How can tension vary along a rod with a uniform mass distribution? From what I understand, if the mass were to be distributed evenly in a rope or a rod, the tension within every cross-section must be equal. But in the case of a rod pivoted at one end being rotated with a uniform angular velocity in a horizontal plane, I am told that the tension varies along the length of the rod. How can we get a varying tension if the mass is uniformly distributed? I can't quite wrap my head around it.
 A: 
From what I understand, if the mass were to be distributed evenly in a rope or a rod, the tension within every cross-section must be equal.

This is not true. Generally the tension will vary along the rope/rod.
To see this conceptually in your question, think about a little section of the rod. It is being pulled inward by the rod section closer to the center and outward by the rod section farther away from the center. But in order for this mass section to move in a circle, it needs to be accelerating. Therefore, these two forces have to be unequal, and so the tension must be changing along the rod. I'll leave it to you to deduce how it's changing.
An easier system to grasp that hits the same concept is masses connected by massless rods. e.g. take three masses connected by rods and look at what the tension in each rod does in order for each mass to move in a circle at the same angular velocity.
Another system you could play around with is a rope held by one end hanging vertically. The tension will also vary along the rope here. I'll leave that for you to explore.
