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Consider a square (blue) on top of a platform (yellow), on which gravitational force (black) acts. We know that if the blue square is in equilibrium, contact forces from the platform (white) must act in the opposite way to gravity.

I would like to ask whether something can be said about the distribution of such forces on the boundary of the square?

If a force $dF$ acts on a small patch of "surface" around a point $x$, we must clearly have $\int dF = F_g$, i. e., the net force is the gravitational force. Also, we must have $\int (a-x)dF = F_g \cdot r$, which results from torque balance. Gravity produces a torque $F_g \cdot r$, which has to be balanced by the torques from the contact forces.

But is there something more that can be said about the distribution of forces? The two integrals by themselves give an infinite number of solutions. The force could be concentrated at one specific point, or spread out over the whole surface (which sounds more realistic)... Which is it and how exactly is it spread out?

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  • $\begingroup$ Your question ist not clear. is your square of homogeneous material, why should the border be different ? You talk with 3 colors, but five no picture $\endgroup$
    – trula
    Mar 18, 2022 at 21:42

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Without additional assumptions, nothing more can be said about the distribution of forces between two rigid bodies being in contact at a plane (or other finite) surface. The reason is pretty obvious if you think about the real world problem, where the bodies are not rigid but compliant, and even plane surfaces are not exactly plane. Every actual unevenness in the plane will result in point-contacts at these irregularities, which then deform to realize Hertzian contact situation, where contact area and hence pressure depend very sensitively on the displacement perpendicular to the surface. Since you do not know the microscopic irregularities of the surfaces, these hot spots of pressure are very hard to predict. The ambiguity you have correctly identified simply expresses the fact that rigid bodies and mathematical surfaces are just an idealization.

You can, however, just assume in rigid body mechanics, that the pressure is homogeneous across a plane surface. But that is just an artificial assumption. One could say, Bullshit in, Bullshit out, because you should be careful not to apply this to an engineering problem. If you are lucky, the total contact force is high enough so as to flatten all surface irregularities, so that the above homogeneous pressure condition is at least approximately fulfilled.

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