91Ó°ÊÓ

In the context of simple harmonic motion, the equation of motion is essentially a mathematical formula that describes how a particle moves over time. This equation can be expressed in the form: \[ y(t) = A \sin(\omega t + \phi) \] - **\( y(t) \)** represents the position of the particle at time \( t \), along the y-axis in this case.- **\( A \)** is the amplitude of the motion, which is the maximum displacement from the origin. Here, it's given as 3.0 cm.- **\( \omega \)** is the angular frequency, which we'll discuss in more detail next.- **\( \phi \)** is the phase constant, indicating the initial state of the motion at \( t = 0 \). In this problem, since we're dealing with simple harmonic motion along the y-axis, understanding the parameters allows us to construct this equation which is essential to predict the particle's future positions.

Angular frequency \( \omega \) is a crucial component in the equation of motion for simple harmonic motion. It represents how many radians per second the particle oscillates through during its motion. Angular frequency and frequency \( f \) of oscillation are linked by the equation: \[ \omega = 2\pi f \] - For this exercise, the frequency \( f \) is given as 20 Hz.- Using the relationship, we find \( \omega = 2\pi \times 20 = 40\pi \) rad/s. Understanding angular frequency helps describe the tempo of the oscillation. It tells us essentially how rapid the cycles are occurring over time. This rapidity matters when determining how quickly the particle returns to the same position, which is fundamental to plotting its course over time.

In simple harmonic motion, the phase constant \( \phi \) plays a role in determining the initial position of the particle. It's the offset at the starting point of time \( t = 0 \). It aligns the motion with its specific starting condition.For a motion starting at the origin, known as its equilibrium position, the phase constant is often zero. This is because we want sin(\( 0 \)) to also be zero, aligning perfectly with the origin when time \( t = 0 \). In scenarios where the particle needs to start at peak amplitude or at some other point on its path, \( \phi \) would be adjusted. Nonetheless, choosing \( \phi = 0 \) often simplifies problems such as the one given, where the particle starts exactly at the origin at \( t = 0 \). This means there's no phase shift needed in the sine function.

In simple harmonic motion, amplitude and frequency are both fundamental characteristics, yet they describe different aspects of the motion. - **Amplitude (A):** Is the maximum displacement from the equilibrium position. It defines the extent of motion on either side of the origin. In this problem, the amplitude is 3.0 cm. - **Frequency (f):** Measures how often the cycles of the motion repeat per second. It's given by 20 Hz here, defining how quickly the motion repeats. Even though amplitude and frequency are measured independently, they're related in how they define the motion: amplitude provides the size, while frequency provides the pace. Both need to be known to fully understand the nature of a vibrational motion, allowing complete prediction of the particle's oscillation behavior over time.