CKDextHavn
Aww. And here I thought down was towards the center of our galaxy. Or is it towards the center of our local group?
<< 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!)
Dex
I’m sorry that you think so. I was just trying to help.
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.
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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.
CKDextHavn wrote:
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.
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.
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?
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?
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?
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?
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.
The plumb line is normal to an equipotential surface.
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.
ZenBeam 01-10-2000 04:50 PM
Well, the words are there (in my first post), but the context is lost, and words were elided. The meaning is totally gone.
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.
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.
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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.
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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.
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.
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?
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Great. First we get bobbing for apples, now we get bobbing for plumbs.
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:
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.
ZenBeam
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.
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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.
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.
Irishman
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.
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.
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