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 \( \eta \) given the energy of vibrations of a string in its fundamental mode. Let's break down the steps:

### Step 1: Understand the fundamental mode of vibration
In the fundamental mode, the string forms one complete wave (one loop). The length of the string \( l \) is equal to half the wavelength \( \lambda \):
\[ \frac{\lambda}{2} = l \implies \lambda = 2l \]

### Step 2: Determine the frequency of the vibrating string
The speed \( v \) of the wave on the string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( \mu \) is the mass per unit length of the string, defined as:
\[ \mu = \frac{m}{l} \]
Thus, substituting \( \mu \) into the equation for \( v \):
\[ v = \sqrt{\frac{T \cdot l}{m}} \]

The frequency \( f \) of the wave can be related to the speed and wavelength:
\[ f = \frac{v}{\lambda} = \frac{v}{2l} = \frac{1}{2l} \sqrt{\frac{T \cdot l}{m}} = \frac{1}{2} \sqrt{\frac{T}{m \cdot l}} \]

### Step 3: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is related to the frequency \( f \) by:
\[ \omega = 2\pi f \]
Substituting for \( f \):
\[ \omega = 2\pi \cdot \frac{1}{2} \sqrt{\frac{T}{m \cdot l}} = \pi \sqrt{\frac{T}{m \cdot l}} \]

### Step 4: Calculate the total energy of the vibrating string
The total energy \( E \) stored in the string vibrating in its fundamental mode is given by:
\[ E = \frac{1}{4} m \omega^2 a^2 \]
Substituting \( \omega \):
\[ E = \frac{1}{4} m \left(\pi \sqrt{\frac{T}{m \cdot l}}\right)^2 a^2 \]
Calculating \( \omega^2 \):
\[ \omega^2 = \pi^2 \frac{T}{m \cdot l} \]
Thus, substituting back into the energy equation:
\[ E = \frac{1}{4} m \cdot \pi^2 \frac{T}{m \cdot l} a^2 = \frac{\pi^2 T a^2}{4l} \]

### Step 5: Compare with the given energy expression
The problem states that the energy is given by:
\[ E = \frac{\pi^2 a^2 T}{\eta l} \]
Setting the two expressions for energy equal to each other:
\[ \frac{\pi^2 T a^2}{4l} = \frac{\pi^2 a^2 T}{\eta l} \]

### Step 6: Solve for \( \eta \)
By comparing the coefficients, we can deduce:
\[ \frac{1}{4} = \frac{1}{\eta} \implies \eta = 4 \]

### Final Answer
Thus, the value of \( \eta \) is:
\[ \eta = 4 \]