In SHM why do we use $\omega^2$? My question is why do we use $\omega^2$? in the equation $A=-\omega^2x$.
Why we just don't call it $A=-k/m×x$
I want to know the actual reason behind this.
 A: The author used hindsight: after playing around with the equations he/she found out that $\omega=\sqrt{k/m}$ is a nice parameter and instead of first deriving the solution and then define $\omega$ the author is making the derivation nicer by defining it now.
By defining $\omega$ the solution looks like
$$x(t)=A_1\cos(\omega t)+A_2\sin(\omega t)$$
A: For one dimensional motion one way of defining simple harmonic motion is to say that
(a) the acceleration, $\vec a = a \,\hat x$, is proportional to the displacement, $\vec x = x \, \hat x$, from a fixed point and
(b) the acceleration is always directed towards the fixed point.
From condition (a) one can write $\vec a \propto \vec x \Rightarrow a\,\hat x \propto x\,\hat x \Rightarrow a\propto x$ where $a$ and $x$ are components and
from condition (b) one can write $a = -\omega^2 x$ knowing that $\omega^2$ is always going to be a positive number.
So if the component of the displacement is positive then the component of the acceleration is negative, ie the direction of the acceleration is towards the origin and if the component of the displacement is negative then the component of the acceleration is positive, ie the direction of the acceleration is towards the origin.
There are also useful linear relationships between $\omega$ and the frequency of the motion, $f$, and the period of the motion, $T$, viz $\omega = 2 \pi f$ and $\omega = 2\pi/T$
