Problem 12 - Precalculus Mathematics for Calculus

Question

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is zero at time \(t=0\) amplitude \(24 \mathrm{ft},\) period \(2 \mathrm{min}\)

Step-by-Step Solution

Answer

The function modeling the harmonic motion is \( f(t) = 24 \sin(\pi t) \).

Step 1: Identify the Function Type

The exercise describes a simple harmonic motion, so we will use a sine function, typically in the form \( f(t) = A \sin(\omega t + \phi) \) where \( A \) is amplitude, \( \omega \) is angular frequency, and \( \phi \) is phase shift. Since displacement is zero at \( t = 0 \), we'll use the sine function without phase shift initially (\( \phi = 0 \)).

Step 2: Determine the Amplitude

The amplitude \( A \) of the motion is the maximum displacement from the equilibrium position given as \( 24 \text{ ft} \). Thus, \( A = 24 \).

Step 3: Calculate the Angular Frequency

The period \( T \) is the time it takes for one full cycle, given as \( 2 \text{ min} \). The angular frequency \( \omega \) is calculated using \( \omega = \frac{2\pi}{T} \). Substituting, \( \omega = \frac{2\pi}{2} = \pi \).

Step 4: Construct the Harmonic Motion Function

Now we have all the components: \( A = 24 \), \( \omega = \pi \), and \( \phi = 0 \). Substitute these into the sine function: \( f(t) = 24 \sin(\pi t) \).

Key Concepts

Amplitude

In simple harmonic motion, the amplitude refers to the maximum distance that an object moves from its equilibrium position. Imagine a pendulum swinging back and forth; at its highest point, the distance from the midpoint is the amplitude.
The amplitude is always a positive value and represents the peak of the motion. In the context of this exercise, a displacement of zero at time \( t = 0 \) means the object starts at its central or neutral point of motion.

• Helps determine the maximum speed
• Crucial for understanding the extent of motion

For the exercise, the amplitude \( A \) is given as \( 24 \text{ ft} \). This tells us how far, at most, the object moves from its original point during its cycle. It's a vital part of constructing the function that models this motion.

Angular Frequency

Angular frequency, denoted by \( \omega \), is a measure of how quickly an object completes its cycles of motion. It's essentially the rate of change of the phase of the sine function over time and is expressed in radians per unit time.
While the amplitude tells us how high or low the wave peaks, angular frequency tells us how often these peaks occur within a set time frame. In simple harmonic motion, angular frequency connects to the period (\( T \)) with the formula:

• \( \omega = \frac{2\pi}{T} \)

Here, the period is the duration of one complete cycle (given as \( 2 \text{ min} \) in the exercise). Thus, using the formula above, we calculate \( \omega = \pi \), meaning the motion completes one cycle in \( 2 \) minutes. This is a pivotal element in forming the equation for harmonic motion.

Period

The period in simple harmonic motion is the time taken to complete one full cycle. It's the duration required for the motion to reset to its starting point. Imagine the cycle of a swing or a vibrating spring; each complete set of motion returning to the start position indicates one period.
Period \( T \) is crucial as it directly links to angular frequency. In this exercise, the period given is \( 2 \text{ min} \). Knowing the period helps us compute the angular frequency, which is an essential factor in the equation of motion.
With a period of \( 2 \text{ min} \), we understand that our object in motion returns to its original state every two minutes, allowing us to project and model the motion accurately over time.

Sine Function

The sine function is integral in modeling simple harmonic motion. It characterizes how the displacement varies over time in a smooth, oscillatory pattern. A sine wave follows the pattern of a smooth, repetitive oscillation, making it perfect for describing many types of wave forms, including simple harmonic motion.
The general form of a sine function modeling harmonic motion is \( f(t) = A \sin(\omega t + \phi) \), where:

• \( A \) is the amplitude
• \( \omega \) is the angular frequency
• \( \phi \) is the phase shift (with \( \phi = 0 \) as given in this exercise)

In the exercise's context, the sine function becomes \( f(t) = 24 \sin(\pi t) \), showing the displacement at any time as a function of the wave parameters calculated earlier.
This function is instrumental in predicting the dynamic behavior of systems undergoing simple harmonic motion.