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Angular frequency is a crucial concept in understanding simple harmonic motion. It is denoted by the symbol \( \omega \) and is defined as the rate at which an object oscillates in radians per second. To compute the angular frequency of a particle undergoing simple harmonic motion, you can use the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of the motion.
In our exercise, the period \( T \) is given as 1.5 seconds. By substituting this value into our formula, we find that \( \omega = \frac{2\pi}{1.5} \). This value of \( \omega \) helps us determine how rapidly the particles oscillate along their path. Understanding angular frequency ties together the concepts of time and position and allows us to precisely predict the positions of the particles at any given moment.

Phase difference refers to the amount by which one oscillation leads or lags behind another. In the context of simple harmonic motion, it describes when two waves, such as those of our particles, start or differ in their oscillation cycles.
For two particles in this scenario, they differ in phase by \( \frac{\pi}{6} \) radians. This phase difference means that while one particle reaches a maximum or minimum point, the other does so later or earlier, depending on whether it is leading or lagging.

• A phase difference of zero would mean that the particles move in perfect synchrony.
• Phase differences allow us to analyze how the oscillations compare at a given moment in time.

Understanding the phase difference is essential for calculating the exact positions of particles relative to each other at any time, such as at \( t = 0.5 \) seconds in our exercise.

The equation of motion in simple harmonic oscillation allows us to predict the position of an object at any time. In mathematical terms, it is expressed as \( x(t) = A \cos(\omega t + \phi) \), where \( x(t) \) is the position of the particle, \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase angle.
For the two particles in our problem, we set their equations of motion as:

• Leading particle: \( x_1(t) = A \cos(\omega t) \)
• Lagging particle: \( x_2(t) = A \cos(\omega t + \frac{\pi}{6}) \)

By substituting \( t = 0.5 \) seconds and the computed \( \omega \) value, we can find the positions of each particle along their path at that specific time. This calculation allows us to understand not only their motion but also their relative position and distance from one another.

Trigonometric expressions come into play through the cosine functions in simple harmonic motion. In this scenario, we use the expressions \( \cos(\omega t) \) and \( \cos(\omega t + \frac{\pi}{6}) \) to describe the oscillations of our particles.
These expressions are crucial as they help us determine the specific positions of our particles at any moment. When solving problems like these, it often involves simplifying complex trigonometric equations to find clear, concise answers.
Calculating the distance between the particles at a specific time involves:

• Substituting the given values of \( t \), \( \omega \), and phase \( \phi \)
• Evaluating the cosine expressions
• Subtracting the positions \( |x_1 - x_2| \) for the desired outcome.

Understanding these trigonometric expressions and the method to work with them will give insights into how the particles are moving relative to one another.