Problem 70 - 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 \(t=0\) ). $$d=6 \sin \frac{2 \pi}{3} t$$

Step-by-Step Solution

Answer

Amplitude is 6 cm, period is 3 seconds, and frequency is \(\frac{1}{3}\) Hz. The point starts at 0 cm, goes to 6 cm, and returns to 0 cm in 3 seconds.

Step 1: Identify the Amplitude

The given equation is \(d = 6 \sin \frac{2\pi}{3} t\). In this equation, the amplitude is the coefficient in front of the sine function. Hence, the amplitude is 6 cm.

Step 2: Determine the Equation Format

The standard equation for harmonic motion is \(d = A \sin(Bt + C) + D\). Here, \(A\) is the amplitude, \(B\) relates to the period, \(C\) determines the phase shift, and \(D\) is the vertical shift. Our equation is \(d = 6 \sin\frac{2\pi}{3} t\), with \(A = 6\), \(B = \frac{2\pi}{3}\), \(C = 0\), and \(D = 0\).

Step 3: Calculate the Period

The period \(T\) of a sine wave is determined by the formula \(T = \frac{2\pi}{B}\). For our equation, \(B = \frac{2\pi}{3}\), so we calculate the period as follows: \(T = \frac{2\pi}{\frac{2\pi}{3}} = 3\) seconds.

Step 4: Calculate the Frequency

The frequency \(f\) is the reciprocal of the period, defined as \(f = \frac{1}{T}\). Since we found the period to be 3 seconds, the frequency is \(f = \frac{1}{3}\) Hz.

Step 5: Describe the Motion During One Oscillation

Starting at \(t=0\), \(d = 6 \sin 0 = 0\), meaning the motion begins at the equilibrium position. As \(t\) increases to \(\frac{3}{2}\), \(d\) reaches the maximum amplitude of 6 cm. At \(t=3\), the motion completes one full cycle returning to 0 cm.

Key Concepts

Amplitude

The amplitude in harmonic motion refers to the maximum extent of displacement from the equilibrium position. In the equation given, \( d = 6 \sin \frac{2\pi}{3} t \), the amplitude is represented by the coefficient preceding the sine function, which is 6 in this case. This means the maximum distance the point \( P \) moves from its central position is 6 centimeters.
This amplitude tells us how far each oscillation reaches above and below the central axis. In simple terms, the oscillations will peak at 6 cm above the center and 6 cm below.

• A larger amplitude would result in wider swings from the central position.
• A smaller amplitude would indicate tighter oscillations closer to the center.

Understanding amplitude helps us visualize how large or small the motion is in vertical displacement terms.

Period

The period of a harmonic motion is the time it takes for the motion to complete one full cycle. From the formula \( T = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( t \) in the argument of the sine function, we can determine the period.
In our equation, \( B = \frac{2\pi}{3} \). Using the formula, we get:

• \( T = \frac{2\pi}{\frac{2\pi}{3}} = 3 \text{ seconds} \)

This result means the point \( P \) takes 3 seconds to return to its original starting point after completing one oscillation.
The importance of the period:

• A shorter period indicates more rapid oscillations.
• A longer period means slower motion.

Understanding the period is crucial to interpreting how swiftly or slowly the oscillation occurs.

Frequency

Frequency in harmonic motion is the number of complete oscillations per unit time, often measured in Hertz (Hz). It is inversely related to the period, calculated as \( f = \frac{1}{T} \).
Given our period \( T \) of 3 seconds from the previous calculation:

• The frequency \( f = \frac{1}{3} \) Hz, meaning the point completes \( \frac{1}{3} \) of an oscillation per second.

This frequency value provides insight into the speed of oscillation:

• A higher frequency means more oscillations per second, indicating a rapid motion.
• A lower frequency shows fewer oscillations, translating to a slower motion.

By understanding frequency, we can quantify how often the point reaches the peak amplitude within a given time frame.

Sine Wave

A sine wave describes a smooth, periodic oscillation and is generally used to represent harmonic motion. The equation \( d = 6 \sin \frac{2\pi}{3} t \) describes such a motion.
Key characteristics of sine waves include:

• Repeats at a consistent rhythm, demonstrated by equal periods.
• Symmetrical, having the same pattern throughout each cycle.
• Defined by amplitude and frequency, determining the height and number of waves.

The sine wave in our equation has an amplitude of 6 cm and completes a cycle every 3 seconds. This wave shape is fundamental in physics to model waves in sound, light, and other periodic phenomena.
Understanding the sine wave helps predict the future location of objects in oscillatory motion.

Oscillation

Oscillation refers to the repetitive variation or fluctuation around a central value. In the context of our exercise, it's the back and forth movement of point \( P \) along the vertical axis.
The motion begins at the central point (equilibrium position), moves up to the maximum amplitude of 6 cm, returns through the center, dips to a minimum, and cycles back to the start. This describes one complete oscillation.

• Oscillations are consistent in cycle duration, determined by the period.
• The extent of movement in each direction is dictated by the amplitude.

The concept of oscillation is critical to understanding the regular patterns of movement in various physical systems, from pendulums to springs and even circuits.