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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
"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... :-)].

"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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