Problem 61 - Algebra and Trigonometry

Question

A piano tuner uses a tuning fork. If middle \(\mathrm{C}\) has a frequency of 264 vibrations per second, write an equation in the form \(d=\sin \omega t\) for the simple harmonic motion.

Step-by-Step Solution

Answer

The equation in the form \(d = sin \omega t\) for the simple harmonic motion is \(d = sin(528\pi t)\).

Step 1: Understand Simple Harmonic Motion

Simple harmonic motion describes a motion in which the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It can be represented by the equation \(d = sin \omega t\), where \(d\) is the displacement, \(\omega\) is the angular frequency, and \(t\) is the time.

Step 2: Convert Frequency to Angular Frequency

Angular frequency, \(\omega\), is given by \(2\pi\) times the frequency. As we are given that the frequency of vibration for the middle C note is 264 vibrations per second, we can determine the angular frequency. So, \(\omega = 2\pi \times 264 = 528\pi\) radians per second.

Step 3: Substitute into Equation

We substitute the angular frequency that we have calculated into the simple harmonic motion equation. So, the resulting equation is \(d = sin(528\pi t)\). This equation describes the motion of the tuning fork for the middle C note.

Key Concepts

Frequency Conversion

Converting frequency to angular frequency is an essential step in analyzing simple harmonic motion. Frequency refers to the number of complete vibrations or cycles per second and is measured in Hertz (Hz).
To convert it into angular frequency, which tells us how fast an object rotates or oscillates, we use the relation:

• Angular Frequency: \( \omega = 2\pi f \)

Here, \( f \) is the frequency. The factor of \( 2\pi \) accounts for converting the circular motion from cycles to radians.
This is because one complete cycle in circular motion corresponds to \( 2\pi \) radians.
For the middle C note on a piano, given as 264 vibrations per second, the conversion results in:

• \( \omega = 2\pi \times 264 = 528\pi \) radians per second

Thus, frequency conversion is crucial for expressing vibrations in terms that align with rotational motion.

Angular Frequency

Angular frequency is a pivotal concept in the study of oscillations, especially in simple harmonic motion. It describes how quickly something oscillates regarding its phase in radians.
The symbol \( \omega \) represents angular frequency, and it is measured in radians per second.
The formula to derive angular frequency from ordinary frequency is:

• \( \omega = 2\pi f \)

Angular frequency illustrates how fast an object travels through its complete cycle, translating cycles per second into radians.
For the middle C note with a frequency of 264 Hz, angular frequency computation is:

• \( \omega = 2\pi imes 264 = 528\pi \) radians per second

Understanding angular frequency aids in analyzing wave properties and motions in harmonic oscillators.

Harmonic Motion Equation

The harmonic motion equation is central to understanding simple harmonic motion, a type of periodic motion.
The equation formula is generally presented as:

• \( d = \sin \omega t \)

Here, \( d \) stands for displacement from an equilibrium position, \( \omega \) is angular frequency, and \( t \) is time.
For the tuning fork of middle C, based on the known frequency and converted angular frequency, the harmonic motion equation becomes:

• \( d = \sin(528\pi t) \)

This equation reflects the repetitive nature of the tuning fork's vibration.
It emphasizes the principal characteristics:

• The motion is sinusoidal, following a sine curve.
• Displacement varies smoothly around a neutral position.
• Time and angular frequency dictate how motion progresses.

This expression captures how waves, sound vibrations, and objects behave under oscillatory systems.