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wu :: forums    riddles    medium (Moderators: towr, Icarus, Eigenray, ThudnBlunder, Grimbal, SMQ, william wu)    Glass spirals « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Glass spirals  (Read 1607 times) Noke Lieu Uberpuzzler pen... paper... let's go! (and bit of plastic)     Gender: Posts: 1884 Glass spirals   « on: Apr 18th, 2011, 7:35pm » Quote Modify I stuck a 30m long piece of fishing line to a 5cm diameter circular glass.   I then attached a 1 cm wide pencil to the free end of the line. Keeping the line taught and running tangentally to the glass,  I started to draw a spiral-inspired arc, stopping when the pencil touched the glass... How long was it? IP Logged a shade of wit and the art of farce. Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: Glass spirals   « Reply #1 on: Apr 19th, 2011, 7:46am » Quote Modify How much of the line have you lost in the knots?   How elastic is the line?  Don't tell me it is not.  All fishing lines are somewhat elastic.   When you start, is the line already tangential to the glass?   Does the pencil keep its orientation, therefore also winding up some of the rope, or does the pencil turn as you run around, always facing the same side to the glass?  In the first case, do you start with the rope tangential to the pencil.   Are the glass and the pencil perfectly vertical and the line horizontal?   (just searching excuses for not doing the calculations) IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Glass spirals   « Reply #2 on: Apr 19th, 2011, 9:11am » Quote Modify I get about 4775 m IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Noke Lieu Uberpuzzler pen... paper... let's go! (and bit of plastic)     Gender: Posts: 1884 Re: Glass spirals   « Reply #3 on: Apr 19th, 2011, 3:28pm » Quote Modify ah, whoops! I meant 30 cm...! But, yes. IP Logged a shade of wit and the art of farce. towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Glass spirals   « Reply #4 on: Apr 19th, 2011, 10:30pm » Quote Modify In that case I get about 158.1824 cm   length of wire: x(t) = 30 - 3t radius: r(t) = sqrt( (30 - 3t)2 + 32)  --  Assuming wire is tangential to both pencil and glass at all times. derivative: r'(t) = (3t-30))/sqrt(t2-20 t+101) curve length: integral over t=0..10 of sqrt[ r(t)2 + r'(t)2 ] dt ~= 158.1824 « Last Edit: Apr 19th, 2011, 10:32pm by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Immanuel_Bonfils Junior Member     Posts: 114 Re: Glass spirals   « Reply #5 on: May 24th, 2011, 2:40pm » Quote Modify Wasn't it  30 cm long?   And what means 3t ? IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Glass spirals   « Reply #6 on: May 24th, 2011, 9:47pm » Quote Modify It's a function of time. IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Immanuel_Bonfils Junior Member     Posts: 114 Re: Glass spirals   « Reply #7 on: May 25th, 2011, 12:52pm » Quote Modify To a silly question a "good answer"? Obviously the question is what makes 3 overthere? Shouldn't be something X ?   And what about the 30 cm ? IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Glass spirals   « Reply #8 on: May 25th, 2011, 1:17pm » Quote Modify I can't remember why I chose to include a 3 there. It doesn't really matter, you could pick pi or e or phi, but choosing something other than a divider of 30 just gives ugly integration bounds. What's important to note is that speeding up or slowing down will not change the length of the curve. So make choices that simplify the calculation. IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Ant_Man Newbie     Gender: Posts: 23 Re: Glass spirals   « Reply #9 on: Sep 2nd, 2011, 8:41pm » Quote Modify I think in polar coordinates ds=sqrt{ r'(t)2+[r(t) q'(t)]2 }dt, where q(t) is the angle of the (imaginary) line connecting the pencil to the center of the glass (NOT the angle of the string). That's not what towr has written so I'm confused about his method.   I get 90 cm by sticking with a Cartesian parameterization parameterized by the angle, \theta, between the center of the glass and the point where the string leaves the glass. This has a neat form for a string of length L and glass of radius R, namely, s=0.5 L2/R. IP Logged alien2 Uberpuzzler     Gender: Posts: 6991 Re: Glass spirals   « Reply #10 on: Sep 5th, 2011, 12:12pm » Quote Modify on Apr 18th, 2011, 7:35pm, Noke Lieu wrote: How long was it? Size doesn’t matter. IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Glass spirals   « Reply #11 on: Sep 5th, 2011, 1:00pm » Quote Modify on Sep 5th, 2011, 12:12pm, alien2 wrote: Size doesn’t matter. You wouldn't say that after walking a mile in shoes that are a size too small. 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