A pendulum has maximum kinetic energy `K_1` If its length is doubled keeping amplitude same then maximum kinetic energy becomes `K_2` .Then relation between `K_1 and K_2` is

Options

A. `K_2=2K_1`
B. `K_1=2K_2`
C. `K_2=K_1`
D. `K_1=4K_2`

Answer

To solve the problem, we need to analyze the relationship between the maximum kinetic energy of a pendulum and its length.

### Step-by-step Solution:

1. **Understanding Maximum Kinetic Energy (K)**:
The maximum kinetic energy \( K \) of a pendulum can be expressed as:
\[ K = \frac{1}{2} m \omega^2 A^2 \]
where:
- \( m \) is the mass of the pendulum bob,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of the pendulum.

2. **Angular Frequency (\( \omega \))**:
The angular frequency \( \omega \) for a simple pendulum is given by:
\[ \omega = \sqrt{\frac{g}{L}} \]
where:
- \( g \) is the acceleration due to gravity,
- \( L \) is the length of the pendulum.

3. **Substituting \( \omega \) into the Kinetic Energy Formula**:
Substituting the expression for \( \omega \) into the kinetic energy formula gives:
\[ K = \frac{1}{2} m \left(\sqrt{\frac{g}{L}}\right)^2 A^2 = \frac{1}{2} m \frac{g}{L} A^2 \]

4. **Initial Kinetic Energy (\( K_1 \))**:
Let the initial length of the pendulum be \( L \) and the maximum kinetic energy be \( K_1 \):
\[ K_1 = \frac{1}{2} m \frac{g}{L} A^2 \]

5. **Doubling the Length**:
If the length of the pendulum is doubled (i.e., \( L \) becomes \( 2L \)), the new maximum kinetic energy \( K_2 \) can be expressed as:
\[ K_2 = \frac{1}{2} m \frac{g}{2L} A^2 \]

6. **Relating \( K_2 \) to \( K_1 \)**:
We can express \( K_2 \) in terms of \( K_1 \):
\[ K_2 = \frac{1}{2} m \frac{g}{2L} A^2 = \frac{1}{2} \left(\frac{1}{2}\right) m \frac{g}{L} A^2 = \frac{1}{2} K_1 \]

7. **Final Relation**:
Thus, the relationship between \( K_1 \) and \( K_2 \) is:
\[ K_2 = \frac{1}{2} K_1 \]

### Conclusion:
The maximum kinetic energy of the pendulum when its length is doubled (keeping amplitude the same) becomes half of the original maximum kinetic energy.