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Is there some sort of experiment one could perform that would differentiate between the absence of a force and a force with a magnitude of 0? Or are they just different ways of describing the same thing?

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"Absence of force" and "zero force" are just two different ways to describe the same thing.

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Kinematatticaly they are the same but dynamically, there will be stresses built up in the object experiencing a net force of zero. Consider a spring for example, when you pull on its ends, even though the force is zero, there are still stresses in it.

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