You are provided with three similar,but slightly different tuning forks.When A and B are both struck,a beat frequeny of `f_(AB)` is heard.If A and C are both struck,a beat frequency of `f_(AC)` is heard.It was noticed that `f_(AB)` < `f_(AC)` .If B and C are simultaneously struck,what will be the observed beat frequency?

Options

A. `|f_(AB)+f_(AC)|`
B. `|f_(AB)-f_(AC)|`
C. Either `|f_(AB)+f_(AC)| or |f_(AB)-f_(AC)|` will be heard
D. Both `|f_(AB)+f_(AC)| and |f_(AB)-f_(AC)|` will simultaneously be heard

Answer

To solve the problem, we need to analyze the beat frequencies produced when different tuning forks are struck together. Let's denote the frequencies of the tuning forks as follows:

- Frequency of fork A: \( f_A \)
- Frequency of fork B: \( f_B \)
- Frequency of fork C: \( f_C \)

### Step 1: Understanding Beat Frequency
The beat frequency between two tuning forks is given by the absolute difference of their frequencies. Therefore, we have:
- Beat frequency between A and B:
\[ f_{AB} = |f_A - f_B| \]
- Beat frequency between A and C:
\[ f_{AC} = |f_A - f_C| \]

### Step 2: Analyzing Given Information
We know from the problem that:
\[ f_{AB} < f_{AC} \]
This implies that the difference \( |f_A - f_B| \) is less than the difference \( |f_A - f_C| \).

### Step 3: Establishing Relationships Between Frequencies
From the inequality \( f_{AB} < f_{AC} \), we can infer two possible cases regarding the relative sizes of \( f_A \), \( f_B \), and \( f_C \):

**Case 1:**
Assume \( f_A > f_B > f_C \):
- Then, \( f_{AB} = f_A - f_B \) and \( f_{AC} = f_A - f_C \).
- Since \( f_B \) is closer to \( f_A \) than \( f_C \), this case holds true.

**Case 2:**
Assume \( f_C > f_B > f_A \):
- Then, \( f_{AB} = f_B - f_A \) and \( f_{AC} = f_C - f_A \).
- Here, \( f_C \) is greater than \( f_A \), and \( f_B \) is still greater than \( f_A \).

### Step 4: Finding the Beat Frequency Between B and C
Now, we need to find the beat frequency when forks B and C are struck together:
\[ f_{BC} = |f_B - f_C| \]

### Step 5: Using the Cases to Find \( f_{BC} \)
1. **For Case 1**:
- Since \( f_A > f_B > f_C \):
\[ f_{BC} = f_B - f_C \]
We can express \( f_B \) and \( f_C \) in terms of \( f_A \):
- From \( f_{AB} = f_A - f_B \) and \( f_{AC} = f_A - f_C \):
\[ f_B = f_A - f_{AB}, \quad f_C = f_A - f_{AC} \]
Thus,
\[ f_{BC} = (f_A - f_{AB}) - (f_A - f_{AC}) = f_{AC} - f_{AB} \]

2. **For Case 2**:
- If \( f_C > f_B > f_A \):
\[ f_{BC} = f_C - f_B \]
Using the same expressions for \( f_B \) and \( f_C \):
\[ f_{BC} = (f_A + f_{AC}) - (f_A + f_{AB}) = f_{AC} - f_{AB} \]

### Final Answer
In both cases, we arrive at the same conclusion:
\[ f_{BC} = |f_{AC} - f_{AB}| \]

### Summary
The observed beat frequency when B and C are struck simultaneously is:
\[ f_{BC} = |f_{AC} - f_{AB}| \]