Problem 60 - Precalculus Essentials

Question

An object in simple harmonic motion has a frequency of \(\frac{1}{4}\) oscillation per minute and an amplitude of 8 feet. Write an equation in the form \(d=a\) sin \(\omega t\) for the object's simple harmonic motion.

Step-by-Step Solution

Answer

The simple harmonic motion equation for the object is \(d = 8 \sin (\frac{\pi}{2} t)\).

Step 1: Determine the Required Values

Firstly, identify the given frequency and amplitude. Here, the frequency \(f\) is \(\frac{1}{4}\) oscillation per minute and the amplitude \(a\) is 8 feet.

Step 2: Convert the Frequency to Angular Frequency

The formula for converting frequency to angular frequency (\(\omega\)) is \(\omega = 2\pi f \). Now, substitute \(f\) with \(\frac{1}{4}\) into the formula to obtain the value for the angular frequency: \(\omega = 2 \pi \times \frac{1}{4} = \frac{\pi}{2} \) radians per minute.

Step 3: Insert the Values into Simple Harmonic Motion Formula

The equation of simple harmonic motion can be written in the form \(d = a \sin (\omega t)\). Replace \(a\) and \(\omega\) with the values obtained in steps 1 and 2 respectively. Therefore, the equation for the object's simple harmonic motion becomes \(d = 8 \sin (\frac{\pi}{2} t)\).

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) refers to a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. This scenario is akin to a child's swing moving back and forth or a mass attached to a spring when pulled and released.

To describe such motion mathematically, we usually employ an equation involving a sine or cosine function due to the periodic nature of SHM. The standard form of this equation is: \(d = a \sin(\omega t)\) or \(d = a \cos(\omega t)\), where \(d\) represents the displacement of the object at time \(t\), \(a\) is the amplitude, and \(\omega\) is the angular frequency of the oscillation.

Angular Frequency

Angular frequency, often denoted by the Greek letter \(\omega\), gives us a measure of how many radians an object in SHM covers per unit time. It is critical for understanding the rate at which the object oscillates which has the standard unit of radians per second.

To calculate angular frequency from the frequency of oscillation, we use the formula: \(\omega = 2\pi f\). The constant \(2\pi\) reflects the 360 degrees or \(2\pi\) radians in a full circle, bridging traditional frequency (oscillations per time) and the angular counterpart (radians per time). For instance, if an object has a frequency of \(\frac{1}{4}\) oscillation per minute, the angular frequency \(\omega\) would be \(\frac{\pi}{2}\) radians per minute, as seen in our exercise.

Amplitude

Understanding Amplitude in SHM

Amplitude in the context of SHM denotes the maximum displacement of the object from its equilibrium position. It is a measure of how far the object moves from the central position during its motion and is represented by \(a\) in the SHM equation.

The amplitude plays a significant role in determining the energy of the oscillation; a higher amplitude means greater energy. For example, in our exercise's context, an amplitude of 8 feet indicates that the object moves 8 feet away from its equilibrium position at the peak of its motion.

Oscillation

Oscillation describes the motion of an object as it moves back and forth repeatedly through a central or equilibrium position. This motion is characteristic of systems which exhibit SHM.

Each complete movement from one peak, through the equilibrium point, to the opposite peak and back again is known as a cycle. The number of these cycles per unit time is called the frequency of oscillation. In our example, the object performs \(\frac{1}{4}\) of an oscillation per minute, meaning it would take four minutes to complete a full cycle.