Monday, September 9, 2013

Which way is up?

Simple questions sometimes have subtle answers.  Of course, some answers are also pretty simple.  Which way is up starts out simple and then gets pretty subtle. (Note on scientist-speak: subtle = complicated and/or difficult).  This winds up being related to What is a day? as we get a little more complex.  But, while we can, let's go with simple.  Up is the opposite of down.  Slightly less simple, down is the direction a ball falls.

Even less simple: hang a weight on a string.  Hold it still.  This is difficult, so maybe hang it from a nail or off a board.  There's probably still a little swinging back and forth.  So either wait (it'll come to a halt eventually, but who says scientists are always patient?!) or get a large (larger than your weight) cup or bucket of water and bring that up underneath the weight.  Make sure the weight is made of something that doesn't float if you use this approach!  Once the weight comes to a halt, the string gives you a line which points up and down.  The weight is the 'down' side of the line.

By the way -- not only do you not have to be good at math to be good at science, you also don't have to be good at drawing. For me, this is pretty good artwork. Some people are great at drawing, same as some are great at math. Some of us, well, you see my caliber of artwork.

 Now for getting subtle ... which also explains why the earth isn't exactly a sphere.
In keeping our method simple, we assumed that the only thing acting on the weight was gravity.  Since we think of gravity as being the thing which provides weight, that seems reasonable.  And it isn't too bad.  It's just wrong in detail.  What we really need to do is consider all the forces which act on the blob (let's call it something else so that we don't confuse gravity/weight with up/down).  Gravity from the earth is the main force, which is why the earth is nearly a sphere.

But the earth is also rotating.  That means in addition to the force towards the center of the earth, there's a force pushing the blob a little to the side if you're not at the equator or poles.  For another wonderful example of my artwork, not to scale and not showing the stuff holding up the blob:

This is all tremendously not to scale.  If it were, you wouldn't be able to see the blob or the difference between gravity only and the resultant. 

But there's another aspect to this.  The deflection of true down from what it would be if the earth didn't rotate means the earth can't be a perfect sphere.  In particular, it has to be a little squashed at the poles -- smaller distance from the center of earth to the poles than from center of earth to equator.  So the earth's polar radius is only 6351 km, while the equatorial radius is 6378 km (about, in both cases).  Try drawing this accurately to scale. If you succeed, please send me a copy.  I don't think we'll be able to tell the difference from a perfect sphere by eye.

The fact that the earth is squashed along the rotation axis, an oblate spheroid, was a controversial matter for a many years in the early 1700s.  Newton argued for this.  But Cassini, who was an excellent scientist as well, and more of an observationalist (he discovered Cassini's Division in the rings around Saturn) argued that the earth was narrower at the equator, a prolate spheroid.  It wasn't resolved until an observational expedition in 1738 by Maupertuis and Clairaut.  The hard core math types should take a look at Ellipsoidal Figures of Equilibrium by S. Chandrasekhar.


coby said...

I was considering quibbling about your definition of down...

It is appealing to suggest that any centrifugal effects should be ignored and therefore accept that on most of the earth's surface a ball would not actually fall straight down. But then I started thinking about planetary shapes and geometric centers and centers of gravity and can't really decide what definition works best in all situations.

If you lived on a perfect cube it would be tempting to define down as a vector perpendicular to the surface regardless of gravitational attractions. What if you lived on a sphere with a non-homogenous mass distribution causing the center of gravity to be displaced from the geometric center? Two pretty good options come to mind, the better depending on your purpose. An extremely elliptical planet is interesting to think about too.

I suppose "the direction a free weight falls in a vacuum" satisfies the engineers, but what if you have the misfortune of being a mathematician, then what...?

Robert Grumbine said...

Mathematicians have their uses too. Sometimes they lead us to better definitions, which is good. For the case at hand, the professionals use 'surface' and 'down' a little differently than my operational description.

'Down' is the inward normal to the geoid. The geoid is the equipotential surface of the earth. This does include rotation. It also is not exactly an oblate spheroid due to some of those mass-variations you were thinking of. The difference, though, is pretty small (at most, 100 meters or so, vs. the 27 km differences from a perfect sphere).

You can dress this up further, by going to general relativity. Up/Down is then the geodesic line (a nice mathematical term!) through the body. (This, too, includes rotation, but in a very different sense.)