Kinetic energy

Kinetic energy . It is an energy that arises in the phenomenon of movement. It is defined as the work necessary to accelerate a body of a given mass from its equilibrium position to a given speed. Once this energy is achieved during acceleration, the body maintains its kinetic energy regardless of the change in speed. Negative work of the same magnitude may be required for the body to return to its equilibrium state.

Table of Contents

Summary

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1 Description
2 Kinetic energy of a particle
3 Kinetic energy of a rotating rigid solid
4 Kinetic Energy in relativistic mechanics
5 Kinetic energy in quantum mechanics
6 Kinetic energy and temperature
7 Formula
8 Units of Measurements
9 Operational definition
10 Source

Description

This ability to make changes that moving bodies possess is fundamentally due to two factors: the body’s mass and its speed. A body that has a great mass, will be able to produce great effects and transformations due to its movement. An example of the application of this energy is the one used in the Middle Ages, when the attackers of a castle pushed the doors with a heavy battering ram: a large and heavy trunk, reinforced with iron or bronze. Also the speed of the body is decisive for its kinetic energy. This effect can be seen when a bullet, weighing just a few grams, can penetrate thick trunks, when fired at high speed with a rifle. A ball at the top of a hill, for example, has potential energy, but as it rolls down it loses it. Electric, chemical and nuclear energies are forms of potential energy. A heavy object at high speed has kinetic energy that decreases as it rolls down.

The potential energy is stored in the bodies at rest capable of moving.

In determining kinetic energy , only the mass and speed of an object are taken into account, regardless of how the movement originated; instead, potential energy depends on the type of force applied to an object. For this reason there are different types of potential energy . The study of all the aspects with which a chemical system approaches an equilibrium condition, is what is defined as chemical-kinetic.

In chemistry, kinetic theory explains the behavior of matter in its three states: solid, liquid, and gas. The state of a body is determined by the amount of kinetic energy of its atoms and molecules (small particles that make up matter).

State changes occur when the amount of energy varies. The atoms of a gas have more energy than those of a liquid, and the atoms of this more than those of a solid. The temperature, pressure and volume that a gas occupies depends on the kinetic energy of its molecules.

When a body is in motion, it has kinetic energy, since when it collides with another, it can move it and, therefore, produce work.

In order for a body to acquire kinetic or movement energy, that is, to set it in motion, it is necessary to apply a force to it. The longer the force is acting, the greater the speed of the body and, therefore, its kinetic energy will also be greater.

Another factor that influences kinetic energy is the mass of the body. For example, if a glass ball of 5 grams of mass advances towards us at a speed of 2 km / h, no effort will be made to avoid it. However, if with that same speed a truck is advancing towards us, the collision cannot be avoided.

When a body is in motion, it possesses kinetic energy since, by colliding with another, it can move it and, therefore, produce work.

Kinetic energy of a particle

In classical mechanics, the kinetic energy of a point object (a body so small that its dimension can be ignored), or in a rigid solid that does not rotate, is given the equation where m is the mass and v is the speed (or speed) of the body.

In classical mechanics the kinetic energy can be calculated from the equation of work and the expression of a force F given by Newton’s second law: The kinetic energy increases with the square of the speed. Thus the kinetic energy is a measure dependent on the reference system. The kinetic energy of an object is also related to its linear momentum: Kinetic energy in different reference systems [edit] As we have said, in classical mechanics, the kinetic energy of a point mass depends on its mass and its components of movement. These are described by the velocity v of the point mass, thus: In a special coordinate system, this expression has the following forms: Cartesian coordinates (x, y, z): Polar coordinates (r, φ): Cylindrical coordinates (r , φ, z): Spherical coordinates (r, φ, θ): With this, the meaning of a point in a coordinate and its temporal change is described as the temporal derivative of its displacement: In a Hamiltonian formalism we do not work with those components of movement, that is, with their speed, but with their impulse p ( change in momentum). In case of using Cartesian components we obtain: Kinetic energy of particle systems [edit] For a particle, or for a rigid solid that is not rotating, the kinetic energy goes to zero when the body stops. However, for systems that contain many bodies with independent movements, that exert forces on each other and that may (or may not) be rotating; this is not entirely true. This energy is called ‘internal energy’.

An example of this may be the solar system. In the center of mass of the solar system, the sun is (almost) stationary, but planets and planetoids are in motion above it. Thus in a stationary center of mass, kinetic energy is still present. However, recalculating the energy of different frames can be tedious, but there is a trick. The kinetic energy of a system of different inertial frames can be calculated as the simple sum of the energy in a frame with a center of mass and add to the energy the total of the masses of the bodies that move with relative speed between the two frames. This can be easily demonstrated: let V be the relative speed in a system k of a center of mass i: However, let be the kinetic energy in the center of mass of that system, it could be the total moment that is by definition zero in the center of mass and is the total mass:. Substituting we obtain:

The kinetic energy of a system then depends on the inertial reference system and is lower with respect to the referential center of mass, for example: in a reference system in which the center of mass is stationary. In any other frame of reference there is an additional kinetic energy corresponding to the total mass moving at the speed of the center of mass.

Sometimes it is convenient to divide the total kinetic energy of a system by the sum of the centers of mass of the bodies, in their translational kinetic energy and the energy of rotation about the center of mass: where: Ec is the total kinetic energy , Et is the translational kinetic energy and Er is the rotational energy or angular kinetic energy in this system.

So the kinetic energy in a traveling tennis ball has a kinetic energy that is the sum of the energy in its translation and its rotation.

Kinetic energy of a rotating rigid solid

For a rigid solid that is rotating, the total kinetic energy can be decomposed as two sums: the translational kinetic energy (which is associated with the displacement of the body’s center of mass through space) and the rotational kinetic energy (which is the associated with rotational motion with a certain angular velocity ). The mathematical expression for kinetic energy is: Where: Translational energy. Rotation energy. body mass. tensor of (moments of) inertia. angular velocity of the body. transposition of the vector of the angular velocity of the body. linear speed of the body.

The value of the kinetic energy is always positive, and depends on the reference system that is considered when determining the value of the velocity y. The previous expression can be deduced from the general expression: In hydrodynamics. In Hydrodynamics, kinetic energy is very often changed by the density of kinetic energy. This is usually written through a small e or an ε, like so, where ρ describes the density of the fluid.

Kinetic energy in relativistic mechanics

If the speed of a body is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate kinetic energy. In special relativity, we must change the expression for the linear momentum and from it by interaction the expression of the kinetic energy can be deduced: Taking the relativistic expression above, developing it in Taylor series and making the classical limit, the expression of energy is recovered kinetics typical of Newtonian mechanics: The equation shows that the energy of an object approaches infinity when the speed v approaches the speed of light c, so it is impossible to accelerate an object to these magnitudes. This mathematical product is the formula of equivalence between mass and energy, when the body is at rest we obtain this equation: Thus, the total energy E can be partitioned between the energies of the resting masses plus the traditional low velocity Newtonian kinetic energy. When objects move at speeds much lower than light (eg any phenomenon on earth) the first two terms in the series predominate.

The relationship between kinetic energy and momentum is more complicated in this case and is given by the equation: This can also be expanded as a Taylor series, the first term of this simple expression comes from Newtonian mechanics. What this suggests is that the formulas for energy and momentum are neither special nor axiomatic, but some concepts emerge from the equations of mass with energy and from the principles of relativity.

Kinetic energy in quantum mechanics

In quantum mechanics, the expected value of kinetic energy of an electron,, for an electron system describes a wave function that is the sum of an electron, the operator is expected to reach the value of: where i is the mass of an electron and is the Laplacian operator acting on the coordinates of the ith electron and the sum of all other electrons. Note that it is a quantized version of a non-relativistic expression of kinetic energy in terms of momentum: The formalism of the density functional in quantum mechanics requires a knowledge of electron density, for this formal knowledge of the wave function is not required.

Given an electronic density, the exact functional of the kinetic energy of the nth electron is uncertain; however, in a specific case of a one-electron system, kinetic energy can be written like this: where T [ρ] is known as the Von Weizsacker kinetic energy functional.

In quantum theory a physical quantity such as kinetic energy must be represented by a self-attached operator in a suitable Hilbert space. That operator can be constructed by a quantization process, which leads for a particle moving through the three-dimensional Euclidean space to a natural representation of that operator on the Hilbert space given by: that, on a dense domain of said space formed equivalence classes Representable by C functions, it defines a self-attached operator with always positive eigenvalues, which makes them interpretable as physically measurable values of kinetic energy.

Kinetic energy of the rigid solid in quantum mechanics [edit] A rigid solid despite being made up of an infinite number of particles, is a mechanical system with a finite number of degrees of freedom which means that its quantum equivalent can be represented by on an L-type infinite-dimensional Hilbert space on a finite-dimensional configuration space. In this case the configuration space of a rigid solid is precisely the Lie group SO (3) and therefore the relevant Hilbert space and the operator kinetic energy of rotation can be represented by: where μh is the Haar measure invariant of SO (3), are the operators of the angular momentum in the appropriate representation and the scalars Ii are the main moments of inertia.

Kinetic energy and temperature

At the microscopic level the average kinetic energy of the molecules of a gas defines its temperature. According to the Maxwell-Boltzmann law for a classic ideal gas, the relationship between the temperature (T) of a gas and its average kinetic energy is: where κB is the Boltzmann constant, it is the mass of each of the molecules of the gas.

Formula

E c = 1/2 • m • v 2 E c = Kinetic energy m = mass v = speed

When a body of mass m moves with a speed v it possesses a kinetic energy that is given by the formula written above.

In this equation, there must be agreement between the units used. All of them must belong to the same system. In the International System (SI), the mass m is measured in kilogram (kg) and the speed v in meters divided per second (m / s), whereby the kinetic energy is measured in Joule (J).

Units of measurement

As it is an energy, and of course, kinetic energy is measured in the same units as mechanical energy: the joule, the erg and the kilowatt-hour. As an example, we can point out that a body of 2 kilograms of mass, which moves with a speed of 1 m / s, has a kinetic energy of 1 joules.

Operational definition

Operationally, the way to determine the kinetic energy of a body is to multiply half of its mass by the square of its speed. The square of the speed of the body is the speed multiplied by itself. That is: Ec = ½ (m * v2) Ec: Kinetic energy m: mass v: speed v2: speed squared Examples of kinetic energy: EXPERIMENT Materials: · 1 pencil · 30 cms. thin wire · 1 AAA battery · 1 needle motor · 1 motor base · 1 cup · 1 switch · electrical tape · a piece of acrylic Development: · We cut the cables in three parts and the ends of each were stripped, winding them . · With the pencil a hole was made to the tazo, which was placed in the motor needle. · Cable 1 was screwed to the motor leg and the other part of this cable to the switch, it was ordered to be glued with solder. · Cable 2 was twisted to the other leg of the motor and stuck to the battery. · Cable 3 was glued to the switch and the other end was attached to the stack. · Everything was glued with silicone to the acrylic piece.