A string with a frequency `n` and `B` string with a frequency `1//8` that of `A`. If the energy remains the same and the amplitude of `A` is `a`, then amplitude of `B` will be

Options

A. `2a`
B. `8a`
C. `4a`
D. `a`

Explanation

To solve the problem, we need to relate the amplitudes of two strings (A and B) based on their frequencies and the condition that their energies are the same.

### Step-by-Step Solution:

1. **Identify Given Information:**
- Frequency of string A, \( n \)
- Frequency of string B, \( \frac{1}{8} n \) (which is \( n_B = \frac{1}{8} n_A \))
- Amplitude of string A, \( A \)
- Amplitude of string B, \( A_B \) (unknown)

2. **Understand the Energy Relationship:**
- The energy of a wave is given by the formula:
\[ E \propto A^2 n^2 \]
- This means that the energy is proportional to the square of the amplitude and the square of the frequency.

3. **Set Up the Energy Equations:**
- For string A:
\[ E_A \propto A^2 n^2 \]
- For string B:
\[ E_B \propto A_B^2 n_B^2 \]
- Since the energies are the same (\( E_A = E_B \)), we can write:
\[ A^2 n^2 = A_B^2 n_B^2 \]

4. **Substitute for \( n_B \):**
- Substitute \( n_B = \frac{1}{8} n \) into the equation:
\[ A^2 n^2 = A_B^2 \left(\frac{1}{8} n\right)^2 \]

5. **Simplify the Equation:**
- This simplifies to:
\[ A^2 n^2 = A_B^2 \left(\frac{1}{64} n^2\right) \]
- Rearranging gives:
\[ A^2 n^2 = \frac{1}{64} A_B^2 n^2 \]

6. **Cancel \( n^2 \) from Both Sides:**
- Since \( n^2 \) is common and non-zero, we can cancel it:
\[ A^2 = \frac{1}{64} A_B^2 \]

7. **Solve for \( A_B \):**
- Multiply both sides by 64:
\[ 64 A^2 = A_B^2 \]
- Taking the square root gives:
\[ A_B = 8A \]

### Final Answer:
The amplitude of string B is \( A_B = 8A \).