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Step-by-Step Solution
Step 1: Identify the Known Information
A tuning fork of frequency $256\,\text{Hz}$ is sounded along with a piano string. They produce $5$ beats per second initially, so the difference between their frequencies is $5\,\text{Hz}$.
Step 2: Interpret the Beat Frequency Condition
When two sources produce beats, the beat frequency equals the absolute difference in their frequencies. Thus, if $f_{\text{tuning fork}} = 256\,\text{Hz}$ and $f_{\text{piano}}$ is the frequency of the piano string, then:
$|f_{\text{piano}} - 256| = 5$.
This implies:
$f_{\text{piano}} = 256 + 5 = 261\,\text{Hz}$ or $f_{\text{piano}} = 256 - 5 = 251\,\text{Hz}$.
Step 3: Analyze the Effect of Increasing Tension
Increasing the tension of the piano string increases its frequency. Given that the beat frequency decreases from $5$ to $2$, the piano’s frequency must have moved closer to $256\,\text{Hz}$. If the piano’s frequency were initially higher than the tuning fork (i.e., $261\,\text{Hz}$), an increase in tension would push it even higher, causing a larger beat difference rather than a smaller one. Therefore, the only possibility for the beat frequency to decrease is if the piano's initial frequency was below $256\,\text{Hz}$.
Step 4: Conclude the Initial Frequency
Hence, the piano’s initial frequency must be $251\,\text{Hz}$. After tension is increased, its frequency rises and gets closer to $256\,\text{Hz}$, causing the beat frequency to drop from $5$ to $2$.
Answer
The frequency of the piano string before increasing the tension was $251\,\text{Hz}$.
