Problem 25 - Algebra and Trigonometry

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. $$ d=\frac{1}{2} \sin 2 t $$

Step-by-Step Solution

Answer

The maximum displacement is \(0.5\) inches, the frequency is \(\frac{1}{\pi}\) Hz, and the time required for one cycle is \(\pi\) seconds.

Step 1: Compute Maximum Displacement

Maximum displacement, also known as the amplitude, is the coefficient of the sine function in the given equation. Hence, it can be obtained directly from the equation \(d = 0.5 \sin 2t\). The maximum displacement is \(0.5\) inches.

Step 2: Compute Frequency

In simple harmonic motion, the frequency is given by \(\frac{\omega}{2\pi}\), where \(\omega\) is the angular frequency. The angular frequency is the coefficient of the time variable \(t\) in the equation, which is \(2\). Therefore, the frequency \(f\) is \(\frac{2}{2\pi} = \frac{1}{\pi}\) Hz.

Step 3: Compute Time Required for One Cycle

The time required to complete one cycle, or the period \(T\), is the reciprocal of the frequency. The period is therefore \(T = \frac{1}{f} = \frac{1}{\frac{1}{\pi}} = \pi\) seconds.

Key Concepts

Understanding Maximum Displacement

In simple harmonic motion, maximum displacement is the farthest point that an object moves from its equilibrium position. This is also known as amplitude.
As seen in the equation described, the maximum displacement is derived from the coefficient in front of the sine function.
For the equation \(d = \frac{1}{2} \sin 2t\), the amplitude, or maximum displacement, is \(0.5\) inches.
This value represents how far the object moves from its center in both directions, positive and negative.

• It serves as an indicator of the energy in the system.
• Higher amplitude values indicate larger swings in motion.
• Ampitude is essential for understanding the intensity of the vibration or oscillation.

Recognizing amplitude in such equations helps in predicting behavior in systems like springs or pendulums.

Frequency in Simple Harmonic Motion

Frequency is a crucial concept in understanding how often an event repeats over time. In the context of simple harmonic motion, frequency tells us how many cycles occur in a second.
This repetition rate is essential in identifying the nature of oscillating systems.
In the given equation, \(d = \frac{1}{2} \sin 2t\), the coefficient of \(t\) provides the angular frequency which helps to determine frequency.

• Angular frequency (\(\omega\)) is the coefficient of the time variable, in this case, \(2\).
• Frequency \(f\) is calculated by dividing angular frequency by \(2\pi\): \(\frac{2}{2\pi} = \frac{1}{\pi}\) Hz.
• This means the motion completes approximately \(0.318\) cycles per second.

A higher frequency means faster oscillations per second, significant in sound waves and signal broadcast settings.

Period of a Wave

The period of a wave is the time taken for one complete cycle of oscillation. It is the reciprocal of frequency, giving insight into the duration of each cycle.
For the given equation, knowing the period allows us to determine how long it takes for an object to return to its initial position.
By finding the reciprocal of frequency, we compute the period \(T\).
Given that \(f = \frac{1}{\pi}\) Hz, the period is \(T = \pi\) seconds.

• The period \(T\) helps us predict the timing of oscillations.
• Longer periods mean more time for each cycle, seen in slower moving pendulums.
• Essential for designing clocks, musical tunes, and understanding natural rhythms.

Understanding period helps clarify the pace of oscillating systems, making it easier to handle timing in mechanics and wave-related fields.