Animated Solution for Physics - Waves: A tuning fork of known frequency 256 Hz makes 5 beats per second with the vibrating string of a 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

Solving the Piano String Mystery

Imagine you are tuning a grand piano. You strike a tuning fork, and simultaneously, you hit a piano key. Instead of a smooth, continuous sound, you hear a rhythmic pulsing—a "wah-wah-wah" effect. These are beats, a beautiful phenomenon in physics that occurs when two sound waves of slightly different frequencies interfere with each other.

In this classic problem, we are given a tuning fork with a known frequency of $f_1=256\text{ Hz}$. When sounded with a piano string, it produces $5$ beats per second. Our mission is to find the original frequency of the piano string before its tension was altered.

Two Frequencies, One Beat

The fundamental equation for beat frequency is elegantly simple. It is the absolute difference between the two interfering frequencies:

Because of the absolute value, the piano string's frequency ($f_2$) could be either $5\text{ Hz}$ above or $5\text{ Hz}$ below the tuning fork's frequency. This gives us two initial suspects:

To avoid getting tangled in algebra, the best way to solve beat frequency problems is to draw a number line. Place the known frequency ($256\text{ Hz}$) in the center. Then, plot the two possible frequencies ($251\text{ Hz}$ and $261\text{ Hz}$) on either side. The distance from the center represents the beat frequency.

The problem introduces a crucial twist: the tension in the piano string is slightly increased. From the laws of stretched strings, we know that the frequency of a string is directly proportional to the square root of its tension:

Therefore, increasing the tension will always increase the frequency of the string. On our number line, this means the true frequency must shift to the right.

Now, we test our two suspects against the final clue: after increasing the tension, the beat frequency decreases to $2$ beats per second. This means the new frequency must be closer to the center ($256\text{ Hz}$) than it was before.

Case 1: What if the original frequency was $261\text{ Hz}$? If we increase the tension, the frequency increases further (e.g., to $263\text{ Hz}$). This moves it away from $256\text{ Hz}$, causing the beat frequency to increase ($|256-263|=7\text{ Hz}$). This contradicts our clue. So, $261\text{ Hz}$ is impossible.

Case 2: What if the original frequency was $251\text{ Hz}$? If we increase the tension, the frequency increases (e.g., to $254\text{ Hz}$). This moves it closer to $256\text{ Hz}$, causing the beat frequency to decrease ($|256-254|=2\text{ Hz}$). This perfectly matches our clue!

Therefore, the original frequency of the piano string must have been $251\text{ Hz}$, which is mathematically expressed as $(256-5)\text{ Hz}$.