Frequencies of two tuning forks are in the ratio 20:21. When sounded together 8 beats are heard per second. What are their frequencies :

Answer

To solve the problem of finding the frequencies of two tuning forks that are in the ratio of 20:21 and produce 8 beats per second, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Frequencies**:
Let the frequencies of the two tuning forks be \( f_1 \) and \( f_2 \). According to the problem, the ratio of their frequencies is given as:
\[ \frac{f_1}{f_2} = \frac{20}{21} \]
This can be expressed as:
\[ f_1 = \frac{20}{21} f_2 \]

2. **Understand Beat Frequency**:
The beat frequency is defined as the absolute difference between the two frequencies:
\[ \text{Beat Frequency} = |f_2 - f_1| \]
We are given that the beat frequency is 8 beats per second:
\[ |f_2 - f_1| = 8 \]

3. **Substitute \( f_1 \) in the Beat Frequency Equation**:
Substitute \( f_1 \) from Step 1 into the beat frequency equation:
\[ |f_2 - \frac{20}{21} f_2| = 8 \]
Simplifying this gives:
\[ |f_2 - \frac{20}{21} f_2| = |f_2 \cdot (1 - \frac{20}{21})| = |f_2 \cdot \frac{1}{21}| \]
Therefore, we have:
\[ \frac{1}{21} f_2 = 8 \]

4. **Solve for \( f_2 \)**:
Multiply both sides by 21 to solve for \( f_2 \):
\[ f_2 = 8 \times 21 = 168 \text{ Hz} \]

5. **Calculate \( f_1 \)**:
Now that we have \( f_2 \), we can find \( f_1 \) using the equation from Step 1:
\[ f_1 = \frac{20}{21} f_2 = \frac{20}{21} \times 168 \]
Simplifying this gives:
\[ f_1 = \frac{20 \times 168}{21} = \frac{3360}{21} = 160 \text{ Hz} \]

6. **Final Frequencies**:
The frequencies of the two tuning forks are:
\[ f_1 = 160 \text{ Hz}, \quad f_2 = 168 \text{ Hz} \]

### Summary of the Solution:
The frequencies of the two tuning forks are \( f_1 = 160 \text{ Hz} \) and \( f_2 = 168 \text{ Hz} \).