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:

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

Véase también[editar]

Sinusoidal Waveforms

Generation of Sinusoidal Waveforms

In our tutorials about Electromagnetism, we saw how an electric current flowing through a conductor can be used to generate a magnetic field around itself, and also if a single wire conductor is moved or rotated within a stationary magnetic field, an “EMF”, (Electro-Motive Force) will be induced within the conductor due to this movement.

From this tutorial we learnt that a relationship exists between Electricity and Magnetism giving us, as Michael Faraday discovered the effect of “Electromagnetic Induction” and it is this basic principal that electrical machines and generators use to generate a Sinusoidal Waveform for our mains supply.

In the Electromagnetic Induction, tutorial we said that when a single wire conductor moves through a permanent magnetic field thereby cutting its lines of flux, an EMF is induced in it.

However, if the conductor moves in parallel with the magnetic field in the case of points A and B, no lines of flux are cut and no EMF is induced into the conductor, but if the conductor moves at right angles to the magnetic field as in the case of points C and D, the maximum amount of magnetic flux is cut producing the maximum amount of induced EMF.

Also, as the conductor cuts the magnetic field at different angles between points A and C, 0 and 90^o the amount of induced EMF will lie somewhere between this zero and maximum value. Then the amount of emf induced within a conductor depends on the angle between the conductor and the magnetic flux as well as the strength of the magnetic field.

An AC generator uses the principal of Faraday’s electromagnetic induction to convert a mechanical energy such as rotation, into electrical energy, a Sinusoidal Waveform. A simple generator consists of a pair of permanent magnets producing a fixed magnetic field between a north and a south pole. Inside this magnetic field is a single rectangular loop of wire that can be rotated around a fixed axis allowing it to cut the magnetic flux at various angles as shown below.

Basic Single Coil AC Generator

AC generator

As the coil rotates anticlockwise around the central axis which is perpendicular to the magnetic field, the wire loop cuts the lines of magnetic force set up between the north and south poles at different angles as the loop rotates. The amount of induced EMF in the loop at any instant of time is proportional to the angle of rotation of the wire loop.

As this wire loop rotates, electrons in the wire flow in one direction around the loop. Now when the wire loop has rotated past the 180^o point and moves across the magnetic lines of force in the opposite direction, the electrons in the wire loop change and flow in the opposite direction. Then the direction of the electron movement determines the polarity of the induced voltage.

So we can see that when the loop or coil physically rotates one complete revolution, or 360^o, one full sinusoidal waveform is produced with one cycle of the waveform being produced for each revolution of the coil. As the coil rotates within the magnetic field, the electrical connections are made to the coil by means of carbon brushes and slip-rings which are used to transfer the electrical current induced in the coil.

The amount of EMF induced into a coil cutting the magnetic lines of force is determined by the following three factors.

• Speed – the speed at which the coil rotates inside the magnetic field.
• Strength – the strength of the magnetic field.
• Length – the length of the coil or conductor passing through the magnetic field.

We know that the frequency of a supply is the number of times a cycle appears in one second and that frequency is measured in Hertz. As one cycle of induced emf is produced each full revolution of the coil through a magnetic field comprising of a north and south pole as shown above, if the coil rotates at a constant speed a constant number of cycles will be produced per second giving a constant frequency. So by increasing the speed of rotation of the coil the frequency will also be increased. Therefore, frequency is proportional to the speed of rotation, ( ƒ ∝ Ν ) where Ν = r.p.m.

Also, our simple single coil generator above only has two poles, one north and one south pole, giving just one pair of poles. If we add more magnetic poles to the generator above so that it now has four poles in total, two north and two south, then for each revolution of the coil two cycles will be produced for the same rotational speed. Therefore, frequency is proportional to the number of pairs of magnetic poles, ( ƒ ∝ P ) of the generator where P = is the number of “pairs of poles”.

Then from these two facts we can say that the frequency output from an AC generator is:

generator frequency

Where: Ν is the speed of rotation in r.p.m. P is the number of “pairs of poles” and 60 converts it into seconds.

Instantaneous Voltage

The EMF induced in the coil at any instant of time depends upon the rate or speed at which the coil cuts the lines of magnetic flux between the poles and this is dependant upon the angle of rotation, Theta ( θ ) of the generating device. Because an AC waveform is constantly changing its value or amplitude, the waveform at any instant in time will have a different value from its next instant in time.

For example, the value at 1ms will be different to the value at 1.2ms and so on. These values are known generally as the Instantaneous Values, or V_i Then the instantaneous value of the waveform and also its direction will vary according to the position of the coil within the magnetic field as shown below.

Displacement of a Coil within a Magnetic Field

displacement of a coil

The instantaneous values of a sinusoidal waveform is given as the “Instantaneous value = Maximum value x sin θ ” and this is generalized by the formula.

Where, V_max is the maximum voltage induced in the coil and θ = ωt, is the angle of coil rotation.

If we know the maximum or peak value of the waveform, by using the formula above the instantaneous values at various points along the waveform can be calculated. By plotting these values out onto graph paper, a sinusoidal waveform shape can be constructed.

In order to keep things simple we will plot the instantaneous values for the sinusoidal waveform at every 45^o of rotation giving us 8 points to plot. Again, to keep it simple we will assume a maximum voltage, V_MAX value of 100V. Plotting the instantaneous values at shorter intervals, for example at every 30^o (12 points) or 10^o (36 points) for example would result in a more accurate sinusoidal waveform construction.

Sinusoidal Waveform Construction

Coil Angle ( θ )	0	45	90	135	180	225	270	315	360
e = Vmax.sinθ	0	70.71	100	70.71	0	-70.71	-100	-70.71	-0

sinusoidal waveforms

The points on the sinusoidal waveform are obtained by projecting across from the various positions of rotation between 0^o and 360^o to the ordinate of the waveform that corresponds to the angle, θ and when the wire loop or coil rotates one complete revolution, or 360^o, one full waveform is produced.

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From the plot of the sinusoidal waveform we can see that when θ is equal to 0^o, 180^o or 360^o, the generated EMF is zero as the coil cuts the minimum amount of lines of flux. But when θ is equal to 90^o and 270^o the generated EMF is at its maximum value as the maximum amount of flux is cut.

Therefore a sinusoidal waveform has a positive peak at 90^o and a negative peak at 270^o. Positions B, D, F and H generate a value of EMF corresponding to the formula e = Vmax.sinθ.

Then the waveform shape produced by our simple single loop generator is commonly referred to as a Sine Wave as it is said to be sinusoidal in its shape. This type of waveform is called a sine wave because it is based on the trigonometric sine function used in mathematics, ( x(t) = Amax.sinθ ).

When dealing with sine waves in the time domain and especially current related sine waves the unit of measurement used along the horizontal axis of the waveform can be either time, degrees or radians. In electrical engineering it is more common to use the Radian as the angular measurement of the angle along the horizontal axis rather than degrees. For example, ω = 100 rad/s, or 500 rad/s.

Radians

The Radian, (rad) is defined mathematically as a quadrant of a circle where the distance subtended on the circumference equals the radius (r) of the circle. Since the circumference of a circle is equal to 2π x radius, there must be 2π radians around a 360^o circle, so 1 radian = 360^o/2π = 57.3^o. In electrical engineering the use of radians is very common so it is important to remember the following formula.

Definition of a Radian

Radians

Using radians as the unit of measurement for a sinusoidal waveform would give 2π radians for one full cycle of 360^o. Then half a sinusoidal waveform must be equal to 1π radians or just π (pi). Then knowing that pi, π is equal to 3.142 or 22÷7, the relationship between degrees and radians for a sinusoidal waveform is given as.

Relationship between Degrees and Radians

Applying these two equations to various points along the waveform gives us.

sinusoidal waveform radians

The conversion between degrees and radians for the more common equivalents used in sinusoidal analysis are given in the following table.

Relationship between Degrees and Radians

Degrees	Radians	Degrees	Radians	Degrees	Radians
0^o	0	135^o	3π 4	270^o	3π 2
30^o	π 6	150^o	5π 6	300^o	5π 3
45^o	π 4	180^o	π	315^o	7π 4
60^o	π 3	210^o	7π 6	330^o	11π 6
90^o	π 2	225^o	5π 4	360^o	2π
120^o	2π 3	240^o	4π 3

The velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform. As the frequency of the waveform is given as ƒ Hz or cycles per second, the waveform has angular frequency, ω, (Greek letter omega), in radians per second. Then the angular velocity of a sinusoidal waveform is given as.

Angular Velocity of a Sinusoidal Waveform

and in the United Kingdom, the angular velocity or frequency of the mains supply is given as:

in the USA as their mains supply frequency is 60Hz it is therefore: 377 rad/s

So we now know that the velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform and which can also be called its angular velocity, ω. But we should by now also know that the time required to complete one revolution is equal to the periodic time, (T) of the sinusoidal waveform.

As frequency is inversely proportional to its time period, ƒ = 1/T we can therefore substitute the frequency quantity in the above equation for the equivalent periodic time quantity and substituting gives us.

angular velocity

The above equation states that for a smaller periodic time of the sinusoidal waveform, the greater must be the angular velocity of the waveform. Likewise in the equation above for the frequency quantity, the higher the frequency the higher the angular velocity.

Sinusoidal Waveform Example No1

A sinusoidal waveform is defined as: V_m = 169.8 sin(377t) volts. Calculate the RMS voltage of the waveform, its frequency and the instantaneous value of the voltage after a time of 6ms.

We know from above that the general expression given for a sinusoidal waveform is:

Then comparing this to our given expression for a sinusoidal waveform above of V_m = 169.8 sin(377t) will give us the peak voltage value of 169.8 volts for the waveform.

The waveforms RMS voltage is calculated as:

rms voltage

The angular velocity (ω) is given as 377 rad/s. Then 2πƒ = 377. So the frequency of the waveform is calculated as:

sinusoidal waveform frequency

The instantaneous voltage V_i value after a time of 6mS is given as:

instantaneous voltage

Note that the phase angle at time t = 6mS is given in radians. We could quite easily convert this to degrees if we wanted to and use this value instead to calculate the instantaneous voltage value. The angle in degrees will therefore be given as:

phase angle

Sinusoidal Waveform

Then the generalised format used for analysing and calculating the various values of a Sinusoidal Waveform is as follows:

A Sinusoidal Waveform

sinusoidal waveform

In the next tutorial about Phase Difference we will look at the relationship between two sinusoidal waveforms that are of the same frequency but pass through the horizontal zero axis at different time intervals.