A string of length `L` fixed at its at its both ends is vibrating in its `1^(st)` overtone mode. Consider two elements of the string of same small length at position `l_1=0.2L` and `l_2=0.45L` from one end. If `K_1` and `K_2` are their respective maximum kinetic energies then

Options

A. `K_(1) = K_(2)`
B. `K_(1)gt K_(2)`
C. `K_(1)lt K_(2)`
D. it is not possible to decide the relation.

Explanation

To solve the problem, we need to analyze the behavior of a vibrating string in its first overtone mode. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Mode of Vibration
In the first overtone mode of a string fixed at both ends, there are two nodes (points of zero displacement) and one antinode (point of maximum displacement). The string vibrates in such a way that the length of the string is divided into segments.

### Step 2: Identify the Positions
Given:
- \( l_1 = 0.2L \)
- \( l_2 = 0.45L \)

### Step 3: Determine the Amplitude at Each Position
In the first overtone mode:
- The nodes are located at \( 0 \), \( \frac{L}{2} \), and \( L \).
- The antinodes are located at \( \frac{L}{4} \) and \( \frac{3L}{4} \).

At \( l_1 = 0.2L \):
- This position is closer to the node at \( 0 \) than to the antinode, hence it will have a smaller amplitude.

At \( l_2 = 0.45L \):
- This position is closer to the node at \( \frac{L}{2} \) and also further from the antinode, meaning it will have an even smaller amplitude.

### Step 4: Compare the Amplitudes
Since \( l_1 \) is at \( 0.2L \) and \( l_2 \) is at \( 0.45L \), we can conclude that:
- The amplitude at \( l_1 \) is greater than the amplitude at \( l_2 \).

### Step 5: Relate Amplitude to Kinetic Energy
The maximum kinetic energy \( K \) of a particle in a vibrating string is given by:
\[ K \propto A^2 \]
where \( A \) is the amplitude of the vibration at that point.

Since the amplitude at \( l_1 \) is greater than that at \( l_2 \):
\[ A_1 > A_2 \]

Thus, we can write:
\[ K_1 \propto A_1^2 > A_2^2 \propto K_2 \]

### Conclusion
From the above analysis, we conclude that:
\[ K_1 > K_2 \]

### Final Answer
The relationship between the maximum kinetic energies is:
\[ K_1 > K_2 \]