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I could not find an answer by web search. From Why is rolling friction less than the maximum static friction? I understand static balls "experience" static friction, not rolling one.

I'm trying to build a tetrahedron from tennis balls, like here https://en.wikipedia.org/wiki/Tetrahedral_number.

enter image description here

I noted large pyramids are unstable, of about 10 balls edge is very easily breaking/falling/rolling apart during construction (8 balls are ok, 9 is doable but difficult).

My floor is rather smooth, so the structure can be approximated by placing ideal balls with some friction due to the rough surface on a no-friction floor.

The weight of the pyramid grows by the power of 3 whereas the surface of the basement by the power of 2, but static friction is proportional to weight, so still I do not immediately see why larger tetrahedrons should be more unstable. But building real ones shows they are. Why? can some formula be derived for maximum height with friction coefficients, size and weight of the balls?

Mritunjay Kumar
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Martian2020
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1 Answers1

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Andria Rogava, Professor of Physics at Ilia State University in Georgia, Eurasia, has constructed some remarkable towers using tennis balls. For example, a 9 storey tower made of 25 balls and a frustrum of a pyramid made of 46 balls. In both structures some balls overhang so that they are supported from below by only 2 balls instead of 3.

Friction between the balls is essential to provide inward torques which counteract the radial outward torques arising from the balls pushing against each other. Which is why tennis balls are ideal. For the tower the top-most single ball (the 'keystone') is essential - remove it, even with extreme care, and the tower collapses. (See video in link.)

These tall structures have been achieved by careful placement of balls, and the use of temporary 'scaffolding' during construction. This suggests that the limit which you found is far from what can be achieved.

enter image description here

Each layer of balls pushes both outwards and inwards on the layers below and above. If the lowest layer is resting on a horizontal surface, there must be friction between the balls and the that surface in order to generate an inward force. Alternatively the outermost balls in the lowest layer could be welded together, or held in place by a frame; then friction with the floor is not required.

Mathematical Analysis

TO BE COMPLETED LATER.

Except for the simplest cases, such structures of rigid spheres are statically indeterminate and cannot be analysed without introducing other properties (or assumptions) such as elasticity and deformation.

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sammy gerbil
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