Problem 68 - Precalculus: Functions and Graphs

Question

The formula specifles the position of a point \(P\) that is moving harmonically on a vertical axis, where \(t\) is in seconds and \(d\) is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at \(t=0\) ). $$d=\frac{1}{3} \cos \frac{\pi}{4} t$$

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{3}\) cm, Period: 8 sec, Frequency: \(\frac{1}{8}\) Hz; completes one oscillation in 8 seconds.

Step 1: Identify the Formula

The given formula is \( d = \frac{1}{3} \cos \frac{\pi}{4} t \). This is a cosine function that describes harmonic motion.

Step 2: Determine the Amplitude

In the function \( d = a \cos(bt) \), the amplitude \( A \) is the coefficient of the cosine, which is \( \frac{1}{3} \). Therefore, the amplitude is \( \frac{1}{3} \) cm.

Step 3: Find the Period of the Function

The period \( T \) of a cosine function \( \cos(bt) \) is given by \( T = \frac{2\pi}{b} \). Here, \( b = \frac{\pi}{4} \), so the period is \( T = \frac{2\pi}{\frac{\pi}{4}} = 8 \) seconds.

Step 4: Calculate the Frequency

The frequency \( f \) is the reciprocal of the period, given by \( f = \frac{1}{T} \). So, \( f = \frac{1}{8} \) Hz.

Step 5: Describe the Motion of the Point

The point starts at the maximum position \( \frac{1}{3} \) cm (since the cosine function starts at its maximum when \( t=0 \)), moves downward to \(-\frac{1}{3}\) cm (the minimum), and returns back to \( \frac{1}{3} \) cm, completing one full oscillation in 8 seconds.

Key Concepts

Amplitude

Amplitude is a key concept in harmonic motion, describing the maximum extent of the oscillation. In the cosine function given, \(d = \frac{1}{3} \cos \frac{\pi}{4} t\), the amplitude is simply the absolute value of the coefficient of the cosine term.

This coefficient, \(\frac{1}{3}\), tells us how far the point \(P\) moves away from the central resting point on the vertical axis during each cycle. Hence, the amplitude here is \(\frac{1}{3}\) cm.

Amplitude is crucial as it represents the energy or intensity of the motion—larger amplitudes mean greater energy. It's important to visualize it as half the total distance between the highest and lowest point the oscillation reaches, which in this case is from \(\frac{1}{3}\) to \(-\frac{1}{3}\) cm.

Period

The period of a harmonic motion is the time taken for one complete cycle of oscillation. It's a measure of how "long" the wave is, in terms of time. To find the period \(T\), we use the formula \(T = \frac{2\pi}{b}\), where \(b\) is the coefficient of \(t\) in the function \(d = \frac{1}{3} \cos \frac{\pi}{4} t\).

By substituting, we get:

• \(b = \frac{\pi}{4}\)
• \(T = \frac{2\pi}{\frac{\pi}{4}} = 8\) seconds

This means the point \(P\) takes 8 seconds to complete one full oscillation cycle, starting from the maximum, moving to the minimum, and back to maximum.

Periodically, analyzing this value helps determine how "fast" or "slow" the oscillation is perceived, affecting the frequency.

Frequency

The frequency of harmonic motion is the number of cycles the motion completes in one second. It is the reciprocal of the period. Frequency is calculated using: \(f = \frac{1}{T}\). From our calculation, given the period \(T = 8\) seconds, the frequency \(f = \frac{1}{8}\) Hz.

In this context:

• The point \(P\) completes \(1/8\) of a cycle every second.
• A lower frequency means fewer oscillations per unit time, reflecting how spread out the cycles are.

Understanding frequency is vital for applications that require knowledge of the motion's repetition rate, such as in music theory or electronics. Frequency represents how "often" the motion repeats its cycle in a given timeframe.