If the length of a simple pendulum is equal to the radius of the earth, its time period will be

Options

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

Explanation

To find the time period of a simple pendulum whose length is equal to the radius of the Earth, we can follow these steps:

### Step 1: Understand the formula for the time period of a simple pendulum
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: Identify the length of the pendulum
In this case, the length \( l \) of the pendulum is equal to the radius of the Earth \( R_e \). Therefore, we can substitute \( l = R_e \) into the formula.

### Step 3: Substitute the values into the formula
Now, substituting \( l \) with \( R_e \):
\[ T = 2\pi \sqrt{\frac{R_e}{g}} \]

### Step 4: Consider the effect of large length
Since the length of the pendulum is comparable to the radius of the Earth, we need to consider the effect of this large length. In such cases, the formula for the time period changes slightly. We can use the modified formula for large lengths:
\[ T = 2\pi \sqrt{\frac{R_e}{2g}} \]

### Step 5: Final expression for the time period
Thus, the time period of the pendulum when its length is equal to the radius of the Earth is:
\[ T = 2\pi \sqrt{\frac{R_e}{2g}} \]

### Conclusion
This expression gives us the time period of the pendulum under the specified conditions.

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