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Can somebody explain to me why we measure the spring potential energy from the point it's unstretched? I've been puzzled with that and I've tried to prove it mathematically but I don't understand why we should measure it from this particular point .

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When a spring is neither stretched nor compressed it has zero potential energy. It can do no work, it cannot push or pull. Therefor when it is stretched or compressed, giving it potential energy, it cannot expend more energy than was put into stretching or compressing it.

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