I need help calculating gravity and shape of a planet based on a few factors.
If a mass rotates quickly enough, the inertia (or "imaginary centrifugal force") will pull the mass outward "perpendicular" to the rotational axis. This is shown in both Earth (very slightly) and Haumea (a dwarf planet with a greater rotational speed), as well as other examples.
The mass (and therefore gravity) will affect how quickly the object must rotate to begin becoming more and more oblong. The tangential velocity at the equator (where the planet will be most oblong, if rotating quickly enough) will work against the gravity to make it seem "lighter", while gravity would be heavier at the poles. How oblong the planet is (how far the surface of the equator is from the center) will affect the tangential velocity, and in turn the resultant gravity.
I am trying to create a large rocky planet (that will be habitable) with a relatively quick rotational speed. The speed is debatable, and doesn't need to be definite as of now (i am hoping it can be a variable, but am looking at a period of one hour or less, until unrealistically fast). My problem is that I don't know how to calculate how oblong the planet will be with a given mass or rotational speed. I am hoping to be able to set a rotational speed and mass (or gravitational force), and determine and equation that either is a graph to show the shape, or can be used to then make a graph to show the shape.
My desire is to determine the shape and appearance of the planet, as well as the gravitational force at each latitude. Using the equation, I hope to be able to change the rotational speed and mass (or gravity based on mass) to find different results.