Chapter 8.5, Problem 53AYU

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the equation $d=a\sin \left(\omega t\right)$. If a tuning fork for the note $A$ above middle $C$ on an even-tempered scale ( ${\text{A}}_4$, the tone by which an orchestra tunes itself) has a frequency of 440 hertz (cycles per second), find $\omega$. If the maximum displacement of the end of the tuning fork is $0.01$ millimeter, determine the equation that describes the movement of the tuning fork.

Source: David Lapp. Physics of Music and Musical Instruments. Medford, M A:Tufts University, 2003

To determine

$\omega$. If the maximum displacement of the end of the tuning fork is $0.01$ millimeter,determine the equation that describes the movement of the tuning fork.

Answer to Problem 53AYU

$d = 0.01\text{sin}\left(880\pi t\right)$

Explanation of Solution

Given:

The end of a tuning fork moves in simple harmonic motion described by the equation

$\text{d} = \text{a}\text{sinωt}$

If a tuning fork for the note $\text{A}$ above middle $\text{C}$ on an even–tempered scale ($\text{A}$, the tone by which an orchestra 4tunes itself) has a frequency of 440 hertz (cycles per second).find $\text{v}$. If the maximum displacement of the end of the tuning fork is $0.01$ millimeter,

Formula used:

$\text{d} = \text{a}\text{sinωt}$

Calculation:

The maximum displacement is the amplitude. Therefore $a = 0.01$.

The frequency is given by $f = \frac{\omega }{2\pi } = 440.$

Therefore $\omega = 880\pi$ and the movement of the tuning fork is described by the equation

$\text{d} = \text{a}\text{sinωt}$

substituting the values we get,

$d = 0.01 \text{sin}\left(880\pi t\right)$