Problem 28 - Precalculus Essentials

Question

An object moves in simple harmonic motion described by the given equation, where \(t\) is measured in seconds and \(d\) in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. $$d=-4 \sin \frac{3 \pi}{2} t$$

Step-by-Step Solution

Answer

a. The maximum displacement is 4 inches.\nb. The frequency is 0.75 Hz.\nc. The time required for one cycle is 1.33 seconds.

Step 1: Identify maximum displacement (Amplitude)

The maximum displacement is given by the absolute value of the coefficient of the sine function in the equation. In this case, it is |-4| = 4. So, the maximum displacement is 4 inches.

Step 2: Calculate Frequency

The frequency of the motion is given by dividing the coefficient of \(t\) in the equation by \(2\pi\). In this case, the coefficient is \(3\pi/2\), so the frequency is \(\frac{3\pi/2}{2\pi}\) = 0.75 cycles per second. Hence, the frequency of the motion is 0.75 Hz.

Step 3: Find time required for one cycle (Period)

The time period of the motion, which is the time taken for one complete cycle, is given the reciprocal of the frequency. So, the time required for one cycle is \(\frac{1}{0.75}\) = 1.33 seconds.

Key Concepts

Amplitude

In simple harmonic motion, the amplitude is the maximum distance an object moves from its equilibrium position. It indicates the extent of displacement over time.
From the given equation, the amplitude is represented by the coefficient of the sine function, in this case, \( -4 \).
Because amplitude is always a positive value, take the absolute value of this coefficient. Thus, the amplitude is \(|-4| = 4\) inches.

• Equilibrium Position: The central point where the object rests when not in motion.
• Displacement: The distance from the equilibrium position in a specific direction.

The amplitude, therefore, tells us that the object can be displaced a maximum of 4 inches upward or downward from its equilibrium. Understanding this concept helps in visualizing how far an object in harmonic motion is capable of moving away from its starting point.

Frequency

The frequency in simple harmonic motion indicates how many complete cycles occur per unit of time, usually measured in Hertz (Hz), or cycles per second. It gives us a sense of how fast the oscillating motion is.
In our equation, the frequency can be determined using the coefficient of \( t \), which is \( \frac{3\pi}{2} \).
Compute the frequency by dividing this coefficient by \( 2\pi \): \(rac{3\pi/2}{2\pi} = 0.75 \). Hence, the frequency is 0.75 cycles per second, or 0.75 Hz.

• One Cycle: A complete motion returning to the starting point.
• Importance: A higher frequency means more cycles per second, indicating quicker motion.

This means the object completes 0.75 oscillations every second. Grasping frequency helps predict how often the object returns to its specific position in a given timeframe.

Period

The period is crucial in understanding simple harmonic motion. It's the duration of time needed to complete one full cycle of motion, measured in seconds. This provides insight into how long it takes for the object to return to its starting position.
We can find the period by taking the reciprocal of the frequency. Given that the frequency in our example is 0.75 Hz, the period \( T \) is computed as:\[ T = \frac{1}{0.75} = 1.33 \]seconds.

• Complete Cycle: The journey from the start position, to maximum displacement, and back.
• Measurement: Typically recorded in seconds for each cycle.

Knowing the period helps us determine how long it will take for the object to undergo one complete movement before starting the pattern again. With a period of 1.33 seconds, the object returns to its starting position after this time has elapsed.