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Movimiento circular

De Wikipedia, la enciclopedia libre

Movimiento circular.

En cinemática, el movimiento circular (también llamado movimiento circunferencial) es el que se basa en un eje de giro y radio constante, por lo cual la trayectoria es una circunferencia. Si además, la velocidad de giro es constante (giro ondulatorio), se produce el movimiento circular uniforme, que es un caso particular de movimiento circular, con radio y centro fijos y velocidad angular constante.

Índice

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Conceptos[editar]

En el movimiento circular hay que tener en cuenta algunos conceptos que serían básicos para la descripción cinemática y dinámica del mismo:

Eje de giro: es la línea recta alrededor de la cual se realiza la rotación, este eje puede permanecer fijo o variar con el tiempo pero para cada instante concreto es el eje de la rotación (considerando en este caso una variación infinitesimal o diferencial de tiempo). El eje de giro define un punto llamado centro de giro de la trayectoria descrita (O).
Arco: partiendo de un centro fijo o eje de giro fijo, es el espacio recorrido en la trayectoria circular o arco de radio unitario con el que se mide el desplazamiento angular. Su unidad es el radián (espacio recorrido dividido entre el radio de la trayectoria seguida, división de longitud entre longitud, adimensional por tanto).
Velocidad angular: es la variación del desplazamiento angular por unidad de tiempo (omega minúscula, $omega$ ).
Aceleración angular: es la variación de la velocidad angular por unidad de tiempo (alfa minúscula, $alpha$ ).

En dinámica de los movimientos curvilíneos, circulares y/o giratorios se tienen en cuenta además las siguientes magnitudes:

Momento angular (L): es la magnitud que en el movimiento rectilíneo equivale al momento lineal o cantidad de movimiento pero aplicada al movimiento curvilíneo, circular y/o giratorio (producto vectorial de la cantidad de movimiento por el vector posición, desde el centro de giro al punto donde se encuentra la masa puntual).
Momento de inercia (I): es una cualidad de los cuerpos que depende de su forma y de la distribución de su masa y que resulta de multiplicar una porción concreta de la masa por la distancia que la separa al eje de giro.
Momento de fuerza (M): o par motor es la fuerza aplicada por la distancia al eje de giro (es el equivalente a la fuerza agente del movimiento que cambia el estado de un movimiento rectilíneo).

Paralelismo entre el movimiento rectilíneo y el movimiento circular[editar]

Movimiento
Lineal	Angular
Posición	Arco
Velocidad	Velocidad angular
Aceleración	Aceleración angular
Masa	Momento de inercia
Fuerza	Momento de fuerza
Momento lineal	Momento angular

A pesar de las diferencias evidentes en su trayectoria, hay ciertas similitudes entre el movimiento rectilíneo y el circular que deben mencionarse y que resaltan las similitudes y equivalencias de conceptos y un paralelismo en las magnitudes utilizadas para describirlos. Dado un eje de giro y la posición de una partícula puntual en movimiento circular o giratorio, para una variación de tiempo Δt o un instante dt, dado, se tiene:

Arco descrito o desplazamiento angular[editar]

Arco angular o desplazamiento angular es el arco de la circunferencia recorrido por la masa puntual en su trayectoria circular, medido en radianes y representado con la letras griegas $varphi,$ (phi) o $heta,$ (theta). Este arco es el desplazamiento efectuado en el movimiento circular y se obtiene mediante la posición angular ( $varphi_p$ ó $heta_p$ ) en la que se encuentra en un momento determinado el móvil y al que se le asocia un ángulo determinado en radianes. Así el arco angular o desplazamiento angular se determinará por la variación de la posición angular entre dos momentos final e inicial concretos (dos posiciones distintas):

$Deltavarphi = varphi_f - varphi_o qquad mbox{ó} qquad Delta heta = heta_f - heta_o$

Siendo $Deltavarphi$ ó $Delta heta$ el arco angular o desplazamiento angular dado en radianes.

Si se le llama $e,$ al espacio recorrido a lo largo de la trayectoria curvilínea de la circunferencia de radio $R,$ se tiene que es el producto del radio de la trayectoria circular por la variación de la posición angular (desplazamiento angular):

$e = RDeltavarphi = R(varphi_f - varphi_o) qquad mbox{ó} qquad s = RDelta heta = R( heta_f - heta_o)$

En ocasiones se denomina $s,$ al espacio recorrido (del inglés "space"). Nótese que al multiplicar el radio por el ángulo en radianes, al ser estos últimos adimensionales (arco entre radio), el resultado es el espacio recorrido en unidades de longitud elegidas para expresar el radio.

Velocidad angular y velocidad tangencial[editar]

Velocidad angular es la variación del arco angular o posición angular respecto al tiempo. Es representada con la letra $omega,$ (omega minúscula) y viene definida como:

$omega = lim_{Delta t o 0}frac{Delta varphi}{Delta t} = lim_{Delta t o 0}frac{varphi_f - varphi_o}{t_f - t_o} qquad mbox{ ó } qquad omega = frac{d varphi}{d t}$

Siendo la segunda ecuación la de la velocidad angular instantánea (derivada de la posición angular con respecto del tiempo).

Velocidad tangencial de la partícula es la velocidad del objeto en un instante de tiempo (magnitud vectorial con módulo, dirección y sentido determinados en ese instante estudiado). Puede calcularse a partir de la velocidad angular. Si $v_t$ es el módulo la velocidad tangencial a lo largo de la trayectoria circular de radio R, se tiene que:

$v_t = omega,R$

Aceleración angular y tangencial[editar]

La aceleración angular es la variación de la velocidad angular por unidad de tiempo y se representa con la letra: $alpha,$ y se la calcula:

$alpha = frac{d omega }{d t}$

Si a_t es la aceleración tangencial, a lo largo de la circunferencia de radio R, se tiene que:

a_t = R , alpha ;

Período y frecuencia[editar]

El período indica el tiempo que tarda un móvil en dar una vuelta a la circunferencia que recorre. Se define como:

$T=frac{2pi}{omega}$

La frecuencia es la inversa del periodo, es decir, las vueltas que da un móvil por unidad de tiempo. Se mide en hercios o s^-1

$f=frac{1}{T}=frac{omega}{2pi}$

Aceleración y fuerza centrípeta[editar]

Mecánica clásica[editar]

La aceleración centrípeta, también llamada normal o radial, afecta a un móvil siempre que éste realiza un movimiento circular, ya sea uniforme o acelerado. Se define como:

$a_c = a_n = frac{v^2_t}{R}=omega^2R$

La fuerza centrípeta es la fuerza que produce en la partícula la aceleración centrípeta. Dada la masa del móvil, y basándose en la segunda ley de Newton ( $vec {F} = m vec {a}$ ) se puede calcular la fuerza centrípeta a la que está sometido el móvil mediante la siguiente relación:

$F_c=ma_c=frac{mv^2}{R}=momega^2R$

Mecánica relativista[editar]

En mecánica clásica la aceleración y la fuerza en un movimiento circular siempre son vectores paralelos, debido a la forma concreta que toma la segunda ley de Newton. Sin embargo, en relatividad especial la aceleración y la fuerza en un movimiento circular no son vectores paralelos a menos que se trate de un movimiento circular uniforme. Si el ángulo formado por la velocidad en un momento dado es $scriptstyle alpha$ entonces el ángulo $scriptstyle eta$ formado por la fuerza y la aceleración es:

$cos eta = frac{1+cfrac{v^2}{c^2}(1-cos^2alpha)}{sqrt{left(1+cfrac{v^2}{c^2}(1-cos^2alpha) ight)^2+cfrac{v^4}{c^4}cos^2alphasin^2alpha}}$

Para el movimiento rectilineo se tiene que $scriptstyle sin alpha = 0$ y por tanto $scriptstyle eta = 0$ y para el movimiento circular uniforme se tiene $scriptstyle cos alpha = 0$ y por tanto también $scriptstyle eta = 0$ . En el resto de casos $scriptstyle eta e 0$ . Para velocidades muy pequeñas y ángulos expresados en radianes se tiene:

$eta approx frac{v^2}{c^2} cosalpha sinalpha + Oleft(frac{v^4}{c^4} ight)$

http://es.wikipedia.org/wiki/Movimiento_circular

Time-warped Fields

Time-warped Fields use energy within curvatures of spacetime surrounding a rotating mass or energy field to generate containable and controllable fields of closed-timelike curves that can move matter and information forward or backward in time.

Time-warped Field Time Control and Time Travel

David Lewis Anderson, USAF
Officer and Scientist, founder
of time-warped field theory.

As general relativity predicts, rotating bodies drag spacetime around themselves in a phenomenon referred to as frame-dragging. This rotational frame-dragging effect is also known as the Lense-Thirring effect. The rotation of an object alters space and time, dragging a nearby object out of position compared to the predictions of Newtonian physics. The predicted effect is small—about one part in a few trillion.

However, as Dr. David Lewis Anderson proposed in 1987 with his announcement of time-warped field theory, the difference in potential energy between two different areas of twisted spacetime due to frame-dragging is significantly large. Even the smallest twist in spacetime contains enormous energy potential and can be used to create containable and controllable fields of close-timelike curves without the need for significant input power. This makes both forward and reverse time control possible within the limits of technology today.

The key characteristics of the application of time-warped fields for time control and time travel are presented in the picture below. This is followed by more detail describing the science below.

Time-warped Fields Time Control and Time Travel

Frame Dragging Effect Basics

The Anderson Time Reactor operates by accessing the high energy
potential and effects, existing across two regions of twisted spacetime,
to create containable and controllable fields of closed-timelike curves.

Rotational frame-dragging appears in the general principle of relativity and similar theories in the vicinity of rotating massive objects. Under this effect, the frame of reference in which a clock ticks the fastest is one which is rotating around the object as viewed by a distant observer. This also means that light traveling in the direction of rotation of the object will move around the object faster than light moving against the rotation as seen by a distant observer. It is now the best-known effect, partly thanks to the Gravity Probe B experiment.

Linear frame dragging is the similarly inevitable result of the general principle of relativity, applied to linear momentum. Although it arguably has equal theoretical legitimacy to the "rotational" effect, the difficulty of obtaining an experimental verification of the effect means that it receives much less discussion and is often omitted from articles on frame-dragging

Static mass increase is another effect. The effect is an increase in inertia of a body when other masses are placed nearby. While not strictly a frame dragging effect, it is also derived from the same equation of general relativity. It is a tiny effect that is difficult to confirm experimentally.

Mathematical Derivation of Frame Dragging

Frame-dragging may be illustrated most readily using the Kerr metric, which describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J

where r_s is the Schwarzschild radius

and where the following shorthand variables have been introduced for brevity

In the non-relativistic limit where M (or, equivalently, r_s) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates

We may re-write the Kerr metric in the following form

This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius r and the colatitude θ

In the plane of the equator this simplifies to:

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging. Frame-dragging occurs about every rotating mass and at every radius r and colatitude θ.

The Anderson Time Reactor

Twisted spacetime around the earth, or
any rotating body, contains enormous
levels of potential energy. This is due
to the tension in the fabric of spacetime
caused by inertial frame-dragging.

Time-warped field theory shows how a properly configured energy beam can be used to initiate and maintain the coupling of two different areas of slightly twisted spacetime. This enables the discharge of significantly greater levels of stored potential energy and generates controllable fields of closed-timelike curves. The system that couples these two regions of different spacetime potential is common referred to as an Anderson Time Reactor or spacetime battery.

The Anderson Time Reactor is a system that couples two different areas of twisted spacetime, with two different spacetime tensions. The system can access and create a conduit to harvest that stored energy and through the coupling process create dense fields of Closed Timelike Curves (CTCs).

A reactor consists of a region of spacetime, large or small, surrounding a rotating mass, where inertial frame dragging effects are present twisting spacetime between two regions of space.

David Lewis Anderson

A specialized beam emitter, with a localized source nearer to the rotating mass, is directed toward a more distant region of space, across the region of twisted spacetime created by inertial frame-dragging.

A series of power collectors near and surrounding the beam emitter provide a conduit to then channel and control the received power. The resulting effect is that the potential energy in the twisted fabric of spacetime is coupled or bridged from the distant point to the local power collector array. The entire process is initiated and controlled by the system.

The Anderson Time Reactor system achieves this by using the application of Time-warped Field theory to create the ability to leak, tap into and control the greater energy stored in this spacetime tension (or energy potential difference), in between the distant point and the localized point in spacetime.

In the most basic terms, the Time Reactor can be looked at as a simple spacetime battery, accessing the significant potential energy that existing around any rotating body anywhere in spacetime.

Spacetime-Motive Force

Visualization of energy pattern near time reactor

Spectral image of energy pattern
near time reactor emitter and power
collector array showing coupling
and discharge of spacetime-motive
force including energy drift in the
direction of inertial frame dragging
of the Earth. USA, 2008

The coupling of these two points accesses what Dr. Anderson labeled a "spacetime-motive force" with the ability to produce high energy and time-warped fields allowing the containment and controlling of fields of closed-timelike curves.

The force between the localized and distant point is called the open spacetime-motive force. The open spacetime-motive force, even in the minimal effects of inertial frame-dragging, can be extremely large by present-day power generation standard standards. It is estimated that a single next-generation time reactor may have the ability to produce more than all of the worlds combined power generation capabilities today.

The amount of spacetime motive force depends on several factors. These include the mass of the rotating body, its rotation speed, relative orientation of the two point to the axis of rotation, and the medium and distance between the localized and distant points in space. More simply, it is a function of the degree of inertial frame-dragging and the characteristics of the medium through which the Time Reactor must operate between the two regions to open a "discharge path." Also, the amount of energy that is accessed, or time-warped fields generated, can be controlled in several ways through phasing and other characteristics of the emitter and power collector array.

A Practical Approach to Achieving Time Control

Practical time control and time travel requires significantly large energy levels, from some source, to operate effectively. To achieve time control we can attempt to generate this large energy level or, as an alternative, access and channel the energy already existing and inherent in natural processes and the basic makeup or fabric of spacetime surrounding our planet.

As stated above, it is estimated that a single next-generation time reactor may have the ability to produce more than all of the world's combined power generation capabilities today.

Time-warped field theory demonstrates
a practical way to generate the
necessary concentrated CTCs and
high power levels, without high input
power, for practical time control

The fabric of spacetime is elastic and very powerful. It takes a tremendous amount of power to create even the slightest twist in spacetime. One can think of the fabric of spacetime surrounding a rotating mass, like the Earth, to be a spring or a battery.

The rotating mass creates a twist in the fabric of spacetime who's natural state and desire is to unwind, just like a spring, or to discharge, just like a battery. Time-warped field technology uses relatively low input power to open a discharge path for this spacetime battery. This technology itself does not create the energy levels required for time control and time travel. Instead, it relies on and operates using the energy stored within twisted spacetime around a rotating body that is created by the inertial frame-dragging effect. With only a small amount of system input power, time-warped field theory shows how enormous power levels can be accessed.

The coupling and discharge process, initiated and also defined by time-warped field theory and technology, generates significant levels of spacetime-motive force that can be used to generate very concentrated fields of closed-timelike curves near the Time Reactor's emitter and power collector array. These fields of closed-timelike curves are concentrated and controllable and can permit both forward and backwards time control.

Time Control Technologies and Methods

The ability to control time in both a forward and backwards direction is possible within the laws of our mathematics and physics. The chart below (click for larger view) compares ten different technologies an methods. Key characteristics are identified for each and described below.

Under each key characteristic is a column with either a solid or empty circle. A solid circle indicates a key characteristic is supported by the indicated technology or method, an empty circle indicates it is not.

"Time Control" indicates whether travel to future, past, or both are possible. "Matter Transport" is solid if both matter and information can be transported, empty if only information can be transported. "Tech Viability" is solid if the technology or method is viable with present state-of-the-art technology or within two generations. "Possible Without Exotic Materials" is solid if materials required are available today or within two generations. "Relatively Low Input Power" is solid if time control is achievable within power generation capabilities available today or within two generations.

The time control technologies and methods above include the following:

		Quantum Tunneling: is an evanescent wave coupling effect that occurs in quantum mechanics. The correct wavelength combined with the proper tunneling barrier makes it possible to pass signals faster than light, backwards in time.

		Near-Lightspeed Travel: has the ability to significantly dilate time, sending an accelerating traveler rapidly forward in time relative to those left behind before her travel. The closer to the speed of light, the further into the future the travel.

		Alcubierre Warp Drive: stretches spacetime in a wave causing the fabric of space ahead of a spacecraft to contract and the space behind it to expand. The ship can ride the wave to accelerate to high speeds and time travel.

		Faster-than-Light Travel: is a controversial subject. According to special relativity anything that could travel faster-than-light would move backward in time. As the same time, special relativity states that this would require infinite energy.

		Time-warped Fields: use energy within curvatures of spacetime around a rotating mass or energy field to generate containable and controllable fields of closed-timelike curves that can move matter and information forward or backward in time.

		Circulating Light Beams: can be created using gamma and magnetic fields to warp time. The approach can twist space that causes time to be twisted, meaning you could theoretically walk through time as you walk through space.

		Wormholes: are hypothetical areas of warped spacetime with great energy that can create tunnels through spacetime. if traversable would allow a traveler to quickly move through great distances in space and also travel through time.

		Cosmic Strings: are a hypothetical 1-dimensional (spatially) topological defect in the fabric of spacetime left over from the formation of the universe. Interaction could create fields of closed timelike curves permitting backwards time travel.

		Tipler Cylinder: uses a massive and long cylinder spinning around its longitudinal axis. The rotation creates a frame-dragging effect and fields of closed timelike curves traversable in a way to achieve subluminal time travel to the past.

		Casimir Effect: a physical force arising from a quantized field, for example between two uncharged plates. This can produce a locally mass-negative region of space-time that could stabilize a wormhole to allow faster than light travel.