The length of a simple pendulum is made one-fourth. Its time period becomes :

Options

A. four times
B. one-fourth
C. double
D. half

Explanation

To solve the problem, we need to understand the relationship between the length of a simple pendulum and its time period. The time period \( T \) of a simple pendulum is given by the formula:

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

where:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity (which is constant).

### Step 1: Understand the initial condition
Let the initial length of the pendulum be \( L_1 \). The initial time period \( T_1 \) can be expressed as:

\[ T_1 = 2\pi \sqrt{\frac{L_1}{g}} \]

### Step 2: Determine the new length
According to the problem, the length of the pendulum is made one-fourth of its original length. Therefore, the new length \( L_2 \) is:

\[ L_2 = \frac{L_1}{4} \]

### Step 3: Calculate the new time period
Now, we can calculate the new time period \( T_2 \) using the new length \( L_2 \):

\[ T_2 = 2\pi \sqrt{\frac{L_2}{g}} = 2\pi \sqrt{\frac{\frac{L_1}{4}}{g}} = 2\pi \sqrt{\frac{L_1}{4g}} \]

### Step 4: Simplify the expression
We can simplify \( T_2 \):

\[ T_2 = 2\pi \sqrt{\frac{L_1}{g}} \cdot \frac{1}{\sqrt{4}} = 2\pi \sqrt{\frac{L_1}{g}} \cdot \frac{1}{2} \]

This means:

\[ T_2 = \frac{1}{2} T_1 \]

### Conclusion
Thus, the new time period \( T_2 \) is half of the original time period \( T_1 \). Therefore, if the length of the pendulum is made one-fourth, its time period becomes half.

### Final Answer
The time period becomes \( \frac{1}{2} T_1 \).

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