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This question arose from Physics, where the force on an object attached on a spring is proportional to the displacement to the equilibrium (that is, the rest position). Also, if the displacement to the equilibrium is positive, the force will be negative, as it tries to pull the object back (i.e. if you pull a string, the force is opposite to your direction of pull).

Therefore, it can be said that:

$$F \propto -x$$ Where $F$ is the force and $x$ is the displacement from equilibrium

Is this the same as: $$F \propto x$$ According to the relation of proportionality, it should be, but my friend says that putting the second one is not correct.

Both are equivalent right?

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  • $\begingroup$ It depends on definitions/context. Using the definition on wikipedia, yes they are equivalent. $\endgroup$ – David Peterson Apr 27 at 21:36
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Yes, proportionality does not tell anything about the sign of the proportionality constant. This is probably done in physics so that the proportionality constant $k$ can be considered to be always positive. This has a physical interpretation , so it is convenient for it to be positive.

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Mathematically they are equivalent, but physycist may want to differentiate between a force in the same direction as the displacement, and a force opposite to the displacement, as they lead to physically different behavior of the system. Thus the minus sign matters.

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