The time period of a simple pendulum of infinite length is (R=radius of earth).

Options

A. infinite
B. `2pisqrt((R)/(g))`
C. `2pisqrt((g)/(R))`
D. `(1)/(2pi)sqrt((R)/(g))`

Explanation

To find the time period of a simple pendulum of infinite length, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Formula for Time Period**:
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.

2. **Considering Infinite Length**:
In this case, we are considering a pendulum of infinite length, which means \( L \) approaches infinity (\( L \to \infty \)).

3. **Effect of Infinite Length on Time Period**:
When \( L \) is infinite, the term \( \frac{L}{g} \) also approaches infinity. However, we need to consider the gravitational effect at the Earth's surface, which is influenced by the radius of the Earth \( R \).

4. **Modified Time Period Formula**:
For a pendulum whose length is comparable to the radius of the Earth, we can modify the formula for the time period:
\[ T = 2\pi \sqrt{\frac{R}{g}} \]
Here, \( R \) is the radius of the Earth.

5. **Substituting Values**:
The radius of the Earth \( R \) is approximately \( 6400 \) kilometers, which is \( 6.4 \times 10^6 \) meters. The acceleration due to gravity \( g \) is approximately \( 9.8 \, \text{m/s}^2 \).

6. **Calculating the Time Period**:
Now we can substitute the values into the modified formula:
\[ T = 2\pi \sqrt{\frac{6.4 \times 10^6}{9.8}} \]

7. **Performing the Calculation**:
First, calculate \( \frac{6.4 \times 10^6}{9.8} \):
\[ \frac{6.4 \times 10^6}{9.8} \approx 65306.12 \]
Now, take the square root:
\[ \sqrt{65306.12} \approx 255.53 \]
Finally, multiply by \( 2\pi \):
\[ T \approx 2\pi \times 255.53 \approx 1604.22 \, \text{seconds} \]

8. **Converting to Minutes**:
To convert seconds to minutes:
\[ T \approx \frac{1604.22}{60} \approx 26.74 \, \text{minutes} \]

### Final Answer:
The time period of a simple pendulum of infinite length is approximately **26.74 minutes**.