# Is the apparent gravitational force on certain parts of a rotating spherical planet off centre?

For a person standing at the equator, if he sees an object in free fall, he will see that the object accelerates downward at the rate $$a = g - \omega^2R$$ where $$R$$ is the radius of the planet and $$\omega$$ is the spin angular velocity of the planet. I can't really provide a diagram right now but lets assume that the north-south axis is in the vertical direction. At latitude $$\phi$$ then an object would go about a circle of radius $$R\cos\phi$$ with angular velocity $$\omega$$ and so the "horizontal" (horizontal to the north-south axis, not horizontal to a person on the surface at latitude $$\phi$$) component of acceleration is $$a_{\text{hor}} = g\cos\phi - \omega^2R\cos\phi$$ while the vertical component remains $$a_{\text{ver}} = -g\sin\phi$$ and so the net force seems to be off centre. So I guess my question is I know that weight seems to increase as we move towards the poles but does the force also slightly go off center when we move away from the equator or did I make some mistake in my analysis?