We have two tuning forks:

A known fork of frequency $n_2 = 310\,\text{Hz}.$

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 $314$ Hz or $306$ 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 “filing a tuning fork”:

Typically, if you file (remove mass from) the tips of the fork’s prongs, you make the prongs effectively shorter, and this raises the frequency of the fork.

If one wanted to lower the fork’s frequency by filing, that filing would have to be done near the “root” (base) of the prongs, which is less common in many standard problems (unless explicitly stated).

In most textbook treatments, “filing the fork” 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 fork’s frequency increases on filing:

Case A: If $n_1$ was $314$ Hz initially, any further increase would lead to, say, $315$, $316$, … etc. Then the difference $|n_1 - 310|$ would become $5, 6,\dots$, which would not stay at $4$.

Case B: If $n_1$ was $306$ 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 “filing raises the frequency and the beats remain $4$” is that the fork started at $306$ Hz. After filing, its frequency rose (possibly through some intermediate values) until it reached $314$ Hz—still exactly $4$ Hz away from $310$ Hz.

Final Answer

$ \boxed{\text{The initial frequency of the unknown fork is }306\,\text{Hz}.} $

Waves

Continue this paper

Similar Waves PYQs

More practice from this question