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?
The hemisphere is a solid 1/2 sphere of radius 50.
Imagine the long axis of the cylinder is horizontal and the pivot point is where the cylinder meets the half sphere.
Torque of hemisphere:
integral from 0 to 50 of x*A*dx
and A is π*y^2 = π*(50^2 - x^2)
= 2500π* integral x dx - π integral x^3 dx
= 2500π * 2500/2 - π * 50^4 / 4
= (3125000 - 1562500)*π
= 1562500*π
Torque of cylinder. Length is L
= integral from 0 to L of π(50^2 - 49^2)*x*dx
= 99π L^2 / 2
L^2 = 1562500 * 2 / 99 = 31565.656565
L = 177.66726 cm
The center of mass for this construct is the center of the circle where the hemisphere and cylinder meet. So any orientation where the hemisphere touches the table will have the CG directly over the point of contact with the table.
The minimum ceiling height is the radius 50 plus the diagonal distance from the CG to the outer edge of the cylinder.
This diagonal is √(2500 + L^2) = √(2500 + 31565.656565) = 184.5688
Ceiling height min is 234.5688 cm.
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Posted by Larry
on 2025-01-23 11:54:51 |
Solution
