Problem 26 - 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. $$ d=\frac{1}{3} \sin 2 t $$

Step-by-Step Solution

Answer

The maximum displacement is \(\frac{1}{3}\) inches, the frequency is \(\frac{1}{\pi}\) Hz, and the time required for one cycle is \(\pi\) seconds.

Step 1: Determine Maximum Displacement

The maximum displacement also known as the amplitude in a simple harmonic motion is denoted by the coefficient of the sine function. So, in this case, the amplitude is \(\frac{1}{3}\).

Step 2: Find the Frequency

The frequency in the motion is given by the coefficient of \(t\) in the sine function, divided by \(2\pi\). Thus, the frequency \(f\) is given by \(f = \frac{2}{2\pi} = \frac{1}{\pi}\).

Step 3: Find the Time Required for One Cycle

The period of the motion, which is the time required for one complete cycle, is the reciprocal of the frequency. Thus, the time \(T\) is given by \(T=\frac{1}{f} = \pi\) seconds.

Key Concepts

Amplitude in Harmonic Motion

In simple harmonic motion (SHM), amplitude represents the greatest distance an object travels from its equilibrium position. It is a measure of how far the object will move away from this central point. This concept is crucial because it determines the maximum displacement that the object will experience during its motion.

In the exercise, the equation given for SHM is: \[ d = \frac{1}{3} \sin 2t \. \] The amplitude can be obtained from the coefficient of the sine function, which is the term without the trigonometric part and the time variable, thus making it \( \frac{1}{3} \) inches.

The amplitude is significant in numerous applications, such as engineering, because it can tell us about the potential stress and strains on materials under periodic forces. Understanding how to extract the amplitude from an equation like this informs us about the energy involved in the motion, as energy in harmonic systems is proportional to the square of the amplitude.

Frequency of Oscillation

The frequency of oscillation in SHM informs us about how many cycles or vibrations occur in one second. It is a direct measure of how 'fast' the object is moving back and forth. In mathematical terms, frequency is denoted as the number of cycles per second and is measured in Hertz (Hz).

The given SHM equation \[ d = \frac{1}{3} \sin 2t \] contains the term \( 2t \), where the coefficient of \( t \), which is '2' in this case, is related to the angular frequency \( \omega \). To find the actual frequency (f), we divide this coefficient by \( 2\pi \), arriving at \( f = \frac{1}{\pi} \) Hz.

Higher frequency means more cycles per second, which has real-world implications, such as in the tuning of musical instruments, where frequency determines pitch, or in electrical systems, where frequency is crucial for the stability and efficiency of power transmission.

Period of Motion

The period of motion, often represented by the symbol \( T \), is the time it takes for one complete oscillation or cycle in SHM. This concept is the inverse of frequency and provides valuable insights into the duration of each cycle. Simply put, if you know how often something moves back and forth in a second (frequency), the period tells you how long one back-and-forth cycle takes.

In the provided exercise, to calculate the period, we use the reciprocal of the frequency. From the frequency we found earlier, \( f = \frac{1}{\pi} \), so the period \( T \) is \( T = \frac{1}{f} = \pi \) seconds.

Understanding the period of motion is important, especially when timing is crucial. For example, in mechanical clocks, the period of the pendulum governs the accurate keeping of time. In digital electronics, the clock period dictates the speed at which the processor operates. Knowledge of the period also proves fundamental when synchronizing various oscillatory systems to function harmoniously together.