A tuning fork C produces 8 beats per second with another tuning fork D of frequency 340Hz. When the prongs of the tuning fork C are filed a little, the number of beats produced per second decreases to 4. Find the frequency of the tuning fork C before filing its prongs.

To determine the frequency of tuning fork C before its prongs were filed, we can use the concept of beats in sound waves. The number of beats per second is the absolute difference between the frequencies of the two tuning forks. Let's break this down step by step.

Understanding Beats

When two sound waves of slightly different frequencies interact, they produce a phenomenon known as beats. The beat frequency is calculated as:

• Beat Frequency = |f1 - f2|

Here, f1 and f2 are the frequencies of the two tuning forks. In this case, we have:

• f2 (frequency of tuning fork D) = 340 Hz
• Initial beat frequency = 8 beats per second

Setting Up the Equation

Let’s denote the frequency of tuning fork C as f1. The initial condition gives us:

• |f1 - 340| = 8

This absolute value equation can yield two scenarios:

• f1 - 340 = 8 (1)
• 340 - f1 = 8 (2)

Solving the First Scenario

From equation (1):

• f1 = 340 + 8 = 348 Hz

Solving the Second Scenario

From equation (2):

• f1 = 340 - 8 = 332 Hz

Analyzing the Effect of Filing the Prongs

After filing the prongs of tuning fork C, the number of beats produced decreases to 4 beats per second. This means that the new frequency of tuning fork C is either closer to or further from 340 Hz, resulting in a new equation:

• |f1' - 340| = 4

Where f1' is the new frequency of tuning fork C after filing. This again gives us two scenarios:

• f1' - 340 = 4 (3)
• 340 - f1' = 4 (4)

Finding the New Frequency

From equation (3):

• f1' = 340 + 4 = 344 Hz

From equation (4):

• f1' = 340 - 4 = 336 Hz

Determining the Original Frequency

Now, we need to check which of the original frequencies (348 Hz or 332 Hz) could lead to the new frequencies (344 Hz or 336 Hz) after filing. If we consider:

• If f1 = 348 Hz, then after filing, it could decrease to 344 Hz (which fits).
• If f1 = 332 Hz, it could only increase to 336 Hz (which also fits).

Both scenarios are valid, but since we are looking for the frequency of tuning fork C before filing, we can conclude that:

• The frequency of tuning fork C before filing its prongs is either 348 Hz or 332 Hz.

Final Thoughts

In this case, both frequencies are mathematically valid, but typically, filing the prongs would lower the frequency. Therefore, the more likely original frequency of tuning fork C is 348 Hz.