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Angular frequency, represented by \( \omega \), is a fundamental concept in Simple Harmonic Motion (SHM). It defines how fast an object oscillates through its cycle. Think of it as the rate at which the oscillation cycles are completed per unit of time. This is expressed in radians per second. For instance, in this exercise, the needle oscillates with a frequency of 2.5 Hz, which means it completes 2.5 cycles each second. To find the angular frequency, you use the formula \( \omega = 2\pi f \), with \( f \) being the frequency in Hz. This converts our frequency into the angular frequency, resulting in \( \omega = 5\pi \, \mathrm{rad/s} \). The angular frequency is crucial because it helps describe the oscillatory motion accurately in the equations for position, velocity, and acceleration.

The phase constant, denoted as \( \phi \), is a pivotal factor in determining the starting point of the oscillation. In other words, it tells us where the object is in its cycle at \( t=0 \). In many problems, especially involving SHM, we'll have initial conditions that help us find this value. For example, when you know the initial position \( x(0) = 1.1 \, \mathrm{cm} \) and initial velocity \( v(0) = -15 \, \mathrm{cm/s} \), these conditions can be plugged into the SHM equations.By setting \( x(0) = A \cos(\phi) \) and \( v(0) = -A\omega \sin(\phi) \), and knowing \( \omega = 5\pi \), we solve these simultaneous equations to find \( \phi \). The phase constant aligns the mathematical model to the actual motion in space and time, highlighting how starting conditions impact the motion.

Amplitude \( A \) represents the extent of the oscillation from its equilibrium position. It measures the maximum displacement on either side of the equilibrium point. In the context of our sewing machine problem, amplitude will tell us how far the needle moves in its simple harmonic path.To determine \( A \), we use initial conditions provided in the problem: the initial position and velocity. By solving the equations \( x(0) = A \cos(\phi) \) and \( v(0) = -A\omega \sin(\phi) \), we can uncover both \( A \) and \( \phi \). Amplitude doesn't change over time in SHM, making it a fixed value that contributes to defining the motion's scale and energy.

In Simple Harmonic Motion, position, velocity, and acceleration are intricately linked. They describe how an object moves back and forth in a predictable pattern. The position function \( x(t) = A \cos(\omega t + \phi) \) provides the displacement from equilibrium at any time \( t \).The velocity is the time derivative of the position, calculated as \( v(t) = -A\omega \sin(\omega t + \phi) \). This shows how the speed and direction change as the needle passes through its cyclical path. Acceleration comes next as the derivative of velocity, \( a(t) = -A\omega^2 \cos(\omega t + \phi) \), revealing how the rate of velocity changes, influenced directly by both the amplitude and angular frequency squared.Together, these equations demonstrate the periodic nature of SHM: when the position is maximum, velocity is zero, and acceleration reaches its highest negative or positive value. Understanding this relationship is crucial for predicting the behavior of oscillatory systems.