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    Re: Coriolis and gyros
    From: Frank Reed
    Date: 2009 Sep 3, 22:51 -0700

    Sorry to be replying two weeks late to this message. I was travelling for much of August.
    
    John Huth (email apacherunner), you wrote:
    "The term "fictitious force" is intended to imply that it is an apparent force 
    that arises in an accelerating frame of reference, whereas such a force does 
    not exist in an inertial (i.e. non-accelerating) frame.   For this reason, I 
    refer to this as the "Coriolis effect", as opposed to "Coriolis force", to 
    avoid confusion."
    
    
    And yet, it's a force like any other so long as you use it correctly. Myself, 
    I generally refer to it as the "Coriolis acceleration" since, like gravity, 
    it is an acceleration field and these only fit into the framework of 
    Newtonian "forces" if we adopt the "fiction" that they are defined by forces 
    which are proportional to the object's own mass. While Coriolis and 
    centrifugal accelerations are frame-dependent, this makes them no less "real" 
    than local gravity itself. When you drop a pencil, it accelerates downward, 
    but if you observe it in a freefall frame of reference (which is the modern 
    definition of an inertial frame of reference) it does not accelerate at all. 
    Rotating frames of reference are widely used in physics, and the suggestion 
    that they are something to be avoided (not from you, but from a couple of 
    other folks) is misleading. Of course, accelerated frames of reference have 
    also been widely and infamously abused in an almost endless number of popular 
    accounts of physics which undoubtedly explains some of the frustration that 
    people feel when discussing these forces.
    
    And you wrote:
    "It is certainly true that you can express the Coriolis effect as an effective 
    force if you are in a rotating frame of reference."
    
    And a great many phenomena in physics are readily understood by reference to 
    centrifugal and coriolis forces which are rather more difficult to understand 
    in inertial frames of reference, to put it mildly. 
     
    I wrote previously:
    "The Coriolis acceleration is frame-dependent --it depends on the choice of a 
    rotating frame of reference. It's intriguing to note that there is also an 
    extremely tiny physical, non-frame-dependent, version of the Coriolis 
    acceleration which is created by spinning masses. This is known as 
    "frame-dragging" or "gravito-magnetism" and was measured (barely due to 
    various problems with the ultra-sensitive gyroscopes) by a spacecraft known 
    as "Gravity Probe B": http://einstein.stanford.edu/highlights/status1.html."
    
    And John, you replied:
    "I would be careful in separating the frame dragging effect, which is a result 
    of general relativity from the classic Coriolis effect. Although they both 
    result from the rotation of a mass, frame dragging only comes from general 
    relativity and is a very small effect."
    
    Yes. And it is, as I said, "extremely tiny"... In any case, the important 
    point here is that there is an exact correspondence between the relationship 
    between the static gravitational force and centrifugal force, on the one 
    hand, and the gravitomagnetic gravitational force and coriolis force, on the 
    other. In much the same way that the centrifugal force in the rotating frame 
    attached to the Earth is inseparable from static gravitation and leads us to 
    say that the acceleration of gravity varies with latitude in both direction 
    and magnitude, so also the Coriolis acceleration is modified by the dynamic 
    gravitomagnetic field of the Earth in an inseparable fashion. The 
    gravitomagnetic field is a physical, non-frame-dependent field which behaves 
    exactly like a position-dependent version of the Coriolis acceleration. 
    Finally, though this correspondence exists, the big difference (huge 
    difference!) is that the gravitomagnetic field of the Earth is so tiny that 
    it has barely been measured while the static gravitational field of the Earth 
    is obviously all around us. [For anyone following along here, if you're 
    wondering how you could go your whole life without hearing about this 
    "gravitomagnetic field" thing, don't worry. You really can continue your 
    whole life without thinking about it again unless it entertains you or 
    enlightens you --it is a tiny and insignificant phenomenon. But it can be in 
    fact very englightening so don't forget it too quickly... :-)].
    
    John, you added:
    "I'm not sure about why you say that the Coriolis effect has a deep connection 
    to magnetism. Torque causing precession is happens in many systems.   I don't 
    think you're suggesting that magnetic fields are the result of non-inertial 
    frames, or are you?"
    
    No, not that of course. :-)
    
    First there is a simple classical physics connection. To see it, let's model a 
    Foucault pendulum and a gyrocompass with equivalent electric and magnetic 
    fields and with charges instead of masses. Let's assume that we are doing 
    this experiment in freefall (no local gravity). For our pendulum we put a 
    small plastic ball on the end of a string and hang it from the top of a large 
    box. Large here means large compared to the amplitude of the swinging motion 
    after we set the pendulum swinging. The string is just a bit shorter than the 
    height of the box so that the ball can almost reach the base of the box. Put 
    a small positive charge on that ball. The base of the box is a plastic sheet 
    which we cover with a uniform negative charge. This creates a nearly uniform 
    electric field above the base of the box. The ball on the string is attracted 
    to it. If we give it a little push, it will starting swinging back and forth, 
    and by selecting the correct charge density, we can get the pendulum swinging 
    with just the right period to match any pendulum near the surface of the 
    Earth. For the gyrocompass analogue, we make a spinning plastic wheel and 
    hang it from the top of the box so that its center of mass is below its point 
    of suspension, as is the case with a gyrocompass. On the rim of the wheel, we 
    put some positive charge. Whether spinning or not, the wheel will be pulled 
    by the static charge on the base of the box and it will hang "horizontally". 
    So far, so good. We have an electrostatic model that corresponds to the 
    static gravitational field. The pendulum swings back and forth. The gyro 
    hangs horizontally.
    
    Now we need to add an equivalent to the Coriolis acceleration. To do this, we 
    put our box inside a solenoid --a cylinder wrapped with current-carrying 
    wire. This has a nearly constant magnetic field within the cylinder with the 
    magnetic field vector aligned along the axis of the cylinder. The effect of 
    this magnetic field is to deflect moving charges with an acceleration 
    proportional to velocity and perpendicular to the plane containing the 
    velocity vector and the axis of the cylinder and also proportional to the 
    sine of the angle between the velocity and the axis of the cylinder. This is 
    exactly analogous to the Coriolis acceleration where the axis of rotation of 
    the frame of reference corresponds to the symmetry axis of the cylinder. The 
    plane of the swing of the pendulum will now precess as the swinging charge is 
    deflected by the magnetic field. Similarly, the gyroscope will precess about 
    the direction of the magnetic field (though because it is hanging in the 
    static electric field, it only nods back and forth around the direction of 
    the axis of the cylinder. For even more fun, we can tilt the cylinder 
    relative to the base of the box. The angular tilt corresponds to the tilt of 
    the local gravitational surface with respect to the rotation of the Earth, or 
    in other words, the tilt is the same as the observer's latitude on the Earth. 
    And there we have it: a very nice electromagnetic analogue for phenomena in a 
    gravitational field in a rotating frame of reference. But it must always be 
    emphasized (can't do it enough!) that these are analogous systems. There are 
    many similarities, some quite "profound", between gravitation and 
    electromagnetism, but they are distinct force fields with different sources. 
    And of course, electromagnetism is radically different from gravitational 
    systems when we go beyond simple cases because EM charges can be positive, 
    negative (or neutral).
    
    Note that the above electromagnetic model does not work merely because all 
    torques and precession phenomena are similar. A constant magnetic field 
    produces exactly the same motions on systems of charges (all assumed to have 
    the same sign here) as a Coriolis acceleration field produces on systems of 
    masses in a rotating frame of reference. See the PS below for a small 
    difference...
    
    Beyond this classical analogy between constant magnetic fields and the 
    accelerations in a rotating frame of reference, there is a more profound 
    connection. Gravitation itself obeys Maxwell's equations (in the weak field, 
    low velocity limit --no black holes, in other words). That is, the 
    relationships which were discovered in the electromagnetic field in the 
    nineteenth century have turned out to general properties and relationships of 
    all field theories, primarily a ressult of local special relativistic 
    transformation properties. Just as a current of flowing charge generates a 
    field which deflects moving charges and causes current loops to precess near 
    it, so a current of mass (a helluva lot of mass!) generates a similar field 
    which deflects moving masses (any mass) and causes mass loops (gyros) to 
    precess. This is the gravitomagnetic field. Unlike the common magnetic field, 
    the gravitomagnetic field can be eliminated at any point by a suitable 
    acceleration just as the static gravitational field can be eliminated at any 
    point by working in a freefall frame of reference. But in this case, the 
    elimination is done by rotation. The Coriolis acceleration nulls out the 
    gravitomagnetic field at any point just as linear acceleration nulls out the 
    static gravitational acceleration. I should emphasize that there is a lot 
    more to this. It's not exactly "easy" either. For those among us who have 
    studied some general relativity, a readable (but serious mathematical 
    physics) account of some of this material can be found in "Gravitation and 
    Inertia" by Ciufolini and the late, great John A. Wheeler.
    
    -FER
    PS: For completeness, I should add that the charges in this analogous system 
    will produce some electromagnetic radiation which will damp out the motions 
    much as air resistance (or other resistance) damps out the motion of a 
    Foucault pendulum and also damps out the swinging of a gyrocompass until it 
    falls rather quickly on true north. This EM radiation is a distinct 
    difference from the gravitational case since gravitational radiation is many 
    orders of magnitude weaker than EM radiation, and has, in fact, never been 
    observed directly even from the brightest astronomical sources. 
    
    
    
    
    
    
    
    
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