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How do you proof that $F = - kx $? And why is there (-) on the formula(?)

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  • $\begingroup$ There are, of course, easy empirical measurements for simple elastic system that show that the obey Hooke’s law to a good approximation over a suitable range of displacements. But that’s not the same as the idealization known as “the harmonic oscillator”. $\endgroup$ – dmckee Oct 16 at 4:03
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No. There are two points of view on this. First, it's a definition of what a simple harmonic oscillator/ideal spring is. Second, it's an empirical formula that is approximately correct in many circumstances. It is literally the same thing as searching for the minimum of a function by setting the derivative of the function equal to zero, and then approximating the function there with a parabola.

The minus sign is there so that the parabola (in potential energy) is pointed upward instead of downward. In other words, "uphill" is any point away from $x=0$, so the force pushes the object back toward there.

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$F = -kx$ is the definition of a harmonic oscillator. It's not something that's proven. Most real systems can be reduced to a harmonic oscillator for sufficiently small displacements from the equilibrium; the proof of that involves Taylor expansion.

There's a $-$ in the formula because the force is restoring - it always attempts to shift the object back to the equilibrium point.

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