A guy finds himself in a room which is actually a long hallway, but he doesn't know the extent. He has with him a calculator, paper and pen, ruler( one foot and metric marks also) and a guitar string, an E string which is 0.01" in diameter (the smallest string on a guitar) and hears a voice from an unknown source saying, you are free to go if you figure out where you are, or what kind of a place and how big is it you are in.
What he is in is a huge gravity wheel in space, where the 'floor' is where the ceiling would be if we were on earth but centrifugal/centripetal forces are putting him on the inside skin of a rotating wheel one kilometer in radius so the circumference is about 6 Km. Now he can walk all the way round the rim like that but the walls are so black he cannot see down the hall to notice he is in a curved environment. So what can he do to figure out his situation? He can walk all the way round the thing in a few hours but he doesn't know he has come back to the starting point, he doesn't want to lose his stuff, so he has to do some thinking. What would that be?
Originally posted by AThousandYoungThat's true, but his keeper want's something more quantitative, like the size of the wheel if and when he figures out he is in a wheel. Can you think of a way he could figure out he is in a wheel, without using the ruler and wire? Can you think of a way he can figure out he is in a wheel USING the ruler, wire, calculator, pen and paper?
Shouting and listening to the echo might help. If it's a straight endless hallway there should be no echo, etc.
One thing, if he does yell, and it is 6 odd Km in circumference, could he maybe hear his own voice coming out at him from behind? At about 300 meters per second, the sound could be heard around 20 seconds later if it didn't attenuate to undetectability which maybe it wouldn't, being in a kind of whispering gallery. Interesting point. We will add another thing to his kit of stuff, a nice Casio Chronograph like the G shock series. What can he do with that besides timing his round trip voice if he could hear it?
Also, I might add, he feels he is in a one G environment and has no idea he is in a wheel, so how many RPM would the 1 Km radius wheel rotate at to give him a 1 G envornment?
Originally posted by sonhouselay the ruler on the ground. The ground is curved so there should be a measurable gap in the centre of the ruler. All he has to do is measure the gap (using the piano wire) and then calculate the circumference\diameter of the circle
I changed the ruler length, one foot won't do, it's really a two meter ruler.
Originally posted by uzlessDoes he really know the shape of the circumference? What if it is an ellips? What will be the difference?
lay the ruler on the ground. The ground is curved so there should be a measurable gap in the centre of the ruler. All he has to do is measure the gap (using the piano wire) and then calculate the circumference\diameter of the circle
Originally posted by uzlessSo you do some measures to find out that it's not a perfect circle, you do some more to find out that it isn't even a perfect ellips, then you cannot know anything about the circumference.
of course you'd have to take a few additional measurements. the consistency would become immediately clear.
Originally posted by uzlessAre you sure the ground is curved enough? I thought of this too and dismissed it as impractical due to the gentleness of the curve.
lay the ruler on the ground. The ground is curved so there should be a measurable gap in the centre of the ruler. All he has to do is measure the gap (using the piano wire) and then calculate the circumference\diameter of the circle
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The wheel is 1000m in radius, the ruler is 2m in length.
So, we can form a right angled triangle, W, c, e where W is the centre of the wheel, c is the centre of the ruler (lying on the ground across the corridor) and e is the end of the ruler.
We know
1000^2 = 1^2 + {Wc}^2
so {Wc} = sqrt(1000^2 - 1)
And the gap under the ruler is:
gap = 1000 - sqrt(1000^2 - 1)
That is about 0.0005m
i.e 0.05cm
i.e 0.5mm
The guitar string is diameter 0.01 inches, which is 0.254 mm So our guitar string should fit under it, but wouldn't if the ruler was half as long, so that's why it has to be 2m. It has to be a very stiff ruler too.
Originally posted by iamatigerYep, that's right, I thought up the problem before I did the calc's and found you got a half mm only with a 2 meter ruler, with a good straight edge. There is one other thing you are forgetting about.
The wheel is 1000m in radius, the ruler is 2m in length.
So, we can form a right angled triangle, W, c, e where W is the centre of the wheel, c is the centre of the ruler (lying on the ground across the corridor) and e is the end of the ruler.
We know
1000^2 = 1^2 + {Wc}^2
so {Wc} = sqrt(1000^2 - 1)
And the gap under the ruler is:
gap = 1000 - s ...[text shortened]... he ruler was half as long, so that's why it has to be 2m. It has to be a very stiff ruler too.
Originally posted by sonhouseyou could measure the ceiling too....
Yep, that's right, I thought up the problem before I did the calc's and found you got a half mm only with a 2 meter ruler, with a good straight edge. There is one other thing you are forgetting about.
i thought i'd leave that one to someone else. But since no one mentioned it, you'd hold your ruler at one end and measure the gap from the other end of the ruler to the ceiling.
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Originally posted by sonhouseI should have said he *lays* the ruler along the corridor. Essentially he wants to get it as flat as it can go, and if it is still not touching the floor in the middle he knows the corridor is curving up at all points around him (He doesn't know that it will carry on doing that and come back on itself though)
Yep, that's right, I thought up the problem before I did the calc's and found you got a half mm only with a 2 meter ruler, with a good straight edge. There is one other thing you are forgetting about.