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. **Understanding the Problem**:
- We have two pendulums. The first pendulum completes 9 oscillations while the second pendulum completes 7 oscillations in the same time period.

2. **Defining Variables**:
- Let \( T_1 \) be the time period of the first pendulum.
- Let \( T_2 \) be the time period of the second pendulum.
- Let \( L_1 \) be the length of the first pendulum.
- Let \( L_2 \) be the length of the second pendulum.

3. **Relating Time and Oscillations**:
- The total time taken by the first pendulum for 9 oscillations is \( 9T_1 \).
- The total time taken by the second pendulum for 7 oscillations is \( 7T_2 \).
- Since both pendulums start swinging simultaneously and complete their respective oscillations in the same total time, we can equate the two:
\[ 9T_1 = 7T_2 \]

4. **Finding the Ratio of Time Periods**:
- Rearranging the equation gives:
\[ \frac{T_1}{T_2} = \frac{7}{9} \]

5. **Using the Relationship Between Time Period and Length**:
- The time period of a pendulum is related to its length by the formula:
\[ T \propto \sqrt{L} \]
- Therefore, we can write:
\[ T_1 \propto \sqrt{L_1} \quad \text{and} \quad T_2 \propto \sqrt{L_2} \]

6. **Setting Up the Ratio of Lengths**:
- From the proportionality, we can express the lengths in terms of the time periods:
\[ \frac{\sqrt{L_1}}{\sqrt{L_2}} = \frac{T_1}{T_2} \]
- Substituting the ratio we found:
\[ \frac{\sqrt{L_1}}{\sqrt{L_2}} = \frac{7}{9} \]

7. **Squaring Both Sides**:
- To find the ratio of lengths, we square both sides:
\[ \frac{L_1}{L_2} = \left(\frac{7}{9}\right)^2 = \frac{49}{81} \]

8. **Conclusion**:
- Thus, the ratio of the lengths of the two pendulums is:
\[ \frac{L_1}{L_2} = \frac{49}{81} \]

### Final Answer:
The ratio of the lengths of the two pendulums is \( \frac{49}{81} \).

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