On a certain planet, which is perfectly spherically symmetric, the free-fall acceleration has magnitude g = go at the north pole and g = ago at the equator (with 0 < a < 1). find g(9), the freefall acceleration at colatitude 9 as a function of 9.

The reason why there is a difference between free-fall acceleration is a centrifugal force.
I attached a diagram that shows how this force aligns with the force of gravity.
From the diagram we can see that:

Where g' is the free-fall acceleration when there is no centrifugal force, r is the radius of the planet, and w is angular frequency of planet's rotation. is the latitude.
We can calculate g' and wr^2 from the given conditions in the problem.

Our final equation is:

Colatitude is:

The answer is:

Did you ever figue this