Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Harmonic Motion For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{64} \sin 792 \pi t$$

Step-by-Step Solution

Answer

The maximum displacement (amplitude) is 1/64, the frequency is 396, the value of \(d\) at \(t = 5\) is some value in the interval \([-1/64, 1/64]\) and the least positive value of \(t\) at which \(d = 0\) is \(1/792\).

Step 1: Finding the maximum displacement

The maximum displacement or amplitude of a normal sine function is 1 but in this function it is multiplied by \(1/64\). Thus, the maximum displacement is \(1/64\).

Step 2: Finding the frequency

The frequency of the function is found by dividing the coefficient of \(t\) by \(2\pi\). Here, the coefficient of \(t\) is \(792\pi\). Thus the frequency can be found as \( \frac{792\pi }{2\pi} \), cancelling out \( \pi \), we get 396.

Step 3: Finding the value of \(d\) when \(t=5\)

We substitute \(t = 5\) into the function to get the displacement at that time: \(d = \frac{1}{64} \sin (792 \pi * 5)\). The sine function oscillates between -1 and 1, so the value is some number in the interval \([-1/64, 1/64]\).

Step 4: Finding the least positive value of \(t\) for which \(d=0\)

For \(d\) to be zero, the value inside the sine function needs to be a multiple of \(\pi\). The earliest this happens is when \(t\) is \(1/792\), since \(\sin(792\pi \times 1/792) = \sin(\pi)\), and \(\sin(\pi) = 0\). So, the least positive value of \(t\) for which \(d=0\) is \(1/792\).

Step 5: Verify the results using graphing utility

To ensure that the findings are accurate, it is recommended to plug the function \(d = 1/64\sin(792\pi t)\) into a graphing utility and check its behaviour. This step confirms the maximum displacement, frequency and points where the displacement \(d = 0\).

Key Concepts

Amplitude

Amplitude refers to the maximum displacement of a wave from its rest position. It's akin to how tall ocean waves are from the sea level.
For simple harmonic motion (SHM), which is a repetitive back-and-forth motion passed by objects in oscillation, the amplitude is a key indicator of how "strong" the motion is.
In trigonometric functions like the sine function, the regular amplitude is 1 unless altered by a scaling factor.

• In our given equation for harmonic motion, \( d = \frac{1}{64} \sin(792 \pi t) \), the amplitude is scaled by \(\frac{1}{64}\).
• The amplitude is thus \(\frac{1}{64}\), indicating a relatively small range of movement compared to when the amplitude is 1 in a regular sine wave.

This modest amplitude has practical implications such as representing slight vibrations or small displacements in engineering contexts.

Frequency

Frequency describes how often the wave oscillates over a unit of time. In the context of SHM, it refers to the number of complete cycles or repetitions of the oscillation per second.

• It gives us insight into how "fast" the waves are occurring.
• Technically, it is calculated as the coefficient of \(t\), divided by \(2\pi\).

For the equation provided, \( d = \frac{1}{64} \sin(792 \pi t) \), the coefficient of \(t\) is \(792 \pi\).
By computing \( \frac{792\pi }{2\pi} \), the \(\pi\) cancels out leaving us with 396.
This means that the frequency is 396 cycles per second, a rapid series indicating high-frequency oscillation.
Understanding frequency helps in various fields such as signal processing or acoustic engineering, where tuning of vibrations or sound waves is critical.

Graphing Utility

A graphing utility is a digital tool that allows the visualization of mathematical functions and their behavior.
It's an essential resource for students and professionals to understand complex functions like those involving trigonometric operations.

• By plotting the function \(d = \frac{1}{64}\sin(792\pi t)\), one can observe the oscillations, verify frequency, amplitude, and identify zero-points efficiently.
• It confirms predictions made by mathematical calculations.

Such utilities are available as software applications on various platforms, enabling users to explore functions interactively.
By manipulating variables and observing changes in real-time, users can deeply understand wave behaviors, enhancing problem-solving skills.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to displacement and acts in the direction opposite to that of displacement.
Classic examples include the oscillation of a pendulum, vibration of a spring, or sound waves.
For our function, \( d = \frac{1}{64}\sin(792\pi t) \), we see a sine function which is central to describing SHM.

• The sine function's periodic nature makes it an ideal tool for modeling the sinusoidal oscillations characteristic of SHM.

Understanding SHM is fundamental in physics and engineering as it forms the basis for analyzing cyclic motions and designing systems influenced by repetitive forces.
Whether in designing natural frequency systems or minimizing vibrational impact, SHM is crucial.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are mathematical functions of an angle. They relate the angles of a triangle to the lengths of its sides and are foundational in studying wave patterns.
They provide a powerful model for oscillatory phenomena.
In our exercise, the sine function is prominent in modeling harmonic motion with the equation \(d = \frac{1}{64}\sin(792\pi t)\).

• The sine wave model aligns perfectly with the periodic nature of waves and oscillations.
• Through trigonometry, we represent periodic events that repeat in regular intervals.

Trigonometric functions are essential in diverse fields, from engineering and physics to music and seismology, making them indispensable tools for mathematically modeling cycles and amplitudes.