force can be defined as A = F∙s. The modulus of force according to Newton's second law is F = m∙a, and the displacement module s with uniformly accelerated rectilinear motion, it is associated with the modules of the initial υ 1 and final υ 2 speeds and accelerations but\(~s = \frac(\upsilon^2_2 - \upsilon^2_1)(2a)\) .

Hence, to work, we get

\(~A = F \cdot s = m \cdot a \cdot \frac(\upsilon^2_2 - \upsilon^2_1)(2a) = \frac(m \cdot \upsilon^2_2)(2) - \frac (m \cdot \upsilon^2_1)(2)\) . (one)

A physical quantity equal to half the product of the body's mass and the square of its speed is called kinetic energy of the body.

Kinetic energy is denoted by the letter E k .

\(~E_k = \frac(m \cdot \upsilon^2)(2)\) . (2)

Then equality (1) can be written in the following form:

\(~A = E_(k2) - E_(k1)\) . (3)

Kinetic energy theorem

the work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body.

Since the change in kinetic energy is equal to the work of the force (3), the kinetic energy of the body is expressed in the same units as the work, i.e., in joules.

If the initial velocity of the body mass m is zero and the body increases its speed to the value υ , then the work of the force is equal to the final value of the kinetic energy of the body:

\(~A = E_(k2) - E_(k1)= \frac(m \cdot \upsilon^2)(2) - 0 = \frac(m \cdot \upsilon^2)(2)\) . (4)

The physical meaning of kinetic energy

The kinetic energy of a body moving at a speed υ shows how much work the force acting on a body at rest must do to give it this speed.

Potential energy

Potential energy is the energy of the interaction of bodies.

The potential energy of a body raised above the Earth is the energy of interaction between the body and the Earth by gravitational forces. The potential energy of an elastically deformed body is the energy of interaction of individual parts of the body with each other by elastic forces.

Potential called strength, whose work depends only on the initial and final position of a moving material point or body and does not depend on the shape of the trajectory.

With a closed trajectory, the work of the potential force is always zero. Potential forces include gravitational forces, elastic forces, electrostatic forces, and some others.

Forces, whose work depends on the shape of the trajectory, are called non-potential. When moving a material point or body along a closed trajectory, the work of a non-potential force is not equal to zero.

Potential energy of interaction of a body with the Earth

Find the work done by gravity F t when moving a body with a mass m vertically down from a height h 1 above the Earth's surface to a height h 2 (Fig. 1). If the difference h 1 – h 2 is negligible compared to the distance to the center of the Earth, then the force of gravity F m during the motion of the body can be considered constant and equal to mg.

Since the displacement coincides in direction with the gravity vector, the work done by gravity is

\(~A = F \cdot s = m \cdot g \cdot (h_1 - h_2)\) . (five)

Consider now the motion of a body along an inclined plane. When moving a body down an inclined plane (Fig. 2), gravity F t = m∙g does the job

\(~A = m \cdot g \cdot s \cdot \cos \alpha = m \cdot g \cdot h\) , (6)

where h is the height of the inclined plane, s- displacement modulus equal to the length of the inclined plane.

Body movement from a point IN exactly FROM along any trajectory (Fig. 3) can be mentally represented as consisting of movements along sections of inclined planes with different heights h’, h'' etc. Work BUT gravity all the way out IN in FROM is equal to the sum of work on individual sections of the path:

\(~A = m \cdot g \cdot h" + m \cdot g \cdot h"" + \ldots + m \cdot g \cdot h^n = m \cdot g \cdot (h" + h"" + \ldots + h^n) = m \cdot g \cdot (h_1 - h_2)\) , (7)

where h 1 and h 2 - heights from the Earth's surface, on which the points are located, respectively IN And FROM.

Equality (7) shows that the work of gravity does not depend on the trajectory of the body and is always equal to the product of the modulus of gravity and the difference in heights in the initial and final positions.

When moving down, the work of gravity is positive, when moving up, it is negative. The work of gravity on a closed trajectory is zero.

Equality (7) can be represented as follows:

\(~A = - (m \cdot g \cdot h_2 - m \cdot g \cdot h_1)\) . (8)

The physical quantity equal to the product of the mass of the body by the modulus of the acceleration of free fall and the height to which the body is raised above the surface of the Earth is called potential energy interaction between the body and the earth.

The work of gravity when moving a body with a mass m from a point at a height h 2 , to a point located at a height h 1 from the surface of the Earth, along any trajectory is equal to the change in the potential energy of interaction between the body and the Earth, taken with the opposite sign.

\(~A = - (E_(p2) - E_(p1))\) . (nine)

Potential energy is denoted by the letter E p .

The value of the potential energy of a body raised above the Earth depends on the choice of the zero level, i.e., the height at which the potential energy is assumed to be zero. It is usually assumed that the potential energy of a body on the surface of the Earth is zero.

With this choice of the zero level, the potential energy E p of a body at a height h above the Earth's surface, is equal to the product of the mass m of the body and the modulus of the free fall acceleration g and distance h it from the Earth's surface:

\(~E_p = m \cdot g \cdot h\) . (10)

The physical meaning of the potential energy of the interaction of the body with the Earth

The potential energy of a body on which gravity acts is equal to the work done by gravity when moving the body to the zero level.

Unlike the kinetic energy of translational motion, which can only have positive values, the potential energy of a body can be either positive or negative. body mass m at the height h, where h < h 0 (h 0 - zero height), has a negative potential energy:

\(~E_p = -m \cdot g \cdot h\) .

Potential energy of gravitational interaction

Potential energy of gravitational interaction of a system of two material points with masses m And M located at a distance r one from the other is equal to

\(~E_p = G \cdot \frac(M \cdot m)(r)\) . (eleven)

where G is the gravitational constant, and the zero of the potential energy reference ( E p = 0) is accepted for r = ∞.

Potential energy of gravitational interaction of a body with mass m with the earth where h is the height of the body above the earth's surface, M e is the mass of the Earth, R e is the radius of the Earth, and the zero of the potential energy is chosen at h = 0.

\(~E_e = G \cdot \frac(M_e \cdot m \cdot h)(R_e \cdot (R_e +h))\) . (12)

Under the same condition of choosing the reference zero, the potential energy of the gravitational interaction of a body with a mass m with Earth for low altitudes h (h « R e) is equal to

\(~E_p = m \cdot g \cdot h\) ,

where \(~g = G \cdot \frac(M_e)(R^2_e)\) is the gravitational acceleration modulus near the Earth's surface.

Potential energy of an elastically deformed body

Let us calculate the work done by the elastic force when the deformation (elongation) of the spring changes from some initial value x 1 to final value x 2 (Fig. 4, b, c).

The elastic force changes as the spring deforms. To find the work of the elastic force, you can take the average value of the modulus of force (because the elastic force depends linearly on x) and multiply by the displacement modulus:

\(~A = F_(upr-cp) \cdot (x_1 - x_2)\) , (13)

where \(~F_(upr-cp) = k \cdot \frac(x_1 - x_2)(2)\) . From here

\(~A = k \cdot \frac(x_1 - x_2)(2) \cdot (x_1 - x_2) = k \cdot \frac(x^2_1 - x^2_2)(2)\) or \(~A = -\left(\frac(k \cdot x^2_2)(2) - \frac(k \cdot x^2_1)(2) \right)\) . (fourteen)

A physical quantity equal to half the product of the rigidity of a body and the square of its deformation is called potential energy elastically deformed body:

\(~E_p = \frac(k \cdot x^2)(2)\) . (15)

From formulas (14) and (15) it follows that the work of the elastic force is equal to the change in the potential energy of an elastically deformed body, taken with the opposite sign:

\(~A = -(E_(p2) - E_(p1))\) . (16)

If x 2 = 0 and x 1 = X, then, as can be seen from formulas (14) and (15),

\(~E_p = A\) .

The physical meaning of the potential energy of a deformed body

the potential energy of an elastically deformed body is equal to the work done by the elastic force when the body goes into a state in which the deformation is zero.

Potential energy characterizes interacting bodies, and kinetic energy characterizes moving bodies. Both potential and kinetic energy change only as a result of such an interaction of bodies, in which the forces acting on the bodies do work that is different from zero. Let us consider the question of energy changes during the interactions of bodies forming a closed system.

closed system is a system that is not acted upon by external forces or the action of these forces is compensated. If several bodies interact with each other only by gravitational and elastic forces and no external forces act on them, then for any interactions of bodies, the work of the elastic or gravitational forces is equal to the change in the potential energy of the bodies, taken with the opposite sign:

\(~A = -(E_(p2) - E_(p1))\) . (17)

According to the kinetic energy theorem, the work of the same forces is equal to the change in kinetic energy:

\(~A = E_(k2) - E_(k1)\) . (eighteen)

Comparison of equalities (17) and (18) shows that the change in the kinetic energy of bodies in a closed system is equal in absolute value to the change in the potential energy of the system of bodies and is opposite in sign:

\(~E_(k2) - E_(k1) = -(E_(p2) - E_(p1))\) or \(~E_(k1) + E_(p1) = E_(k2) + E_(p2) \) . (19)

The law of conservation of energy in mechanical processes:

the sum of the kinetic and potential energy of the bodies that make up a closed system and interact with each other by gravitational and elastic forces remains constant.

The sum of the kinetic and potential energies of bodies is called full mechanical energy.

Let's take a simple experiment. Throw up a steel ball. Having reported the initial speed υ beginning, we will give it kinetic energy, because of which it will begin to rise upwards. The action of gravity leads to a decrease in the speed of the ball, and hence its kinetic energy. But the ball rises higher and higher and acquires more and more potential energy ( E p= m∙g∙h). Thus, kinetic energy does not disappear without a trace, but it is converted into potential energy.

At the moment of reaching the top point of the trajectory ( υ = 0) the ball is completely deprived of kinetic energy ( E k = 0), but at the same time its potential energy becomes maximum. Then the ball changes direction and moves down with increasing speed. Now there is a reverse transformation of potential energy into kinetic energy.

The law of conservation of energy reveals physical meaning concepts work:

the work of gravitational and elastic forces, on the one hand, is equal to an increase in kinetic energy, and on the other hand, to a decrease in the potential energy of bodies. Therefore, work is equal to energy converted from one form to another.

Mechanical Energy Change Law

If the system of interacting bodies is not closed, then its mechanical energy is not conserved. The change in the mechanical energy of such a system is equal to the work of external forces:

\(~A_(vn) = \Delta E = E - E_0\) . (twenty)

where E And E 0 are the total mechanical energies of the system in the final and initial states, respectively.

An example of such a system is a system in which, along with potential forces, non-potential forces act. Friction forces are non-potential forces. In most cases, when the angle between the friction force F r body is π radians, the work of the friction force is negative and equal to

\(~A_(tr) = -F_(tr) \cdot s_(12)\) ,

where s 12 - the path of the body between points 1 and 2.

Friction forces during the motion of the system reduce its kinetic energy. As a result, the mechanical energy of a closed non-conservative system always decreases, turning into the energy of non-mechanical forms of motion.

For example, a car moving along a horizontal section of the road, after turning off the engine, travels a certain distance and stops under the action of friction forces. The kinetic energy of the forward motion of the car became equal to zero, and the potential energy did not increase. During the braking of the car, the brake pads, car tires and asphalt heated up. Consequently, as a result of the action of friction forces, the kinetic energy of the car did not disappear, but turned into the internal energy of the thermal motion of molecules.

The law of conservation and transformation of energy

in any physical interaction, energy is converted from one form to another.

Sometimes the angle between the force of friction F tr and elementary displacement Δ r is zero and the work of the friction force is positive:

\(~A_(tr) = F_(tr) \cdot s_(12)\) ,

Example 1. May an external force F acts on the bar IN, which can slide on the trolley D(Fig. 5). If the trolley moves to the right, then the work of the sliding friction force F tr2 acting on the cart from the side of the bar is positive:

Example 2. When the wheel is rolling, its rolling friction force is directed along the movement, since the point of contact of the wheel with the horizontal surface moves in the direction opposite to the direction of the wheel movement, and the work of the friction force is positive (Fig. 6):

Literature

1. Kabardin O.F. Physics: Ref. materials: Proc. allowance for students. - M.: Enlightenment, 1991. - 367 p.
2. Kikoin I.K., Kikoin A.K. Physics: Proc. for 9 cells. avg. school - M .: Pro-sveshchenie, 1992. - 191 p.
3. Elementary textbook of physics: Proc. allowance. In 3 volumes / Ed. G.S. Landsberg: v. 1. Mechanics. Heat. Molecular physics. – M.: Fizmatlit, 2004. – 608 p.
4. Yavorsky B.M., Seleznev Yu.A. A reference guide to physics for applicants to universities and self-education. – M.: Nauka, 1983. – 383 p.

In connection with a number of features, and also in view of the special importance, the question of the potential energy of the forces of universal gravitation must be considered separately and in more detail.

We encounter the first feature when choosing the reference point for potential energies. In practice, one has to calculate the motion of a given (trial) body under the action of universal gravitational forces created by other bodies of different masses and sizes.

Let us assume that we have agreed to consider the potential energy equal to zero in a position in which the bodies are in contact. Let the test body A, when interacting separately with balls of the same mass, but different radii, first be removed from the centers of the balls at the same distance (Fig. 5.28). It is easy to see that when the body A moves before it comes into contact with the surfaces of the bodies, the gravitational forces will do different work. This means that we must consider the potential energies of the systems to be different for the same relative initial positions of the bodies.

It will be especially difficult to compare these energies with each other in cases where the interactions and movements of three or more bodies are considered. Therefore, for the forces of universal gravitation, such an initial level of counting of potential energies is sought, which could be the same, common, for all bodies in the Universe. It was agreed to consider such a common zero level of potential energy of the forces of universal gravitation the level corresponding to the location of bodies at infinitely large distances from each other. As can be seen from the law of universal gravitation, the forces of universal gravitation themselves vanish at infinity.

With such a choice of the energy reference point, an unusual situation is created with the determination of the values ​​of potential energies and the performance of all calculations.

In the cases of gravity (Fig. 5.29, a) and elasticity (Fig. 5.29, b), the internal forces of the system tend to bring the bodies to zero. As bodies approach the zero level, the potential energy of the system decreases. The zero level really corresponds to the lowest potential energy of the system.

This means that for all other positions of the bodies, the potential energy of the system is positive.

In the case of universal gravitational forces and when choosing zero energy at infinity, everything happens the other way around. The internal forces of the system tend to move the bodies away from the zero level (Fig. 5.30). They do positive work when the bodies move away from the zero level, i.e., when the bodies approach each other. At any finite distances between the bodies, the potential energy of the system is less than at In other words, the zero level (at corresponds to the highest potential energy. This means that for all other positions of the bodies, the potential energy of the system is negative.

In § 96, it was found that the work of the forces of universal gravitation when moving a body from infinity to a distance is equal to

Therefore, the potential energy of the universal gravitational forces must be considered equal to

This formula expresses another feature of the potential energy of the forces of universal gravitation - the relatively complex nature of the dependence of this energy on the distance between bodies.

On fig. 5.31 shows a graph of dependence on for the case of attraction of bodies by the Earth. This graph has the form of an isosceles hyperbola. Near the surface of the Earth, the energy changes relatively strongly, but already at a distance of several tens of Earth radii, the energy becomes close to zero and begins to change very slowly.

Any body near the Earth's surface is in a kind of "potential well". Whenever it turns out to be necessary to free the body from the action of the forces of the earth's gravity, special efforts must be made in order to "pull" the body out of this potential hole.

In the same way, all other celestial bodies create such potential holes around themselves - traps that capture and hold all not very fast moving bodies.

Knowing the nature of the dependence on makes it possible to significantly simplify the solution of a number of important practical problems. For example, you need to send a spacecraft to Mars, Venus, or any other planet in the solar system. It is necessary to determine what speed should be reported to the ship when it is launched from the surface of the Earth.

In order to send a ship to other planets, it must be removed from the sphere of influence of the forces of earth's gravity. In other words, you need to raise its potential energy to zero. This becomes possible if the ship is given such kinetic energy that it can do work against the forces of gravity, equal to where the mass of the ship,

mass and radius of the earth.

It follows from Newton's second law that (§ 92)

But since the speed of the ship before launch is zero, we can simply write:

where is the speed reported to the ship at launch. Substituting the value for A, we get

Let us use for an exception, as already done in § 96, two expressions for the force of terrestrial attraction on the surface of the Earth:

Hence - Substituting this value into the equation of Newton's second law, we obtain

The speed required to bring the body out of the sphere of influence of the forces of the earth's gravity is called the second cosmic velocity.

In the same way, one can pose and solve the problem of sending a ship to distant stars. To solve such a problem, it is already necessary to determine the conditions under which the ship will be taken out of the sphere of influence of the forces of attraction of the Sun. Repeating all the arguments that were carried out in the previous problem, we can obtain the same expression for the speed reported to the ship at launch:

Here a is the normal acceleration that the Sun informs the Earth and which can be calculated from the nature of the Earth's motion in orbit around the Sun; radius of the earth's orbit. Of course, in this case it means the speed of the ship relative to the Sun. The speed required to take a ship out of the solar system is called the third escape velocity.

The method we have considered for choosing the origin of potential energy is also used in calculations of the electrical interactions of bodies. The concept of potential wells is also widely used in modern electronics, solid state theory, atomic theory, and atomic nucleus physics.

« Physics - Grade 10 "

What is the gravitational interaction of bodies?
How to prove the existence of the interaction of the Earth and, for example, a physics textbook?

As you know, gravity is a conservative force. Now let's find an expression for the work of the gravitational force and prove that the work of this force does not depend on the shape of the trajectory, i.e. that the gravitational force is also a conservative force.

Recall that the work done by a conservative force in a closed loop is zero.

Let a body of mass m be in the Earth's gravitational field. Obviously, the size of this body is small compared to the size of the Earth, so it can be considered a material point. The gravitational force acts on the body

where G is the gravitational constant,
M is the mass of the Earth,
r is the distance at which the body is located from the center of the Earth.

Let the body move from position A to position B along different trajectories: 1) along the straight line AB; 2) along the curve AA "B" B; 3) along the DIA curve (Fig. 5.15)

1. Consider the first case. The gravitational force acting on the body is continuously decreasing, so consider the work of this force on a small displacement Δr i = r i + 1 - r i . The average value of the gravitational force is:

where r 2 сpi = r i r i + 1 .

The smaller Δri, the more valid is the written expression r 2 сpi = r i r i + 1 .

Then the work of the force F cpi , on a small displacement Δr i , can be written as

The total work of the gravitational force when moving a body from point A to point B is:

2. When the body moves along the trajectory AA "B" B (see Fig. 5.15), it is obvious that the work of the gravitational force in sections AA "and B" B is zero, since the gravitational force is directed towards the point O and is perpendicular to any small movement along arc of a circle. Consequently, the work will also be determined by expression (5.31).

3. Let's determine the work of the gravitational force when the body moves from point A to point B along the trajectory DIA (see Fig. 5.15). The work of the gravitational force on a small displacement Δs i is equal to ΔА i = F срi Δs i cosα i ,..

It can be seen from the figure that Δs i cosα i = - Δr i , and the total work will again be determined by formula (5.31).

So, we can conclude that A 1 \u003d A 2 \u003d A 3, i.e., that the work of the gravitational force does not depend on the shape of the trajectory. It is obvious that the work of the gravitational force when moving the body along a closed trajectory AA "B" BA is equal to zero.

The force of gravity is a conservative force.

The change in potential energy is equal to the work of the gravitational force, taken with the opposite sign:

If we choose the zero level of potential energy at infinity, i.e. E pV = 0 as r В → ∞, then, consequently,

The potential energy of a body of mass m, located at a distance r from the center of the Earth, is equal to:

The law of conservation of energy for a body of mass m moving in a gravitational field has the form

where υ 1 is the speed of the body at a distance r 1 from the center of the Earth, υ 2 is the speed of the body at a distance r 2 from the center of the Earth.

Let us determine what minimum speed must be given to a body near the Earth's surface so that in the absence of air resistance it can move away from it beyond the limits of the forces of Earth's gravity.

The minimum speed at which a body, in the absence of air resistance, can move beyond the limits of the forces of gravity is called second cosmic velocity for the Earth.

A gravitational force acts on a body from the side of the Earth, which depends on the distance of the center of mass of this body to the center of mass of the Earth. Since there are no non-conservative forces, the total mechanical energy of the body is conserved. The internal potential energy of the body remains constant, since it does not deform. According to the law of conservation of mechanical energy

On the surface of the Earth, the body has both kinetic and potential energy:

where υ II is the second cosmic velocity, M 3 and R 3 are the mass and radius of the Earth, respectively.

At an infinitely distant point, i.e., at r → ∞, the potential energy of the body is zero (W p \u003d 0), and since we are interested in the minimum speed, the kinetic energy should also be equal to zero: W k \u003d 0.

From the law of conservation of energy follows:

This speed can be expressed in terms of free fall acceleration near the Earth's surface (in calculations, as a rule, this expression is more convenient to use). Insofar as then GM 3 = gR 2 3 .

Therefore, the desired speed

A body falling to the Earth from an infinitely high height would acquire exactly the same speed if there were no air resistance. Note that the second cosmic velocity is twice as large as the first one.