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I'm trying to solve this problem:

Two wheels are one next to another and linked by a conveyor belt. How to express the length of the radius of the smaller wheel in terms of the length of the belt? Note that the size of the radius of A is three times the radius of B.

Sketch of the problem (not in scale)

N. F. Taussig
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Chris Steinbeck Bell
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1 Answers1

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Let the length of the belt be $b$.

enter image description here

\begin{align} d&=\sqrt{(4r)^2-(2r)^2} =2\sqrt{3}r ,\\ \phi&=\arccos\tfrac{2r}{4r}=\tfrac\pi3 ,\\ b&=2\,((\pi-\phi)\cdot3r+d+\phi\,r) \\ &=2(2\pi r+2\sqrt{3}r+\tfrac\pi3 r) \\ &=(\tfrac{14}3 \pi +4\sqrt{3})r ,\\ r&=\frac{b}{\tfrac{14}3 \pi +4\sqrt{3}} . \end{align}

g.kov
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  • Mind sharing what software did you used to sketch the graphics and label the angles and the lenghts?. Plus, how can be proved that the $PO_{1}O_{2}$ angle formed on the bigger wheel and $BO_{2}$ in the smaller one are the same?. – Chris Steinbeck Bell Oct 04 '17 at 02:56
  • @ChrisSteinbeckBell The software appears to be $\LaTeX$ – gen-ℤ ready to perish Oct 04 '17 at 02:57
  • @ChaseRyanTaylor I am not very fond of Latex use in geometry. Do you happen to know any documentation where can I learn how geometry is supported on Latex?. Thanks in advance. – Chris Steinbeck Bell Oct 04 '17 at 03:00
  • @ChrisSteinbeckBell Share$\LaTeX$ – gen-ℤ ready to perish Oct 04 '17 at 03:02
  • @ChaseRyanTaylor The documentation provided does not seem to give insight on how to draw euclidean geometry, however there is this site http://latexdraw.sourceforge.net/ which does seem to support graphics but I'm not sure if its what the answerer has used in his answer. – Chris Steinbeck Bell Oct 04 '17 at 03:10
  • @Chris Steinbeck Bell: The drawing was made with the Asymptote, "a powerful descriptive 2D and 3D vector graphics language that provides a natural coordinate-based framework for technical drawing. Labels and equations are typeset with LaTeX, for high-quality PostScript output." It is also part of the TeXLive distribution. – g.kov Oct 04 '17 at 03:21
  • @Chris Steinbeck Bell: $O_1A || O_2 B$. – g.kov Oct 04 '17 at 03:24
  • @g.kov Thanks that explained a lot. I was not aware such software existed. It will come in handy for next time if required. – Chris Steinbeck Bell Oct 04 '17 at 03:46
  • @g.kov Can you take a look into this question https://math.stackexchange.com/questions/2458271/how-to-find-the-perimeter-of-a-triangle-which-connects-three-circles? The answer there is not sufficient to let me understand. I would appreciate your help with that one. – Chris Steinbeck Bell Oct 09 '17 at 03:39