Question 21: Waves

Question

A tuning fork of unknown frequency produces 4 beats with tuning fork of frequency $$310 \mathrm{~Hz}$$. It gives the same number of beats on filing. The initial frequency of a tuning fork is

Accepted Answer

We have two tuning forks:
A known fork of frequency $n_2 = 310\,	ext{Hz}.$
An unknown fork of frequency $n_1$.
When they are sounded together (initially), they produce $4$ beats per second.
Therefore,
$    |n_1 - 310| \;=\; 4     \quad\Longrightarrow\quad    n_1 = 310 \pm 4.    $
So $n_1$ could be either $314$Ã¢ÂÂ¯Hz or $306$Ã¢ÂÂ¯Hz initially.
The fork of unknown frequency is then
filed
, and it is noted that the beat frequency with the $310$-Hz fork
remains
$4$.
Important physical fact about Ã¢ÂÂfiling a tuning forkÃ¢ÂÂ
:
Typically, if you file (remove mass from) the
tips
of the forkÃ¢ÂÂs prongs, you make the prongs effectively shorter, and this
raises
the frequency of the fork.
If one wanted to
lower
the forkÃ¢ÂÂs frequency by filing, that filing would have to be done near the Ã¢ÂÂrootÃ¢ÂÂ (base) of the prongs, which is less common in many standard problems (unless explicitly stated).
In most textbook treatments, Ã¢ÂÂfiling the forkÃ¢ÂÂ is taken to mean removing a small amount of mass at or near the free ends, thereby
increasing
its frequency.
Figuring Out Which Value $n_1$ Was Initially
We want to see which initial frequency ($306$ or $314$) is consistent with the beat frequency remaining $4$
after
the fork is filed.
If the forkÃ¢ÂÂs frequency
increases
on filing:
Case A
: If $n_1$ was $314$Ã¢ÂÂ¯Hz initially, any further increase would lead to, say, $315$, $316$, Ã¢ÂÂ¦ etc. Then the difference $|n_1 - 310|$ would become $5, 6,\dots$, which would
not
stay at $4$.
Case B
: If $n_1$ was $306$Ã¢ÂÂ¯Hz initially, increasing it could take it up to $307, 308, \dots, 314,\dots$. If finally it lands at $314$, then the difference with $310$ is again $4$. Hence the beat frequency would still be $4$.
Therefore, the scenario consistent with Ã¢ÂÂfiling raises the frequency and the beats remain $4$Ã¢ÂÂ is that the fork
started
at $306$ Hz. After filing, its frequency rose (possibly through some intermediate values) until it reached $314$ HzÃ¢ÂÂstill exactly $4$Ã¢ÂÂ¯Hz away from $310$ Hz.
Final Answer
$ \boxed{	ext{The initial frequency of the unknown fork is }306\,	ext{Hz}.} $

Answer

620 Hz

Answer

306 Hz

Answer

314 Hz

Answer

310 Hz

Waves

A tuning fork of unknown frequency produces 4 beats with tuning fork of frequency $310 \mathrm{~Hz}$. It gives the same number of beats on filing. The initial frequency of a tuning fork is

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