At the point where the substitution is made, no assumption is made about what omega will represent in the real world. When the solution is obtained, it becomes clear that it corresponds to angular speed. Of course, the choice of omega as the symbol, and ##\omega^2## as the specific substitution, is because the author knows that's how it will turn out.ThatDude said:
Simple harmonic motion is a type of periodic motion where an object oscillates back and forth around a central equilibrium point. This motion is characterized by a restoring force that is directly proportional to the displacement of the object from its equilibrium position.
Some examples of simple harmonic motion include the swinging of a pendulum, the motion of a mass attached to a spring, and the vibrations of a guitar string.
The formula for simple harmonic motion is x = A sin(ωt + φ), where x is the displacement from equilibrium, A is the amplitude, ω is the angular frequency, and φ is the phase angle.
Simple harmonic motion is different from other types of motion because it is a type of periodic motion with a constant amplitude and a restoring force that is directly proportional to the displacement. This results in a sinusoidal or circular motion around an equilibrium point.
The period and frequency of simple harmonic motion are affected by the mass of the object, the spring constant, and the amplitude of the motion. Increasing the mass or the spring constant will increase the period and decrease the frequency, while increasing the amplitude will increase the frequency and decrease the period.