Problem 27 - 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. $$d=-5 \sin \frac{2 \pi}{3} t$$

Step-by-Step Solution

Answer

a. The maximum displacement is 5 inches. b. The frequency is \(1/3\) Hz. c. The time required for one cycle is 3 seconds.

Step 1: Find the maximum displacement

The maximum displacement is the absolute value of the coefficient of the sine function. In the equation \(d=-5 \sin \frac{2 \pi}{3} t\), the maximum displacement \(A=|-5|=5\) inches.

Step 2: Find the frequency

The term in the argument of the sine function, i.e., \(B = \frac{2\pi}{3}\), is equal to \(2\pi f\). Solving this equation for \(f\), the frequency, gives \(f = \frac{B}{2\pi} = \frac{1}{3}\) Hz.

Step 3: Find the time required for one cycle

The period \(T\), or time required for one full cycle, is the reciprocal of the frequency \(f\): \(T = \frac{1}{f} = 3\) seconds.

Key Concepts

Maximum Displacement

In the realm of simple harmonic motion (SHM), the maximum displacement, often referred to as the amplitude, is a key concept.
This is the farthest the object moves from its equilibrium position in either direction.
Let's look at the given equation: \[d = -5 \sin \left(\frac{2\pi}{3} t\right)\]
Here, the coefficient of the sine function is \(-5\).
The absolute value of this coefficient represents the maximum displacement, which is also known as the amplitude.

• The amplitude here is calculated as \(A = |-5| = 5\) inches.

This means the object can reach 5 inches away from its equilibrium position, both in the positive and negative directions.
Understanding maximum displacement is crucial because it tells us the extent of the motion.

Frequency

Frequency in simple harmonic motion refers to how often an object completes a full cycle of motion in one second.
It is measured in Hertz (Hz), which is equivalent to cycles per second.In our example, the equation is:\[d = -5 \sin \left(\frac{2\pi}{3} t\right)\]The frequency can be extracted from the coefficient of \(t\) inside the sine function.
This coefficient is \(\frac{2\pi}{3}\), often represented by \(B\).To find the frequency \(f\):

• The formula connecting \(B\) to frequency is \(B = 2\pi f\).
• Rearranging gives \(f = \frac{B}{2\pi} = \frac{1}{3}\) Hz.

This means the object completes \( \frac{1}{3} \) of a full cycle each second.
Understanding frequency is essential as it directly influences how rapidly a system oscillates.

Period of Motion

The period of motion is the time it takes for one complete cycle of simple harmonic motion.
It is denoted by the symbol \(T\) and is closely related to the frequency.From our equation:\[d = -5 \sin \left(\frac{2\pi}{3} t\right)\]We have already determined the frequency \(f\) as \(\frac{1}{3}\) Hz.To calculate the period:

• The period \(T\) is the reciprocal of the frequency: \(T = \frac{1}{f}\).
• Substituting the frequency gives \(T = \frac{1}{\frac{1}{3}} = 3\) seconds.

This means it takes 3 seconds for the object to return to its starting position after completing one full oscillation.
Knowing the period helps predict the timing of the motion.