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Which side of the balance is the power arm? Should I put the object here or the weight there?
It's all nonsense. Power arm and resistance arm are just "named" for convenience of explanation, and there is no real physical difference.

When a heavy object and two sides of an object are balanced, do you think the weight of the handle of the object is pressed down or the object balances the weight?

There is only the concept of "force arm" in real physics, and the power/resistance arm is a more convenient statement for people in their work, which is a bit far-fetched for the balance. If you want to think of it as a moving object with weight, put the weight there.