I'm studying a dynamic problem and am trying to break it into the vertical and angular components - please help me!

I have a uniform rod, of mass M, length L that is rotating around the point 'x'. The leftmost end of the rod is A and the rightmost is B. The pivot point 'x' is closer to A than to B (i.e. it lies in the leftmost half of the rod). For the purpose of this discussion let's say that the pivot point is 1/4 of the way along the rod. The pivot point is free to move vertically and to rotate. The centre of mass 'c' lies closer to B than to A (i.e. the centre of mass lies in the righmost half of the rod). Let's say, for the sake of argument that the centre of mass is at the 3/4 point along the rod.

I have two forces acting upon the rod, vertical forces and rotational. The rod moving downawrds is considered positive, the rod twisting clockwise is positive.

When the rod rotates I know that, because the pivot point is not on the centre of mass, the pivot point will rise and fall vertically as the rod rotates. When I say I "know" this - I know this instinctively. I cannot work out where this comes from mathematically. If someone could please help me then I definitely owe you a beer!

Greenwood - " Principles of Dynamics" , Check for detailed discussion on something very similar.

Also this is very similar to pitching and plunging motion of airfoils.