On static equilibrium of a hemispheroid

Extract

In the course of a coffee-table conversation with my friends regarding the nature of static equilibrium of different solid objects the situation involving a uniform hemisphere came up. Intuition (and perhaps experience) tells that a uniform hemisphere as shown in Figure 1 resting on a flat surface will be at stable equilibrium, and so will an oblate hemispheroid as shown in Figure 2. Things get complicated when we move to a prolate hemispheroid like the one shown in Figure 3, for the nature of its equilibrium is less obvious. The intuition does come to mind though that if the prolate hemispheroid is made indefinitely taller, keeping its equatorial radius fixed, then the equilibrium should eventually become unstable. Intrigued, we decided to probe into the matter quantitatively.