Question

Two tuning forks $X$ and $Y$ are of frequencies $280\, Hz$ and $284\, Hz$. A third tuning fork $Z$ is of unknown frequency, when $X$ and $Z$ are sounded together certain beats are heard per second. When $Y$ and $Z$ are sounded together beat frequency is found to be thrice as great. The frequency of $Z$ is

• A. $282\, Hz$
• B. $286\, Hz$
• C. $280\, Hz$
• D. $278\, Hz$

Answer 1

Answer: (D)

Given, frequency of tuning fork $X$ and $Y$ are,
$n_{X}=280\, Hz$
$n_{Y}=284\, Hz$
$n_{Z}=?$
When tuning forks $X$ and $Z$ are sounded together, then beats produced is $b$ (assume).
$\therefore n_{z}-n_{X}=b...$ (i)
Again, when $Y$ and $Z$ are sounded together, then
$n_{z}-n_{Y}=3 b...$ (ii)
Dividing Eq. (i) by Eq. (ii), we have
$\frac{n_{Z}-n_{X}}{n_{Z}-n_{Y}}=\frac{b}{3 b}$
$\Rightarrow \frac{n_{Z}-280}{n_{Z}-284}=\frac{1}{3}$
$\Rightarrow 3 n_{z}-840=n_{z}-284$
$\Rightarrow 3 n_{z}-n_{Z}=840-284$
$\Rightarrow 2 n_{Z}=556$
$\Rightarrow n_{Z}=278 Hz$