Chapter 8.5, Problem 54AYU

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the equation $d=a\sin (\omega t)$. If a tuning fork for the note $E$ above middle $C$ on an even-tempered scale $\left(E_4\right)$ has a frequency of approximately $329.63$ hertz (cycles per second), find $\omega$. If the maximum displacement of the end of the tuning fork is $0.025$ millimeter, determine the equation that describes the movement of the tuning fork.

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

To determine

$\omega$ and determine the equation that describes the movement of the tuning fork.

Answer to Problem 54AYU

$d = 0.025\text{sin}\left(659.26\pi t\right)$

Explanation of Solution

Given:

The end of a tuning fork moves in simple harmonic motion described by the equation $\text{d} = \text{a} \text{sin(ωt)}$. The tuning fork for the note $\text{E}$ above middle $\text{C}$ on an even–tempered scale (E ) has a frequency of approximately $329.63$ hertz(cycles per second). The maximum displacement of the end of the tuning fork is $0.025$ millimeter.

Formula used:

$\text{d} = \text{a} \text{sinωt}$

Calculation:

The maximum displacement is the amplitude. Therefore $a = 0.025$. The frequency is given by $f = \frac{\omega }{2\pi } = \mathrm{329.63.}$

Therefore $\omega = 659.26\pi$ and the movement of the tuning fork is described by the equation

$\text{d} = \text{a} \text{sinωt}$

$d = 0.025\text{sin}\left(659.26\pi t\right)$