A pendulum is executing simple harmonic motion and its maximum kinetic energy is `K_(1)`. If the length of the pendulum is doubled and it perfoms simple harmonuc motion with the same amplitude as in the first case, its maximum kinetic energy is `K_(2)` Then:

Options

A. `K_(2) =2K_(1)`
B. `K_(2) = (K_(1))/(2)`
C. `K_(2) =(K_(1))/(2)`
D. `K_(2) =K_(1)`

Answer

To solve the problem, we need to analyze the relationship between the maximum kinetic energies \( K_1 \) and \( K_2 \) of a pendulum executing simple harmonic motion (SHM) when its length is doubled while keeping the amplitude the same.

### Step-by-Step Solution:

1. **Understanding Maximum Kinetic Energy in SHM:**
The maximum kinetic energy \( K \) of a pendulum in simple harmonic motion is given by the formula:
\[ K = \frac{1}{2} m \omega^2 A^2 \]
where \( m \) is the mass of the pendulum bob, \( \omega \) is the angular frequency, and \( A \) is the amplitude of the motion.

2. **Identifying Initial Conditions:**
Let the initial length of the pendulum be \( L \) and the initial maximum kinetic energy be \( K_1 \). Therefore, we have:
\[ K_1 = \frac{1}{2} m \omega_1^2 A^2 \]
where \( \omega_1 = \sqrt{\frac{g}{L}} \).

3. **Changing the Length of the Pendulum:**
When the length of the pendulum is doubled, the new length \( L' \) becomes:
\[ L' = 2L \]
The amplitude \( A \) remains the same.

4. **Calculating New Angular Frequency:**
The new angular frequency \( \omega_2 \) for the doubled length is:
\[ \omega_2 = \sqrt{\frac{g}{L'}} = \sqrt{\frac{g}{2L}} = \frac{1}{\sqrt{2}} \sqrt{\frac{g}{L}} = \frac{\omega_1}{\sqrt{2}} \]

5. **Calculating New Maximum Kinetic Energy:**
The new maximum kinetic energy \( K_2 \) can be expressed as:
\[ K_2 = \frac{1}{2} m \omega_2^2 A^2 \]
Substituting \( \omega_2 \):
\[ K_2 = \frac{1}{2} m \left(\frac{\omega_1}{\sqrt{2}}\right)^2 A^2 = \frac{1}{2} m \frac{\omega_1^2}{2} A^2 = \frac{1}{2} K_1 \]

6. **Conclusion:**
Therefore, the relationship between the maximum kinetic energies is:
\[ K_2 = \frac{K_1}{2} \]

### Final Answer:
Thus, the maximum kinetic energy \( K_2 \) when the length of the pendulum is doubled is:
\[ K_2 = \frac{K_1}{2} \]