Problem 13 - Algebra and Trigonometry

Question

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 60 ft, period 0.5 min

Step-by-Step Solution

Answer

The function is \( f(t) = 60 \cos(4\pi t) \).

Step 1: Understanding Simple Harmonic Motion

Simple harmonic motion (SHM) describes oscillating systems like springs or pendulums. The displacement over time can be modeled using the equations for sine or cosine functions. Here, since the displacement is at its maximum at time \(t = 0\), we will use the cosine function: \[ f(t) = A \cos(\omega t) \] where \(A\) is the amplitude and \(\omega\) is the angular frequency.

Step 2: Calculating Angular Frequency

The angular frequency \(\omega\) relates to the period \(T\) of the function. The period \(T\) is given by the relation: \[ \omega = \frac{2\pi}{T} \] Here, the period \(T\) is 0.5 min, so: \[ \omega = \frac{2\pi}{0.5} = 4\pi \] rad/min.

Step 3: Writing the Function

Now that we have the amplitude \(A = 60\) ft and \(\omega = 4\pi\) rad/min, we can write the function modeling the simple harmonic motion: \[ f(t) = 60 \cos(4\pi t) \] This function represents the displacement as a function of time and incorporates the given amplitude and period.

Key Concepts

Amplitude

Amplitude in the context of simple harmonic motion refers to the maximum displacement from the equilibrium position. Imagine a swinging pendulum; its amplitude is the farthest point it reaches from the center of its swing. In the given exercise, the amplitude is 60 feet, meaning the maximum vertical distance the object will move from its midpoint is 60 feet. Amplitude provides:

• The measure of how "strong" or "great" the motion is.
• A scalar quantity because it only has magnitude.

If you visualize a wave, either in the ocean or a sound wave, the amplitude is the height from the central axis to the peak or trough of the wave. Understanding amplitude helps you comprehend the extent of oscillation without considering time. It's the primary factor in determining how large or noticeable the simple harmonic oscillation is.

Period

The period of a simple harmonic oscillator is the time it takes for the motion to complete one full cycle. If you imagine a pendulum swinging back and forth, the period is the time it takes for it to return to the starting point. In this exercise, the period is specified as 0.5 minutes. This means every 0.5 minutes, the oscillating system (or the modeled motion) resets its cycle. Important aspects of period include:

• The measurement of time in seconds or minutes.
• Determining the frequency of the oscillation as the inverse value.

The period helps us understand how often the cycle of motion repeats, and it links directly to the concept of frequency, where frequency is \(1/T\). The shorter the period, the more rapid the cycles of the oscillating motion, indicating faster repeated movements.

Angular Frequency

Angular frequency is closely related to the period but expresses how fast the object is oscillating in terms of radians per unit time. In this context, the angular frequency \(\omega\) tells us how many radians the oscillator moves through per minute. The formula \(\omega = \frac{2\pi}{T}\) is used to find it. Our step-by-step solution uses this, where \(T = 0.5\) min:

• We calculate \(\omega = \frac{2\pi}{0.5} = 4\pi\) rad/min.

The significance of angular frequency is to track the motion using angular displacement per time unit, similar to how we manage circular motion. A higher angular frequency indicates quicker oscillations, which means the object completes more cycles in less time. Essentially, it provides a dynamic view of the oscillation rate.

Cosine Function

The cosine function is commonly used in simple harmonic motion, especially when the initial displacement is at its maximum when time \(t = 0\). Our function \(f(t) = A \cos(\omega t)\) models this kind of motion. The choice of cosine over sine is due to:

• Maximal initial position, as cosine achieves its peak at \(t = 0\).

Thus, the cosine function is ideal because it aligns with the observed motion characteristics in this exercise. As with any trigonometric function used to describe oscillation, it helps us:

• Predict the position of the oscillator at any given time.
• Understand the phase of the motion, which tells us where in the cycle the motion currently is.

By using cosine, along with amplitude and angular frequency, we derive a model that perfectly mirrors the motion's initial condition and the described characteristics through its full cycle.