[Physics] I was learning SHM and I came across equation $k = mw^2$

harmonic-oscillatorhomework-and-exercisesnewtonian-mechanics

I tried searching for proofs here and there but couldn't find.. Some people used $a= w^2 x$ to prove this.. But how do you prove that?

Best Answer

Newton's second law states the force on a system is proportional to its acceleration. For linear restoring force $-kx$, we have

$$F = ma$$ $$ -kx = m\ddot{x}$$

where $\ddot{x}$ is the second time-derivative of position, i.e. acceleration. Then you have

$$ \ddot{x} = a = -\frac{k}{m}x = -\omega^2 x$$

where we define $\omega = \sqrt{\frac{k}{m}}$.