Two apdulums begin to swing simultaneosuly. The first pendulum makes `9` full oscillations when the other makes `7`. Find the ratio of length of the two pendulums

Options

A. `(49)/(81)`
B. `(7)/(9)`
C. `(50)/(81)`
D. `(1)/(2)`

Answer

To solve the problem of finding the ratio of the lengths of two pendulums based on their oscillations, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Time Periods**:
Let \( T_1 \) be the time period of the first pendulum and \( T_2 \) be the time period of the second pendulum.

2. **Set Up the Relationship**:
The first pendulum makes 9 oscillations while the second makes 7. Therefore, the total time taken for these oscillations can be expressed as:
\[ 9 T_1 = 7 T_2 \]

3. **Express Time Periods in Terms of Length**:
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity. Thus, we can express \( T_1 \) and \( T_2 \) as:
\[ T_1 = 2\pi \sqrt{\frac{L_1}{g}} \quad \text{and} \quad T_2 = 2\pi \sqrt{\frac{L_2}{g}} \]

4. **Substitute Time Periods into the Relationship**:
Substituting \( T_1 \) and \( T_2 \) into the equation from step 2 gives:
\[ 9 \left(2\pi \sqrt{\frac{L_1}{g}}\right) = 7 \left(2\pi \sqrt{\frac{L_2}{g}}\right) \]

5. **Cancel Common Terms**:
The \( 2\pi \) and \( \sqrt{g} \) terms can be canceled from both sides:
\[ 9 \sqrt{L_1} = 7 \sqrt{L_2} \]

6. **Rearrange to Find the Ratio**:
Rearranging gives:
\[ \frac{\sqrt{L_1}}{\sqrt{L_2}} = \frac{7}{9} \]

7. **Square Both Sides**:
Squaring both sides to eliminate the square roots results in:
\[ \frac{L_1}{L_2} = \left(\frac{7}{9}\right)^2 = \frac{49}{81} \]

8. **Final Answer**:
The ratio of the lengths of the two pendulums is:
\[ \frac{L_1}{L_2} = \frac{49}{81} \]