Simple Static 3-Particle Model

[aasnrsd]

Homework Statement

If we have three particles in the configuration shown in the image, how do I show that pressure should not influence the locked configuration? That it is really based on the angle in particle C. It's a very simplified system: 1D, neglecting drag and friction, equal masses, uniform applied pressure on each surface of particle. I have attempted to solve it, but not sure how correct this is! Any help would be appreciated!

Relevant Equations

Forces

Diagram

Integrate pressure for each particle:

Add normal force and weight

Sum all the forces in the x and y direction to get:

 

[Lnewqban]

Welcome, @aasnrsd !

It seems to me that the last equality is not correct regarding units (force/pressure).
Besides, that angle is fixed and determined by the geometry of the system formed by the three identical particles (of cylindrical or spherical shape).

Consider that for any change in pressure to unlock the shown configuration, particles A and B must move away from each other in the direction of the only degree of freedom they have: horizontally.

Also consider that such situation could happen only if any pressure-induced vertical downward force from particle C becomes able to overcome the pressure-induced horizontal forces exerted by particles A and B in opposite directions.

I would recommend drawing a free body diagram, considering the magnitude and directions of those three forces.
Then, drawing a force vectors addition and see what situation or unbalance of forces is keeping the configuration locked regardless changes in pressure.

Hints:
1) A line joining the centers of the three particles form an equilateral triangle.
2) The exterior surface exposed to pressure for particles A and B is only 70% of the exterior surface exposed to pressure for particle C.

 

[haruspex]

The way you have drawn the diagram the angles are completely determined by the geometry. Plugging that into your final equation gives a relationship between mg and P.

You have treated the particles as infinitely long cylinders. Is that what you want?

The dimensions would balance if you were to include a variable for the radii.

In your force balance equations, I think you have mixed up $\cos(\phi)$ and $\sin(\phi)$.

Since you allow no force between A and B, I assume you are looking for the condition in which they are barely staying together. But I also see no reference to the normal forces from the ground. I think for A and B you can only use the horizontal force balance.

You can simplify the analysis by using symmetry.