Solved Problem on Acoustics

A sprocket has 48 teeth and rotates 330 times per minute. Calculate the frequency of the sound produced by the vibration of a blade touching it.

Problem data:

Number of teeth in sprocket: n = 48;
Frequency of rotation: f = 330 rpm.

Problem diagram:

The sprocket rotates at a frequency of 330 rpm, making the blade vibrate.

Figure 1

Solution

First, we must convert the number of rotations of the sprocket from revolutions per minute (rpm) to hertz (Hz) used in the International System of Units (SI)

\[ \begin{gather} f=330\times\frac{\;\mathrm{revolutions}}{1\;\cancel{\mathrm{minute}}}\times\frac{1\;\cancel{\mathrm{minute}}}{60\;\mathrm{seconds}}=5.5\;\text{Hz} \end{gather} \]

The sprocket makes 330 revolutions in one minute or 5.5 revolutions in one second, as each revolution passes 48 teeth through the blade in 5.5 revolutions pass x, using the cross multiplication

\[ \begin{gather} \frac{1\;\mathrm{lap}}{48\;\mathrm{teeth}}=\frac{5.5\;\mathrm{laps}}{x\;\mathrm{teeth}}\\ x\;\mathrm{teeth}=\frac{48\;\mathrm{teeth}\times 5.5\;\cancel{\mathrm{laps}}}{1\;\cancel{\mathrm{lap}}}\\ x\;\mathrm{teeth}=264\;\mathrm{teeth} \end{gather} \]

Thus the blade is vibrated 264 times in 1 second, \( 264\;\dfrac{1}{\mathrm{s}} \) or

\[ \bbox[#FFCCCC,10px] {f=264\;\mathrm{Hz}} \]

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