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I was reading on rolly-polly dolls when the following questions crept into my mind

  1. How exactly does center of gravity play a role which helps rolly-polly doll to stand up on its own?

  2. Can a rolly-polly doll be made to stand on a slanting mountain surface so that it may lean forward but not fall down? (I mean roll down.).

  3. If yes, how does the slant angle and weight of its base related?

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  • $\begingroup$ This should really be on homework help. All questions you asked are pretty trivial. Hint: The most stable position of the doll is one in which its potential energy is minimum. Use this fact to solve all three problems. $\endgroup$
    – gautam1168
    Sep 18 '15 at 12:44
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    $\begingroup$ How can you ask a question like that and not give us a (small) picture, I never heard of them:) $\endgroup$
    – user81619
    Sep 18 '15 at 14:37
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The following diagram should give you some insight:

enter image description here

The black dot is the center of mass. If the center of mass is below the center of curvature of the bottom, then when you tilt the doll the c.o.m. will be displaced relative to the point of contact with the surface such that there is a torque that will attempt to right the doll again.

If you place the whole thing on a slope, the same thing happens, but the point of support moves closer. It should be easy to see how the point where the doll becomes unstable is now a function of the distance $d$ relative to the radius of curvature $r$.

enter image description here

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