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
re: soln by Larry)
Thanks for the stability comments, Larry. I think my worries were
a little simpler than your discussion and maybe also needless. I was
thinking of instability as in: if you started it tipping, say from
upright, would it accelerate? (My question is limited to the plane
holding both the force vector applied to the side of the cylinder the
cylinder axis. (Think of a ladder falling over.) I guess if the COM stays
supported, the structure will gently glide, slowed only by rolling friction
at the base. This is opposed to a situation where the COM finds itself
hanging over thin air and falls (again, like the ladder).
I think this is why massive telescopes that are correctly
counterweighted, like the 500 ton Mt Palomar telescope,
can actually be steered to place almost by hand
(a 1/12 HPW motor) , if so desired.
Edited on January 27, 2025, 5:27 am
re(2): soln
