🚀 go-pugleaf

On Tue, 25 Apr 1995, Barry S. Riehle wrote:

> Donald,
>
> Thank you for the explanation of the tippy top - other diagrams and the like I
> will be able to augment with John Perry's "Spinning Tops and Gyroscopic
> Motion".
>
> Regarding the tides, I gather you have a significant disdain for rationales
> presented on why we have tides on the side of the earth opposite the moon.
> Well, unfortunately, what I have read is all I know and must confess that I
> was never convinced of the explanation of inertia myself but have no clue as
> to a more accurate explanation.  Any enlightenment or place to look for it?
>
> Barry S. Riehle
>

Please excuse my delayed response. Also, to the group: Please excuse the
poor proofreading of some of my recent posts. It's end of term here, and
every time I'm 'on a roll' composing an e-mail, some faculty or student
comes in with a problem which has to be attended to *now*. So I hastily
type a few lines and press *mail* without adequate proofing. I shall try
to do better in the future.

A student of mine is doing a report next week on pitfalls of teaching
about tides. As I was checking references last night I looked in Arons'
book on Teaching Introductory Physics. He has a brief off-hand reference
to tides, which I found very disappointing. It left the reader with the
impression that motion about the center of mass was responsible (or
largely responsible) for the tides. Now I have great respect for Arnold,
and think this book should be on the library of every teacher of physics,
but here he was too hasty.

Then I looked in a stack of mechanics books which a colleague had just
returned to me (borrowed for nearly a year!). In it was my copy of Vernon
D. Barger and martin G. Olsson, _Classical Mechanics, a Modern
Perspective_, Mc. Graw-Hill, 1973. This book was recommended here for its
good analysis of the Tippy-Top. I find it also has a good discussion of
tides (Sec. 7.2). It is nearly 8 pages long, with equations and nice
diagrams, and I don't think the words 'inertia' and 'centripetal force'
are mentioned once. It develops the topic sufficiently to include numeric
results, which are compared with actual measurements.

This treatment begins: "... to calculate the gross features of the
midocean tides, we begin with a static theory in which the rotation of
the earth about its own axis is initially neglected. The daily rotation
of the earth will be invoked later to explain the propagation of the
tides." As you read this section you find that the revolution of the moon
about the earth is not considered either, nor need it be, since the bulk
of the tidal effect (the deformation of the oceans and the earth itself)
and the fact that there are high tides on both sides of the earth, the
lunar side and the opposite side, do not depend on rotation.

Later: "The tide for an ocean devoid of continents has a football-like
shape, with the major axis in the direction of the distant mass. The
calculation of such an ideal tide was first made by Newton in 1687." Then
follows the calculation of the ratio of the maximum heights of the lunar
and solar tides, the height ratio being 2.2 (lunar/solar). Then: "The
tidal range due to the moon along the earth-moon axis can be
calculated ... 0.56 m. This figure compares roughly with the measured
tidal difference in midocean."

And finally, "The tides are really more complicated than described above.
Along coastal regions the configuration of the land masses and the ocean
bottom cause considerable amplification or suppression of the tidal
range. Over the world, tidal ranges vary by amounts over an order of
magnitude 0 to 20 m.  The friction of the moving tidal waves against
ocean bottoms and the collisions of the tides with the continental
shorelines disspate energy at a rate estimated at 2 billion horsepower.
To compensate for this energy loss, the earth's rotation about its axis
must slow down at a rate of 4.4 E-8 s per rotation. The cumulative loss
of time over a century is about 28 s. This gradual lengthening of the day
is confirmed by the observation that various astronomical events such as
eclipses and motions of planets seem to run systematically 28 s ahead of
calculations based on observations a century ago."

A caution. When one asks for an explanation of the tides in an elementary
course, one doesn't expect treatment of all the small nuances and fine
details. Nor does one expect analysis of shoreline and ocean depth
effects, though we should acknowledge their importance, and perhaps cite
data on the variation of their size and the reasons for them. We usually
simply want to explain the deformation of the earth's shape (and the
ocean's shape) due to the gravitational forces from a distant mass, and
the reason for the *two* tides on the opposite side of the earth. This can
be dealt with on a static model (but requires calculus to get numeric
results). Drag effects due to viscosity and friction (oceans) tensile
strength ('solid' earth), and inertia are next in importance. Dynamic
effects (motion effects) involving the tilt of the earth's axis, and the
tilt of the moon's orbit ought to be discussed, but are easy to grasp.
Centripetal force dynamic effects are relatively small perturbations on
the size of the tides.

        -- Donald