I posted a problem called Pressure Profile a few days ago. When I wrote the problem, I assumed that when you push on a box without causing it to slide (there is friction), the normal reaction pressure varies linearly with the position along the box.
And sure enough, you can uniquely solve for a linear pressure profile which maintains the box in stasis. This is done by writing and solving two equations which describe the conditions for stasis (2 equations and 2 unknowns).
I then wondered what would happen if we assumed a quadratic pressure variation of the form . Now there are three unknown parameters, but still only two equations. I've found that if you simply choose the parameter and solve for the other two, you can easily come up with a quadratic profile which has the following properties:
1) Satisfies the two equations for stasis
2) Results in positive upward pressure for all values of x between 0 and L
And there are presumably infinitely many such solutions, since I've already just casually found three quadratic profiles that have the above two properties. I suppose it would be possible to find solutions of even higher order as well (5th, 11th, 27th, etc.).
So my questions are these:
1) In the real world, would the pressure profile indeed be linear, as I originally assumed when posting the problem?
2) If so, how does nature know not to choose any higher-order variations in the pressure, since they appear to work, as far as I can tell?
3) Is there another constraint that forces a linear variation that I have simply not considered?
For reference, here's the derivation for the general quadratic pressure profile, of which a linear profile is a special case.
By choosing the parameter and solving the two equations for and , we can generate numerous quadratic solutions. By setting to zero and solving for the other two, we can uniquely generate a linear profile. Here are three example profiles that seem to work (pressure is positive between 0 and L and the two equations for stasis are satisfied):