The masses and length of two simple pendulums are given as respectively `m_(A), m_(B), L_(A) and L_(B)`. If frequency of A is double that of B then relation between `L_(A) and l_(B)` is given as

Options

A. `L_(A) = (L_(B))/(2)`
B. `L_(A) = 2L_(B)`
C. `L_(A) = (4)/(L_(B))`
D. `L_(A) = (L_(B))/(4)`

Explanation

To solve the problem, we need to establish the relationship between the lengths of the two pendulums based on their frequencies.

### Step-by-Step Solution:

1. **Understanding the Frequency of a Simple Pendulum**:
The frequency \( f \) of a simple pendulum is related to its length \( L \) by the formula:
\[ f = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]
where \( g \) is the acceleration due to gravity.

2. **Relating the Frequencies of Pendulums A and B**:
Given that the frequency of pendulum A is double that of pendulum B, we can express this as:
\[ f_A = 2 f_B \]

3. **Substituting the Frequency Formula**:
Using the frequency formula for both pendulums, we have:
\[ \frac{1}{2\pi} \sqrt{\frac{g}{L_A}} = 2 \left( \frac{1}{2\pi} \sqrt{\frac{g}{L_B}} \right) \]

4. **Simplifying the Equation**:
We can simplify this equation by canceling \( \frac{1}{2\pi} \) from both sides:
\[ \sqrt{\frac{g}{L_A}} = 2 \sqrt{\frac{g}{L_B}} \]

5. **Squaring Both Sides**:
To eliminate the square roots, we square both sides:
\[ \frac{g}{L_A} = 4 \frac{g}{L_B} \]

6. **Canceling \( g \)**:
Since \( g \) is a common factor, we can cancel it from both sides:
\[ \frac{1}{L_A} = \frac{4}{L_B} \]

7. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ L_B = 4 L_A \]

8. **Final Relation**:
Rearranging the equation, we find the relation between the lengths:
\[ L_A = \frac{L_B}{4} \]

### Conclusion:
Thus, the relation between the lengths of the two pendulums is:
\[ L_A = \frac{L_B}{4} \]