The sample problem in my physics book that I do not understand is here

The only thing that I do not understand is the LAST sentance...

I'm pretty sure that they are talking about the centripetal acceleration but they do not show how they got it. The book stops there. Any ideas of how they got 3g/4?

What equation are you referring to? While the rod has a single angular acceleration, the linear acceleration varies along the length (as Daniel illustrated): [itex]a = \alpha R[/itex], where R is the distance from the pivot.

L/2 is used in determining the torque due to the weight of the rod, which acts at the center of mass.