Simple harmonic motion and positive constant w^2?

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SUMMARY

The discussion centers on the equation of simple harmonic motion (SHM), specifically the relationship defined by acceleration a = -w² * x, where w² is a positive constant. The necessity of using the square of w, rather than w itself, is rooted in the mathematical derivation of acceleration as the second derivative of displacement. This ensures that the acceleration remains a real and positive quantity, which is essential for the behavior of SHM, where a restoring force acts towards the origin. The discussion also emphasizes that expressing a real quantity as a square guarantees its positivity.

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  • Understanding of basic calculus, particularly derivatives
  • Familiarity with the concepts of displacement, velocity, and acceleration
  • Knowledge of simple harmonic motion principles
  • Basic physics concepts related to forces and motion
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  • Study the derivation of the simple harmonic motion equation in detail
  • Explore the implications of restoring forces in SHM
  • Learn about the role of positive constants in physical equations
  • Investigate the mathematical properties of squares and their applications in physics
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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?
 
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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
 
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
 
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