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Re: Bulges




On Sun, 2 Mar 1997, Mark Sylvester wrote:

Note also that the procedure is NOT mathematically correct if the two
bodies are "nailed to the fabric of space". In this case it's like
having two blobs of jelly (in the English sense - I can't think of
the American word) staked through their centres: they'll bulge
towards each other, yes, but not away from each other. Or perhaps it
depends on just how the bodies are nailed. I think the point is that
you get TWO tidal bulges when the bodies are in free fall,
accelerating towards each other, regardless of whether there is
circular motion, but the nailing is a red herring. Centripetal
effects are not essential, but I don't see how one can avoid the
centre of mass (Plait doesn't avoid it btw as he is taking the force
of the moon on the centre of the earth and subtracting the local
force of the moon), as the whole thing depends on the fact that the
body has the same acceleration everywhere, determined by the position
of its centre of mass, but the gravitational pull on it differs from
place to place, so the body has to deform - in both directions,
making two bulges.

Again, my intent was misread. Must not have stated it clearly enough.

By 'nailed down' I meant that 'some point' on earth and one on the moon
was fixed, and it doesn't have to be the center of mass. Imagine a huge
framework with a stake to it through the earth and one through the moon.
For convenience of discussion, imagine the stake through the earth passing
along the earth's polar axis. Also, for convnience, do this at a time when
the earth is above the equator. Now imagine that both earth and moon have
some elasticity, that they can be deformed by gravitational forces. Keep
both earth and moon at rest.

Under these conditions the tidal forces (gradient of the gravitational
force) on each piece of each body, are such that both bodies are distorted
from their initial spherical shape into, what is the term, oblate
spheroids. Anyway, slightly elongated along the axis joining the two
bodies. With respect to the polar axis (the stake, remember) there are two
bulges on the earth, one toward the moon, one on the other side. This fact
of there being *two* bulges will be the case even with motion, rotation,
or fussing about the center of mass. Now, of course, if you want to get
quantitative about the exact relative size of the bulges, you'll want to
talk about centers of mass. Then you'll unstake the planets and let them
rotate, which will introduce centripetal effects which will further modify
the bulges. Drag forces will ensure that the bulges lag, so they no longer
are directly along the earth-moon line, etc. etc.

Yes, it is amazing how much misunderstanding of this there is out there.

A very good discussion of this will be found in the following paper:

Paul Qunicey. "Why We Are Unmoved As Oceans Ebb And Flow", _Skeptical
Inquirer_, _18_, 509-515. (Fall 1994)

Paul Quincey is a physicist at the National Physical Laboratory in the
U.K.

His article has nice computer-generated diagrams of the forces on each
part of a spherical body.

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Prof. of Physics Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745 CIS: 73147,2166
Home page: http://www.lhup.edu/~dsimanek FAX: 717-893-2047
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