# Is there any proof that $F=-kx$?

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

• 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”. – dmckee Oct 16 at 4:03

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.
$$F = -kx$$ 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.