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Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. The equations governing SHM are foundational for understanding the motion of particles in such a system. The position of a particle in SHM can be described by the equation:
\[x(t) = A \sin(\omega t + \phi)\]
where \(x(t)\) is the displacement at time \(t\), \(A\) is the maximum displacement from the equilibrium position known as the amplitude, \(\omega\) is the angular velocity, and \(\phi\) is the initial phase angle. This equation provides us with the displacement of the particle at any given time, which is essential when analyzing the motion of the two particles in the given problem.

The time period of a particle in SHM, denoted by \(T\), is the amount of time it takes to complete one full oscillation or cycle. This is a crucial concept as it relates to the frequency of the motion with which the particle goes back and forth. The time period is inversely proportional to the angular frequency \(\omega\) and is given by the formula:
\[T = \frac{2\pi}{\omega}\]
For the two particles described in the exercise, knowledge of the time period is valuable because it allows us to compare the positions of the particles at various instances within their cycles, leading us to determine when and where they will cross paths.

Amplitude, often represented by the symbol \(A\), is the maximum extent of displacement of the particle from its equilibrium position in SHM. It determines the range within which the particle oscillates and is a measure of the energy in the motion. The amplitude is a key indicator of how far the particle will travel from the center position before reversing direction. In the context of our exercise, both particles have the same amplitude, which simplifies the analysis because their maximum displacements are equal, although their starting positions and phase angles may differ.

The concept of phase difference, represented as \(\Delta \phi\), is used to describe the relative positions in the cycles of two oscillating particles. It is defined as the difference in their phase angles. Phase difference dictates whether the particles are in phase, out of phase, or somewhere in between. When particles have a phase difference of \(\pi\) radians (180 degrees), they are in antiphase, meaning when one is at maximum positive displacement, the other is at an equivalent negative displacement. This concept is central to predicting when particles will be at the same position, as their phase difference will indicate if and when their paths intersect, a situation explored in the exercise at hand.

Angular velocity, denoted by \(\omega\), is a measure of how quickly the particle is oscillating in SHM. It is calculated by the equation \(\omega = \frac{2\pi}{T}\), where \(T\) is the time period. Angular velocity gives us the rate at which the angle (in terms of radians) changes with time, revealing the speed of the oscillation. In our exercise, knowing the angular velocity is necessary to plug into our SHM equations to describe the particles' motion as they travel across their paths, ultimately helping to solve for the time \(t\) and position \(x\) at which they cross.