dorsal/arxiv
View SchemaRocking and rolling: a can that appears to rock might actually roll
| Authors | Manoj Srinivasan, Andy Ruina |
|---|---|
| Categories | |
| ArXiv ID | physics/0702028 |
| URL | https://arxiv.org/abs/physics/0702028 |
| DOI | 10.1103/PhysRevE.78.066609 |
| Journal | Phys. Rev. E 78, 066609 (2008) |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
A beer bottle or soda can on a table, when slightly tipped and released, falls to an upright position and then rocks up to a somewhat opposite tilt. Superficially this rocking motion involves a collision when the flat circular base of the container slaps the table before rocking up to the opposite tilt. A keen eye notices that the after-slap rising tilt is not generally just diametrically opposite the initial tilt but is veered to one side or the other. Cushman and Duistermaat (2006) recently noticed such veering when a flat disk with rolling boundary conditions is dropped nearly flat. Here, we generalize these rolling disk results to arbitrary axi-symmetric bodies and to frictionless sliding. More specifically, we study motions that almost but do not quite involve a face-down collision of the round container's bottom with the table-top. These motions involve a sudden rapid motion of the contact point around the circular base. Surprisingly, like for the rolling disk, the net angle of motion of this contact point is nearly independent of initial conditions. This angle of turn depends simply on the geometry and mass distribution but not on the moment of inertia about the symmetry axis. We derive simple asymptotic formulas for this "angle of turn" of the contact point and check the result with numerics and with simple experiments. For tall containers (height much bigger than radius) the angle of turn is just over $\pi$ and the sudden rolling motion superficially appears as a nearly symmetric collision leading to leaning on an almost diametrically opposite point on the bottom rim.
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"abstract": "A beer bottle or soda can on a table, when slightly tipped and released,\nfalls to an upright position and then rocks up to a somewhat opposite tilt.\nSuperficially this rocking motion involves a collision when the flat circular\nbase of the container slaps the table before rocking up to the opposite tilt. A\nkeen eye notices that the after-slap rising tilt is not generally just\ndiametrically opposite the initial tilt but is veered to one side or the other.\nCushman and Duistermaat (2006) recently noticed such veering when a flat disk\nwith rolling boundary conditions is dropped nearly flat. Here, we generalize\nthese rolling disk results to arbitrary axi-symmetric bodies and to\nfrictionless sliding. More specifically, we study motions that almost but do\nnot quite involve a face-down collision of the round container\u0027s bottom with\nthe table-top. These motions involve a sudden rapid motion of the contact point\naround the circular base. Surprisingly, like for the rolling disk, the net\nangle of motion of this contact point is nearly independent of initial\nconditions. This angle of turn depends simply on the geometry and mass\ndistribution but not on the moment of inertia about the symmetry axis. We\nderive simple asymptotic formulas for this \"angle of turn\" of the contact point\nand check the result with numerics and with simple experiments. For tall\ncontainers (height much bigger than radius) the angle of turn is just over\n$\\pi$ and the sudden rolling motion superficially appears as a nearly symmetric\ncollision leading to leaning on an almost diametrically opposite point on the\nbottom rim.",
"arxiv_id": "physics/0702028",
"authors": [
"Manoj Srinivasan",
"Andy Ruina"
],
"categories": [
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],
"doi": "10.1103/PhysRevE.78.066609",
"journal_ref": "Phys. Rev. E 78, 066609 (2008)",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Rocking and rolling: a can that appears to rock might actually roll",
"url": "https://arxiv.org/abs/physics/0702028"
},
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