Total mechanical energy of a particle. How are the work of a force and the total mechanical energy of a particle related? As well as other works that may interest you
The increment of the kinetic energy of each particle is equal to the work of all forces acting on the particle: ΔK i = A i . Therefore, work A, which is performed by all forces acting on all particles of the system, when its state changes, can be written as follows: TO, or
(1.6.9)
where K is the total kinetic energy of the system.
So, the increment of the kinetic energy of the system is equal to the work done by all the forces acting on all the particles of the system:
Note that the kinetic energy of a system is an additive quantity: it is equal to the sum of the kinetic energies of the individual parts of the system, regardless of whether they interact with each other or not.
Equation (1.6.10) is valid both in inertial and non-inertial frames of reference. It should only be remembered that in non-inertial reference systems, in addition to the work of interaction forces, it is necessary to take into account the work of inertial forces.
Now let's establish a connection between the kinetic energies of a system of particles in different frames of reference. Let the kinetic energy of the system of particles of interest to us be K in a fixed frame of reference. The speed of the i-th particle in this frame can be represented as, , where is the speed of this particle in a moving frame of reference, a is the speed of the moving system relative to the fixed frame of reference Then the kinetic energy of the system
where is the energy in the moving system, T is the mass of the entire system of particles, is its momentum in the moving reference frame.
If the moving reference frame is connected to the center of mass (C-frame), then the center of mass is at rest, which means that the last term is zero and the previous expression takes the form
where is the total kinetic energy of particles in the C-system, called the self-kinetic energy of the particle system
Thus, the kinetic energy of a system of particles is the sum of its own kinetic energy and the kinetic energy associated with the motion of the system of particles as a whole. This is an important conclusion, and it will be repeatedly used in what follows (in particular, in studying the dynamics of a rigid body).
From formula (1.6.11) it follows that the kinetic energy of the system, particles is minimal in the C-system. This is another feature of the C-system.
The work of conservative forces.
Using formula (1.6.2) and
graphical way of defining work,
Let's calculate the work of some forces.
1.Work done by gravity
The force of gravity is directed
vertically down. Let's choose the z axis,
pointing vertically upwards and
project force onto it.
Let's build a graph
depending on z (Fig.1.6.3). The work of gravity
when moving a particle from a point with a coordinate to a point with a coordinate is equal to the area of the rectangle
As can be seen from the expression obtained, the work of gravity is equal to a change in a certain quantity that does not depend on the particle trajectory and is determined up to an arbitrary constant
2.The work of the elastic force.
The projection of the elastic force on the x-axis indicating the direction of deformation,
We have shown that the work done to move a particle from position 1 to position 2 can be expressed in terms of the increment in kinetic energy:
In the general case, both potential and nonpotential forces can act on a particle. Thus, the resulting force acting on the particle is:
.
The work of all these forces is used to increase the kinetic energy of the particles:
.
But, on the other hand, the work of potential forces is equal to the decrease in the potential energy of particles:
hence,
The value is called the total mechanical energy of the particle. Let's denote it by E.
Thus, the work of nonpotential forces goes to the increment of the total mechanical energy of the particle.
The increment of the total mechanical energy of a particle in a stationary field of potential forces when moving it from point 1 to point 2 can be written as:
.
If > 0, then the total mechanical energy of the particle increases, and if< 0, то убывает. Следовательно, полная механическая энергия частицы может измениться под действием только непотенциальных сил. Отсюда непосредственно вытекает закон сохранения механической энергии одной частицы. Если непотенциальные силы отсутствуют, то полная механическая энергия частицы в стационарном поле потенциальных сил остается постоянной.
In real processes, where resistance forces act, there is a deviation from the law of conservation of mechanical energy. For example, when a body falls to the Earth, the kinetic energy of the body first increases as the speed increases. The resistance force also increases, which increases with increasing speed. Over time, it will compensate for gravity, and in the future, with a decrease in potential energy relative to the Earth, the kinetic energy does not increase. The work of resistance forces leads to a change in body temperature. The heating of bodies under the action of friction is easy to detect by rubbing the palms together.
The value that equates to half of the product of the mass of a given body and the speed of this body squared is called in physics the kinetic energy of the body or the energy of action. The change or inconstancy of the kinetic or driving energy of the body for some time will be equal to the work that has been done for a given time by a certain force acting on a given body. If the work of any force along a closed trajectory of any type is equal to zero, then a force of this kind is called a potential force. The work of such potential forces will not depend on the trajectory along which the body moves. Such work is determined by the initial position of the body and its final position. The starting point or zero for the potential energy can be chosen absolutely arbitrarily. The value that will be equal to the work done by the potential force to move the body from a given position to the zero point is called in physics the potential energy of the body or the energy of the state.
For various kinds forces in physics, there are various formulas for calculating the potential or stationary energy of a body.
The work done by potential forces will be equal to the change in this potential energy, which must be taken in the opposite sign.
If you add the kinetic and potential energy of the body, you get a value called the total mechanical energy of the body. In a position where a system of several bodies is conservative, the law of conservation or constancy of mechanical energy is valid for it. A conservative system of bodies is such a system of bodies that is subject to the action of only those potential forces that do not depend on time.
The law of conservation or constancy of mechanical energy is as follows: "During any processes that occur in a certain system of bodies, its total mechanical energy always remains unchanged." Thus, the total or all mechanical energy of any body or any system of bodies remains constant if this system of bodies is conservative.
The law of conservation or constancy of total or all mechanical energy is always invariant, that is, its form of writing does not change, even when the starting point of time is changed. This is a consequence of the law of homogeneity of time.
When dissipative forces begin to act on the system, for example, such as, then a gradual decrease or decrease in the mechanical energy of this closed system occurs. This process is called energy dissipation. A dissipative system is a system in which the energy can decrease over time. During dissipation, the mechanical energy of the system is completely converted into another. This is fully consistent with the universal law of energy. Thus, there are no completely conservative systems in nature. One or another dissipative force will necessarily take place in any system of bodies.
The work of the force to move the particle goes to increase the energy of the particle:
dA =( , ) = ( , d ) = (d , )=dE
217. What is bond energy? Explain with the example of the nucleus of an atom.
The binding energy is the difference between the energy of the state in which the constituent parts of the system are infinitely distant from each other and are in a continuous state of active rest and the total energy of the bound state of the system
Where is the total energy of the ith component in the disconnected system, and E is the total energy of the bound system
EXAMPLE:
The nuclei of atoms are strongly bound systems of a large number of nucleons. To completely split the nucleus into its constituent parts and remove them over long distances from each other, it is necessary to expend a certain amount of work A . By bond energy called the energy equal to the work that must be done to split the nucleus into free nucleons
Ebonds = -A
According to the law of conservation, the binding energy is simultaneously equal to the energy that is released during the formation of a nucleus from individual nucleons
What is a macroscopic body, a thermodynamic system?
A macroscopic body is a large body consisting of many molecules.
A thermodynamic system is a set of macroscopic bodies that can interact with each other and other bodies (the external environment) - exchange energy and matter with them.
Why is the dynamic method of description inapplicable to systems consisting of a large number of particles?
Apply the dynamic method (write down the equations of motion and initial conditions for all atoms and molecules and clean out the position of all particles at each moment of time) is impossible, because to study a system consisting of a large number of atoms and molecules, information must be of a generalized nature and refer not to individual particles, but to the whole set.
What is a thermodynamic method for studying a thermodynamic system?
A method for studying systems of a large number of particles, operating with quantities that characterize the system as a whole (p, V, T) during various energy transformations occurring in the system, without taking into account the internal structure of the bodies under study and the nature of individual particles.
What is a statistical method for studying a thermodynamic system?
A method for studying systems of a large number of particles, operating with regularities and average values of physical quantities characterizing the entire system
What are the basic postulates of thermodynamics?
0: Existence and transitivity of thermal equilibrium:
A and C are in equilibrium with each other, B is a thermometer
The equilibrium state of the thermometer is detected by thermometric parameters.
1: The heat received by the thermodynamic system is equal to the sum of the work of the system on the environment. environment and changes in internal energy.
Q=A+
2: Modern formulation: in a closed system, the change in entropy does not decrease (S ≥ 0)
12.4. Energy of a relativistic particle
12.4.1. Energy of a relativistic particle
The total energy of a relativistic particle is the sum of the rest energy of the relativistic particle and its kinetic energy:
E \u003d E 0 + T,
Equivalence of mass and energy(Einstein's formula) allows us to determine the rest energy of a relativistic particle and its total energy as follows:
- rest energy -
E 0 \u003d m 0 c 2,
where m 0 is the rest mass of a relativistic particle (the mass of the particle in its own frame of reference); c is the speed of light in vacuum, c ≈ 3.0 ⋅ 10 8 m/s;
- total energy -
E \u003d mc 2,
where m is the mass of the moving particle (the mass of a particle moving relative to the observer with a relativistic velocity v); c is the speed of light in vacuum, c ≈ 3.0 ⋅ 10 8 m/s.
Relationship between masses m 0 (mass of a particle at rest) and m (mass of a moving particle) is given by
Kinetic energy relativistic particle is determined by the difference:
T = E - E 0 ,
where E is the total energy of the moving particle, E = mc 2 ; E 0 - rest energy of the indicated particle, E 0 = m 0 c 2 ; the masses m 0 and m are related by the formula
m = m 0 1 − v 2 c 2 ,
where m 0 is the mass of the particle in the frame of reference relative to which the particle is at rest; m is the mass of the particle in the frame of reference relative to which the particle moves at a speed v; c is the speed of light in vacuum, c ≈ 3.0 ⋅ 10 8 m/s.
explicitly kinetic energy relativistic particle is defined by the formula
T = m c 2 − m 0 c 2 = m 0 c 2 (1 1 − v 2 c 2 − 1) .
Example 6. The speed of a relativistic particle is 80% of the speed of light. Determine how many times the total energy of the particle is greater than its kinetic energy.
Solution . The total energy of a relativistic particle is the sum of the rest energy of the relativistic particle and its kinetic energy:
E \u003d E 0 + T,
where E is the total energy of the moving particle; E 0 - rest energy of the specified particle; T is its kinetic energy.
It follows that the kinetic energy is the difference
T = E − E 0 .
The desired value is the ratio
E T = E E − E 0 .
To simplify the calculations, we find the reciprocal of the desired:
T E = E − E 0 E = 1 − E 0 E ,
where E 0 \u003d m 0 c 2; E = mc 2 ; m 0 - rest mass; m is the mass of the moving particle; c is the speed of light in vacuum.
Substituting the expressions for E 0 and E into the relation (T /E ) gives
T E = 1 − m 0 c 2 m c 2 = 1 − m 0 m .
The relationship between the masses m 0 and m is determined by the formula
m = m 0 1 − v 2 c 2 ,
where v is the speed of the relativistic particle, v = 0.80c.
Let's express the mass ratio from here:
m 0 m = 1 − v 2 c 2
and substitute it into (T /E ):
T E = 1 − 1 − v 2 c 2 .
Let's calculate:
T E \u003d 1 - 1 - (0.80 s) 2 c 2 \u003d 1 - 0.6 \u003d 0.4.
The desired value is the inverse ratio
E T \u003d 1 0.4 \u003d 2.5.
The total energy of a relativistic particle at the indicated speed exceeds its kinetic energy by a factor of 2.5.