2008-03-20

Two Very Long Steel Wires

In his garage, Bengt has two long steel wires, wound up into balls. They are very, very long indeed.
One of them is so long it could go all the way around the world. The other one is 1 m longer. They are 1 mm thick.
(Assume that the Earth is spherical, and that there is no air in the balls.)

  1. Bengt imagines laying out the two threads along the equator. One of them is touching the ground, the other one is a small distance away from it (same distance everywhere).
    How far from the ground is the second wire?

  2. Bengt imagines pulling up on the long wire, so it is taut against the ground on the other side. How far from the ground can he pull the thread?

  3. What fraction of its length would be touching the ground?

  4. Bengt imagines laying out the threads (like in question a) and then moving the shorter thread to the north, until it is the same distance from the ground as the other one. How far does he move it?

  5. What is the diameter of the balls?

  6. The balls are lying next to each other. What gravity force do they exert on each other?

  7. The floor of the garage has zero friction. Unfortunately it is not entirely horizontal, so Bengt has had to tie the balls to the wall. If he lets one of them loose, and it is held in place by the gravity from the other, what is the maximum possible angle of the floor?

2008-03-10

Pacman's Adventures in Curved Space

Bengt and his friend Alban are playing Pacman on Bengt's new (and very expensive) calculator. The screen is 1x1 dm. Alban is, unlike Bengt, younger than Pacman, and he has played a lot of computer games. Neither of them knows much about the theory of relativity, but Bengt has a habit of discussing things that he doesn't understand.
"This is a bit like the theory of relativity." says Bengt.
"Really? How?" says Alban.
"Well, when Pacman leaves through the hole in the wall in the middle of the top edge of the screen, he immediately reappears at the corresponding hole at the bottom, and the other way around."
"You mean that he has moved faster than light?"
"No, that's not what I mean. Don't try to tell me what I mean. No one but me knows what I mean."
"True... so, what do you mean?"
"That he comes back, it's like if space is curved. As if he lived on a cylinder. The Y coordinate repeats itself; it's a cyclic coordinate."
"Uh... that's not what 'cyclic coordinate' means. It means a coordinate which the lagrangian is independent of, or something like that."
"I don't care. If a coordinate repeats itself cyclically, I will call it a cyclic coordinate. And apart from cyclic coordinates, there are also limited coordinates, which end, and adifatic coordinates, which neither end nor repeat themselves. On a cylinder, one coordinate is cyclic, and the other is limited."
"Okay, if you say so..."
"If there were holes on the left and right edges as well, so he could go around in the X coordinate as well, Pacman might get the impression that the world was round. Spherical. Or that he was living on on a doughnut. One might also imagine parallel universes, in the shape of doughnuts linked as a chain. I call the theory "the catananese doughnut theory".
"Blast! Now I died again. Could you please stop talking geometry? I'm trying to concentrate."
"No I won't. It is also possible to let him leave and reappear not only in the middle, but anywhere on the top or bottom edge, and we can imagine that the screen is infinite in the X dimension, so that that coordinate is adifatic. Then it would be like an infinitely long cylinder, and if Pacman seems to move on a straight line on the screen, he is in fact moving in a spiral on the cylinder surface, and depending on his direction the time it takes to get back to the same Y position varies. It's just like in string theory."
"Sure, sure, a spiral, on a doughnut..."
"No, that wouldn't work. One can also imagine that he can leave anywhere on any edge, but it will become neither a sphere nor a doughnut."
"Cut it out, I'm getting a headache."
"Okay. Beer?"
"Yes please."

  1. If Pacman lives on a finite cylinder, what is the surface area and volume?

  2. What if it is a sphere? What is the surface area and volume then?

  3. Since the screen is a square, is it really possible that he lives on a doughnut? How big is the hole?

  4. If Pacman is spiralling along an infinite cylinder, what is the time until he returns to the same Y coordinate, as a function of the angle between his movement and the X axis?

  5. If he can leave anywhere on any edge, under what conditions will he ever come back to the same point (without turning)? How far will he then have moved?

  6. If he can leave anywhere on any edge, why is it impossible that he is on a sphere or a doughnut?

  7. In that case, if he is moving on the surface of some shape, what kind of shape is it? How many dimensions does it have?

  8. What is it's volume? (That is, the equivalent of volume in however many dimensions it has.)

2008-03-02

Rapunzel

Bengt is a big fan of fairy tales. Unfortunately, he is the kind of person who always ruins them by pointing out things that are not logical.
One example is the story of Rapunzel. She was a girl whose hair was very long and beautiful, and grew very fast. Every time she cut her hair, it grew twice as fast.
She was imprisoned by a witch, who put her in a tower. The tower was as tall as twelve men. It had no stairs and no door, so the only way to get to Rapunzel was to call for her to let down her hair through the window, and then use the hair to climb up. Eventually a prince comes along, and uses this method to enter the tower.
There are many things about this that makes Bengt think.
"Is it possible to climb up someone's hair like that? It would be pretty uncomfortable for the girl... but I guess she could just wrap it a couple of times around something at the end near her head, before letting it out."
He assumes that all men, like himself, weigh about 100 kg, and are about 180 cm tall. He also assumes that Rapunzel is only 120 cm, and weighs 40 kg. This is clearly a rather short girl, but Bengt happens to like short girls.
Bengt pulls out one of his own hairs, to test its strength. It turns out it can carry the weight of ten grams, and can be extended by about 5% of its normal length.
Next, he wants to see how thick his hair is. He gets out a red laser pointer, and points it at the single hair. On the wall two meters away, he sees the diffraction pattern. The first black band is 15 mm from the center.
Bengt decides Rapunzel was probably blonde. His own hair is some sort of brownish colour. He happens to know that blonde people have about 50% more hairs, and they are about half as thick. On the other hand, they both have rather straight hair, which means that the cross section of the hair is circular.
Bengt realises he needs to do one more test. He immerses his hair in water, and somehow manages to determine that the volume of the displaced water is 0.8 ml. He also decides that "displace" is a funny word, and makes a mental note to start using it instead of "misplace", if only to annoy people.
He dries his hair, happy that it is only a decimeter long, so no one will be climbing in it any time soon.

  1. If Bengt was to cut his hair only when it reached the floor, and then cut off all of it, he would have to cut his hair once every 15 years. If Rapunzel, with her ever-increasing hair growth speed, did the same, what would eventually happen?

  2. How long could she keep doing that? (Assume that she starts off with the same growth speed as Bengt.)

  3. Suppose instead that her hair grows at normal speed. How long would it take for the hair to reach the ground outside the tower?

  4. Sometimes the prince might need to make a hasty exit. He can then grab a hold of the hair, and jump out through the window. Assuming that he can hold on to the hair, for how tall towers would this be possible?

  5. Without any jumping, how many princes could hang on Rapunzel's hair at once?

  6. For best result, should they be spaced equally, or all hanging at the end?