Is the function y = [1-cos(2wt)] ÷ 2 represents the simple harmonic oscillatiory motion?

The function \( y = \frac{1 - \cos(2wt)}{2} \) does indeed represent a form of motion that can be related to simple harmonic oscillatory motion, but it's essential to break down what this means and how it connects to the principles of harmonic motion.

Understanding the Function

At first glance, the equation resembles a transformation of the cosine function, which is fundamental in describing oscillatory behavior. The term \( \cos(2wt) \) indicates that the function oscillates with a frequency related to \( w \), which is the angular frequency. The factor of 2 in the argument of the cosine function suggests that the oscillation occurs at a frequency that is double that of a standard cosine wave.

Breaking Down the Components

• Cosine Function: The cosine function oscillates between -1 and 1. In this case, \( \cos(2wt) \) will also oscillate within this range.
• Transformation: The expression \( 1 - \cos(2wt) \) shifts the range of the cosine function. Instead of oscillating between -1 and 1, it now oscillates between 0 and 2.
• Division by 2: Finally, dividing by 2 scales the oscillation down, resulting in a function that oscillates between 0 and 1.

Relating to Simple Harmonic Motion

Simple harmonic motion (SHM) is characterized by a restoring force that is proportional to the displacement from an equilibrium position, typically represented by equations of the form \( y = A \cos(\omega t + \phi) \) or \( y = A \sin(\omega t + \phi) \). The function \( y = \frac{1 - \cos(2wt)}{2} \) can be interpreted as a transformation of a cosine function, which is indeed a hallmark of SHM.

Key Characteristics of SHM

To further illustrate the connection, consider the following:

• Equilibrium Position: In this case, the equilibrium position is at \( y = 0.5 \), which is the midpoint of the oscillation range.
• Amplitude: The amplitude of the oscillation is 0.5, as the function varies from 0 to 1.
• Frequency: The angular frequency \( w \) determines how quickly the oscillation occurs. The factor of 2 in \( 2wt \) indicates that the oscillation completes a cycle more rapidly than a standard cosine function.

Conclusion

In summary, while the function \( y = \frac{1 - \cos(2wt)}{2} \) is not a standard representation of simple harmonic motion, it is fundamentally related to it through its oscillatory nature and the properties of the cosine function. It captures the essence of oscillation, albeit with a specific transformation that shifts and scales the typical behavior of SHM. Thus, it can be considered a valid representation of oscillatory motion, reflecting the principles underlying simple harmonic motion.