Problem 18 - Algebra and Trigonometry

Question

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 6.25 in, frequency 60 \(\mathrm{Hz}\)

Step-by-Step Solution

Answer

The function is \( y(t) = 6.25 \cos(120\pi t) \).

Step 1: Identify the Form of the Function

Simple harmonic motion can be modeled by the function \( y(t) = A \cos(2\pi f t) \) or \( y(t) = A \sin(2\pi f t) \). Since the displacement is at its maximum at \( t=0 \), we use \( y(t) = A \cos(2\pi f t) \).

Step 2: Define the Amplitude

The amplitude \( A \) is the maximum displacement from the rest position. Here, \( A = 6.25 \) inches.

Step 3: Determine the Angular Frequency

Frequency \( f \) is given as 60 Hz, and angular frequency \( \omega \) is related as \( \omega = 2\pi f \). Therefore, \( \omega = 2\pi \times 60 = 120\pi \).

Step 4: Substitute Values into the Function

Substituting \( A = 6.25 \) and \( \omega = 120\pi \) into the cosine function gives us the model for the motion: \( y(t) = 6.25 \cos(120\pi t) \).

Key Concepts

Amplitude

In simple harmonic motion, amplitude is a crucial measurement. It represents the maximum displacement from the object's equilibrium or rest position. This distance measures how far the object moves from its center spot during vibration or oscillation.
For example, in the exercise provided, the amplitude is 6.25 inches. This means that at its peak, the object moves 6.25 inches away from its rest position.
Didn't get it? Imagine pushing a swing. The height it reaches from its middle point on either side is the amplitude.

• Amplitude indicates energy: Larger amplitudes mean more energy is needed to create the displacement.
• It is usually measured from the center, not from one extreme point to another.
• Amplitude remains constant as long as there is no damping.

Knowing the amplitude helps in determining the intensity and reach of the harmonic motion. It's a key factor to calculate when modeling these systems.

Frequency

Frequency refers to how often an object completes a full cycle of motion in one second. It is measured in hertz (Hz). When the frequency is given, you know how many oscillations occur per second.
In our problem, the frequency is 60 Hz, which means the object completes 60 cycles every second.

• Frequency determines the speed of oscillation: Higher frequency means the object moves quicker.
• It is expressed in cycles per second.
• Frequency is essential in defining the periods of waves and vibrations.

Understanding frequency is vital for analyzing the performance of oscillating systems. Knowing the frequency helps anticipate how fast changes in position occur over time.

Angular Frequency

Angular frequency, denoted by \(\omega\), is related to the frequency, but it describes how quickly the angle changes in radians per second. This is especially useful in circular or oscillating motions.
In the provided solution, the frequency was 60 Hz, leading to an angular frequency of \(120\pi\) radians per second. The formula connecting them is \( \omega = 2\pi f \).

• Angular frequency is expressed in radians per second.
• It simplifies calculations in wave physics and engineering when the oscillations are periodic.
• Angular frequency gives insight into the rotational velocity in circular motions.

Understanding angular frequency helps visualize the speed of rotation or oscillation in an intuitive manner, often leading to simpler solutions for modeling motion.