> Gravitational potential energy

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

Gravitational energy- potential energy associated with gravitational force.

Learning challenge

• 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 gravity backwash - each point universal mass attracts another with the help of a force acting directly proportional to their masses and inversely proportional to the square of their distance.
• Gravity is the net force of the ground surface that pulls objects towards the center. Created by rotating.

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 be seen in the formula:

If we add mass and earth radius.

Gravitational energy reflects the potential energy associated with the force of gravity, because it is necessary to overcome the earth's gravity in order to do the work on lifting objects. If an object falls from one point to another within the gravitational field, then gravity will do a positive job, 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, certain outside interference works against the gravitational force. If it falls, then it is the work of gravity. Therefore, the process of falling reflects 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 influenced by the height relative to a particular point, the mass and the strength of the gravitational field. So the book on the table is inferior in gravitational potential energy to the heavier book below. Remember that altitude cannot be used to calculate gravitational potential energy unless gravity is constant.

Local approximation

Location affects the strength of the gravitational field. If the change in distance 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 ascending force is equated to weight, so the work is correlated with mgh, resulting in the formula: U = mgh (U is the 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 transmitted as

General formula

However, if we are faced with major changes in distance, then g cannot remain constant and we have to apply calculus and the mathematical definition of work. To calculate 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 equated to zero. Here the potential energy turns to zero when r is infinite.

« Physics - Grade 10 "

How is the gravitational interaction of bodies expressed?
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 we will 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, that is, that the gravitational force is also a conservative force.

Recall that the work of the conservative force in a closed loop is equal to zero.

Let a body of mass m be in the gravitational field of the Earth. Obviously, the dimensions of this body are small in comparison with the dimensions of the Earth, therefore it can be considered a material point. The body is affected by the force of gravity

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

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

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

where r 2 cpi = r i r i + 1.

The smaller Δri, the more true the written expression r 2 cpi = r i r i + 1.

Then the work of the force F сpi, at a small displacement Δr i, can be written in the form

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

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 the sections AA "and B" B is equal to zero, since the gravitational force is directed to the point O and is perpendicular to any small displacement along arc of a circle. Therefore, the work will also be determined by the expression (5.31).

3. Let's define the work of the gravitational force when the body moves from point A to point B along the trajectory ACB (see Fig. 5.15). The work of gravity at small displacement Δs i is equal to ΔA i = F avi Δs i cos α i, ..

The figure shows that Δs i cos α i = - Δr i, and the total work will again be determined by the formula (5.31).

So, we can conclude that A 1 = A 2 = A 3, that is, that the work of the gravitational force does not depend on the shape of the trajectory. Obviously, the work of the gravitational force when the body moves along the closed trajectory AA "B" BA is equal to zero.

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., Е пВ = 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 energy conservation law 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 imparted to a body near the Earth's surface so that in the absence of air resistance it can move away from it beyond the gravity forces.

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

On the body from the side of the Earth, the force of gravity acts, 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, a 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, that is, at r → ∞, the potential energy of the body is zero (W p = 0), and since we are interested in the minimum velocity, the kinetic energy should also be zero: W k = 0.

It follows from the law of conservation of energy:

This speed can be expressed through the acceleration of gravity 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 required speed

Exactly the same speed would be acquired by a body that fell to the Earth from an infinitely great height, if there was no air resistance. Note that the second cosmic velocity is times greater than the first.

Speed

Acceleration

Called tangential acceleration magnitude

Are called tangential acceleration characterizing the change in speed along direction

Then

V. Geisenberg,

Dynamics

Force

Inertial frames of reference

Frame of reference

Inertia

Inertia

Newton's laws

Newton's th law.

inertial systems

Newton's th law.

3rd Newton's Law:

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

Material points system

Internal forces:

External forces:

The system is called closed system if the bodies of the system external forces do not act.

Material point momentum

Impulse conservation law:

If and wherein hence

Galileo's transformations, the principle regarding Galileo

center of mass .

Where is the mass of i - that particle

Center mass speed

6)

Work in mechanics

)

potential .

non-potential.

The first includes

Complex: called kinetic energy.

Then Where are the external forces

Keene. the energy of the system of bodies

Potential energy

Equation of moments

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 the force acting on the point relative to the same axis.

The sum of all internal forces relative to any point is equal to zero. That's why

Thermal coefficient of performance (COP) of the cycle Heat engine.

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

Thermodynamic CRD:

Heat machine: when converting thermal energy into mechanical work. The main element of a heat engine is the work of bodies.

Energy cycle

Refrigeration machine.

26) Carnot cycle, efficiency of the Carnot cycle. Second started thermodynamics. Its 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 supplies heat.

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

3-4: Isothermal process of gas compression in this case heat is removed and the temperature is equal to T 2

4-1: The adiabatic process of gas compression while the gas temperature develops from the cooler to the heater.

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

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

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

Second started thermodynamics

The second law of thermodynamics determines the direction of flow of heat engines. It is not possible to construct a thermodynamic cycle operating a heat engine without a refrigerator. In this cycle, the energy of the system will see….

In this case, the efficiency

Its various formulations.

1) The first formulation: "Thomson"

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

2) The second formulation: "Clausis"

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 ideal gas processes. Clausius inequality. The main property of entropy (formulation of the second law of thermodynamics through entropy). The statistical meaning of the second principle.

Clausius inequality

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

The equal sign of the reversible cycle and process, respectively.

Most likely

The speed of molecules, respectively, the maximum value of the distribution function is called the most probable probability.

Einstein's postulates

1) Einstein's principle of relativity: all physical laws are the same in all inertial reference frames, 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 in all IFR is the same and equal to the speed of an electromagnetic wave in vacuum and does not depend on the direction of its propagation, not on the movement of the source and receiver.

Consequences from the Lorentz transformations

Lorentzian length contraction

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

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

34) Relativistic dynamics. Newton's second law applied to large
speeds. Relativistic energy. The connection between mass and energy.

Relativistic dynamics

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

Relativistic energy

A particle at rest has energy

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

The connection 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 characterizing the change in speed along magnitude

Along a normal trajectory at a given point

Are called tangential acceleration characterizing the change in speed along direction

Then

The 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 movement of a point. In the case of a nonclassical consideration of the motion of microparticles, the concept of the trajectory of their motion does not exist, but one can speak of the probability of finding a particle in one or another 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

V. Geisenberg, where h = 1.05 ∙ 10 -34 J ∙ s (Planck's constant), which determines the errors of simultaneous measurement of the coordinate and momentum

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

Dynamics- this is a branch of physics, it studies the movement of bodies in connection with the 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), as well as characterizing the amount of matter

Force- a measure of borrowing between bodies.

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

Frame of reference- inertial: any other motion relative to heliocentrism is uniform and straight, 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 is, the “harder” it is to change it v. The quantitative measure of inertness is body mass, as a measure of body inertia.

Newton's laws

Newton's th law.

There are such frames of reference called inertial systems, in which the material point is in a state of rest or uniform linear motion until the action from other bodies brings it out of this state.

Newton's th law.

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

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

1) If body A is acted upon by a force from body B, then body B is acted upon by force A. These forces F 12 and F 21 have the same physical nature

2) The force interact between bodies, does not depend on the speed of the bodies

Material points system: it is such a system contained by points that are 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 interact on the points of the system from the side of bodies that are not included in the system are called external forces.

The system is called closed system if the bodies of the system external forces do not act.

Material point momentum is called the product of mass by the speed of a point Material point system momentum: The momentum of a system of material points is equal to the product of the mass of the system by the speed of movement of the center of masses.

Impulse conservation law: For a closed system, bodies interact, the total impulse 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 momentum conservation law can be used under closed conditions, even if the system is not closed.

If and wherein hence

The law of conservation of momentum also works in a micrometer, when classical mechanics does not work, the momentum is preserved.

Galileo's transformations, the principle regarding Galileo

Suppose we have 2 inertial reference frames, one of which moves relative to the second, with a constant speed v o. Then, in accordance with the Galileo 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 taking place in them.

2) We put all inertial systems as properties equivalent to each other.

3) No mechanical experiments inside the system can establish the system at rest or it moves uniformly or rectilinearly.

The relativity of mechanical motion and the sameness of the laws of mechanics in different inertial reference frames are called Galileo's principle of relativity

5) The system of material points. The center of mass of the system of material points. A 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, the positions of which relative to the inertial reference frame are characterized by vectors, respectively, then, by definition, the position center of mass 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 a given coordinate system,

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

Center mass speed

The momentum of a system of material points is equal to the product of the mass of the system by the speed of movement of the center of masses.

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

1) The center of mass of the system of motion so if the entire mass of the system was concentrated in the center of mass, and all forces act 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 forces acting on the body of the system.

3) If (acceleration = 0) then the impulse of the system will not change.

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

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

Work is an algebraic quantity ( )

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

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

From the point of view of production work, it will mark potential and non-potential bodies. Forces, for everyone:

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.

The forces are convenient to these conditions is called potential .

The forces are not convenient to these conditions is called non-potential.

The first includes and only by the shear force of friction is it non-potential.

7) Kinetic energy of a material point, a system of material points. The theorem on the change in kinetic energy.

Complex: called kinetic energy.

Then Where are the external forces

The theorem on the change in kinetic energy: change kin. the energy of a point is equal to the algebraic sum of the work of all forces applied to it.

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

Keene. the energy of the 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- physical wilichina, 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 is doing work, the potential energy decreases.

Change in 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) The work of the force of elasticity

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

Let the movement only along the x-axis

Similarly, let the movement only be along the y or z axis, we got

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

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

10) The law of conservation of energy. Absolutely not resilient and absolutely resilient central ball impacts.

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.

*) Mechanical energy conservation law: The mechanical energy of the system is conserved if the work of all non-potential forces (both internal and external) is equal to zero.

In this case, it is possible to merge the transition of potential energy into kinetic energy and vice versa, the total energy is constant:

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

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

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

Energy obeys the conservation law, 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 velocities and mutual arrangement of bodies. It is equal to the sum of kinetic and potential energies.

Kinetic energy

Consider the case when a body with a mass m there is a constant force \ (~ \ vec F \) (it can be the resultant of several forces) and the force vectors \ (~ \ vec F \) and displacements \ (~ \ vec s \) are directed along one straight line in one direction. In this case, the work of 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 is associated with the modules of the initial υ 1 and final υ 2 speeds and accelerations a expression \ (~ s = \ frac (\ upsilon ^ 2_2 - \ upsilon ^ 2_1) (2a) \).

From here, for 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) \). (1)

A physical quantity equal to half the product of the mass of a body by 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 as follows:

\ (~ 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, that is, in joules.

If the initial speed of movement of a body with a mass m is equal to zero and the body increases its speed to the value υ , then the work of 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 with a speed υ shows what work must be done by a force acting on a body at rest in order to impart this speed to it.

Potential energy

Potential energy Is the energy of 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 are 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 a material point or body moves along a closed trajectory, the work of the non-potential force is not zero.

Potential energy of interaction of the 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 surface of the Earth 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 movement of the body can be considered constant and equal mg.

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

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

Let us now consider the movement of a body along an inclined plane. When the body moves down an inclined plane (Fig. 2), the force of gravity F t = m ∙ g doing work

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

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

Body movement from a point V exactly WITH along any trajectory (Fig. 3) can be mentally represented as consisting of displacements along sections of inclined planes with different heights h’, h'' Etc. Work A gravity all the way from V v WITH is equal to the sum of work on separate sections of the track:

\ (~ 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 surface of the Earth, at which the points are located, respectively V and WITH.

Equality (7) shows that the work of the force of gravity does not depend on the trajectory of the body and is always equal to the product of the modulus of the force of gravity by 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 path is zero.

Equality (7) can be represented as follows:

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

A physical quantity equal to the product of the mass of the body by the modulus of acceleration of gravity and by the height to which the body is lifted above the surface of the Earth is called potential energy interaction of the body and the Earth.

The work of gravity when moving a body with mass m from a point located 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 indicated 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, that is, the height at which the potential energy is taken to be zero. It is usually assumed that the potential energy of a body on the Earth's surface is zero.

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

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

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

the potential energy of the body, which is acted upon by the force of gravity, is equal to the work done by the force of gravity when the body moves 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 both positive and negative. Body mass m at altitude h, where h < h 0 (h 0 - zero height), has 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 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 ( E p = 0) is adopted at r = ∞.

Potential energy of gravitational interaction of a body with mass m with the Earth where h- body height 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 for choosing the zero reference, the potential energy of the gravitational interaction of a body with mass m with Earth for low heights h (h « R e) equals

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

where \ (~ g = G \ cdot \ frac (M_e) (R ^ 2_e) \) is the modulus of gravitational acceleration 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 the force (since the elastic force linearly depends 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 of the product of the stiffness of a body by the square of its deformation is called potential energy an 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 = NS, 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 performed by the elastic force during the transition of the body to 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 interaction of bodies, in which the forces acting on the bodies perform work other than zero. Let us consider the question of energy changes during the interactions of bodies that form a closed system.

Closed system Is a system that is not affected by external forces or the action of these forces is compensated... If several bodies interact with each other only by gravitational forces and elastic forces and no external forces act on them, then for any interactions of bodies, the work of elastic forces 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)

From a comparison of equalities (17) and (18), it can be seen 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 to it in sign:

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

Energy conservation law 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 the forces of gravity and the forces of elasticity, remains constant.

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

Here is the simplest experience. Let's throw up a steel ball. Having informed the initial speed υ start, we will give it kinetic energy, because of which it will begin to rise upward. 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 its direction of motion and moves downward with increasing speed. Now the reverse transformation of potential energy into kinetic one takes place.

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

the work of gravitational forces 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. Consequently, work is equal to energy that has changed from one type to another.

Mechanical Energy Change Act

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 - total mechanical energies of the system in the final and initial states, respectively.

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

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

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

Frictional forces during movement of the system reduce its kinetic energy. As a result of this, 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 influence of friction forces. The kinetic energy of the vehicle's translational motion became zero, and the potential energy did not increase. During the braking of the vehicle, heating of the brake pads, vehicle tires and asphalt occurred. 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 friction force 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... Let, external force F acts on the bar V that can slide on the cart D(fig. 5). If the carriage moves to the right, then the work of the sliding friction force F tr2 acting on the trolley 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 movement of the wheel, and the work of the friction force is positive (Fig. 6):

Literature

1. O.F. Kabardin Physics: Ref. materials: Textbook. manual for students. - M .: Education, 1991 .-- 367 p.
2. Kikoin I.K., Kikoin A.K. Physics: Textbook. for 9 cl. wednesday shk. - M .: Pro-sveshenie, 1992 .-- 191 p.
3. Elementary physics textbook: Textbook. allowance. In 3 volumes / Ed. G.S. Landsberg: vol. 1. Mechanics. Heat. Molecular physics. - M .: Fizmatlit, 2004 .-- 608 p.
4. Yavorskiy B.M., Seleznev Yu.A. A reference guide to physics for university applicants and self-education. - M .: Nauka, 1983 .-- 383 p.