Thursday, September 23, 2010

Why rounded stones don’t like to sit on top of each other – technical Edition

There are several possibilities to explain why you can’t easily fit two “round-shaped” rocks on top of each other and even press against the upper stone without him sliding/falling off the lower stone.

However, I will explain it from the view of technical mechanics explaining three physical “properties” while I do it.

First, I have to tell you that you could set two stones above each other in such a way that even a high force wouldn’t force them off each other. However, the stones would have to be perfect.


However, this changes, when we have to consider real geometry. Unlike this special geometry, real geometry is not perfectly (or ideally) round. And in 95% of all cases the roundness deviation will not be compensated along the circumference.

Instead we have two stones lying on top of each other, both have a center of gravity which is not on the same axis with the contact point and thus we have not only a contact force at the contact point, but also a torque, which translates in a rotation movement of the body on this contact point.

image As you can see the right half of the picture shows the real situation with real rocks. Even if you don’t press on them, they have a center of gravity and thus a force which is pulling them down along the Axis of gravity. Now, the axis of gravity of the blue and violet rock are not symmetric like in the ideal example. This has of course a consequence for the stability. Without an external force, the rocks would just sit on each other (think space), but on Earth they will experience some force. The distance between the two centers of gravity introduces a torque around the contact point, which is the support for the upper stone.

The normal distance between the two axis of gravity may be called the distance d. This distance is the lever of the Force F with regard to the Support Point / Contact Point.

T1 = d * F


Since the Support F is only fixed in downwards direction, the rock can (and will rotate around the contact point) and thus roll of the lower rock.

This means the rock was in a meta-stable and statically undefined position. When the torque applied to the rock, we have a rolling contact between rock 1 and 2. The contact point herein has the velocity 0, the center of gravity of the constrained Rock 2 has the speed resulting from the moment in tangential direction.

So, ultimately with this small example, I can explain three concepts:

  • Center of Gravity and Support
  • Torques and lever
  • Rolling Contact and Center of Rotation

Actually, I also explained the difference between reality and mathematics. Although even mathematicians disagree on this solution, because of convergence problems ;)

It was certainly nothing special, but just wanted to share it.

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