Simple harmonic motion and positive constant w^2?

[gunparashar]

in defining the Equation of simple harmonic motion taking origin as fixed point and the line of motion as x axis. a(acceleration) = - w^2 * x. where w^2 is positive constant. what is the reason behind taking square of w as constant not just w?

 

[PhysicoRaj]

Because it is not just 'taken'. Acceleration is the rate of change of velocity. Velocity is rate of change of displacement. So the double derivative of the equation for displacement in SHM gives you the equation for acceleration. During the process of deriving it, you get an w^2.
Here's how it is done: http://physicscatalyst.com/wave/shm_0.php

 

[bhillyard]

Another way of looking at is:
SHM is where a restoring force always acts towards the origin. So when x is positive the force has to be in the negative direction, when negative the force has to be in the positive direction.
The acceleration will be in the direction of the force.
So when x is positive the force is negative, when x is negative the force is positive.
For acceleration = - const X x to keep the minus the constant MUST be positive. That means it must be a square (guaranteed to be positive).

More generally when you require a real quantity to be positive - express it as a square i.e x²
When required to be negative - express it as the negative of a square i.e. - x²