|Jan/Feb 2000 Miscellany|
If you travel frequently enough you may know the phenomenon: you awaken, glance about the hotel, and wonder -- Where am I? The disorientation is usually transitory, but suppose you could extend that confused state over time, even play with it? How would you determine where you are, or at least pin down the city?
You could, of course, run to the window and look for clues: the style of architecture, messages on billboards, the signage on public transit. You could even pick up the phone and call the concierge. But just to make the problem more difficult, let's seal the window, rip out the phone, and while we're at it, remove the radio and television. Assuming you could replace those things with the appropriate measuring devices, would it be possible to deduce your location, using only your wits and the laws of physics?
There is one determination you could make rather quickly, with only a Foucault pendulum (essentially a swinging weight suspended by a wire). Due to the Earth's rotation, the pendulum's plane of swing would appear to turn clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere. Thus, in only a few minutes, you could narrow the possibilities to half the planet.
Greater precision would require additional equipment. Recommended is: a calendar watch, an astronomical almanac, a scale accurate to .000005 pounds1, and a known weight.
Like every other object on Earth, the known weight -- let's make it a steel sphere -- would weigh slightly more at the North or South Pole than at the equator.
Thus, if you know it weighs 100 pounds at the poles, you can calculate how much it should weigh at the Equator (99.5 lbs.) or at various points between. A reading midway in the possible range would suggest that your room is midway between pole and equator. 2 That would be the 45th Parallel, the approximate latitude of Paris or Milan or St. Paul.
Now all you need is a longitude.
This will take a little longer, but your present equipment should suffice. Over time, you will note that the weight of your sphere undergoes infinitesimal fluctuations as the sun and moon pass overhead in their daily cycles. As they do so, their gravity affects everything on Earth, including your sphere, which will register its lightest readings when the sun or moon is at zenith. (The difference for the moon alone is about .00002 pounds for every 100 pounds. For the sun the difference is slightly less than half that amount.)
Since the moon and sun can appear at their respective zeniths from only one longitude line at a time, you can now use the calendar watch and almanac to determine the longitude of your room. You will then have the only two coordinates you need to pinpoint any location on the planet. Naturally, the Earth's asymmetries would deny you enough certainty to place yourself at, say, 42nd and Broadway but, with reasonable controls for error (see footnotes), you should at least be able to tell if you are in New York State.
But there's a fly in the ointment.
Perhaps you have caught the fact that all the foregoing deductions are based on the assumption that your windowless room is on Earth. How do you know that you are not in a vibration-free rocket ship, accelerating at 32 feet per second2 (the speed at which objects fall on Earth)? Wouldn't that provide a perfect simulation of gravity? Is there any experiment you could perform from inside the room which would confirm or eliminate such a possibility?
Happily there is.
Once again, you'll need special equipment: two marbles will do, along with some precise marble-tracking lasers. The purpose of the lasers is to determine if the marbles "fall" in exactly parallel lines when you drop them, or if there is a slight convergence. If the lines are exactly parallel you are experiencing artificial gravity, such as might be produced by an accelerating rocket. If they converge, you can assume you are on Earth (or another planetary body). This is because objects on planets do not really fall toward the surface, but toward the center of gravity, a point roughly in the geometric center. From the surface of Earth, two marbles, dropped five feet apart, would fall out of parallel by .0000136 degrees.
Thought experiments like the ones above may be unlikely to produce practical breakthroughs, but at least they require no funding. So, as promised, I have proposed the following five gravity puzzles, all of which begin in the same windowless room. All have at least one solution (and possibly others the author has not considered). The rules are these: you are guaranteed enough information to come to at least one solution. You may import any apparatus you require as long as it does not violate the laws of physics. But all you will truly need are: some marbles, marble-tracking lasers, and a scale. (Answers are at the end.)
You have won a free stay at the Hilton Space Station. It has been explained to you that space travel is so uncomfortable you will have to be transported under general anesthesia. You awaken in your windowless room, thrilled to think you are in outer space and amazed by the feel of artificial gravity.
Then a paranoid thought occurs to you: how do you know this isn't a hoax? For all you know this could be a sealed room in Provo, Utah. The brochure showed you a great rotating wheel, hurtling in orbit. Is there any experiment you could perform from within the windowless room to determine whether or not you are inside this wheel-like structure?
This time you awaken to find yourself weightless. This proves you are in space, right?
After enjoying a few pinwheels and loop-de-loops, a disturbing notion comes to mind: Suppose you and the entire room are plummeting toward the surface of a planet. Wouldn't that also leave you weightless? Before you are smashed to bits, is there any experiment you could perform from inside the room to confirm or dispel this possibility?
You are in the same room, still weightless, but your previous experiment has left you satisfied that you are not plummeting toward a planetary surface. The on-board computer assures you that you are in Earth orbit. But later, when you catch the computer cheating at chess, you realize that you cannot trust anything it has told you. Is there any experiment you could perform that would confirm you are in planetary orbit, as opposed to drifting aimlessly among the galaxies?
NOTE: In the two following problems I have already performed the experiments and given the results. You only have to explain what they mean.
You are in the same room. Awakening on your back, you are surprised to discover that your chair and writing desk are stuck to the ceiling. Or could it be that you are stuck to the ceiling and the furniture is where it is supposed to be?
You stand up feeling a little light-headed, but since you are moored to the "floor," you assume that gravity is behaving normally. You drop a marble and it falls to your feet. Then you toss some marbles upward, and depending on the strength of your toss, they either veer off and stick to one of the walls, or in some cases, stick to the "ceiling" (which is still supporting the furniture). Suddenly, you realize what is going on. To confirm your suspicions, you determine the exact center of the room, and place a marble there. As you suspected, it stays put. What have you proven?
Same room, but now everything is floating. You try releasing a few marbles, which initially appear to float weightlessly beside you. Then you notice that the marbles (along with all other objects in the room) are slowly moving toward the room's center. You repeat the previous experiment, placing one of the marbles in the exact center of the room, and once again it stays there. Move it away, and it slowly moves back to the center. In time, it passes through the center, slows down, then continues to oscillate back and forth. All of which suggests something truly disturbing. What is it?
* a). Drop a marble. If you are truly on a space station it will not fall in a perfectly vertical line, but will veer opposite the direction of the wheel's rotation.
b). Weigh something from various heights in the room. On a space station the weight will increase as you approach the floor, where the centrifugal force is greatest.
* Drop two marbles. Since you are weightless they will not "fall." But if you are plummeting through a gravitational field the marbles will slowly converge as they approach the common center of gravity. If you release the marbles at 100,000 miles from the center of gravity, with the marbles five feet apart, their angle out of parallel would be only .000000543 degrees. That may seem negligible, but by the time you fell 50,000 miles, you could discern, even with your naked eye, that the marbles had moved considerably closer (2.5 feet) -- in which case, you had better brace yourself.
* Place a marble in each corner of the room and watch for changes in their relative positions. If you are in planetary orbit, the marbles will also be in orbit, just as if the rest of the room were not there. The marbles nearest the planet would therefore be orbiting at a slightly greater speed than those further away.
* The room is rotating. Only the center is stationary, and that is why the marble stays where it is.
* You are in the center of the Earth (or some other massive body). Things appear weightless because gravity is pulling outward, almost equally, from all directions. Objects drift toward the center of the room, because that is the center of gravity.
* For the purpose of this article we are talking about an idealized scale, not subject to local turbulences which could overwhelm extremely small measurements. A physicist would prefer that we use a gravimeter, a device which measures gravity by acceleration rather than weight.
* In reality, of course, things are a little more complicated as factors such as altitude above sea level and local variations in the Earth's density can make the meaning of your measurements uncertain. If you use an altimeter to correct for altitude, and are not situated in a geologically unusual area, such as on top of an oil field, you should be able to determine latitude within five degrees.