A string of length L, fixed at its both ends is vibrating in its `1^(st)` overtone mode. Consider two elements of the string of the same small length at positions `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)gtK_(2)`
C. `K_(1)ltK_(2)`
D. it is not possible to decide the relation

Answer

To solve the problem, we need to analyze the positions of the two points on the string and their corresponding kinetic energies in the first overtone mode of vibration.

### Step-by-Step Solution:

1. **Understanding the First Overtone Mode**:
- The first overtone mode of a string fixed at both ends has a specific pattern of nodes and antinodes. In this mode, there are two segments of the string vibrating, with nodes at the ends and one node in the middle.
- The positions of the nodes and antinodes can be determined by the wavelength and the length of the string.

2. **Identifying Positions**:
- Given the string length \( L \), the positions are:
- \( l_1 = 0.2L \)
- \( l_2 = 0.45L \)
- The midpoint of the string (where the node is located) is at \( L/2 \).

3. **Locating the Nodes and Antinodes**:
- In the first overtone, the nodes are located at \( 0 \), \( L/2 \), and \( L \).
- The antinodes are located at \( L/4 \) and \( 3L/4 \).
- Position \( l_1 = 0.2L \) is closer to the node at \( 0 \) than to the antinode.
- Position \( l_2 = 0.45L \) is closer to the node at \( L/2 \).

4. **Kinetic Energy in Vibrating Strings**:
- The kinetic energy \( K \) of a segment of the string is maximum at antinodes and minimum at nodes.
- Since \( l_1 \) is at \( 0.2L \) (closer to the node), it will have a higher kinetic energy than \( l_2 \) which is at \( 0.45L \) (closer to the node at \( L/2 \)).

5. **Comparison of Kinetic Energies**:
- Since \( l_1 \) is further away from the node compared to \( l_2 \), we conclude that:
\[ K_1 > K_2 \]
- Thus, the maximum kinetic energy at position \( l_1 \) is greater than that at position \( l_2 \).

### Conclusion:
The correct answer is that the kinetic energy at position \( l_1 \) (0.2L) is greater than the kinetic energy at position \( l_2 \) (0.45L).