Question

A tunning fork of known frequency $256Hz$ makes 5 beats per second with the vibrating string of the piano. The beat frequency decreases to 2 beats per second when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was
A. $256+5\,Hz$
B. $256+2\,Hz$
C. $256-2\,Hz$
D. $256-5\,Hz$

Answer 1

The difference in the frequency of the tunning fork and the string of the piano will give us the number of beats per second.

Formula used: ${{f}_{2}}-{{f}_{1}}=n$
Where,
${{f}_{1}}$ is the frequency of the tunningfork
${{f}_{2}}$ is the frequency of the Piano
$n$ is the number of beats per second or the beat frequency

Complete step by step solution:
Given that, the frequency of the tunning fork, ${{f}_{1}}=256\,\,Hz$
And the strings of the piano 5 beats per second
Let ${{f}_{2}}$ be the frequency of string of the piano
Therefore, ${{f}_{2}}-{{f}_{1}}=\pm 5$
Or, we can say that ${{f}_{2}}=\left( 256+5 \right)Hz\,\,or\,\,\left( 256-5 \right)Hz$
We know that $f\propto \sqrt{Tension}$

Now, according to the question, when the tension in the string increases, the beat frequency decreases by 2 beats per second. And this is possible when the string frequency is increasing but the beat frequency is decreasing.
So if we suppose ${{f}_{2}}=261\,\,Hz$, then on increasing the tension, string frequency will be something more than 261 Hz, let the value be $265\,\,Hz$
Now this simply means that $265\,-256>261-256$, Which is not possible as the no. of beats is decreasing.

Thus, the value of string frequency of the piano before increasing the tension will be $251\,\,Hz$or can be written as ${{f}_{2}}=256-5\,\,Hz$
The correct answer is (D).

Additional information: A tunning is basically a U-shaped instrument which is in the form of two pronged forks. It is used to produce a specific constant pitch when striked against any surface or body.
It emits a pure musical tone once the high overtone fades out.

Note: The frequency of the string of a piano is directly proportional to the square root of the tension in the string. It means more the tension in the string higher will be the frequency of the string.