The frequencies of three tuning forks A, B and C have a relation `n_(A)gt n_(B) gt n_(C)`. When the forks A and B are sounded together the number of beats produced is `n_(1)` When A and C are sounded together the number of beats produced is `n_(2)` then the number of beats produced when B and C are sounded together is

Options

A. `n_(1)+n_(2)`
B. `(n_(1)+n_(2))/(2)`
C. `n_(2)-n_(1)`
D. `n_(1)-n_(2)`

Answer

To solve the problem, we need to find the number of beats produced when tuning forks B and C are sounded together, given the relations between their frequencies and the number of beats produced when other pairs are sounded together.

### Step-by-Step Solution:

1. **Understanding the Frequency Relations**:
- We have three tuning forks A, B, and C with frequencies \( n_A > n_B > n_C \).

2. **Beats Between A and B**:
- When forks A and B are sounded together, the number of beats produced is \( n_1 \).
- The beat frequency is given by the absolute difference of their frequencies:
\[ n_A - n_B = n_1 \quad \text{(Equation 1)} \]

3. **Beats Between A and C**:
- When forks A and C are sounded together, the number of beats produced is \( n_2 \).
- Similarly, the beat frequency is:
\[ n_A - n_C = n_2 \quad \text{(Equation 2)} \]

4. **Finding the Beat Frequency Between B and C**:
- We need to find the number of beats produced when forks B and C are sounded together.
- The beat frequency for B and C is:
\[ n_B - n_C = n \quad \text{(Equation 3)} \]

5. **Expressing \( n_B \) and \( n_C \)**:
- From Equation 1, we can express \( n_B \):
\[ n_B = n_A - n_1 \]
- From Equation 2, we can express \( n_C \):
\[ n_C = n_A - n_2 \]

6. **Substituting into Equation 3**:
- Substitute the expressions for \( n_B \) and \( n_C \) into Equation 3:
\[ n = (n_A - n_1) - (n_A - n_2) \]
- Simplifying this gives:
\[ n = n_2 - n_1 \]

7. **Conclusion**:
- Therefore, the number of beats produced when tuning forks B and C are sounded together is:
\[ n = n_2 - n_1 \]

### Final Answer:
The number of beats produced when B and C are sounded together is \( n_2 - n_1 \).

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