Ponder This Challenge - June 2016 - Tilting a truncated icosahedron

Ponder This Challenge:

This month's challenge is from Bert Dobbelaere. (Thanks!)
The marketing boys behind a large beverage brand decide to take advantage of a much-hyped soccer tournament by changing the shape of their plastic canisters. The new canisters are shaped as a truncated icosahedrons (achieved by taking an icosahedron and cutting off one-third of each edge on both ends; see http://mathworld.wolfram.com/TruncatedIcosahedron.html), which are often associated with the sport. The canisters have a small tap just above the bottom face. A pentagon was chosen as bottom, as this accentuates the symmetry of the shape. There were some concerns about the stability of the canister, which is typically placed near the edge of a table top.

On the assumption that the empty weight of the canister is negligible and its walls are infinitely thin, what is the maximum tilt angle of the table, so that the canister will never lose its balance, independent of the direction of the tilt and the filling level of the canister? If the maximum tilt angle is marginally exceeded and the canister starts to tumble, what fraction of its volume is filled with fluid?

Solution

• Click here to view the solution

  We don't have any elegant solution for this month's challenge (but we'd be happy to hear from you if you have one). We present here John Snyder's elaborate solution as is:(See attached file: TippingCanister.pdf)

  Thanks for David Greer for pointing us to Brian Mirtich's paper from 1996 "Fast and Accurate Computation of Polyhedral Mass Properties" (http://www.math.uiuc.edu/~redavid2/PW/TimsPaper.pdf)

Solvers

• Shirish Chinchalkar (05/06/2016 IDT)
• John Snyder (06/06/2016 05:32 PM IDT)
• Rashid Naimi (09/06/2016 02:43 AM IDT)
• Graham Hemsley (10/06/2016 04:00 PM IDT)
• Hugo Pfoertner (13/06/2016 10:07 AM IDT)
• Siddharth Joshi (14/06/2016 04:52 PM IDT)
• Mathias Schenker (14/06/2016 07:27 PM IDT)
• Jesse Rearick (20/06/2016 06:22 AM IDT)
• Jimmy Waters (20/06/2016 06:53 PM IDT)
• Tim Lewis (21/06/2016 04:20 PM IDT)
• Sergey Koposov (28/06/2016 04:57 AM IDT)
• David Greer (29/06/2016 04:15 AM IDT)
• Motty Porat (30/06/2016 02:45 PM IDT)
• Erik Wünstel (30/06/2016 07:06 PM IDT)