The ratio of frequencies of two pendulums are 2 : 3, then their length are in ratio

Options

A. `sqrt((2)/(3))`
B. `sqrt((3)/(2))`
C. `(4)/(9)`
D. `(9)/(4)`

Explanation

To solve the problem, we need to find the ratio of the lengths of two pendulums given the ratio of their frequencies. Here’s a step-by-step solution:

### Step 1: Understand the relationship between frequency and length
The frequency \( f \) of a simple pendulum is given by the formula:
\[ f = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]
where \( g \) is the acceleration due to gravity and \( L \) is the length of the pendulum.

### Step 2: Set up the frequency ratio
Let the frequencies of the two pendulums be \( f_1 \) and \( f_2 \). According to the problem, the ratio of their frequencies is given as:
\[ \frac{f_1}{f_2} = \frac{2}{3} \]

### Step 3: Write the expressions for the frequencies
Using the formula for frequency:
\[ f_1 = \frac{1}{2\pi} \sqrt{\frac{g}{L_1}} \quad \text{and} \quad f_2 = \frac{1}{2\pi} \sqrt{\frac{g}{L_2}} \]

### Step 4: Substitute the expressions into the ratio
Substituting the expressions for \( f_1 \) and \( f_2 \) into the frequency ratio gives:
\[ \frac{\frac{1}{2\pi} \sqrt{\frac{g}{L_1}}}{\frac{1}{2\pi} \sqrt{\frac{g}{L_2}}} = \frac{2}{3} \]

### Step 5: Simplify the equation
The \( \frac{1}{2\pi} \) and \( \sqrt{g} \) terms cancel out, leading to:
\[ \frac{\sqrt{L_2}}{\sqrt{L_1}} = \frac{2}{3} \]

### Step 6: Square both sides to eliminate the square roots
Squaring both sides results in:
\[ \frac{L_2}{L_1} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

### Step 7: Find the ratio of lengths
To find the ratio of the lengths \( L_1 \) to \( L_2 \), we take the reciprocal:
\[ \frac{L_1}{L_2} = \frac{9}{4} \]

### Final Answer
Thus, the ratio of the lengths of the two pendulums is:
\[ \frac{L_1}{L_2} = \frac{9}{4} \]