Yes, the weight travels up a frictionless, smooth curve. But to calculate such a curve requires one to break it down into small integrated steps and then average the steps together.
The fall is most definitely 0.15705 meters. This is not false. Most everyone here on the forum and in the scientific world knows that the only way for gravity to do work such as extend a spring is for a weight to fall. You think that because the weight follows a constant upward path that it is not also falling. Maybe we should use different words? The fact is that gravity pushes the weight downward as the arm is lifted upward. The weight falls relative to the end of the arm that is rising.
This might seem backwards because gravity it pushing down on the weight as the arm at the same time is raising the weight. The net result is the weight rises and the spring gets stretched. Both happen simultaneously. The arm tilts upward as gravity pushes the weight downward (falls) and the path is a constant rise, until about 70º.
If gravity did not push the weight down (fall) against the force of the spring then the torque applied to the arm would lift the weight higher than it does. Thus the question here is not so much semantics of words, but rather it is whether the force to lift the weight is greater when the weight follows the outer path that causes energy to be added to the the spring as it is extended. Clearly the force is greater and the spring does not get stretched without adding more energy than is needed to just lift the weight.
I don't see how I can make it any clearer.
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