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Angular frequency is a critical concept in simple harmonic motion, representing how fast an object oscillates. It is denoted by the symbol \(\omega\) and is related to the frequency \(f\) of the oscillation. The relationship is given by the equation:\[\omega = 2 \pi f\]This means the angular frequency is what you'd get if you translated cycles per second (which is what frequency \(f\) measures) into radians per second, since a full cycle of oscillation is \(2 \pi\) radians.
For the sewing machine problem, with a frequency of 2.5 Hz, the angular frequency would be calculated by substituting into the equation, yielding \(\omega = 5\pi\). This tells us how quickly the point of the needle is oscillating back and forth in terms of radians per second.

Amplitude, symbolized by \(A\), measures the maximum extent of the oscillation from its equilibrium position in simple harmonic motion. It reflects how far the needle reaches on either side of the axis. In the context of our exercise, finding the amplitude involves knowing both the initial position and velocity.
Initially, using the equation for position:\[x(0) = A \cos(\phi) = 1.1\]The amplitude can be derived once the phase constant \(\phi\) is known. Amplitude reflects the energy of the motion; larger amplitudes mean the point moves further from the center, which usually requires more initial energy being imparted to the system.

The phase constant \(\phi\) in simple harmonic motion provides information on where the oscillation begins at \(t = 0\). The phase constant essentially shifts the entire wave function along the time axis and can be critical in determining initial conditions.
To find \(\phi\) in our task, use both the given position and velocity at \(t=0\). With:\[x(0) = A \cos(\phi) = 1.1\]and\[v(0) = -A \omega \sin(\phi) = -15\]By solving these simultaneous equations, you determine \(\phi\). This accounts for the starting point of the needle's oscillation, capturing any phase shifts it might have.

Velocity and acceleration are dynamic components of simple harmonic motion that describe how quickly the object's position changes over time. They’re derived from the basic position function, and each adds dimensions to understanding the motion.The basic equations are:

• Velocity: \(v(t) = -A \omega \sin(\omega t + \phi)\)
• Acceleration: \(a(t) = -A \omega^2 \cos(\omega t + \phi)\)

The velocity is the derivative of position with respect to time, expressing how fast and in what direction the needle moves. It's directly influenced by both the amplitude \(A\) and angular frequency \(\omega\), explaining how fast and in what direction the needle is instantaneously moving.
The acceleration, being the derivative of velocity, provides insight into the forces in play to change that velocity. It incorporates the square of angular frequency, emphasizing how fast the oscillations increasingly or decreasingly act upon the needle. Understanding these relationships helps in predicting the needle's behavior over time, allowing for precise control in practical applications like sewing.