A rope rests on two platforms which are both inclined at an angle θ (which you are free to pick), as shown. The rope has uniform mass density, and its coeffciient of friction with the platforms is 1. The system has left-right symmetry. What is the largest possible fraction of the rope that does not touch the platforms? What angle θ allows this maximum value?
(3/05) This problem has been solved correctly (25/2/05) by Chetan Mandayam Nayakar, a student at India Insttitute of Technology, Madras, India (e-mail mn_chetan@yahoo.com), (26/2/05) by Sebastian Popescu, a lecturer at the Physics Department at "Al. I. Cuza" University in Iasi, Romania (e-mail seba@uaic.ro), (5/3/05) by Ioannis Florakis, from University of Athens, Greece (e-mail iflorakis@yahoo.gr).
This problem and its solution appear at the web site of David Morin from the Physics Department of Harvard University, Cambridge, Mass., USA. You can see the solution in the PDF file.
(8/05) Slightly simpler solution has been suggested (15/8/05) by Yaniv Hefetz from the Faculty of Applied Mathematics and Theoretical Physics at the University of Cambridge, England (e-mail physics@zahav.net.il). His solution can be found in the following PDF file.
The answer:
The maximal possible fraction of the rope that is hanging is (√2-1)2 ≈ 0.172, which is obtained for θ = 22.5 degrees.
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