A uniform thin rod of length L and mass M is pivoted at one end. The pivot is attached to the top of a car accelerating at rate A.

a.) What is the equilibrium value of the angle theta between the rod and the top of the car?

Honestly, I'm not entirely sure what the question even wants me to find. Playing the problem out in my mind, it seems like the angle would just continue to increase at a constant rate as the car accelerates with a constant rate A, but clearly this is not the case.

b.) Suppose that the rod is displaced a small angle phi from equilibrium. What is its motion for small phi?

I would assume that the motion would be towards equilibrium, although the vertical component of motion for the rod itself should still be the same, as the approximation cos(phi)=1 should hold for small values of phi.