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

Options

A. 72 and 50
B. 60 and 38
C. 50 and 28
D. 80 and 58

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the lengths of the two pendulums be \( l_1 \) and \( l_2 \). According to the problem, the difference in lengths is given as:
\[ l_1 - l_2 = 22 \text{ cm} \tag{1} \]

### Step 2: Relate the Number of Oscillations to Angular Frequencies
The problem states that one pendulum makes 30 oscillations while the other makes 36 oscillations in the same time interval. The angular frequencies \( \omega_1 \) and \( \omega_2 \) can be defined as:
\[ \omega_1 = 30 \quad \text{and} \quad \omega_2 = 36 \]

### Step 3: Use the Time Period Formula
The time period \( T \) of a pendulum is given by:
\[ T = 2\pi \sqrt{\frac{l}{g}} \]
where \( g \) is the acceleration due to gravity. The relationship between the time periods of the two pendulums can be expressed as:
\[ \frac{T_1}{T_2} = \frac{\sqrt{l_1}}{\sqrt{l_2}} \tag{2} \]
Also, since \( T = \frac{2\pi}{\omega} \), we can write:
\[ \frac{T_1}{T_2} = \frac{\omega_2}{\omega_1} \tag{3} \]

### Step 4: Set Up the Equation
From equations (2) and (3), we can equate them:
\[ \frac{\omega_2}{\omega_1} = \frac{\sqrt{l_1}}{\sqrt{l_2}} \]
Substituting the values of \( \omega_1 \) and \( \omega_2 \):
\[ \frac{36}{30} = \frac{\sqrt{l_1}}{\sqrt{l_2}} \]
This simplifies to:
\[ \frac{6}{5} = \frac{\sqrt{l_1}}{\sqrt{l_2}} \]

### Step 5: Square Both Sides
Squaring both sides gives:
\[ \left(\frac{6}{5}\right)^2 = \frac{l_1}{l_2} \]
This results in:
\[ \frac{36}{25} = \frac{l_1}{l_2} \tag{4} \]

### Step 6: Express \( l_1 \) in Terms of \( l_2 \)
From equation (4), we can express \( l_1 \) as:
\[ l_1 = \frac{36}{25} l_2 \]

### Step 7: Substitute into the Length Difference Equation
Now substitute \( l_1 \) into equation (1):
\[ \frac{36}{25} l_2 - l_2 = 22 \]
This can be rewritten as:
\[ \left(\frac{36}{25} - 1\right) l_2 = 22 \]
Calculating the left side:
\[ \left(\frac{36 - 25}{25}\right) l_2 = 22 \]
\[ \frac{11}{25} l_2 = 22 \]

### Step 8: Solve for \( l_2 \)
Multiplying both sides by \( \frac{25}{11} \):
\[ l_2 = 22 \times \frac{25}{11} = 50 \text{ cm} \]

### Step 9: Find \( l_1 \)
Now, substitute \( l_2 \) back into equation (1) to find \( l_1 \):
\[ l_1 = l_2 + 22 = 50 + 22 = 72 \text{ cm} \]

### Final Answer
The lengths of the pendulums are:
\[ l_1 = 72 \text{ cm}, \quad l_2 = 50 \text{ cm} \]