Question

The diagram below shows the basic idea behind a disk siren. It consists of a disk in which there are $16$ equally spaced holes, all at the same distance from its axle. When a jet of air is directed at the holes and the disc is rotated at a particular constant rate, the frequency of the note produced is $320Hz$. When a disk containing $24holes$ is rotated at $4/3$ times the rate, then frequency of note produced is $320nHz$ Find the value of $n$.

Answer 1

Step 1: Find the angular speed of the disc when 16 holes are in the disc.

Given, there are 16 equally spaced holes in the disc,
Frequency of the note produced, $f=320Hz$

Angular separation between any two holes,
. $\theta =\frac{2\pi }{16}rad$

Let angular velocity of the disc be $\omega$,

Time passed when disk rotates by

$\frac{2\pi }{16}rad,$ $t=\frac{\theta }{\omega }=\frac{2\pi }{16\omega }$

Frequency of sound

$f=\frac{16\omega }{2\pi }=320Hz$

$\Rightarrow \omega =40\pi rad/s$

Step 2: Find the value of $n$.

Given, disc rotates at the rate $\frac{4}{3}\omega$ when 24 holes are in the disc,

Angular separation between two holes

$\theta \text{'}=\frac{2\pi }{24rad}$

Time passed

$t\text{'}=\frac{2\pi }{24\times \frac{4}{3}\omega }$

Frequency

$f=\frac{24\times 4\omega }{6\pi }=320n$

$\frac{24\times 4\times 40\pi }{6\pi }=320n$

$n=2$

Final answer: $2$