Problem 3 - Precalculus Mathematics for Calculus

Question

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=2 \sin 3 t$$

Step-by-Step Solution

Answer

Amplitude = 2, Period = \(\frac{2\pi}{3}\), Frequency = \(\frac{3}{2\pi}\).

Step 1: Identify the Function's Form

Given the function \(y = 2 \sin 3t\), recognize its form \(y = A \sin(Bt)\), where \(A\) is the amplitude and \(B\) is used to find the period. Here, \(A = 2\) and \(B = 3\).

Step 2: Find the Amplitude

The amplitude of the function is the coefficient of the sine function. Hence, the amplitude is \(2\).

Step 3: Calculate the Period

The period of a sine function given by \(y = A \sin(Bt)\) is calculated as \( \frac{2\pi}{B} \). Substitute \(B = 3\) to get the period: \[ \text{Period} = \frac{2\pi}{3} \].

Step 4: Determine the Frequency

The frequency is the reciprocal of the period. Therefore, the frequency is \( \frac{1}{\text{Period}} = \frac{3}{2\pi} \).

Step 5: Graph the Function Over One Period

To sketch the graph of the function for one complete period, plot points for \( t \) ranging from \(0\) to \(\frac{2\pi}{3}\). At \( t=0 \), \( y=0 \). At \( t = \frac{\pi}{6} \), \( y = 2 \). At \( t = \frac{\pi}{3} \), \( y = 0 \). At \( t = \frac{\pi}{2} \), \( y = -2 \). Lastly, at \( t = \frac{2\pi}{3} \), \( y = 0 \).

Key Concepts

Amplitude

In simple harmonic motion, the amplitude is a measure of how far the object moves from its central position. It represents the maximum displacement of the object.
In the context of the sine function, like in our example with the equation \(y = 2 \sin 3t\), the amplitude corresponds to the absolute value of the coefficient in front of the sine term.

• For \(y = 2 \sin 3t\), the amplitude is \(2\).
• This value indicates the object moves 2 units away from its central (equilibrium) position in both the positive and negative direction.
• Amplitude demonstrates the peak reach of the object's motion on both sides of this mean point.

It is essential to understand that amplitude does not change over time in simple harmonic motion. Despite oscillating back and forth, the object always moves to the same maximum displacement.

Period

The period in simple harmonic motion describes the time it takes for the object to return to its starting position, completing one full cycle of movement. For functions of the form \(y = A \sin(Bt)\), the formula to determine the period is \( \frac{2\pi}{B} \).

• In the equation \(y = 2 \sin 3t\), \(B\) is \(3\).
• Thus, the period is \( \frac{2\pi}{3} \).
• This means it takes this amount of time for one complete oscillation or cycle to occur.

Understanding the period is key to predicting how the motion evolves over time. The shorter the period, the faster the object completes its cycles, while a longer period indicates slower motion.

Frequency

Frequency is a concept closely related to the period. It tells us how many complete cycles of motion occur in a unit of time. The relationship between period \(T\) and frequency \(f\) is reciprocal: \(f = \frac{1}{T}\).

• In our scenario, with the period as \( \frac{2\pi}{3} \), the frequency is \( \frac{3}{2\pi} \).
• This indicates how frequently the object repeats its cycle within a given time frame.
• Think of frequency as the rate of oscillations per time unit.

A higher frequency represents quicker oscillations, implying the object moves back and forth more times within the same duration. Conversely, a lower frequency suggests fewer oscillations in that span.