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

Hint: 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.