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        riddles >> medium >> Glass spirals
        
(Message started by: Noke Lieu on Apr 18^th, 2011, 7:35pm)

Title: Glass spirals
Post by Noke Lieu on Apr 18^th, 2011, 7:35pm

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?

Title: Re: Glass spirals
Post by Grimbal on Apr 19^th, 2011, 7:46am

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)

Title: Re: Glass spirals
Post by towr on Apr 19^th, 2011, 9:11am

I get about [hide]4775 m[/hide]

Title: Re: Glass spirals
Post by Noke Lieu on Apr 19^th, 2011, 3:28pm

ah, whoops! I meant 30 cm...!
But, yes.

Title: Re: Glass spirals
Post by towr on Apr 19^th, 2011, 10:30pm

In that case I get about [hide]158.1824 cm[/hide]

[hide]
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
[/hide]

Title: Re: Glass spirals
Post by Immanuel_Bonfils on May 24^th, 2011, 2:40pm

Wasn't it 30 cm long?

And what means 3t ?

Title: Re: Glass spirals
Post by towr on May 24^th, 2011, 9:47pm

It's a function of time.

Title: Re: Glass spirals
Post by Immanuel_Bonfils on May 25^th, 2011, 12:52pm

To a silly question a "good answer"? Obviously the question is what makes 3 overthere? Shouldn't be something X http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif?

And what about the 30 cm ?

Title: Re: Glass spirals
Post by towr on May 25^th, 2011, 1:17pm

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.

Title: Re: Glass spirals
Post by Ant_Man on Sep 2^nd, 2011, 8:41pm

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 [hide]90 cm[/hide] 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, [hide]s=0.5 L²/R[/hide].

Title: Re: Glass spirals
Post by alien2 on Sep 5^th, 2011, 12:12pm

on 04/18/11 at 19:35:36, Noke Lieu wrote:

How long was it?

Size doesn’t matter.

Title: Re: Glass spirals
Post by towr on Sep 5^th, 2011, 1:00pm

on 09/05/11 at 12:12:00, alien2 wrote:

Size doesn’t matter.

You wouldn't say that after walking a mile in shoes that are a size too small. ::)