The formula for the potential energy of the gravitational interaction of bodies. Potential energy of gravitational interaction of two bodies. Lorentz length contraction

Gravitational energy

Gravitational energy- potential energy of a system of bodies (particles), due to their mutual gravitation.

Gravity-bound system- a system in which the gravitational energy is greater than the sum of all other types of energies (in addition to the rest energy).

The generally accepted scale is that for any system of bodies located at finite distances, the gravitational energy is negative, and for infinitely distant, that is, for gravitationally non-interacting bodies, the gravitational energy is zero. The total energy of the system, equal to the sum of gravitational and kinetic energy, is constant. For an isolated system, gravitational energy is the binding energy. Systems with positive total energy cannot be stationary.

In classical mechanics

For two gravitating point bodies with masses M and m gravitational energy is:

, - gravitational constant ; - distance between the centers of mass of bodies.

This result is obtained from Newton's law of gravitation, provided that for infinitely distant bodies the gravitational energy is 0. The expression for the gravitational force is

- force of gravitational interaction

On the other hand, according to the definition of potential energy:

The constant in this expression can be chosen arbitrarily. She is usually chosen zero so that as r tends to infinity, it tends to zero.

The same result is true for a small body located near the surface of a large one. In this case, R can be considered equal to , where is the radius of the body with mass M, and h is the distance from the center of gravity of the body with mass m to the surface of the body with mass M.

On the surface of the body M we have:

If the dimensions of the body are much larger than the dimensions of the body, then the formula for gravitational energy can be rewritten in the following form:

where the value is called acceleration free fall. In this case, the term does not depend on the height of the body above the surface and can be excluded from the expression by choosing the appropriate constant. Thus, for a small body located on the surface of a large body, following formula

In particular, this formula is used to calculate the potential energy of bodies located near the Earth's surface.

IN GR

In the general theory of relativity, along with the classical negative component of the gravitational binding energy, a positive component appears due to gravitational radiation, that is, the total energy of the gravitating system decreases with time due to such radiation.

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 at uniformly accelerated rectilinear motion associated with the modules of the initial υ 1 and final υ 2 speeds and accelerations a\(~s = \frac(\upsilon^2_2 - \upsilon^2_1)(2a)\) .

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

\(~A = F \cdot s = m \cdot g \cdot (h_1 - h_2)\) . (5)

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 AT exactly With 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 AT in With 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 AT and With.

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:

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

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:

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

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:

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

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:

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

If a 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 opposite to it in sign:

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

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 initial speedυ start, we will give it kinetic energy, because of which it will begin to rise up. 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. Change mechanical energy of such a system is equal to the work of external forces:

\(~A_(vn) = \Delta E = E - E_0\) . (20)

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 internal energy thermal motion molecules.

The law of conservation and transformation of energy

for any physical interactions energy changes 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 AT, 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

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

Speed

Acceleration

called tangential acceleration size

Are called tangential acceleration, which characterizes the change in speed according to direction

Then

W. Heisenberg,

Dynamics

Force

Inertial frames of reference

Reference system

Inertia

inertia

Newton's laws

th Newton's law.

inertial systems

th Newton's law.

Newton's 3rd law:

4) System of material points. Internal and external forces. The momentum of a material point and the momentum of a system of material points. Law of conservation of momentum. Conditions for its applicability of the law of conservation of momentum.

System of material points

internal forces:

External Forces:

The system is called closed system, if on the bodies of the system no outside forces.

momentum of a material point

Law of conservation of momentum:

If a and wherein hence

Galilean transformations, principle relative to Galileo

center of gravity .

Where is the mass of i - that particle

Center of Mass Velocity

Work in mechanics

)

potential .

non-potential.

The first applies

Complex: called kinetic energy.

Then Where are the external forces

Kin. energy system of bodies

Potential energy

Moment equation

The derivative of the angular momentum of a material point with respect to a fixed axis with respect to time is equal to the moment of force acting on the point with respect to the same axis.

The total of all internal forces relative to any point is equal to zero. So

Thermal efficiency (COP) of a cycle Thermal engine.

The measure of the efficiency of converting the heat supplied to the working fluid into the work of a heat engine on external bodies is efficiency thermal machine

Thermodynamic KRD:

heat engine: when thermal energy is converted into mechanical work. The main element of the heat engine is the work of bodies.

energy cycle

Refrigeration machine.

26) Carnot cycle, Carnot cycle efficiency. Second started by thermodynamics. His various
wording.

Carnot cycle: this cycle consists of two isothermal processes and two adiabats.

1-2: Isothermal process of gas expansion at heater temperature T 1 and heat input.

2-3: Adiabatic process of gas expansion while the temperature drops from T 1 to T 2 .

3-4: Isothermal process of compressing the gas while removing heat and the temperature is T 2

4-1: An adiabatic process of compressing a gas while the temperature of the gas develops from the cooler to the heater.

Affects for the Carnot cycle, the general efficiency factor exists for the manufacturer

In a theoretical sense, this cycle will maximum among possible efficiency for all cycles operating between temperatures T 1 and T 2 .

Carnot's theorem: The useful power factor of the Carnot thermal cycle does not depend on the type of worker and the device of the machine itself. And only determined by the temperatures T n and T x

Second started by thermodynamics

The second law of thermodynamics determines the direction of flow of heat engines. It is impossible to construct a thermodynamic cycle that would operate a heat engine without a refrigerator. During this cycle, the energy of the system will see ....

In this case, the efficiency

Its various formulations.

1) First wording: “Thomson”

A process is impossible, the only result of which is the performance of work due to the cooling of one body.

2) Second formulation: “Clausus”

A process is impossible, the only result of which is the transfer of heat from a cold body to a hot one.

27) Entropy is a function of the state of a thermodynamic system. Calculation of entropy change in processes ideal gas . Clausius inequality. The main property of entropy (formulation of the second law of thermodynamics in terms of entropy). Statistical meaning of the second law.

Clausius inequality

The initial condition of the second law of thermodynamics, the Clausius relation was obtained

The equal sign corresponds to the reversible cycle and process.

Most likely

The maximum value of the distribution function, corresponding to the speed of molecules, is called the most certain probability.

Einstein's postulates

1) Einstein's principle of relativity: all physical laws are the same in all inertial frames of reference, and therefore they must be formulated in a form that is invariant with respect to coordinate transformations, reflecting the transition from one IFR to another.

2)
The principle of constancy of the speed of light: there is a limiting speed of propagation of interactions, the value of which is the same in all ISOs and is equal to the speed electromagnetic wave in vacuum and does not depend on the direction of its propagation, not on the motion of the source and receiver.

Consequences from the Lorentz transformations

Lorentz length contraction

Consider a rod located along the axis OX' of the system (X', Y', Z') and fixed relative to this coordinate systems. own rod length the value is called, that is, the length measured in the reference system (X, Y, Z) will be

Therefore, the observer in the system (X,Y,Z) finds that the length of the moving rod is several times less than its own length.

34) Relativistic dynamics. Newton's second law as applied to large
speeds. relativistic energy. Relationship between mass and energy.

Relativistic dynamics

The connection between the momentum of a particle and its speed is now given by

Relativistic energy

A particle at rest has an energy

This quantity is called the rest energy of the particle. The kinetic energy is obviously equal to

Relationship between mass and energy

total energy

Insofar as

Speed

Acceleration

Along the tangent trajectory at its given point Þ a t = eRsin90 o = eR

called tangential acceleration, which characterizes the change in speed according to size

Along a normal trajectory at a given point

Are called tangential acceleration, which characterizes the change in speed according to direction

Then

Limits of applicability of the classical way of describing the motion of a point:

All of the above refers to the classical way of describing the motion of a point. In the case of a non-classical consideration of the motion of microparticles, the concept of the trajectory of their motion does not exist, but we can talk about the probability of finding a particle in a particular region of space. For a microparticle, it is impossible to simultaneously specify the exact values of the coordinate and velocity. In quantum mechanics, there is uncertainty relation

W. Heisenberg, where h=1.05∙10 -34 J∙s (Planck's constant), which determines the errors in the simultaneous measurement of position and momentum

3) Dynamics of a material point. Weight. Force. Inertial reference systems. Newton's laws.

Dynamics- this is a branch of physics that studies the movement of bodies in connection with reasons that return one or the force of the nature of the movement

Mass is a physical quantity that corresponds to the ability of physical bodies to maintain their translational motion (inertia), and also characterizes the amount of matter

Force is a measure of interaction between bodies.

Inertial frames of reference: There are such frames of reference of the relative, in which the body is at rest (moves in a straight line) until other bodies act on it.

Reference system– inertial: any other movement relative to heliocentrism uniformly and directly is also inertial.

Inertia- This is a phenomenon associated with the ability of bodies to maintain their speed.

inertia- the ability of a material body to reduce its speed. The more inert the body, the “harder” it is to change it v. A quantitative measure of inertia is the mass of the body, as a measure of the inertia of the body.

Newton's laws

th Newton's law.

There are systems of reference called inertial systems, in which the material point is in a state of rest or uniform semi-linear motion until the impact from other bodies takes it out of this state.

th Newton's law.

The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force.

Newton's 3rd law: the forces with which two m. points act on each other in IFR are always equal in absolute value and directed in opposite directions along the straight line connecting these points.

1) If a force acts on body A from body B, then force A acts on body B. These forces F 12 and F 21 have the same physical nature

2) Force interact between bodies, does not depend on the speed of movement of bodies

System of material points: this is such a system contained by points, which is rigidly connected to each other.

Internal forces: The forces of interaction between the points of the system are called internal forces

External Forces: The forces interacting on the points of the system from the bodies that are not included in the system are called external forces.

The system is called closed system, if on the bodies of the system no outside forces.

momentum of a material point is called the product of the mass and the speed of the point Momentum of the system of material points: The momentum of a system of material points is equal to the product of the mass of the system and the speed of the center of mass.

Law of conservation of momentum: For a closed system interacting bodies, the total momentum of the system remains unchanged, regardless of any interacting bodies with each other

Conditions for its applicability of the law of conservation of momentum: The law of conservation of momentum can be used under closed conditions, even if the system is not closed.

If a and wherein hence

The law of conservation of momentum also works in the micromeasure, when classical mechanics does not work, the momentum is conserved.

Galilean transformations, principle relative to Galileo

Let we have 2 inertial frames of reference, one of which moves relative to the second, with a constant speed v o . Then, in accordance with the Galilean transformation, the acceleration of the body in both frames of reference will be the same.

1) The uniform and rectilinear movement of the system does not affect the course of the mechanical processes occurring in them.

2) All inertial systems we set the property equivalent to each other.

3) No mechanical experiments inside the system can establish whether the system is at rest or moves uniformly or in a straight line.

Relativity mechanical movement and the identity of the laws of mechanics in different inertial frames of reference is called Galileo's principle of relativity

5) System of material points. The center of mass of the system of material points. The theorem on the motion of the center of mass of a system of material points.

Any body can be represented as a collection of material points.

Let it have a system of material points with masses m 1 , m 2 ,…,m i , whose positions relative to the inertial frame of reference are characterized by vectors respectively , then by definition the position center of gravity system of material points is determined by the expression: .

Where is the mass of i - that particle

– characterizes the position of this particle relative to the given coordinate system,

- characterizes the position of the center of mass of the system relative to the same coordinate system.

Center of Mass Velocity

The momentum of the system of material points is equal to the product of the mass of the system and the speed of the center of mass.

If then the system we say that the system as a center is at rest.

1) The center of mass of the system of motion as if the entire mass of the system was concentrated in the center of mass, and all forces acting on the bodies of the system were applied to the center of mass.

2) The acceleration of the center of mass does not depend on the points of application of the forces acting on the body of the system.

3) If (acceleration = 0) then the momentum of the system does not change.

6) Work in mechanics. The concept of the field of forces. Potential and non-potential forces. Potentiality criterion for field forces.

Work in mechanics: The work of the force F on the displacement element is called scalar product

Work is an algebraic quantity ( )

The concept of the field of forces: If at each material point of space a certain force acts on the body, then they say that the body is in the field of forces.

Potential and non-potential forces, criterion of potentiality of field forces:

From the point of view of the work produced, it will mark out potential and non-potential bodies. Forces, for each:

1) The work does not depend on the shape of the trajectory, but depends only on the initial and final position of the body.

2) Work, which is equal to zero along closed trajectories, is called potential.

Forces that are comfortable with these conditions are called potential .

Forces not comfortable with these conditions are called non-potential.

The first applies and only by the friction force is nonpotential.

7) Kinetic energy of a material point, systems of material points. Theorem on the change in kinetic energy.

Complex: called kinetic energy.

Then Where are the external forces

Kinetic energy change theorem: change kin. the energy of a m. point is equal to the algebraic sum of the work of all the forces applied to it.

If several external forces simultaneously act on the body, then the change in the net energy is equal to the “allebraic work” of all forces that act on the body: this formula of the theorem of kinetic kinetics.

Kin. energy system of bodies called amount of kin. energies of all bodies included in this system.

8) Potential energy. Change in potential energy. Potential energy of gravitational interaction and elastic deformation.

Potential energy- a physical quantity, the change of which is equal to the work of the potential force of the system taken with the “-” sign.

We introduce some function W p , which is the potential energy f(x,y,z), which we define as follows

The “-” sign shows that when this potential force does work, the potential energy decreases.

Change in the potential energy of the system bodies, between which only potential forces act, is equal to the work of these forces taken with the opposite sign during the transition of the system from one state to another.

Potential energy of gravitational interaction and elastic deformation.

1) Gravitational force

2) Work force of elasticity

9) Differential relationship between potential force and potential energy. Scalar field gradient.

Let the displacement be only along the x-axis

Similarly, let's move only along the y or z axis, we get

The “-” sign in the formula shows that the force always changes in the direction of the potential energy, but the opposite is the gradient W p .

The geometric meaning of points with the same value of potential energy is called an equipotential surface.

10) The law of conservation of energy. Absolutely inelastic and absolutely elastic central impacts of the balls.

The change in the mechanical energy of the system is equal to the sum of the work of all non-potential forces, internal and external.

*) Law of conservation of mechanical energy: The mechanical energy of a system is conserved if the work done by all non-potential forces (both internal and external) is zero.

In this case, only the transition of potential energy into kinetic energy is possible, and vice versa, the field energy is constant:

*)General physical law energy conservation: Energy is neither created nor destroyed; it either passes from the first form to another state.

« 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 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 pB = 0 as r B → ∞, 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.

The formula for the potential energy of the gravitational interaction of bodies. Potential energy of gravitational interaction of two bodies. Lorentz length contraction

In classical mechanics

IN GR

see also

See what "gravitational energy" is in other dictionaries:

Kinetic energy

Kinetic energy theorem

The physical meaning of kinetic energy

Potential energy

Potential energy of interaction of a body with the Earth

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

Potential energy of gravitational interaction

Potential energy of an elastically deformed body

The physical meaning of the potential energy of a deformed body

Mechanical Energy Change Law

The law of conservation and transformation of energy

Literature