A simple pendulum of length `l_(1)` has a time period of 4 s and another simple pendulum of length `l_(2)` has a time period 3 s. Then the time period of another pendulum of length `(l_(1)-l_(2))` is

Options

A. `sqrt(3)s`
B. `1 s`
C. `sqrt((3)/(4))s`
D. `sqrt(7)s`

Explanation

To solve the problem, we need to find the time period of a pendulum with length \( l_1 - l_2 \) given the time periods of two other pendulums. Here’s a step-by-step solution:

### Step 1: Understand the Time Period Formula
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.

### Step 2: Calculate \( l_1 \) from \( T_1 \)
We know the time period \( T_1 \) for the first pendulum (length \( l_1 \)) is 4 seconds:
\[ T_1 = 4 \text{ s} \]
Using the formula:
\[ 4 = 2\pi \sqrt{\frac{l_1}{g}} \]
Squaring both sides:
\[ 16 = 4\pi^2 \frac{l_1}{g} \]
Rearranging gives:
\[ l_1 = \frac{16g}{4\pi^2} = \frac{4g}{\pi^2} \]

### Step 3: Calculate \( l_2 \) from \( T_2 \)
For the second pendulum (length \( l_2 \)), the time period \( T_2 \) is 3 seconds:
\[ T_2 = 3 \text{ s} \]
Using the formula:
\[ 3 = 2\pi \sqrt{\frac{l_2}{g}} \]
Squaring both sides:
\[ 9 = 4\pi^2 \frac{l_2}{g} \]
Rearranging gives:
\[ l_2 = \frac{9g}{4\pi^2} \]

### Step 4: Calculate \( l_1 - l_2 \)
Now we need to find \( l_1 - l_2 \):
\[ l_1 - l_2 = \frac{4g}{\pi^2} - \frac{9g}{4\pi^2} \]
Finding a common denominator:
\[ l_1 - l_2 = \frac{16g}{4\pi^2} - \frac{9g}{4\pi^2} = \frac{(16g - 9g)}{4\pi^2} = \frac{7g}{4\pi^2} \]

### Step 5: Calculate the Time Period \( T_3 \)
Now we find the time period \( T_3 \) for the pendulum with length \( l_1 - l_2 \):
\[ T_3 = 2\pi \sqrt{\frac{l_1 - l_2}{g}} = 2\pi \sqrt{\frac{\frac{7g}{4\pi^2}}{g}} \]
Simplifying:
\[ T_3 = 2\pi \sqrt{\frac{7}{4\pi^2}} = 2\pi \cdot \frac{\sqrt{7}}{2\pi} = \sqrt{7} \]

### Final Answer
The time period of the pendulum with length \( l_1 - l_2 \) is:
\[ T_3 = \sqrt{7} \text{ seconds} \]