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#1
01-04-2000, 01:33 AM
 RM Mentock Guest Join Date: Mar 1999
I disagree with the answer given in the mailbag: http://www.straightdope.com/mailbag/mdownup.html

The question was:
Quote:
 What is down? If the Earth were a perfect sphere, then a line perpendicular to the tangent of any point on the Earth's surface would go through the core and would certainly be down. But since the Earth is not a perfect sphere, is down the perpendicular to the tangent or the most direct line to the core?
I agree with the first paragraph of the answer, when they define "down" as the direction that an object goes when dropped. But, the second paragraph says that down is therefore directly towards the center of mass of the earth. That is incorrect. The same effect that causes the Earth to be oblate also causes objects to fall away from the center of the earth.

In the third paragraph, they claim that defining down as as the direction perpendicular to the tangent to the earth won't work. This is also incorrect. When treating the Earth as an oblate spheroid, down *is* the perpendicular (or normal) to the tangent. If it were otherwise, a flat puddle of water at your feet would flow.

.
#2
01-04-2000, 06:56 AM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
I don't think you read the article very carefully, since Karen very specifically addresses WHY the normal (perpendicular to the tangent) is NOT the same as the "down" that you get from a plumb line in all situations.
#3
01-04-2000, 10:12 AM
 Irishman Guest Join Date: Dec 1999
RM, try reading this thread. It pretty much addresses the points you make (and agrees with you).
http://www.straightdope.com/ubb/Forum6/HTML/000125.html

And just to be thorough, try
http://www.straightdope.com/ubb/Forum6/HTML/000130.html

CK, RM did understand the article, and Karen's answer didn't fully explain his question. The linked thread above covers those details.

Essentially, what makes the earth oblate also pulls the plumb bob.
#4
01-04-2000, 01:22 PM
 RM Mentock Guest Join Date: Mar 1999
CKDextHavn

There is a technical term called "deviation of the vertical" that seems to be similar to what is described in the article, but the description in the article is wrong. Here are a couple links:
http://164.214.2.59/GandG/geolay/80003006.GIF http://164.214.2.59/GandG/geolay/TR80003A.HTM

Notice, in the illustration, that neither line goes through the center of the earth.

Irishman

Thanks for the links! When I tried to search for 'plumb bob' a few days ago, I had some problems with the search engine, apparently.

.
#5
01-04-2000, 10:37 PM
 Karen Too Guest Join Date: Dec 1999
Yeesh, you'd think a person with a PhD in physics would get a little more respect.

I did neglect the effects of the rotation of the earth in my discussion of perpendicular-to-the-tangent and direction-of-gravity. I also neglected the fact that the earth is NOT a perfect oblate spheroid. I also neglected mountains, basements, etc. My main point was that the direction that gravity acts locally is most relevant, and plumb bobs measure that, whereas it is impossible to measure the perpendicular to the tangent (there being no "tangent" to a bumpy, not-quite oblate spheroid). The direction a plumb bob takes is NOT the perpendicular-to-the tangent.

So, what this boils down to is: if you neglect rotation, and assume the earth is a perfectly smooth oblate spheroid, my original answer stands. If you wish to add the addition effects of rotation only, and assume that the earth is perfectly smooth oblate spheroid, made of an elastic liquid, at equilibrium with its rotation, then the plumb bob measures the perpendicular-to-the-tangent. However if you continue to add effects, such as the non-oblateness and the mountains and basements, then the plumb bob again does NOT measure the perpendicular-to-the-tangent.

As for the "flat" puddle of water at your feet, you need to be careful of circular definitions. I would define "flat" as the perpendicular to gravity, so a "flat" puddle, by definition, does not flow on any shape planet.

------------------
#6
01-05-2000, 12:44 AM
 RM Mentock Guest Join Date: Mar 1999
Karen Too

I'm struggling with your last answer. What do you mean by the effects of "non-oblateness" and "basements"?

Thanks

.
#7
01-05-2000, 01:55 PM
 DSYoungEsq Guest Join Date: Jul 1999
You know, all this time, all this thought, and I can't believe you don't understand by now that, to define up, you take your right hand and, curling the fingers...
#8
01-05-2000, 02:25 PM
 RM Mentock Guest Join Date: Mar 1999
Karen Too

The amount that a plumb bob will miss the center of the Earth is the same value that you calculated in your sixth paragraph, except you said that it was the normal (perpendicular to the tangent) that would miss the center by that far. The tops of those two hypothetical poles would actually be separated by zero feet.

Because the Earth is a not a perfect sphere, the original questioner asked: is down a) the perpendicular to the tangent, or b) the most direct line to the core?

Your answer was b), as you explained in your second paragraph. The actual answer is a). One doesn't have to calculate the normal, though, because the plumb bob does it for you.

.
#9
01-05-2000, 02:50 PM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
I guess I'm being dense, because I'm still not seeing the objections to Karen's original comments.

The tangent to a 2-dimensional surface (for the sake of this discussion, the surface of the earth) is a thing that happens at a point. Thus, if you are standing on the side of a hill, the tangent plane is nowhere near "flat" to an observer who is not standing on that hill; and hence perpendicular to the tangent is not the direction that objects fall or roll.

Indeed, if you are on the side of a straight-edged incline, and drop an object, it will fall "down" in the direction of the tangent (not orthogonally to the tangent) -- that is, it will roll along the hill -- rather than fall towards the center of the earth.

OK, I'm speculating here because I'm not able to do the math while I'm at work, but... assuming that the earth is a smooth surface (ellipsoid, say), it it NOT THE CASE that the perpendicular to the tangent points to the gravitational center of the ellipsoid. Draw an ellipse for yourself, draw a tangent, see where the orthogonal goes.

Or am I missing something obvious here?
#10
01-05-2000, 03:25 PM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
I didn't need to be so waffly in my prior post. The perpendicular to the tangent in an ellipse does NOT got through the center of the ellipse.

A further example: we have assumed that mass is evenly spread. But suppose that mass is concentrated -- suppose that the center of gravity is NOT the center of the earth, but a different inner point. Then the shape of the earth's surface (that is, the slope of the tangent plane at a point and hence the orthogonal to the tangent plane) is irrelevant to the center of gravity. If "down" is towards the center of gravity, then Karen's plumb line method does the trick... and the tangents are irrelevant.
#11
01-05-2000, 05:40 PM
 RM Mentock Guest Join Date: Mar 1999
Dex

Quote:
 OK, I'm speculating here because I'm not able to do the math while I'm at work, but... assuming that the earth is a smooth surface (ellipsoid, say), it it NOT THE CASE that the perpendicular to the tangent points to the gravitational center of the ellipsoid. Draw an ellipse for yourself, draw a tangent, see where the orthogonal goes.
That's right, but the plumb bob follows that perpendicular. Karen Too admits as much in the second paragraph of her post there on 1/4/2000, and I'm pretty sure that her claim in the last sentence of that post is wrong--the the effect of basements, for instance, do deform the shape of the earth's surface, but the net effect is that the plumb bob follows the normal.

Quote:
 plumb line method does the trick... and the tangents are irrelevant.
The plumb line method certainly does do the trick, but it is the same thing as the perpendicular to the tangent, contrary to the mailbag answer.

There are many definitions of the "surface" of the earth. Is it the top of the atmosphere, or the bottom of the oceans? The usual definition is what is known as the geoid--or "sea-level". The geoid is not a perfect oblate spheroid, as Karen Too points out. It is deformed by mass concentrations such as mid-ocean ridges--in fact, satellite images of the ocean surface can actually detect subsurface topography by noticing where the ocean water is "piled up". But, a plumb bob follows the normal to the geoid even there.

If you use the rock surface as your definition, you will run into the problems you describe. That is also contrary to the intent of the original question, and also to the way it was treated in the original mailbag answer.

.
#12
01-05-2000, 08:37 PM
 Karen Too Guest Join Date: Dec 1999
Non-oblateness: the deviation of the earth's shape from a true oblate spheroid; e.g., the north pole is farther from the center of the earth than is the south pole. There are also additional bulges and flattened spots on the earth which cause some people to say the earth is pear-shaped [1]. I believe these non-oblateness features are on the order of 20 meters, tiny
compared to the 21 km oblateness, which is tiny compared to the 6000 km radius of the earth.

mountains and basements: a mountain is a natural formation of extra mass at larger radius. Basements are artificial formations of removed mass at smaller radius. My "mountains and basements" phrase is short for "mountains and valleys and buildings and basements" by which I mean the natural and artificial bumpiness of the earth's surface. I was thinking of an anecdote from a "Fifth Force" seminar I heard years ago [2].

Dex: I think the objections to my original answer are better stated in the other thread: the same "forces" that fling the earth into oblateness also fling the plumb bob, such that the plumb bob is flung into line with the normal. This may be true for an elastic liquid perfectly oblate spheroid earth, but I am not going to answer that question (we particle physicists hate non-inertial reference frames.) I prefer to evaluate the assumptions by rephrasing the question: are the "forces" that fling the plumb bob the same ones that cause the shape of the earth? The answer is an unequivocal NO. If the rotational and self-gravitational forces are the only effects determining the shape of the earth, then the earth WOULD be a perfect oblate spheroid. We KNOW the earth is not a perfect oblate spheroid; people have measured the difference between the north pole radius and the south pole radius; people have measured the bulges and flattened spots; I myself have seen a mountain or two. Thus, we are forced to conclude that there are additional effects at
work, presumably the rigidity of the earth's crust, plate tectonics, tidal effects from the sun and the moon, and maybe the Chandler wobble for all I know. (There is also considerable earth-shaping caused by those bulldozers that wake me every morning with their bleepity bleep beeping.) The shape of
the earth is in fact a hot topic of current research; I'm sure the GPS folks tear their hair out in frustration everyday trying to model it.

Therefore the earth's shape is much more complex than a simple oblate spheroid, and as such we cannot expect that a plumb bob would ever line up with the normal. I will admit that the plumb bob is probably flung a bit by the earth's rotation, as well as the local mass distribution. Nonetheless,
the plumb bob measures what you typically want to know -- the direction
stuff will fall, aka "down".

If you insist that the normal of an oblate spheroid goes through its center, then I can't help you.

Footnotes:

[1] I believe it is more correct to say that the average radius at a specific latitude (i.e., integrated over longitude) resembles the cross section of half a pear.

[2] As you know, gravitational and electromagnetic forces are very long range, while the weak and strong forces are very short range. This irks some people and about 15 years ago, scientists were all hot to find a medium range "fifth" force. To measure this hypothetical force, you have to
account for very subtle effects of gravity, and the scientists therefore had to know the distribution of mass in their location, so they went down the street knocking on the doors of serious corporations, in truly Cecilian fashion, asking the dimensions of their basements. As you can imagine, the
corporations were suspicious of such inquiries, and the scientists had to do a lot of explaining before the corporations would give them their basement blueprints.
#13
01-05-2000, 08:52 PM
 Karen Too Guest Join Date: Dec 1999
I never said that the plumb bob follows the normal to the surface of the earth. I strongly disagree. I have a couple of remaining points:

1) The actual shape of the earth is not the same as the geoid. Nor is the actual shape of the earth the same as the idealized oblate sphereoid.

2) Unless the geoid takes into account basements, then a (very sensitive) plumb bob is not going to be perpendicular to the geoid either. There are plenty of instances where the gravity from local mass distributions affect scientific measurements. For instance, the Fifth Force experiment I mentioned in my earlier post. Also, the particle accelerator at CERN is quite sensitive to the rainfall, and thus the level of water in Lake Geneva, and the resulting changes in the gravitational pull on the particles in the accelerator.

3) If the geoid takes into account the basements and water level in Lake Geneva, that is, the geoid is defined to be the perpendicular to the plumb bob, then we have another circular definition.

------------------
#14
01-05-2000, 09:24 PM
 RM Mentock Guest Join Date: Mar 1999
Karen Too

Quote:
 If you insist that the normal of an oblate spheroid goes through its center, then I can't help you.
I didn't.

We both agree that the plumb bob should define down. The original question asked whether down was the direct line to the center of the earth, or it was the normal. Your answer was the line to the center, whereas the normal is what the plumb bob follows.

In other words, I'm saying that neither the normal, nor the plumb bob, point to the center. They both point in the same direction, away from the center.

.
#15
01-05-2000, 11:14 PM
 RM Mentock Guest Join Date: Mar 1999
Karen Too:
Quote:
 I never said that the plumb bob follows the normal to the surface of the earth.
I didn't say you did. I was replying to Dex's remarks about a smooth ellipsoid, and was referring to your post where you said (1/4/2000 10:37 PM):
Quote:
 If you wish to add the addition effects of rotation only, and assume that the earth is perfectly smooth oblate spheroid, made of an elastic liquid, at equilibrium with its rotation, then the plumb bob measures the perpendicular-to-the-tangent.
Even if you use a smooth ellipsoid to approximate the surface of the earth, rather than the actual geoid, the difference between that ellipsoid's normal and the plumb bob is an order of magnitude smaller than the difference between them and the line to the center of the earth. So, the answer to the original question should not be the line to the center of the earth, it should be the perpendicular to the tangent.

The geoid[1] is what is measured by earth satellites (and earth-based gravity measurements), and is not exactly an oblate spheroid. One of the first such deviations to be measured, by satellites in the early sixties, was the so-called "pear-shape." Also, the equator is "pinched": one equatorial radius is longer than another at right angles to it. These deviations are a result of deep mass heterogeneities, probably related to mantle convection and plate tectonics. The deep mass actually changes the shape of the earth: ocean water "piles" up higher in some places (180 meters higher) than in others.

The pinching of the equator is a degree two effect. Imagine a sine wave wrapped around the equator, following the geoid surface. It would have two highs, and two lows--so, two periods, or degree two. The pear-shape is degree three: follow the outline of the pear from pole to pole and back. There are three highs, and three lows--so, degree three. Over the decades, models of the geoid have been published out to degree 360, which requires over 13,000 coefficients[2], but they are not exactly the geoid. The geoid is perpendicular to the plumb bob--that is not just a circular definition, because the geoid conforms to sea-level also. It is an equi-potential surface. Geodesists don't tear their hair out in frustration anymore, but they might if they had any left.

An interesting consequence of all of this is that the center of mass of a body is identical to its center of figure of its geoid. So, Dex's example (01-05-2000 03:25 PM) of shifting centers of gravity is...impossible. Consider a circle shifted from the center of reference. As you moved around the circle, you'd have one high then one low, and back to the start. So, it is called "degree one." Degree one shifts are called "forbidden" by geodesists, in imitation of the forbidden transitions of physical chemistry. This is more than just flattery--the functions that are used to model the geoid are the same spherical harmonics that are used in physical chemistry.

[1] http://164.214.2.59/GandG/geolay/TR80003A.HTM#ZZ5
[2] Each degree n has 2n+1 spherical harmonics, so in a degree 360 approximation, you'd have to use 361^2 different spherical harmonics.

.
#16
01-07-2000, 07:24 AM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
RM says: << Your answer was the line to the center, whereas the normal is what the plumb bob follows >>

Here's the confusion in the nutshell. If by "normal" you mean the orthogonal to the tangent plane at a point, then I strenuously disagree. The plumb bob does NOT follow the orthogonal to the tangent. The counterexamle of your assertion that it does is the side of the hill situation.

Now, if you means something different by "normal", that's a different story altogether.
#17
01-07-2000, 10:47 AM
 RM Mentock Guest Join Date: Mar 1999
Dex

I don't think that is the confusion at all. Do you think that Andrew Mattison, the person who posed the original question to the mailbag, was talking about perpendiculars to the tangents of the side of the hill?

I don't think he was. And I'm not.

I think he was asking about perpendiculars to the tangent of the spheroid. And that's where the calculations about the poles as tall as the Empire State building come from for the example in the mailbag answer, so apparently the mailbag thought so also.

If you take any plumb bob anywhere in the United States, it is closer to that perpendicular to the tangent than it is to the line to the center of the earth by [em]at least[/em] a factor of ten. If you adjust the geoid for mountains and basements, they line up even better.
#18
01-08-2000, 12:49 AM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
Let me be even more simplistic. If you want to determine which direction is "down", the simplest thing to do is to drop something. Whichever way it falls, that's "down".

Ergo, the plumb line.

And, RM, I really don't understand what it is you're trying to say. On the one hand, you say that the center of gravity (center of mass) must be the same as the center of the geoid. (That's physics, not math, so I don't pretend to understand it, I'll accept it.) On the other hand, you say that the plumb line (which presumably points to the center of the mass) is not as accurate as the orthogonal to the tangent (which does NOT point to the center of the geoid).

Well, the fact is, that these are all wrong. "Down" is actually towards the sun, that's where all the planets and stuff would fall if they didn't have orbital velocity.

I'm goin' for another beer, and I'll drop a plumb line in it, and see how long I can stay orthogonal.
#19
01-07-2000, 02:07 PM
 Irishman Guest Join Date: Dec 1999
Orthogonal to what? If Karen Too is there, I'm thinking I want to get parallel. Or is that coincident?
#20
01-07-2000, 04:04 PM
 RM Mentock Guest Join Date: Mar 1999
Dex

Quote:
 And, RM, I really don't understand what it is you're trying to say. On the one hand, you say that the center of gravity (center of mass) must be the same as the center of the geoid. (That's physics, not math, so I don't pretend to understand it, I'll accept it.) On the other hand, you say that the plumb line (which presumably points to the center of the mass) is not as accurate as the orthogonal to the tangent (which does NOT point to the center of the geoid).
Thanks for sticking with this. (Not you Irishman! I'll see you at the pub.) I think I see the light at the end of the tunnel.

Your presumption is wrong, the plumb line does [em]not[/em] point to the center of mass. It points along the "orthogonal to the tangent."

.
#21
01-09-2000, 12:03 AM
 Kyberneticist BANNED Join Date: Sep 1999 Posts: 1,324
CKDextHavn
Quote:
 Well, the fact is, that these are all wrong. "Down" is actually towards the sun, that's where all the planets and stuff would fall if they didn't have orbital velocity.
Aww. And here I thought down was towards the center of our galaxy. Or is it towards the center of our local group?
#22
01-10-2000, 12:02 AM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
<< Your presumption is wrong, the plumb line does [em]not[/em] point to the center of mass. It points along the "orthogonal to the tangent." >>

I guess that's where I lose you... I see no reason that the plumb line would point orthoganally to the tanget. It WOULDN'T on the side of a hill, I see no reason why it would anywhere. The plumb line is dropped, it will fall towards the center of mass. It will fall down. Tangent plane is irrelvant to "down" ... and the proof of that, whether you like or it not, is the tangent plane to the side of a hill is clearly irrelevant to the direction something will fall.

When I earlier posted the idea of a construct where the center of gravity was not in the center of the geoid, you said it was impossible. Well, perhaps impossible in planetary terms, for instance, after billions of years... but not impossible in a theoretic construct. Let's say I'm building an interstellar ship that I want to be large enough to generate gravity so the passengers will ride on the surface. And I put the center of gravity (mass) at a specific point, NOT the center of the "geoid"...

Is it not clear that, when you drop an object, it will fall "down" towards the center of mass? That the tangent to the surface at the point where you are standing is irrelevant?

So, I think this discussion is done. Further debate is just plain silly, I think I'm caught by one of those arguing-for-its-sake loops. (That's not an argument!)
#23
01-09-2000, 05:44 PM
 RM Mentock Guest Join Date: Mar 1999
Dex

Quote:
 So, I think this discussion is done. Further debate is just plain silly, I think I'm caught by one of those arguing-for-its-sake loops. (That's not an argument!)
I'm sorry that you think so. I was just trying to help.

Quote:
 Is it not clear that, when you drop an object, it will fall "down" towards the center of mass?
No, it is not clear, even though the mailbag answer says that it is. In fact, it is wrong. That is the only reason that I posted my criticism.

.
#24
01-10-2000, 08:26 AM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
Well, life is full of choices. I suppose this could be moved to great debates at this point -- does an object fall towards the center of mass, or away from it?

Karen, on the Straight Dope Science Advisory Staff, and a physicist, says objects fall towards the center of mass.

RM says not, he says << objects ... fall away from the center of the earth. >>

So, at this point, I guess it's faith. You can pick which one you'd care to believe.

While we all (except Cecil) make mistakes, and errors or misinterpretations have from time to time crept into Mailbag articles, there is a point at which discussion is silly. If you start with different assumptions, you wind up with different conclusions, and there's not much more to be said than that.
#25
01-10-2000, 04:50 PM
 ZenBeam Charter Member Join Date: Oct 1999 Location: I'm right here! Posts: 6,880
CKDextHavn wrote:

Quote:
 RM says not, he says << objects ... fall away from the center of the earth. >>
Well, this is rude. I didn't see anywhere that RM says this.

When Karen says "down is directly towards the center of mass" this is strictly true only for a spherical mass distribution. An extreme example where this fails: the center of mass of the Earth-moon system lies within the Earth, but if you're on the moon, down is more-or-less towards the center of the moon. Proximity to the masses involved matters.

I suspect some of the confusion comes from what Karen means by "normal". CKDextHavn, perhaps you're thinking of the normal to the surface of the planet, so that if you're on the side of a mountain, this direction could be 30, 45 or more degrees away from the direction an object wold fall when dropped. When Karen writes "The normal misses the earth's geometrical center by only 21.5 km", she obviously doesn't mean this.

The only other relevant normal I can think of is the normal to an equipotential surface. The surface of calm, motionless (not spinning) water wold be an equipotential surface. I think RM is correct that this normal is identical with the direction a plumb bob would give. And both do not necessarily point to the center of mass of the Earth. Karen's last paragraph makes sense if the two directions she is comparing are the plumb bob direction, and the direction towards the center of mass of the Earth.

Quote:
 Karen, on the Straight Dope Science Advisory Staff, and a physicist, says objects fall towards the center of mass.
I doubt Karen doesn't understand all this, but that doesn't mean she couldn't have incorrectly translated her thoughts to words.

------------------
It is too clear, and so it is hard to see.
#26
01-10-2000, 05:06 PM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
Zen: Me? Rude? The full quote is in RM's first post in this topic: << The same effect that causes the Earth to be oblate also causes objects to fall away from the center of the earth. >>

Look, "down" is obviously relative, and the examples of "down is towards the sun" or "down is toward the center of the galaxy" are sarcastic but evident. We're talking about on the surface of the earth (not at the core of the earth).

And I'm assuming the word "normal" is used in the sense of "orthogonal" or "perpendicular", and sorry, I just don't see that has any bearing on the direction an object falls. Dropping a plumb line IS, in fact, dropping an object and letting if fall down, and the little cord shows you which way is "down."

Where's Deceased Equine when I need her?
#27
01-10-2000, 05:30 PM
 Ed Zotti Gormless Wienie Administrator Join Date: Feb 1999 Posts: 1,672
CK, Karen, I've been thinking about this, and I think RM is right. In an ideal NON-ROTATING planet-sized ellipsoid of uniform density, the plumb bob falls toward the center of mass. Once you add rotation, however, you've got acceleration to think about, and that pulls the plumb bob's line of fall away from the line that points toward the center of mass. I'm taking RM's word for it that the plumb bob's line is (more or less) perpendicular to the tangent, but that sounds about right. RM, do I understand you correctly? Since we're talking about the Coriolis effect here, I'm also thinking that the line of fall may be slightly curved, am I right?
#28
01-10-2000, 06:40 PM
 Kyberneticist BANNED Join Date: Sep 1999 Posts: 1,324
Um. Why would acceleration affect the plumb bob?

The earth has velocity. As far as I know, it is also deaccelerating slightly (stuff like the moon's tidal drag), if that's what you're referring too.

In any case, if you're referring to the earth's rotational velocity, since the plumb bob and the earth are moving at the same speed, does it make any difference?
#29
01-10-2000, 06:44 PM
 Kyberneticist BANNED Join Date: Sep 1999 Posts: 1,324
Ooops. Never mind. Coriolis effect. Gottcha.

Um. Waitaminute. Coriolis effect... Rotation... how much could that do if the plumb bob isn't moving?

Isn't that like demonstrating Focault's pendulum with an unmoving weight? Rather uninteresting?
#30
01-10-2000, 07:28 PM
 ZenBeam Charter Member Join Date: Oct 1999 Location: I'm right here! Posts: 6,880
Quote:
 And I'm assuming the word "normal" is used in the sense of "orthogonal" or "perpendicular"
Orthogonal or perpendicular to what, precisely? This is important. The local surface of the Earth, an equipotential surface, the spherical approximation to the Earth, and the elipsoidal approximation to the Earth all have different normals. Without being specific, I don't know what you're saying.

Quote:
 Dropping a plumb line IS, in fact, dropping an object and letting if fall down, and the little cord shows you which way is "down."
The plumb line is normal to an equipotential surface.

Quote:
 In an ideal NON-ROTATING planet-sized ellipsoid of uniform density, the plumb bob falls toward the center of mass.
This is only true for spherically symmetric mass distributions, not for ellipsoidal distributions in general. The plumb bob won't point towards the center of mass.

------------------
It is too clear, and so it is hard to see.
#31
01-10-2000, 10:28 PM
 RM Mentock Guest Join Date: Mar 1999
ZenBeam 01-10-2000 04:50 PM

Quote:
 Well, this is rude. I didn't see anywhere that RM says this.
Well, the words are there (in my first post), but the context is lost, and words were elided. The meaning is totally gone.

Quote:
 When Karen writes "The normal misses the earth's geometrical center by only 21.5 km", she obviously doesn't mean [the side of a mountain].
That's exactly right. And the plumb bob misses the earth's geometrical center by the same amount, whereas she claims that it points to the center.

Quote:
 Karen's last paragraph makes sense if the two directions she is comparing are the plumb bob direction, and the direction towards the center of mass of the Earth.
But what the article says is that the two directions are the normal, and the plumb bob. Actually, the normal and the plumb bob line up, and they both miss the center of the earth.

.
#32
01-10-2000, 11:26 PM
 RM Mentock Guest Join Date: Mar 1999
Ed

You do understand me correctly.

You can avoid taking my word for it by this thought experiment. The rotating earth is deformed into a rough ellipsoid by rotation and uneven mass distributions within and on the earth. If the Earth were flooded with water (70% of it already is), the water surface would match that rough ellipsoid. If the water surface weren't perpendicular to a plumb bob, water would flow beneath the plumb bob. When it stopped flowing, it would be perpendicular.

That was the analogy I referred to in my first post, last sentence.

The line of fall would be slightly curved, but almost imperceptibly. I notice that you say the "plumb bob's line of fall"--if by that you mean the string, the deviation is even smaller, because the string has very little mass, and it's pulled taut by the weight of the plumb bob.

The original question posted to the mailbag asked if "down" was to the center of the earth, or perpendicular to the tangent. The differences between the two possibilities are slight (well, about 20 kilometers, at the distance of the radius of the earth), but the reason for asking the question clearly was to distinguish between the two. In that context, down is not to the center of the earth.

I was also disappointed to read in the mailbag the opinion that no one would ever need to know the normal. It's important to map makers, geophysicists, geodesists, and astronomers. With the advent of GPS, it's becoming even more important. Geographic latitude is not the angle from the Earth's center, it is the angle of the normal to a best-fit ellipsoid.

.
#33
01-10-2000, 11:29 PM
 C K Dexter Haven Right Hand of the Master Administrator Join Date: Feb 1999 Location: Chicago north suburb Posts: 14,674
The Coriolis effect is too weak to impact bathtub water rotation, but it can affect a plumb line?

Also, if what we're talking about is the effect of rotation, then we might as well take into account the effect of revolution as well, and the gravitational pull of the sun and the planet Jupiter. And using a relativistic model, the moon falls "down" in four-dimensions when it revolves around the earth.

Seems to me that the definition of "down" all depends on what kinds of assumptions you make.... which gets it back to a matter of faith.
#34
01-11-2000, 12:19 AM
 RM Mentock Guest Join Date: Mar 1999
Dex

The plumb bob is not whirling down a drain. This thread? Maybe. I am getting dizzy.

By doing an actual measurement, we do take into account every single effect you mention. They might all affect the measurement. The actual tests show that the plumb bob does not point to the center of the earth--it misses it by tens of kilometers, at the latitude Karen used for her calculation.

Quote:
 Seems to me that the definition of "down" all depends on what kinds of assumptions you make.... which gets it back to a matter of faith.
Faith? This is science, and we all agree on the definition of down: the plumb bob. Unfortunately, that wasn't the original question. It asked, essentially, does the plumb bob point to the center of the earth or not?

.
#35
01-11-2000, 12:38 AM
 tomndebb Mod Rocker Moderator Join Date: Mar 1999 Location: N E Ohio Posts: 34,343
Great. First we get bobbing for apples, now we get bobbing for plumbs.
#36
01-11-2000, 08:54 AM
 ZenBeam Charter Member Join Date: Oct 1999 Location: I'm right here! Posts: 6,880
Quote:
 But what the article says is that the two directions are the normal, and the plumb bob. Actually, the normal and the plumb bob line up, and they both miss the center of the earth.
I'm still not clear what Karen means by normal. If she means normal to the equipotential surface, then your statement above is correct. If she means normal to an oblate spheroidal approximation to the Earth, then this would be different than the plumb bob direction. Prseumably this difference is what you're referring to in an earlier post:

Quote:
 If you take any plumb bob anywhere in the United States, it is closer to that perpendicular to the tangent than it is to the line to the center of the earth by at least a factor of ten.
So the normal to the oblate spheroidal surface approximation and the plumb bob direction differ by less than about 2 km at the center of the Earth?

------------------
It is too clear, and so it is hard to see.
#37
01-11-2000, 09:45 AM
 RM Mentock Guest Join Date: Mar 1999
ZenBeam

Quote:
 So the normal to the oblate spheroidal surface approximation and the plumb bob direction differ by less than about 2 km at the center of the Earth?
Yes, usually much less. That difference is known as the "deflection of the vertical." I posted some links to an illustration of it in my post 01-04-2000 01:22 PM.

.
#38
01-12-2000, 04:35 PM
 Irishman Guest Join Date: Dec 1999
RM, I think you've been all over this, and everyone has gotten around to saying the same thing. I also think Karen gave up convincing you she knows all that.

The difference is the number of unstated assumptions in Karens column. I believe she is using normal to an ideal oblate spheroid, flattened by the rotation of the Earth. You refer to the geoid, or equipotential spheroid, and normal to that. And I think that is the difference you keep pointing out.

1st order approximation, it falls to center of mass. Center of mass is slightly off from geometric center (earth is egg-shaped).
2nd order approx, earth spins, causing a reduction in grav force at equator, and deflects the plumb bob at all locations but equator and poles.
3rd order approx, mountains and valleys and basements and buildings and lions and tigers and bears, oh my.
4th order approx, moon, sun, Jupiter, et al.
5th order approx, this thread if it were printed out and stacked in a pile.
#39
01-12-2000, 07:19 PM
 ZenBeam Charter Member Join Date: Oct 1999 Location: I'm right here! Posts: 6,880
Quote:
 The difference is the number of unstated assumptions in Karen's column. I believe she is using normal to an ideal oblate spheroid, flattened by the rotation of the Earth. You refer to the geoid, or equipotential spheroid, and normal to that. And I think that is the difference you keep pointing out.
Look at the message right above yours. The difference between the normal to the oblate spheroid and the gravitational gradient is less than about 2 km at the center of the Earth, not the 21 km in the article. It also follows, then, that the distance between the tops of the two poles as tall as the Empire state building is less than half a foot, not five feet. This is RM's point.

You only get 21 km and 5 feet for those two cases respectively if you compare the normal to the oblate spheroid with the direction to the center of the Earth.

To reiterate things which have been said, but not all in the same place:
plumb bob direction = direction of gravitational gradient = normal to geoid = normal to equipotential surface

Normal to oblate spheroid differs from normal to geoid by less than about 2 km at center of Earth.

All of the above miss the center of the Earth by about 21 km.

------------------
It is too clear, and so it is hard to see.
#40
01-12-2000, 10:43 PM
 RM Mentock Guest Join Date: Mar 1999
Irishman

Quote:
 The difference is the number of unstated assumptions in Karens column. I believe she is using normal to an ideal oblate spheroid, flattened by the rotation of the Earth.
I believe she is, too. That's also what I thought the original question meant also. In that case, a) the distance between those two poles is zero, not the five feet that she calculated, b) the plumb bob does not point to the center of the earth, it misses it by those tens of kilometers that she says the normal misses, and c) down is perpendicular to the tangent, contrary to the mailbag answer.

If the unstated assumptions are different, I'd like to see them. Right now, I don't see any way to make the mailbag answer make sense, under any of those five versions of yours.

Quote:
 You refer to the geoid, or equipotential spheroid, and normal to that. And I think that is the difference you keep pointing out.
No, it is not. The plumb bob direction has that difference in any of the versions, except the first one. The problem with using the first version, though, is, under those assumptions, the normal points to the center of the earth, also. Again, the tops of the two poles would be zero feet apart.

Unless I am unclear by what you mean by egg-shaped. Why is the earth egg-shaped? I've heard it referred to as "pear-shaped," but I explained that above, I think, and that's not a first approximation.

.
#41
01-14-2000, 04:40 PM
 Irishman Guest Join Date: Dec 1999
Well I guess I'm just all confused and going to have to give up on this thread.

from RM Mentock"
Quote:
 Why is the earth egg-shaped? I've heard it referred to as "pear-shaped,"
Egg-shaped, pear-shaped, whatever. The point was the shape is slightly unsymmetrical such that more volume (as well as mass) is above the equator than below. Maybe it's the moon that's egg-shaped and the earth is pear-shaped. I'm not playing so picky on the cross sectional geometry.
#42
01-14-2000, 09:25 PM
 RM Mentock Guest Join Date: Mar 1999
Quote:
 Egg-shaped, pear-shaped, whatever. The point was the shape is slightly unsymmetrical such that more volume (as well as mass) is above the equator than below. Maybe it's the moon that's egg-shaped and the earth is pear-shaped. I'm not playing so picky on the cross sectional geometry.
OK, I'll accept egg-shaped as synonymous with pear-shape--I'm don't want to be picky.

As for the point--it's not true. Egg-shape or pear-shape, there is not more mass above the equator than below. The top of the "pear" is taller, but skinnier. The net effect is no difference between the two hemispheres.

.
#43
01-15-2000, 01:36 AM
 Irishman Guest Join Date: Dec 1999
My dictionary says the equator is everywhere equidistant from the two poles. That would mean it can't be taller, because that moves the equator higher. Thus, the lump is big on one "half" and small on the other "half". ;-)

Of course the distinction between mass and volume could be relevant, because the mass contained in the liquid core if off-center, I believe.

But now I'm just making things up. So while I'm at it, I'll make up a new answer for the original question.

Down is soft and fluffy, and some people are allergic.
#44
01-16-2000, 03:26 AM
 RM Mentock Guest Join Date: Mar 1999
Here is an physical analogy that may help with understanding the concept of a geoid.

Fill a water container to the one liter mark. What if the water contains ice? The surface of the water would seem to be ambiguous, because some of the ice protrudes above the water level. However, if the ice were to melt, the water level would not change. So, it is reasonable to take that level as the actual measurement.

For the earth, 70% of the earth is covered with water. The surface of that water is known as the geoid. The continents "float" in the material of the earth's mantle much like ice cubes do--the effect is called "isostasy." Mountains are sticking up because their roots are less dense--just like ice cubes in water. If a mountain and its root were to "melt", the result would be reduced to the geoid level.

When we say, a mountain is 2000m above sea level, we mean that the top of it is 2000m above the geoid. Such measurements have increased in accuracy over the years, but there are refinements still being made.

.
#45
01-17-2000, 10:51 AM
 ZenBeam Charter Member Join Date: Oct 1999 Location: I'm right here! Posts: 6,880
It's been bothering me that we've been neglecting, or at least glossing over, the effect of the Earth's rotation. I get this effect to be 11.05 km at the Earth's center at 45 degrees latitude; fully half the difference between down and the direction to the Earth's center is due to the Earth's rotation. Presumably, the other half comes from the Earth's mass distribution.

For the effect of the Moon, I get less than a meter difference at the Earth's center, and even this amount would tend to average to zero over time, so it can be neglected. Since the Moon causes the biggest tide, the other solar system masses are even more negligible.

If you're not interested in the math, skip the rest of this message:

grot = 3.39 cm/s^2cos(theta) away from axis of rotation (rhodially? )

The angle gmass + grot pulls is atan(gmass sin(theta) / ((gmass+grot)cos(theta))
= atan(1.00347 tan(theta))
= 45.09927 degrees at theta = 45 degrees

So the difference of 0.09927 degrees is the effect on "down" due to the Earth's rotation. Converting to the difference at the Earth's center gives 6378 * tan(0.09927) = 11.05 km.

The effect of the moon's gravity will be smaller than the effect of the Earth by the factor

(Mmoon/MEarth) * (Rearth/Dmoon)^2 * (2 * REarth / Dmoon)

where Mmoon, MEarth are the moon and Earth masses, REarth is the Earth Radius, and Dmoon is the distance to the moon. The last factor
comes from the acceleration we care about being the difference between the acceleration at the Earth center and our location on the surface. This is the worst case number.

Using Mmoon/Mearth = 0.013, Dmoon/Rearth = 60, the Moon's acceleration relative to Earth is smaller by 1.2E-7, less than a meter difference at the Earth's center.

------------------
It is too clear, and so it is hard to see.
#46
01-17-2000, 11:01 AM
 Irishman Guest Join Date: Dec 1999
RM:
Quote:
 When we say, a mountain is 2000m above sea level, we mean that the top of it is 2000m above the geoid.
Of course the sea isn't level. Wind and currents stir up waves, altering the coastal height at different locations. Thus the Panama Canal having a long series of locks.
#47
01-17-2000, 11:57 AM
 Cecil Adams Perfect Master Administrator Join Date: Mar 1999 Location: Chicago, IL, USA Posts: 165
Irishman writes:
Quote:
 Of course the sea isn't level. Wind and currents stir up waves, altering the coastal height at different locations. Thus the Panama Canal having a long series of locks.
No, no, no. As I wrote in Triumph of the Straight Dope, p. 46, the difference in height between the Pacific and Atlantic oceans at the entrances to the Panama Canal is only about eight inches. The locks are needed because the land is 85 feet higher in the middle.
#48
01-18-2000, 09:32 AM
 RM Mentock Guest Join Date: Mar 1999
ZenBeam

That is a very good point.

We haven't necessarily been neglecting the effect of the rotation of the Earth--that it was important was the gist of my original post.

I did have a little trouble following your calculations, but it makes sense to me, if the plus sign in (gmass+grot) is changed to a minus.

We have relied upon Karen's original calculation, but it seems accurate enough. If you were to take the equation for an ellipse, x^2/a^2 + z^2/c^2 = 1, where a is the Earth's equatorial radius, and c is the polar radius, and find the point at which the slope of the normal is 45 degrees (latitude 45 degrees), you'd find that the normal at that point misses the center of that ellipse by about 21 kilometers. That appears to be what Karen calculated.

Cecil
Welcome to the party! Good to see you up and about after the Y2K thing.

.
#49
01-18-2000, 09:42 AM
 ZenBeam Charter Member Join Date: Oct 1999 Location: I'm right here! Posts: 6,880
Quote:
 it makes sense to me, if the plus sign in (gmass+grot) is changed to a minus.

------------------
It is too clear, and so it is hard to see.
#50
01-18-2000, 10:45 AM
 Irishman Guest Join Date: Dec 1999
Quote:
 Irishman writes: quote: Of course the sea isn't level. Wind and currents stir up waves, altering the coastal height at different locations. Thus the Panama Canal having a long series of locks. No, no, no. As I wrote in Triumph of the Straight Dope, p. 46, the difference in height between the Pacific and Atlantic oceans at the entrances to the Panama Canal is only about eight inches. The locks are needed because the land is 85 feet higher in the middle.
[sheepishly]Oops. Guess I missed that one.[/sheepishly]

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