2012-04-17

The brachistochrone, the catenoid, and a skateboard ramp

Bengt has started a new construction project in his garage. He's been working for weeks, when Alban comes by to see the finished product.
"It's a... skateboarding ramp. Bengt, have you taken up skateboarding?"
"Ha! Hell no. Do I look like a skateboarder?"
"So why do you have a ramp?"
"For experiments, obviously. That's the only reason any sane person would build a skateboard ramp in their garage."
Alban ignores the issue of sanity, and Bengt continues.
"It's not any ordinary ramp, you see. It's a brachistochrone."
"Which means...?"
"It means that if you let something slide down one side - a ball, perhaps, or why not a skateboard, as long as it's infinitesimally small - it will always reach the bottom in the same time, regardless of how high up the ramp it starts."
"How do you know?"
"I can measure the time. I have a clock chain, look."
Bengt holds up a long thin chain. He holds it with both ends at the same height, so the chain forms a U-shaped curve. This curve is called the catenoid. There is no clock on the chain.
"There is no clock on the chain." says Alban.
"No, no. The chain is the clock. Look, you hold it up by one end, and dangle it. It swings with a certain frequency, so it can be used to measure the time. You can use a shorter chain if you want a higher frequency. Since each link has the same mass, you can use it to measure mass as well! It's a multi-purpose chain indeed. Also, it has 60 links, one for each second of a minute."
"That makes no sense."
"That there are sixty seconds to a minute? No, it doesn't. But that's not my fault, you'll have to blame the Babylonians."
"That's not what I meant."
"I know that. Anyway, there should be 16 hours to a day, 16 minutes to an hour, and 16 seconds to a minute."
Alban sighs. "That would make pretty long seconds, tho, wouldn't it?"
"Yes. So we need to add another unit. I think I'll call it a ters. There are 16 terses to a second."
"So if I make a terse statement, will I be able to finish it within one ters?"
"You? I doubt it."

  1. What curve is it that Bengt has built?
  2. When he holds up his chain, what curve does that form?
  3. The word "brachistochrone" means "shortest time". So why is the brachiosaurus not a short lizard?
  4. To measure mass with the chain, Bengt wants to take it apart, in such that he can combine any number of links from 1 up to 60. How many links does he need to open to achieve that?
  5. Now suppose he wants to be able to form whole chains from the parts, at any length. So the difference is that all combinations have to have enough open links that he can put them back together without opening any more.
  6. If a link has mass m and length l, and you hold it so that links are doing a pendulum motion, what is the frequency? You can assume that the chain is homogeneous.
  7. How long is a ters, according to Bengt's definition?