tag:blogger.com,1999:blog-5303307482158922565.post4599085300080719092..comments2022-07-08T08:48:29.479-07:00Comments on Math Mama Writes...: Logarithms and Ropes (as found in Mathematician's Delight)Sue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-5303307482158922565.post-70849693368624059992011-04-24T08:41:39.682-07:002011-04-24T08:41:39.682-07:00The last comment just brought me back here. I'...The last comment just brought me back here. I've just now printed out Dr. Drang's post to study it. I would really like to understand this, and I don't yet.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-34152539548136927442011-04-21T08:35:09.915-07:002011-04-21T08:35:09.915-07:00Hope to have one copy of that Math Delight book to...Hope to have one copy of that Math Delight book to be able to see how I could use it to explore more my knowledge in math. Thanks for the link.Math Software For Kidshttp://www.mathsoftwareforkids.comnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-55699093368269813112010-06-17T22:42:26.411-07:002010-06-17T22:42:26.411-07:00Hi, I'm the author of the post Orawnzva linked...Hi, I'm the author of the post Orawnzva linked to. As you figured, the behavior <i>is</i> multiplicative, and Orawnzva's explanation is quite good.<br /><br />It's actually not that difficult to model theoretically; you just have to have some experience with the notion of slicing objects up into differential chunks and applying Newton's laws—and in this case, Coulomb's law—to a typical chunk. Once you've done that, it's just a differential equations problem.<br /><br />Although my original article was about a single pole, it's easy to extend the analysis to several poles, which I <a href="http://www.leancrew.com/all-this/2010/06/a-physicist-walks-into-a-bar/" rel="nofollow">just did</a>.Dr. Dranghttps://www.blogger.com/profile/00494924187814021132noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-74004578740213692712010-06-17T12:08:11.340-07:002010-06-17T12:08:11.340-07:00Incidentally, my LJ handle is related to my real n...Incidentally, my LJ handle is related to my real name by a <a href="http://www.rot13.com/" rel="nofollow">wekll-known substitution cipher</a>.<br /><br />I'm a CS graduate student interested in teaching, having been so far blessed and challenged by my work as a teaching assistant and frustrated and underwhelmed by my work as a research assistant. I added your blog (and other math teaching blogs) to my reading list recently, so you'll probably be seeing more comments from me on scattered posts, current and past, until I catch up.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-90441575186425815782010-06-16T17:52:05.160-07:002010-06-16T17:52:05.160-07:00Thanks, Orawnzva. I'm going to print out the p...Thanks, Orawnzva. I'm going to print out the post at that site, and study it.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-3565223425309479472010-06-16T09:18:44.277-07:002010-06-16T09:18:44.277-07:00There's a good treatment of this with free bod...There's a good treatment of this with free body diagrams and all that physics stuff over at http://www.leancrew.com/all-this/2010/04/aye-aye-capstan/<br /><br />Joshua writes: <i>Maybe it's clear enough that the change in tension in the rope as it wraps around a bit of pole is proportional to the tension?</i><br /><br />Yeah, this is the key to the argument that the relationship must be multiplicative. The friction from each bit of pole is what allows the tension to change along the length of the rope (otherwise the rope would relax so as to equalize the tension). But the friction is proportional to the normal force between the rope and the pole, which is proportional to the tension. And a function which is proportional to its own derivative is exponential.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-15319546517570898622010-06-13T03:58:07.084-07:002010-06-13T03:58:07.084-07:00I was convinced it was multiplicative by imagining...I was convinced it was multiplicative by imagining what happens with a rope going 0 turns around. Obviously that multiplies the force by 1, not 0.<br /><br />But then I thought, wait a minute, that's assuming it's multiplicative already! <br /><br />Maybe it's clear enough that the change in tension in the rope as it wraps around a bit of pole is proportional to the tension?Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-31474017479129508162010-06-01T17:40:12.560-07:002010-06-01T17:40:12.560-07:00Rhett has done the experiment, and posted at his b...Rhett has done the experiment, and posted at <a href="http://scienceblogs.com/dotphysics/2010/06/experimental_rope_logarithms.php" rel="nofollow">his blog</a>. (Thanks, Rhett!!)<br /><br />It looks like it's probably multiplicative. Ben, I have no good instincts for physics. I want this to be either additive or multiplicative, and the data come closer to multiplicative. Does it feel to you like it might be more complex? Can you say why?<br /><br />Rhett, you said, "The normal model for friction says that the frictional force is proportional only to the force the two surfaces are pushing against each other. Not sure if that works here." Can you say more about how friction works?Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-37456635999104442392010-05-28T10:13:46.780-07:002010-05-28T10:13:46.780-07:00I agree that this may be difficult to model theore...I agree that this may be difficult to model theoretically. The only thing left is experiment. I will try this out next week and see what I get.rallainhttps://www.blogger.com/profile/12956503928971828628noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-33877273442882369702010-05-28T08:41:44.056-07:002010-05-28T08:41:44.056-07:00I am actually highly skeptical that the physical r...I am actually highly skeptical that the physical relationship being described here really is multiplicative. It works as a thought experiment but I feel like it's fudging the physical reality.<br /><br />You could get an honestly multiplicative model with pulleys, but it wouldn't have the virtue of supporting the fractional exponents.<br /><br />I haven't experimented with it myself but it actually seems to me that your historical-motivation intro to logs has the potential to be enormously powerful. In order to have this idea deliver fully, I think you'd need to figure out a sequence of questions you could ask that would lead your students to generate the idea that multiplication can be reduced to addition, or at least to become incipiently aware of it without needing to be told directly. For example, giving them an extensive set of multiplication problems involving only powers of 10? And helping them realize that they can solve them quickly just by amalgamating the powers? Like, "what's 100 x 1,000? what's 1,000 x 100,000?" etc. And then highlighting for them the fact that in order to solve these problems really they are solving addition problems. And then maybe another sequence of problems involving powers of 2? So that 32 x 8 becomes a simple matter of 5 + 3? Or something. I'm just brainstorming here. But the idea would be to get them to recognize, in the context of whole-number multiplication they can already do other ways, that if you can see numbers as powers of a certain base, then you can multiply them by adding.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-5881441801469476242010-05-26T19:41:01.452-07:002010-05-26T19:41:01.452-07:00Well, I just don't understand these things. It...Well, I just don't understand these things. It makes sense for surface area to be involved, but I also figure force is involved, and maybe angles make their way into that... makes sense that it would all work out the way you said, but I don't understand these things.Davidhttps://www.blogger.com/profile/10565910956857563935noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-59545928797336110152010-05-25T19:53:32.583-07:002010-05-25T19:53:32.583-07:00I'm picturing one rope, similar to the situati...I'm picturing one rope, similar to the situation pictured. Two half turns would give the same amount of surface for friction as one whole turn, right?Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-72410082095350452662010-05-25T18:44:39.215-07:002010-05-25T18:44:39.215-07:00Hmm, neat!
But also: Why should two half turns (o...Hmm, neat!<br /><br />But also: Why should two half turns (on different ropes) have the same effect as a whole turn?Davidhttps://www.blogger.com/profile/10565910956857563935noreply@blogger.com