Potential energy of gravitational attraction of two material points. Potential energy

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. It is usually chosen equal to zero, so that when 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 the free fall acceleration. 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, the following formula is true

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.

The potential energy of bodies, due to their gravitational. interaction. G. e. sustainable space. object (stars, clouds of interstellar gas, star clusters) by abs. twice as large as cf. kinetic the energies of its constituent particles (bodies; this is ... ... Natural science. encyclopedic Dictionary

- (for a given state of the system) the difference between the total energy of the bound state of a system of bodies or particles and the energy of the state in which these bodies or particles are infinitely distant from each other and are at rest: where ... ... Wikipedia

This term has other meanings, see Energy (meanings). Energy, Dimension ... Wikipedia

gravity energy- gravitacinė energija statusas T sritis Standartizacija ir metrologija apibrėžtis Gravitacinio lauko energijos ir jo veikiamų kitų objektų energijos kiekių suma. atitikmenys: engl. gravitational energy vok. Gravitationsenergie, f rus.… … Penkiakalbis aiskinamasis metrologijos terminų žodynas

- (Greek energeia, from energos active, strong). Perseverance, found in the pursuit of a goal, the ability of the highest exertion of forces, in conjunction with a strong will. Dictionary of foreign words included in the Russian language. Chudinov A.N.,… … Dictionary of foreign words of the Russian language

- (Jeans instability) increase with time of spatial fluctuations of the velocity and density of matter under the action of gravitational forces (gravitational perturbations). Gravitational instability leads to the formation of inhomogeneities (clots) in ... Wikipedia

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 the level corresponding to the location of bodies at infinitely large distances from each other as such a common zero level of the potential energy of the forces of universal gravitation. 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.

Speed

Acceleration

called tangential acceleration size

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

Then

W. Heisenberg,

Dynamics

Strength

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 Consequently

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. That's why

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 ideal gas processes. 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 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

Lorentz length contraction

Consider a rod located along the axis OX' of the system (X', Y', Z') and fixed with respect to this coordinate system. 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

Because the

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 movement of microparticles, the concept of the trajectory of their movement 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. Strength. 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

Strength 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 Consequently

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.

The relativity of mechanical motion and the sameness 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 reference system 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 so 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 the 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 of conservation of energy: Energy is neither created nor destroyed; it either passes from the first form to another state.

Ticket 1

1. . The change in the kinetic energy of the system is equal to the work of all internal and external forces acting on the bodies of the system.

2. Angular moment of a material point with respect to the point O is determined by the vector product

Where is the radius vector drawn from the point O, is the momentum of the material point. J*s

Ticket 2

1. Harmonic oscillator:

The kinetic energy is written as

And the potential energy is

Then the total energy has a constant value Let us find pulse harmonic oscillator. Differentiate the expression by t and, multiplying the result obtained by the mass of the oscillator, we obtain:

2. The moment of force relative to the pole is a physical quantity determined by the vector product of the radius of the vector drawn from the given pole to the point of application of force on the force vector F. newton meter

Ticket 3

1. ,

2. Oscillation phase total - the argument of a periodic function that describes an oscillatory or wave process. Hz

Ticket number 4

Expressed in m/(s^2)

Ticket number 5

, F = –grad U, where .

Potential energy of elastic deformation (springs)

Find the work done when the elastic spring is deformed.
Elastic force Fupr = –kx, where k is the coefficient of elasticity. The force is not constant, so the elementary work is dA = Fdx = –kxdx.
(The minus sign indicates that work has been done on the spring). Then , i.e. A = U1 - U2. Assume: U2 = 0, U = U1, then .

On fig. 5.5 shows a diagram of the potential energy of a spring.

Rice. 5.5
Here E = K + U is the total mechanical energy of the system, K is the kinetic energy at the point x1.

Potential energy in gravitational interaction

The work of the body during the fall A = mgh, or A = U - U0.
We agreed to assume that on the Earth's surface h = 0, U0 = 0. Then A = U, i.e. A = mgh.

For the case of gravitational interaction between masses M and m, located at a distance r from each other, the potential energy can be found by the formula .

On fig. 5.4 shows a diagram of the potential energy of the gravitational attraction of the masses M and m.

Rice. 5.4
Here the total energy is E = K + E. From here it is easy to find the kinetic energy: K = E – U.

Normal acceleration is a component of the acceleration vector directed along the normal to the motion trajectory at a given point on the body motion trajectory. That is, the normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in the direction and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory. ( m/s 2)

Ticket number 6

Ticket 7

1)Moment of inertia of the Rod -

Hoop - L = m*R^2

Disk -

2) According to the Steiner theorem (Huygens-Steiner theorem), the moment of inertia of the body J relative to an arbitrary axis is equal to the sum of the moment of inertia of this body Jc relative to the axis passing through the center of mass of the body parallel to the considered axis, and the product of the body mass m per square distance d between axles:

where m- total body weight.

Ticket 8

1) The equation describes the change in the motion of a body of finite dimensions under the action of a force in the absence of deformation and if it moves forward. For a point, this equation is always true, so it can be considered as the basic law of motion of a material point.

Ticket 9

1) 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 unchanged.

2) - a curve in phase space made up of points representing a state dynamic system consecutively moments of time during the entire time of evolution.

Ticket 10

1. Moment of impulse- vector physical quantity equal to the product of the radius vector drawn from the axis of rotation to the point of application of the impulse, by the vector of this impulse

2. Angular velocity of rotation of a rigid body relative to a fixed axis- limit (at Δt → 0) of the ratio of small angular displacement Δφ to a small time interval Δt

Measured in rad/s.

Ticket 11

1. Center of mass of a mechanical system (MC)- a point whose mass is equal to the mass of the entire system, the acceleration vector of the center of mass (in the inertial frame of reference) is determined only by external forces acting on the system. Therefore, when finding the law of motion of a system of points, we can assume that the vector of the resultant external forces is applied to the center of mass of the system.
The position of the center of mass (center of inertia) of a system of material points in classical mechanics is determined as follows

MS momentum change equation:

Law of conservation of momentum MS: in a closed system, the vector sum of the impulses of all bodies included in the system remains constant for any interactions of the bodies of this system with each other.

2. Angular acceleration of rotation of a rigid body relative to a fixed axis- pseudovector physical quantity equal to the first derivative of the pseudovector of the angular velocity with respect to time.

Measured in rad / s 2.

Ticket 12

1. Potential energy of attraction of two material points

Potential energy of elastic deformations - stretching or compressing the spring leads to the storage of its potential energy of elastic deformation. The return of the spring to the equilibrium position leads to the release of the stored energy of elastic deformation.

2. Impulse of mechanical system- vector physical quantity, which is a measure of the mechanical movement of the body.

measured in

Ticket 13

1. Conservative forces. The work of gravity. Elastic force work.
In physics, conservative forces (potential forces) are forces whose work does not depend on the type of trajectory, the point of application of these forces and the law of their motion, and is determined only by the initial and final positions of this point.
The work of gravity.
Work of elastic force

2. Define the relaxation time of damped oscillations. Specify the unit for this quantity in SI.
The relaxation time is the time interval during which the amplitude of damped oscillations decreases by a factor of e (e is the base of the natural logarithm). Measured in seconds.

3. A disk with a diameter of 60 cm and a mass of 1 kg rotates around an axis passing through the center perpendicular to its plane with a frequency of 20 rpm. What work must be done to stop the disk?

Ticket 14

1. Harmonic vibrations. Vector diagram. Addition of harmonic oscillations of one direction of equal frequencies.

Harmonic oscillations are oscillations in which a physical quantity changes over time according to a harmonic (sinusoidal, cosine) law.

There is a geometric way to represent harmonic vibrations, which consists in depicting vibrations as vectors on a plane. The circuit thus obtained is called a vector diagram (Fig. 7.4).

Let's choose an axis. From the point O, taken on this axis, we set aside the length vector, which forms an angle with the axis. If we bring this vector into rotation with an angular velocity , then the projection of the end of the vector onto the axis will change with time according to the law . Therefore, the projection of the end of the vector onto the axis will make harmonic oscillations with an amplitude equal to the length of the vector; with a circular frequency equal to the angular velocity of rotation, and with an initial phase equal to the angle formed by the vector with the axis X at the initial time.

The vector diagram makes it possible to reduce the addition of oscillations to the geometric summation of vectors.

Consider the addition of two harmonic oscillations of the same direction and the same frequency, which have the following form:

Let's represent both fluctuations with the help of vectors and (fig. 7.5). Let's build the resulting vector according to the vector addition rule. It is easy to see that the projection of this vector onto the axis is equal to the sum of the projections of the terms of the vectors. Therefore, the vector represents the resulting oscillation. This vector rotates with the same angular velocity as the vectors , so that the resulting motion will be a harmonic oscillation with frequency , amplitude and initial phase . According to the law of cosines, the square of the amplitude of the resulting oscillation will be equal to

2. Define the moment of force about the axis. Specify the units of this quantity in SI.

The moment of force is a vector physical quantity equal to the vector product of the radius vector drawn from the axis of rotation to the point of application of the force by the vector of this force. It characterizes the rotational action of a force on a rigid body. The moment of force relative to an axis is a scalar value equal to the projection onto this axis of the vector moment of force relative to any point on the axis. SI: measured in kg * m 2 / s 2 = N * m.

3. A projectile weighing 100 kg flies out of a gun weighing 5 tons when fired. The kinetic energy of the projectile at the departure of 8 MJ. What is the kinetic energy of the gun due to recoil?

Ticket 15

1. The law of conservation of mechanical energy of a mechanical system.

The total mechanical energy of a closed system of bodies, between which only conservative forces act, remains constant.

In a conservative system, all forces acting on a body are potential and, therefore, can be represented as

where is the potential energy of a material point. Then Newton's second law:

where is the mass of the particle, is the vector of its velocity. Scalarly multiplying both sides of this equation by the particle velocity and taking into account that , we obtain

By elementary operations, we obtain

It follows from this that the expression under the sign of differentiation with respect to time is preserved. This expression is called the mechanical energy of a material point.

2. Define the kinetic energy of a rigid body as it rotates around a fixed axis. Specify the units of this quantity in SI.

3. A ball weighing m=20 g is introduced with an initial speed V=20 m/s into a very massive target with sand, which moves towards the ball with a speed U=10 m/s. Estimate how much heat is released during full braking of the ball.

Ticket 16

1. Moment of force about the axis- a vector physical quantity equal to the vector product of the radius vector drawn from the axis of rotation to the point of application of the force by the vector of this force. there is

Angular momentum of the MS relative to the fixed axis- a scalar value equal to the projection onto this axis of the angular momentum vector, defined relative to an arbitrary point 0 of this axis. The value of angular momentum does not depend on the position of point 0 on the z-axis.

The basic equation of the dynamics of rotational motion

2. Acceleration vector - a vector quantity that determines the rate of change in the speed of the body, that is, the first derivative of the speed with respect to time and shows how much the speed vector of the body changes when it moves per unit time.

Measured in m/s 2

Ticket 17

1) The moment of force is a vector physical quantity equal to the vector product of the radius vector drawn from the axis of rotation to the point of application of the force by the vector of this force. Characterizes the rotational action of force on a rigid body.

The angular momentum relative to the fixed axis z is the scalar value Lz, which is equal to the projection onto this axis of the angular momentum vector, determined relative to an arbitrary point 0 of this axis, characterizes the amount of rotational motion.

2) The displacement vector is a directed straight line segment connecting the initial position of the body with its final position. Displacement is a vector quantity. The displacement vector is directed from the starting point of the movement to the end point. The displacement vector module is the length of the segment that connects the start and end points of the movement. (m).

Ticket 18

Uniform rectilinear movement called the movement in which a material point for any equal intervals of time makes the same movement along a given given straight line. The speed of uniform movement is determined by the formula:

Radius of curvature RR trajectories at a point AA is the radius of the circle along the arc of which the point is moving at a given time. The center of this circle is called the center of curvature.

The physical quantity characterizing the change in speed in the direction, - normal acceleration.

The physical quantity characterizing the change in speed modulo, - tangential acceleration.

Ticket 21

Ticket number 22

The coefficient of sliding friction is the ratio of the friction force to the normal component of the external forces acting on the surface of the body.

The coefficient of sliding friction is derived from the formula for the force of sliding friction

Since the support reaction force is the mass multiplied by the free fall acceleration, the coefficient formula is:

Dimensionless quantity

Ticket number 23

The space in which conservative forces act is called the potential field. Each point of the potential field corresponds to a certain value of the force F acting on the body, and a certain value of the potential energy U. This means that there must be a connection between the force F and U, on the other hand, dA = -dU, therefore Fdr = -dU, hence:

Projections of the force vector on the coordinate axes:

The force vector can be written in terms of projections: , F = –grad U, where .

A gradient is a vector showing the direction of the fastest change in a function. Therefore, the vector is directed towards the fastest decrease in U.

If only conservative forces act in the system, then we can introduce the concept potential energy. Let the body mass m finds-

in the gravitational field of the Earth, whose mass M. The force of interaction between them is determined by the law of universal gravitation

F(r) = G Mm,

where G= 6.6745 (8) × 10–11 m3/(kg × s2) - gravitational constant; r is the distance between their centers of mass. Substituting the expression for the gravitational force into formula (3.33), we find its work when the body passes from a point with the radius vector r 1 to a point with a radius vector r 2

r 2 dr

⎟

A 12 = ò dA= ò F(r)dr= -GMmò r

= GMm⎜⎝r

1 r 1 r 1 2 2 1

We represent relation (3.34) as the difference between the values

A 12 = U(r 1) – U(r 2), (3.35)

U(r) = -G Mm+ C

for different distances r 1 and r 2. In the last formula C is an arbitrary constant.

If the body approaches the earth, which is considered immovable, then r 2 < r 1, 1/ r 2 – 1/ r 1 > 0 and A 12 > 0, U(r 1) > U(r 2). In this case, the force of gravity does positive work. The body passes from some initial state, which is characterized by the value U(r 1) functions (3.36), to the final, with a smaller value U(r 2).

If the body moves away from the earth, then r 2 > r 1, 1/ r 2 – 1/ r 1 < 0 и A 12 < 0,

U(r 1) < U(r 2), i.e., the gravitational force does negative work.

Function U= U(r) is a mathematical expression of the ability of the gravitational forces acting in the system to perform work and, according to the definition given above, is the potential energy.

Note that the potential energy is due to the mutual gravitation of bodies and is a characteristic of a system of bodies, and not a single body. However, when considering two or more bodies, one of them (usually the Earth) is considered to be stationary, while the others move relative to it. Therefore, they often talk about the potential energy of these bodies in the field of forces of a motionless body.

Since in the problems of mechanics it is not the magnitude of the potential energy that is of interest, but its change, the value of the potential energy can be counted from any initial level. The latter determines the value of the constant in formula (3.36).

U(r) = -G Mm.

Let the zero level of potential energy correspond to the surface of the Earth, i.e. U(R) = 0, where R is the radius of the earth. Let us write formula (3.36) for the potential energy when the body is at a height h above its surface in the following form

U(R+ h) = -G Mm

R+ h

+ C. (3.37)

Assuming in the last formula h= 0, we have

U(R) = -G Mm+ C.

From here we find the value of the constant C in formulas (3.36, 3.37)

C= -G Mm.

After substituting the value of the constant C into formula (3.37), we have

U(R+ h) = -G Mm+ G Mm= GMm⎛- 1

1 ⎞= G Mm h.

R+ hR

⎝⎜ R+ hR⎟⎠ R(R+ h)

Let's rewrite this formula as

U(R+ h) = mg h,

where gh

R(R+ h)

Acceleration of free fall of a body at a height

h above the surface of the earth.

Approaching h« R we obtain the well-known expression for the potential energy if the body is at a small height h above the surface of the earth

Where g= G M

U(h) = mgh, (3.38)

Acceleration of free fall of a body near the Earth.

In expression (3.38), a more convenient notation is adopted: U(R+ h) = U(h). It shows that the potential energy is equal to the work done by the gravitational force when moving the body from a height h above

Earth on its surface corresponding to the zero level of potential energy. The latter serves as a basis to consider expression (3.38) as the potential energy of the body above the Earth's surface, speak of the potential energy of the body, and exclude the second body, the Earth, from consideration.

Let the body mass m is on the surface of the earth. For it to rise to the occasion h above this surface, it is necessary to apply an external force to the body, which is opposite to gravity and infinitesimally different from it in absolute value. The work performed by the external force is determined by the following relation:

R+ h

R+ hdr

⎡1 ⎤R+ h

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Potential energy of gravitational attraction of two material points. Potential energy

In classical mechanics

IN GR

see also

See what "gravitational energy" is in other dictionaries:

Portal for the student. Self-training

In classical mechanics

IN GR

see also

See what "gravitational energy" is in other dictionaries:

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