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how to fix an object at either 90degree or 0 degree

You're looking for a detent mechanism. Figure 1. A 3D printed detent mechanism. Image source: YouTube. In your case you would need to modify the design for a single 90° detent. To achieve this you ...
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How to find the displacement of an arbitrary point on a spring?

You have to know (or make assumptions about) how the spring is constructed. If the spring is a nice uniform spring of the sort found in textbook illustrations, then you can figure that each section of ...
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