A modern aluminum sculpture consists of a hollow cylinder that is capped on one end by a solid hemisphere. The cylinder has an outer diameter of 100 cm and thickness of 1 cm, and the hemisphere has the same diameter as the outside of the cylinder.
If, on a level surface, the sculpture balances in stable equilibrium at any point on its hemispherical surface, how long is the cylinder, and what is the minimum ceiling height in the museum to permit the sculpture to assume any stable position?
(In reply to
soln by Steven Lord)
Regarding "unstable equilibrium", I'm not exactly sure how to define this idea precisely. I would assume setting this object on the ground would tend to remain in that location with the same degree of stability as would be a complete sphere. i.e. not much stability.
Even if it is on its side, the axis of the cylinder might rotate up .
Suppose you put the cylinder flat, then tried to roll it rotating about the long axis of the cylinder: I think it would be just as happy to rotate up and the table would be in contact with part of the sphere as it would to remain as a rolling cylinder.
I think changing from a static to a dynamic moving system, the moment of inertia around different axes might come into play. If you gave it a push, it might roll like a cylinder for a while, then rotate up as a sphere up and over then roll again like a cylinder for a while.
Someone with a 3 D printer needs to build this and test it out.
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Posted by Larry
on 2025-01-23 18:50:09 |
re: soln
