Guess: The main reason why Alison’s calculation was off was because
The correction used for rotational KE assumed the ball was rolling without slipping for the entire decent, whereas in reality, the top of the ramp is steep enough that the ball mostly slides in the initial decent and thus gains less rotational KE than it would otherwise (resulting in more translational KE and a faster exit speed).
This looks a lot like a typical high school/college freshman physics problem, and I guess the moral of the story is that it leads us to think that we should solve it that way. But if you were to work it out,
I think the ball’s rotational energy would be a much smaller number than the gravitational potential energy of falling a few feet. The rotational energy of a solid sphere is 15mr2ω2, where m and r are the mass and radius of the ball and ω is the angular velocity of rotation. Meanwhile, the gravitational potential energy is mgh, where h≫r. There are some quantities whose values we don’t know, like ω, but looking at the set-up, I seriously doubt that rotational energy, or lack thereof because the ball doesn’t stick to the track, is going to matter.
Fun fact: Galileo didn’t drop weights off the Leaning Tower of Pisa; he rolled balls down slopes like this. He completely ignored/didn’t know about rotational energy, and that was an error in his measurements, but it was small enough to not change the final result. He also used his heartbeat as a stopwatch.
I think the biggest effect here is the bendy track. It’s going to absorb a lot (like O(1)) of the energy, and can’t be ignored. Alison uses the questioner’s motives as data (“Calculating the effect of the ramp’s bendiness seems unreasonably difficult and this workshop is only meant to take an hour or so, so let’s forget that.”), which she shouldn’t.
It might not matter in the grand scheme of things, but my comment above has been on my mind for the last few days. I didn’t do a good job of demonstrating the thing I set out to argue for, that effect X is negligible and can be ignored. That’s the first step in any physics problem, since there are infinitely many effects that could be considered, but only enough time to compute a few of them in detail.
The first respondent made the mistake of using the challenger’s intentions as data—she knew it was a puzzle that was expected to be solvable in a reasonable amount of time, so she disregarded defects that would be too difficult to calculate. That can be a useful criterion in video games (“how well does the game explain itself?”), it can be exploited in academic tests, though it defeats the purpose to do so, and it’s useless in real-world problems. Nature doesn’t care how easy or hard a problem is.
I didn’t do a good job demonstrating that X is negligible compared to Y because I didn’t resolve enough variables to put them into the same units. If I had shown that X’ and Y’ are both in units of energy and X’ scales linearly with a parameter that is much larger than the equivalent in Y’, while everything else is order 1, that would have been a good demonstration.
If I were just trying to solve the problem and not prove it, I wouldn’t have bothered because I knew that X is negligible than Y without even a scaling argument. Why? The answer physicists give in this situation is “physics intuition,” which may sound like an evasion. But in other contexts, you find physicists talking about “training their intuition,” which is not something that birds or clairvoyants do with their instincts or intuitions. Physicists intentionally use the neural networks in their heads to get familiarity with how big certain quantities are relative to each other. When I thought about effects X and Y in the blacked-out comment above, I was using familiarity with the few-foot drop the track represented, the size and weight of a ball you can hold in your hand, etc. I was implicitly bringing prior experience into this problem, so it wasn’t really “getting it right on the first try.” It wasn’t the first try.
It might be that any problem has some overlap with previous problems—I’m not sure that a problem could be posed in an intelligible way if it were truly novel. This article was supposed to be a metaphor for getting AI to understand human values. Okay, we’ve never done that before. But AI systems have some incomplete overlap with how “System 1” intelligence works in human brains, some overlap with a behavioralist conditioned response, and some overlap with conventional curve-fitting (regression). Also, we somehow communicate values with other humans, defining the culture in which we live. We can tell how much they’re instinctive versus learned by how isolated cultures are similar or different.
I think this comment would get too long if I continue down this line of thought, but don’t we equalize our values by trying to please each other? We (humans) are a bit dog-like in our social interactions. More than trying to form a logically consistent ethic, we continually keep tabs on what other people think of us and try to stay “good” in their eyes, even if that means inconsistency. Maybe AI needs to be optimized on sentiment analysis, so when it starts trying to kill all the humans to end cancer, it notices that it’s making us unhappy, or whimpers in response to a firm “BAD DOG” and tap on the nose...
Sorry—I addressed one bout of undisciplined thinking (in physics) and then tacked on a whole lot more undisciplined thinking in a different subject (AI alignment, which I haven’t thought about nearly as much as people here have).
I could delete the last two paragraphs, but I want to think about it more and maybe bring it up in a place that’s dedicated to the subject.
ω is just v/r (v = rω), and translational KE is ½mv² or ½mr²ω², so if rotational KE is ⅕mr²ω², then rotational KE is 10⁄35 or 29% of total KE.
I guess if we assume the ball is rolling without slipping as it exits the track, then the ratio of translational KE to rotational KE is fixed regardless of what happened earlier in the drop, so maybe it doesn’t matter after all.
Sorry that I didn’t notice your comment before. You took it the one extra step of getting kinetic and rotational energy in the same units. (I had been trying to compare potential and rotational energy and gave up when there were quantities that would have to be numerically evaluated.)
Yeah, I follow your algebra. The radius of the ball cancels and we only have to compare 12 and 15. Indeed, a uniformly solid sphere (an assumption I made) rolling without sliding without change in potential energy (at the end of the ramp) has 29% rotational energy and 71% linear kinetic energy, independently of its radius and mass. That’s a cute theorem.
It also means that my “physics intuition trained on similar examples in the past” was wrong, because I was imagining a “negligible” that is much smaller than 29%. I was imagining something less than about 5% or so. So the neural network in my head is apparently not very well trained. (It’s been about 30 years since I did these sorts of problems as a physics major in college, if that can be an excuse.)
As for your second paragraph, it would matter for solving the article’s problem because if you used the ball’s initial height and assumed that all of the gravitational potential energy was converted into kinetic energy to do the second part of the problem, “how far, horizontally, will the ball fly (neglecting air resistance and such)?” you would overestimate that kinetic energy by almost a third, and how much you overestimate would depend on how much it slipped. Still, though, the floppy track would eat up a big chunk, too.
Guess: The main reason why Alison’s calculation was off was because
The correction used for rotational KE assumed the ball was rolling without slipping for the entire decent, whereas in reality, the top of the ramp is steep enough that the ball mostly slides in the initial decent and thus gains less rotational KE than it would otherwise (resulting in more translational KE and a faster exit speed).
This looks a lot like a typical high school/college freshman physics problem, and I guess the moral of the story is that it leads us to think that we should solve it that way. But if you were to work it out,
I think the ball’s rotational energy would be a much smaller number than the gravitational potential energy of falling a few feet. The rotational energy of a solid sphere is 15mr2ω2, where m and r are the mass and radius of the ball and ω is the angular velocity of rotation. Meanwhile, the gravitational potential energy is mgh, where h≫r. There are some quantities whose values we don’t know, like ω, but looking at the set-up, I seriously doubt that rotational energy, or lack thereof because the ball doesn’t stick to the track, is going to matter.
Fun fact: Galileo didn’t drop weights off the Leaning Tower of Pisa; he rolled balls down slopes like this. He completely ignored/didn’t know about rotational energy, and that was an error in his measurements, but it was small enough to not change the final result. He also used his heartbeat as a stopwatch.
I think the biggest effect here is the bendy track. It’s going to absorb a lot (like O(1)) of the energy, and can’t be ignored. Alison uses the questioner’s motives as data (“Calculating the effect of the ramp’s bendiness seems unreasonably difficult and this workshop is only meant to take an hour or so, so let’s forget that.”), which she shouldn’t.
It might not matter in the grand scheme of things, but my comment above has been on my mind for the last few days. I didn’t do a good job of demonstrating the thing I set out to argue for, that effect X is negligible and can be ignored. That’s the first step in any physics problem, since there are infinitely many effects that could be considered, but only enough time to compute a few of them in detail.
The first respondent made the mistake of using the challenger’s intentions as data—she knew it was a puzzle that was expected to be solvable in a reasonable amount of time, so she disregarded defects that would be too difficult to calculate. That can be a useful criterion in video games (“how well does the game explain itself?”), it can be exploited in academic tests, though it defeats the purpose to do so, and it’s useless in real-world problems. Nature doesn’t care how easy or hard a problem is.
I didn’t do a good job demonstrating that X is negligible compared to Y because I didn’t resolve enough variables to put them into the same units. If I had shown that X’ and Y’ are both in units of energy and X’ scales linearly with a parameter that is much larger than the equivalent in Y’, while everything else is order 1, that would have been a good demonstration.
If I were just trying to solve the problem and not prove it, I wouldn’t have bothered because I knew that X is negligible than Y without even a scaling argument. Why? The answer physicists give in this situation is “physics intuition,” which may sound like an evasion. But in other contexts, you find physicists talking about “training their intuition,” which is not something that birds or clairvoyants do with their instincts or intuitions. Physicists intentionally use the neural networks in their heads to get familiarity with how big certain quantities are relative to each other. When I thought about effects X and Y in the blacked-out comment above, I was using familiarity with the few-foot drop the track represented, the size and weight of a ball you can hold in your hand, etc. I was implicitly bringing prior experience into this problem, so it wasn’t really “getting it right on the first try.” It wasn’t the first try.
It might be that any problem has some overlap with previous problems—I’m not sure that a problem could be posed in an intelligible way if it were truly novel. This article was supposed to be a metaphor for getting AI to understand human values. Okay, we’ve never done that before. But AI systems have some incomplete overlap with how “System 1” intelligence works in human brains, some overlap with a behavioralist conditioned response, and some overlap with conventional curve-fitting (regression). Also, we somehow communicate values with other humans, defining the culture in which we live. We can tell how much they’re instinctive versus learned by how isolated cultures are similar or different.
I think this comment would get too long if I continue down this line of thought, but don’t we equalize our values by trying to please each other? We (humans) are a bit dog-like in our social interactions. More than trying to form a logically consistent ethic, we continually keep tabs on what other people think of us and try to stay “good” in their eyes, even if that means inconsistency. Maybe AI needs to be optimized on sentiment analysis, so when it starts trying to kill all the humans to end cancer, it notices that it’s making us unhappy, or whimpers in response to a firm “BAD DOG” and tap on the nose...
Sorry—I addressed one bout of undisciplined thinking (in physics) and then tacked on a whole lot more undisciplined thinking in a different subject (AI alignment, which I haven’t thought about nearly as much as people here have).
I could delete the last two paragraphs, but I want to think about it more and maybe bring it up in a place that’s dedicated to the subject.
ω is just v/r (v = rω), and translational KE is ½mv² or ½mr²ω², so if rotational KE is ⅕mr²ω², then rotational KE is 10⁄35 or 29% of total KE.
I guess if we assume the ball is rolling without slipping as it exits the track, then the ratio of translational KE to rotational KE is fixed regardless of what happened earlier in the drop, so maybe it doesn’t matter after all.
Sorry that I didn’t notice your comment before. You took it the one extra step of getting kinetic and rotational energy in the same units. (I had been trying to compare potential and rotational energy and gave up when there were quantities that would have to be numerically evaluated.)
Yeah, I follow your algebra. The radius of the ball cancels and we only have to compare 12 and 15. Indeed, a uniformly solid sphere (an assumption I made) rolling without sliding without change in potential energy (at the end of the ramp) has 29% rotational energy and 71% linear kinetic energy, independently of its radius and mass. That’s a cute theorem.
It also means that my “physics intuition trained on similar examples in the past” was wrong, because I was imagining a “negligible” that is much smaller than 29%. I was imagining something less than about 5% or so. So the neural network in my head is apparently not very well trained. (It’s been about 30 years since I did these sorts of problems as a physics major in college, if that can be an excuse.)
As for your second paragraph, it would matter for solving the article’s problem because if you used the ball’s initial height and assumed that all of the gravitational potential energy was converted into kinetic energy to do the second part of the problem, “how far, horizontally, will the ball fly (neglecting air resistance and such)?” you would overestimate that kinetic energy by almost a third, and how much you overestimate would depend on how much it slipped. Still, though, the floppy track would eat up a big chunk, too.