I want to create a planet that has a frequency of two rotations per second, thus making a "Planet without Night". The planet will be like a flattened disc, due to inertia causing the planet to be an oblate spheroid. Gravity on the planet should also be very different at the "equator" than at the poles. The effects this planet would have on the civilization would be quite intriguing.
My question is a two-parter: firstly, I would like the mass of the planet to be great enough to create a gravitational force that is greater than the tangential speed created by the spin of the planet. Secondly, I need to know how oblate the planet would be with the mass and given rotational speed. An approximate graph (like y = x^2) would be nice for visualization. The gravity at the "equator" would optimally be "low" such that someone probably couldn't accidentally jump off the planet, but is lighter than the moon (if that's possible). The gravity at the pole would optimally be greater than or equal to earth's gravity.
It would seem to me that the mass would affect how oblate the planet is, which will affect the distance from the center of the planet, thus affecting the tangential speed. If the resultant tangential speed was greater than the resultant gravity, the mass would have to be changed. I can't figure out how to figure out how to find these factors, partially because I can only think of doing a trial-and-error method and don't know a "proper" way to find them, and because I don't know how to determine how oblate the planet is based on mass and speed.