Problem 97 - Precalculus

Question

For Exercises \(95-98,\) refer to the following: A weight hanging on a spring will oscillate up and down about its equilibrium position after it is pulled down and released. (IMAGE CAN'T COPY). This is an example of simple harmonic motion. This motion would continue forever if there were not any friction or air resistance. Simple harmonic motion can be described with the function \(y=A \cos (t \sqrt{\frac{k}{m}}),\) where \(|A|\) is the amplitude, \(t\) is the time in seconds, \(m\) is the mass of the weight, and \(k\) is a constant particular to the spring. The frequency of the oscillations in cycles per second is determined by \(f=\frac{1}{p},\) where \(p\) is the period. What is the frequency for the oscillation modeled by \(y=3 \cos \left(\frac{t}{2}\right) ?\)

Step-by-Step Solution

Answer

The frequency is \(\frac{1}{4\pi}\) cycles per second.

Step 1: Identify the components of the equation

The given function is \(y = 3 \cos \left(\frac{t}{2}\right)\). Here, \(A = 3\) is the amplitude, and the argument of the cosine function is \(\frac{t}{2}\). This means \(\omega = \frac{1}{2}\) since the function is in the form \(y = A \cos(\omega t)\).

Step 2: Determine the formula for period

In simple harmonic motion, the period \(p\) is given by \(p = \frac{2\pi}{\omega}\), where \(\omega\) is the angular frequency from the cosine function argument.

Step 3: Calculate the period

Substitute \(\omega = \frac{1}{2}\) into the period formula: \[ p = \frac{2\pi}{\frac{1}{2}} = 4\pi \]. Thus, the period of oscillation is \(4\pi\).

Step 4: Use the period to find the frequency

The frequency \(f\) is the reciprocal of the period: \[ f = \frac{1}{p} = \frac{1}{4\pi} \]. Therefore, the frequency of the oscillation is \(\frac{1}{4\pi}\) cycles per second.

Key Concepts

Amplitude

The amplitude of a simple harmonic motion is the maximum distance the system reaches from its equilibrium position. In our example, the function is given by \( y = 3 \cos \left(\frac{t}{2}\right) \), where the amplitude \( A \) is 3.

• This value represents the farthest point the weight moves from its resting position before reversing direction.
• A greater amplitude means the object oscillates over a larger range.
• The amplitude does not affect the period or frequency of the motion.

Understanding amplitude helps in visualizing how "lively" the oscillation is, but remember, it's only about distance from the center.

Frequency

Frequency in simple harmonic motion tells us how often the oscillation completes a full cycle within a specific unit of time, usually seconds. It is measured in cycles per second, or Hertz (Hz). From the exercise, frequency can be found using \( f = \frac{1}{p} \). Here, the period \( p \) is \( 4\pi \), thus the frequency is \( \frac{1}{4\pi} \) Hz.

• Frequency helps us understand how "fast" the oscillation process repeats itself.
• Higher frequency means quicker oscillations.

Having a sense of frequency is crucial to grasping how active or dynamic the oscillatory motion is.

Angular Frequency

Angular frequency \( \omega \) is a measure of how quickly an object moves through its cycle. In the function \( y = 3 \cos \left(\frac{t}{2}\right) \), the angular frequency is \( \omega = \frac{1}{2} \).

• It's expressed in radians per second, unlike regular frequency which is in cycles per second.
• It's closely related to frequency and period. Specifically, \( \omega = 2\pi f = \frac{2\pi}{p} \).
• This constant is part of the cosine or sine function and dictates the 'speed' of the oscillation.

Angular frequency is an integral part of understanding the mathematical description of a harmonic oscillator's properties.

Period

The period \( p \) is the duration of time it takes to complete one full cycle of oscillation. From our function, the period is calculated through \( p = \frac{2\pi}{\omega} \). With \( \omega = \frac{1}{2} \), the period \( p \) is \( 4\pi \).

• This time span helps us figure out how long one full swing takes in the motion.
• Against common perceptions, period does not depend on amplitude.
• Instead, period is dependent on factors within the system like mass and spring constant (as in spring systems).

Understanding the period of motion is crucial for syncing events or predicting future positions of oscillating objects.

Oscillation

Oscillation refers to the repetitive back-and-forth motion about an equilibrium position. It is a hallmark of simple harmonic motion. In our case, the weight on the spring displays this behavior.

• One complete back-and-forth movement constitutes a single cycle of oscillation.
• It's what makes the motion "harmonic" and "regular" over time.
• Think of it as the rhythmic heartbeat of systems like pendulums or springs.

Understanding oscillation is key to analyzing systems that revert to their resting state and are a centerpiece in classical mechanics.