The time period T is found to depend upon L as

Options

A. `T prop L`
B. `T prop L^(2)`
C. `T^(2) prop L`
D. `T prop sqrt(1/L)`

Answer

To solve the problem of how the time period \( T \) of a simple pendulum depends on its length \( L \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Simple Pendulum**:
- A simple pendulum consists of a mass (bob) attached to a string of length \( L \) that swings back and forth around a fixed point (the hinge).
- The time period \( T \) is the time taken for the pendulum to complete one full oscillation.

2. **Defining the Time Period**:
- The time period \( T \) can be defined as the time taken for the pendulum to move from its mean position to one extreme, back to the mean position, to the other extreme, and back to the mean position again.

3. **Deriving the Formula for Time Period**:
- The formula for the time period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
- Here, \( g \) is the acceleration due to gravity.

4. **Analyzing the Relationship**:
- From the formula, we can see that \( T \) is proportional to the square root of the length \( L \):
\[ T \propto \sqrt{L} \]
- If we square both sides, we get:
\[ T^2 \propto L \]
- This indicates that the square of the time period \( T^2 \) is directly proportional to the length \( L \).

5. **Conclusion**:
- Therefore, we conclude that the time period \( T \) of a simple pendulum increases with an increase in the length \( L \). Specifically, if the length of the pendulum increases, the time period also increases.

### Final Answer:
The time period \( T \) is directly proportional to the square root of the length \( L \) of the pendulum, expressed as:
\[ T^2 \propto L \]