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The Simple Harmonic Motion (SHM) equation is crucial for understanding the oscillatory behavior of an object. When an object undergoes SHM, its motion can be mathematically described using a sinusoidal function. This is because SHM is periodic and repeats itself over fixed intervals. The standard SHM equation when maximum amplitude is involved is given as:\[ x(t) = A \cos(\omega t) \]- Here, \( x(t) \) is the displacement at any time \( t \).- \( A \) stands for the amplitude, which is the maximum extent of motion from the equilibrium position.- \( \omega \) is the angular frequency indicating how fast the oscillations occur.In our exercise, we have the amplitude set as \( A = 6.00 \text{ cm} \). This equation helps us determine the displacement of the object at any specific time.

In the context of SHM, angular frequency is a measure of how quickly an object completes its cycles of motion. It is represented by the symbol \( \omega \) and is calculated using the relation between the period \( T \) and cycles:\[ \omega = \frac{2\pi}{T} \]- The period \( T \) is the time taken for one complete cycle.- \( 2\pi \) radians constitute a full rotation, which applies to circular movement analogy in oscillations.From the problem, \( T = 0.300 \text{ s} \), giving us an angular frequency:\[ \omega = \frac{2\pi}{0.300} \approx 20.94 \text{ rad/s} \]This high value of \( \omega \) indicates rapid oscillations, typical for low period values. Knowing \( \omega \) helps us analyze the timing and dynamics of the object's motions.

In SHM, the cosine function is integral to describing how displacement varies with time. The choice of the cosine function in the SHM equation arises because the motion starts from the maximum amplitude when \( t = 0 \). This is different from using a sine function, which would start from the equilibrium position.- The cosine function, \( \cos(\omega t) \), oscillates between -1 and 1. These values relate directly to the displacement in SHM.- When \( \cos(\omega t) = 1 \), the object is at its maximum positive amplitude. Conversely, \( \cos(\omega t) = -1 \) indicates the maximum negative amplitude.For the problem, using the cosine function helps us find the exact time it reaches certain displacements, such as when \( x(t) = -1.50 \text{ cm} \), by determining the equivalent angle \( \omega t \).

Displacement in SHM refers to the distance of an object from its equilibrium position at any given time. It is an essential aspect as it helps in identifying the position and state of motion that the object is undergoing at a particular instance, governed primarily by the SHM equation.- The displacement \( x(t) \) is measured typically in units of length, like centimeters or meters.- At time \( t = 0 \), the displacement is equal to the amplitude if the motion starts from the maximum position.In the exercise, we explore how the displacement changes from the maximum position \( x = 6.00 \text{ cm} \) to \( x = -1.50 \text{ cm} \). By solving \[ \cos(\omega t) = \frac{-1.50}{6.00} \], we employed trigonometric principles to yield the necessary time \( t = 0.087 \text{ s} \). This computation demonstrates the practical use of these concepts in finding specific states of motion in oscillatory systems.