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On a hemispherical world half the size of Earth (ignoring that the planet should break apart and become a smaller sphere) what would the gravity be like? How would it change as you traveled from pole to pole along the curved side, as well as down the flat side? What other interesting or distinguishing facts would this planet have?

I suspect that if you 'fell' off the edge on the flat side it would at first be like falling off a cliff and then transition more and more into a slope and finally be level in the center. If so, how far would you fall before the slope feels like a 45 degree angle? A 30? A 60?

I also suspect that gravity at the poles would be greater than gravity at the curved side on the equator opposite the flat side. I'm not sure what percentage of Earth's gravity would be experienced at any given point.

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If what you want is a formal description, you might wish to state the problem in terms of electrostatics and go to Jackson. Solve the Laplace equation with density $\varrho(\rho,z)=\Theta(z)\varrho_{Earth}(\rho,z)$, set $\varrho_{Earth}(\rho,z)$ as realistic as you want...and have fun with the Bessel functions! –  c.p. Jun 8 '12 at 0:19
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On the curved side near the equator it would be similar to Earth (or any other spherical planet) then as you head toward the poles you would be going uphill - since the gravity vector would be pointing to near the middle of the hemisphere and so at 45deg to the 'ground'

On the flat side the area near the 'equator' wouldn't be too different and then again as you go toward the pole it becomes steeper until - until you can stand on the edge.

But to do all this you would need oxygen since the atmosphere would form a ring around the 'lower' tropics.

There is a similar interesting puzzle of what would happen on a cubic planet. Essentially the corners would form peaks extending miles out of the atmosphere which would form circular skies over the center of each face

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