In two similar wires of tension 16 N and T, 3 beats are heard, then T=? if wire having tension 16N has a frequency of 4 Hz

Options

A. 49 N
B. 25 N
C. 64 N
D. 80 N

Answer

To solve the problem, we need to find the tension \( T \) in the second wire when the first wire has a tension of 16 N and a frequency of 4 Hz, and 3 beats are heard. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the relationship between frequency and tension
The frequency \( f \) of a wire is related to its tension \( T \) and mass per unit length \( \mu \) by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Since both wires are similar, they have the same length \( L \) and mass per unit length \( \mu \). Thus, we can say that the frequency is directly proportional to the square root of the tension:
\[ f \propto \sqrt{T} \]

### Step 2: Set up the frequency ratio
Let \( f_1 \) be the frequency of the first wire (with tension \( T_1 = 16 \, \text{N} \)) and \( f_2 \) be the frequency of the second wire (with tension \( T_2 = T \)). We can write:
\[ \frac{f_1}{f_2} = \sqrt{\frac{T_1}{T_2}} \]
Substituting the known values:
\[ \frac{4}{f_2} = \sqrt{\frac{16}{T}} \]

### Step 3: Express \( f_2 \) in terms of \( T \)
Squaring both sides gives:
\[ \left(\frac{4}{f_2}\right)^2 = \frac{16}{T} \]
This simplifies to:
\[ \frac{16}{f_2^2} = \frac{16}{T} \]
Thus, we can express \( f_2 \) as:
\[ f_2^2 = T \]

### Step 4: Use the beat frequency information
The problem states that 3 beats are heard, which means the difference in frequencies is 3 Hz:
\[ |f_2 - f_1| = 3 \]
Substituting \( f_1 = 4 \, \text{Hz} \):
\[ |f_2 - 4| = 3 \]
This gives us two possible equations:
1. \( f_2 - 4 = 3 \) → \( f_2 = 7 \, \text{Hz} \)
2. \( 4 - f_2 = 3 \) → \( f_2 = 1 \, \text{Hz} \)

### Step 5: Calculate \( T \) for both cases
1. For \( f_2 = 7 \, \text{Hz} \):
\[ T = f_2^2 = 7^2 = 49 \, \text{N} \]
2. For \( f_2 = 1 \, \text{Hz} \):
\[ T = f_2^2 = 1^2 = 1 \, \text{N} \]

### Step 6: Conclusion
Since the tension in the wire must be positive and reasonable, we take \( T = 49 \, \text{N} \) as the valid solution.

### Final Answer:
\[ T = 49 \, \text{N} \]