A string of mass `'m'` and length `l`, fixed at both ends is vibrating in its fundamental mode. The maximum amplitude is `'a'` and the tension in the string is `'T'`. If the energy of vibrations of the string is `(pi^(2)a^(2)T)/(etaL)`. Find `eta`

Answer

To solve the problem, we need to find the value of \( \eta \) in the given expression for the energy of vibrations of the string.

### Step-by-Step Solution:

1. **Understand the Fundamental Mode of Vibration**:
- A string fixed at both ends vibrating in its fundamental mode forms a standing wave with one antinode in the middle and nodes at both ends.

2. **Identify the Given Parameters**:
- Mass of the string: \( m \)
- Length of the string: \( l \)
- Maximum amplitude: \( a \)
- Tension in the string: \( T \)
- Energy of vibrations: \( E = \frac{\pi^2 a^2 T}{\eta l} \)

3. **Use the Formula for Energy in a Vibrating String**:
- The energy \( E \) of a vibrating string in its fundamental mode can be expressed as:
\[ E = \frac{1}{4} m \omega^2 A^2 \]
- Here, \( \omega \) is the angular frequency and \( A \) is the amplitude (which is \( a \) in our case).

4. **Relate Angular Frequency to Tension and Mass**:
- The angular frequency \( \omega \) for a string is given by:
\[ \omega = 2\pi f \]
- The fundamental frequency \( f \) can be expressed as:
\[ f = \frac{T}{\mu} \quad \text{where} \quad \mu = \frac{m}{l} \]
- Thus, we can express \( f \) as:
\[ f = \frac{T l}{m} \]
- Therefore, substituting for \( \omega \):
\[ \omega = 2\pi \left(\frac{T l}{m}\right) \]

5. **Substitute \( \omega \) into the Energy Formula**:
- Now substituting \( \omega \) into the energy formula:
\[ E = \frac{1}{4} m \left(2\pi \frac{T l}{m}\right)^2 a^2 \]
- Simplifying this gives:
\[ E = \frac{1}{4} m \cdot 4\pi^2 \frac{T^2 l^2}{m^2} a^2 \]
- This simplifies to:
\[ E = \frac{\pi^2 a^2 T l}{m} \]

6. **Compare with Given Energy Expression**:
- We have:
\[ E = \frac{\pi^2 a^2 T}{\eta l} \]
- Setting the two expressions for energy equal to each other:
\[ \frac{\pi^2 a^2 T l}{m} = \frac{\pi^2 a^2 T}{\eta l} \]

7. **Solve for \( \eta \)**:
- Cross-multiplying gives:
\[ \eta l^2 = m \]
- Therefore, we can solve for \( \eta \):
\[ \eta = \frac{m}{l^2} \]

### Final Answer:
\[ \eta = \frac{m}{l^2} \]