Change in gravitational potential energy for a slender

1. The problem statement, all variables and given/known data

I’ll provide a picture for a clearer view:

Suppose that the slender rod starts at rest at theta = 0. For convenience we chose the datum at theta = 0.
Now I want to calculate the gravitational potential energy at a later instant when theta = theta_0. But it’s tricky since some points of the rod have moved a distance (0.6+0.2)sin(theta_0) and some points have not moved at all. So how does one deal with this case? I have a solution but how should I do in the general case?

Just to be clear: I am looking for an analytical approach to it that does not involve some intuition because that can
be dangerous…

3. The attempt at a solution

Think of the slender rod as a huge amounts of points uniformly spread. For each point at one side of the mass center there is a point on the other side of the mass center so that the distance between these are the distance from O to the mass center. Hence it should be mg(0.6+0.2)sin(theta_0)/2

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