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If you have a piece of paper that is furled and unfurled so that it's in the shape of part of a parabola, and knowing that if you leave it, it'll flatten itself after time, would it flatten faster if you leave it as a right-way-up parabola, or an upside-down parabola? More particularly, I was wondering what factors would influence the flattening rate.

Note: It would be $y = kx^2$ and $y = -kx^2 + l$ on a 3-D graph, but extended perpendicularly into the $z$-axis.

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  • $\begingroup$ Well, I can list out the forces which come to my mind as of now, 1. Atmospheric Air pressure 2. gravity force of the paper itself. $\endgroup$ Commented Oct 30, 2011 at 15:48
  • $\begingroup$ ... Moisture in the air? Temperature? History of furling (hysteresis)? Difficult to predict their effect, I imagine. ... Just noticed this was posted 4 years ago. Did you do any experiments? What did you find out? $\endgroup$ Commented Jul 9, 2016 at 3:58

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It should flatten faster is the parabola is a regular. The reason is that in absence of friction there are two forces on the paper, natural restoring internal forces and the force of gravity. In an upwards parabola both forces felt by a differential piece of paper will point in the same direction, but in upside down parabola will point on different directions. In addition, if the surface has friction, the friction in the upward parabola will be zero by symmetry, but in the upside down parabola will feel horizontally towards the inside, trying to prevent the flattening.

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