Two pendulums differ in lengths by `22m`. They oscillate at the same place so that one of then makes 30 oscillations and the other makes 36 oscillations during the same time. The length `(` in `cm)` of the pendulum are `:`

Options

A. 1. 72 and 50
B. 2. 60 and 38
C. 3. 50 and 28
D. 4. 80 and 58

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Define the Variables
Let:
- \( L_1 \) = length of the first pendulum (in cm)
- \( L_2 \) = length of the second pendulum (in cm)
- The difference in lengths: \( L_2 - L_1 = 2200 \) cm (since 22 m = 2200 cm)

### Step 2: Relate Time Periods to Lengths
The time period \( T \) of a pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
From this, we know that the time period is directly proportional to the square root of the length:
\[ T \propto \sqrt{L} \]
Thus, we can write:
\[ \frac{T_1}{T_2} = \sqrt{\frac{L_1}{L_2}} \]

### Step 3: Relate Oscillations to Time Periods
Given that pendulum 1 makes 30 oscillations and pendulum 2 makes 36 oscillations in the same time \( t \), we can express the time periods as:
\[ T_1 = \frac{t}{30} \quad \text{and} \quad T_2 = \frac{t}{36} \]

### Step 4: Substitute Time Periods into the Ratio
Substituting the expressions for \( T_1 \) and \( T_2 \) into the ratio gives:
\[ \frac{\frac{t}{30}}{\frac{t}{36}} = \sqrt{\frac{L_1}{L_2}} \]
This simplifies to:
\[ \frac{36}{30} = \sqrt{\frac{L_1}{L_2}} \]
\[ \frac{6}{5} = \sqrt{\frac{L_1}{L_2}} \]

### Step 5: Square Both Sides
Squaring both sides results in:
\[ \left(\frac{6}{5}\right)^2 = \frac{L_1}{L_2} \]
\[ \frac{36}{25} = \frac{L_1}{L_2} \]
This implies:
\[ 36L_1 = 25L_2 \quad \text{(Equation 1)} \]

### Step 6: Set Up the Length Difference Equation
From the problem, we know:
\[ L_2 - L_1 = 2200 \quad \text{(in cm)} \]
Substituting \( L_2 \) from Equation 1 into this equation:
\[ L_2 = \frac{36}{25}L_1 \]
Substituting this into the length difference equation:
\[ \frac{36}{25}L_1 - L_1 = 2200 \]

### Step 7: Solve for \( L_1 \)
Rearranging gives:
\[ \frac{36L_1 - 25L_1}{25} = 2200 \]
\[ \frac{11L_1}{25} = 2200 \]
Multiplying both sides by 25:
\[ 11L_1 = 55000 \]
Dividing by 11:
\[ L_1 = 5000 \text{ cm} = 50 \text{ cm} \]

### Step 8: Find \( L_2 \)
Using \( L_2 = \frac{36}{25}L_1 \):
\[ L_2 = \frac{36}{25} \times 50 = 72 \text{ cm} \]

### Final Answer
The lengths of the pendulums are:
- \( L_1 = 50 \) cm
- \( L_2 = 72 \) cm