# down and up

I must point out a slight error in the mailbag answer by Karen to “How are ‘down’ and ‘up’ defined?” ( http://www.straightdope.com/mailbag/mdownup.html ).

The answer is correct, except that it ignores the effects of the earth’s rotation. Keeping the math very simple, the apparent local acceleration due to gravity (measured by a plumb bob) will vary by -3.39 cos h, measured in cm/s^2 and h is the latitude (equator = 0, poles = 90). This will cause a fractional change of up to -0.003 from what the acceleration would be if the earth did not rotate. Note that this effect (called the centrifugal force) always reduces the apparent local acceleration.

Furthermore, the earth deviates from a sphere to counteract this effect. So, typically, the local acceleration due to gravity is the normal to the “surface” of the earth.

Reference: Chapter 2: Accelerated Coordinate Systems from Theoretical Mechanics of Particles and Continua, Fetter & Walecka, 1980 McGraw Hill

[Note: This message has been edited by CKDextHavn]

## http://www.straightdope.com/mailbag/mdownup.html thanks for your spin on things pleon, I don’t do math, could you do some cypherin’ and tell me how long a plumb bob would have to be to get a deviation of, ohh, half an inch? (1 lb bob will do). Of course this only applies on Earth in space there is no UP so you can’t stare at the ceiling and there are no down sleeping bags.this makes it hard to fall asleep, at least you won’t have any nightmares. That’s why they say “In space no one can fear their dreams” Had to go alooong way for that one, almost to the center of the earth.

Signitorily yours, Mr John
" Pardon me while I have a strange interlude." -Marx

Keep in mind that “down” is defined as “the direction in which something falls when you drop it.” This is not necessarily a direct line toward the center of the earth, but it is always the direction that is pointed out by a plumb bob.

To put it another way, the direction we call “down” is the direction from which gravitational force is perceived to emanate. It doesn’t matter whether that perceived force is due to gravitation, or acceleration, or some combination thereof; a plumb line will always point down.

For instance, if a passenger in an accelerating airplane holds up a pendulum, it will not point toward the center of the earth; it will point at an angle, leaning toward the back of the aircraft. This is “down” as far as the rider is concerned, because any object he drops will fall down at the same angle.

Aura, I must disagree, out in west Texas if the line is long enough and the mass too little the wind can push the bob out of plumb several inches. That is why I specified such a heavy bob. Why sometimes it’s so windy the cattle have to face into the wind to make a pie. Once it was so windy it was Wednesday 3 days in a row, then the wind died down and ZIP! the rest of week shot by just like that. But I digress, sometimes I get long winded and plumb silly.

John you need a good beating after that one.

A few problems. First with the original answer. Heres a properly linked url http://www.straightdope.com/mailbag/mdownup.html The content was fine, but I am curious why the most obvious answer was neglected. Now a couple of definitions you use are reasonable, normal=down, and direction of gravitaional force=down, but the definition that is most practical in scientific use is that down and up are simply arbitrary vectors. These vectors of +z & -z can be set at any direction that make logical sense (or not for that matter). On Earth the most common answer is relative to the force of gravitaion, unless you digging a hole “down” then down is perpendicular to the level of the soil at that point, or climbing a mountain then “down” is the direction parallel to the slope of the incline. Granted these last two definitions are imprecise and subjective, they still are as applicable as the plum bob explanation. So I submit that the most correct answer to this question is the arbitrary z axis. One arguement to why I say this is so is that if two astronauts are cunducting an experiment in space I suspect that they would use the term down to describe the direction towards the floor of the module, or possibly towards their feet if they are floating.

Pleonast, its the lack of centripetal (not centrifugal) force, centrifugal force doesn’t exist, its merely apparent. And acceleration due to gravity isn’t important here, the direction of it is. g is a scalar constant. The plum bob measures the direction of this force, and the bob will be slightly affected by the lack of centripetal force (angular acceleration) in a direction parallel to the plane of the equator, but on the equator the effect is non-existant. At the equator this force acts in a direction opposite the acceleration due to gravity, and reduces the apparent gravitational force, and depending on what lataitude the measurement is taken at it alters the dirction of the gravitational force by vector addition, but it has no change on the acceleration due to gravity, just its apparent results.

Finally this result is not measured as simply as you imply. You overlook the fact that the rotation of the Earth is about its center of mass, one of the operative definitions centers around the center of the geometrical shape of the Earth. These points are not concurrent. Also since the radius of rotation will decrease as you move away from the equator the effects of the angular acceleration get increasingly low. The flattening of the poles magnifies this reduction effect as well.

Auraseer is correct that the plumb bob acts in the same direction as down by the most practical definition on Earth, and that he angular acceleration is already included in the plumb bobs direction.

Just out of curiosity could some one give an example of when such a precise location of down would be needed? I don’t mean this facetiously ,are there such occaisions?
I think aura hit it for all practical purposes with 'the direction something falls" lets add ‘without interference from out side influences’, that takes care of your mountain slope and the wind. The wind will move a plumb bob, sometimes the plumb is hung inside a pipe (well casing works great), clamped to whatever you are plumbing, with enough line top and bottom to take a measurement.
Omni, let me nit pick before you beat me. ‘down is perpendicular to the level of the soil’ and ‘down parallel to the slope’ if i dig a hole in that slope I have two downs perpendicular to each other. If i am digging a hole to hide in on that mountain I don’t care where down is, if I am falling I KNOW I am falling ‘down’ But if I am drilling a water well on a slope I am going to use a plumb bop .I want to hit the aquifer as soon as I can and not go drilling at an angle and hit it at the same level but 2 miles away horizontally. (this well is not in the mountains but on the plains, which aren’t as flat as most folks think and there is an aquifier under most of it). now those two astronauts. If they were doing an experiment where down was important, wouldn’t they put the module in motion and down would be opposite to the vector of motion? Or spin the module? After reading the next paragraph you posted of course. Well ,well, well I already used ‘gravity of the situation’ somewhere so I guess I am stuck, I feel so let down.

Have you been drinking?

nope, the well went dry, the sucker rod is pumpin’ dust. I hired a couple of astronauts to drill a new one on the hill, but they don’t know "down’ from a hole in the ground, and the bit is chewing gravel two counties away and ain’t hit the aquifer yet. The fools still want to be paid, we’ll probably wind up in arbitration.

## mr john Member posted 08-14-1999 01:07 AM

“Just out of curiosity could some one give an example of when such a precise location of down would be needed? I don’t mean this facetiously ,are there such occaisions?”

``````It seems to me that, generally speaking, it's a good idea for the center of gravity of a structure to be a close to the center of its foundation as possible (although this is not a professional opinion). More precisely, the c.g. should be as close to a vertical (i.e. pointing up) line going through the center of the foundation as possible. The more accurately one knows which way is up, the closer to this line one can put the c.g.
``````

For a more specific example of where it is important to know which way is down, look at the Leaning Tower of Pisa.

-Ryan
" ‘Ideas on Earth were badges of friendship or enmity. Their content did not matter.’ " -Kurt Vonnegut, * Breakfast of Champions *

well, Mr ryan, I used to build things a lot bigger than furniture, I wanted the stresses of a structure as evenly distributed on the foundation as possible. The ‘center of stress’ may coincide with the center of the foundation or it may not. (lets not get into cantilevering, pier and beam, piles, hardpads etc; except a bit of advice, folks, don’t ever buy a house built on a slab foundation) The problem they had with the Tower wasn’t that it was ‘off balance’ and began to tilt, as that it was built partly over a hard substrate and partly over soft. The soft side settled, they began adding courses to the low side, sort of wedge shape courses, and structural braces to the opposite side hoping the far side would support the low. That added more weight so it settled more and now the low side pulled the high with it, now we got the tilt. The engineers attempted to just ‘straighten out’ the tower by adding even more courses to the low side, and hoping the settling would stop. This only added more weight and the building began to tilt more and added a bend to the building. I beleive the entire building proccess took a few hundred years. And the tower is still settling, I think they have it wrapped in steel cables or bands attached to scafolding and hold fasts. (sort of like ropes and tent pegs) last i heard some engineer has come up with a scheme to get rid of that. here we go again.
I think the ‘deviation’ that Pleno, aura, and omni are talking about is something less than a centimeter. the ‘center of gravity’ in a modest high rise varys more than that several times a day.

Many years ago a friend of mine was taking a Physics course. As part of an experiment he had to define “up”. He was having a problem with coming up with an answer the prof would accept. He asked for some help and I shot off what came to mind first. “Up is the way things don’t fall” When he got done laughing, he decided what the hell, and he used that in his Physics class. The prof gave him an A on this experiment. I understand that this is not a perfect answer (after all things don’t fall to my left and my left is not up) It is fairly correct, even if it not terribly rigorous.
Rick

## Omniscient, thanks for your reply (and correcting my mistake in linking the url). I think we’re basically in agreement, but using terms differently. “Centripetal force” is any force which points toward the axis of rotation. “Centrifugal forces” do not exist in an inertial frame of reference. However in a non-inertial frame of reference (which the surface of the earth is, because of the earth’s rotation), it is often extremely convenient to separate the actual inertial forces into several non-inertial components. Centrifugal force is one of these well-defined components, as is the Coriolis force (which is also fictional in the same sense as the centrifugal force). I left out most of the math, and all of the vector math, to make my explanation more accessible to the average reader. Also I used “acceleration due to gravity” in the typical meaning of “what is measured by a plumb-bob”; obviously, this is different from the acceleration due to Newton’s Law of Gravity. So, Omniscient is correct in what he says, though I think my initial explanation is in general more understandable to the population at large. P.S.: my little formula gives the magnitude of the vector displacement of the local acceleration due to the centrifugal force. The direction is perpendicular to and away from the axis of rotation. mr john, to answer your question, at a latitude of say 35 degrees north, a motionless plumb bob 25’7" will deviate 1/2" south from a radial vector emanating from the center of mass of the earth.

The Pleonast he is I.