The graph between time period (`T`) and length (`l`) of a simple pendulum is

Options

A. Straight line
B. Parabolic
C. Hyperbolic
D. Elliptical

Answer

To determine the graph between the time period (T) and length (l) of a simple pendulum, we start with the fundamental relationship governing the time period of a simple pendulum.

### Step-by-Step Solution:

1. **Understand the 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.

2. **Square Both Sides**: To analyze the relationship between \( T \) and \( l \), we can square both sides of the equation:
\[ T^2 = (2\pi)^2 \frac{l}{g} \]
This simplifies to:
\[ T^2 = \frac{4\pi^2}{g} l \]

3. **Identify the Relationship**: From the equation \( T^2 = \frac{4\pi^2}{g} l \), we can see that \( T^2 \) is directly proportional to \( l \). We can express this as:
\[ T^2 = K \cdot l \]
where \( K = \frac{4\pi^2}{g} \) is a constant.

4. **Graphical Representation**: The equation \( T^2 = K \cdot l \) represents a linear relationship between \( T^2 \) and \( l \). When we plot \( T^2 \) on the y-axis and \( l \) on the x-axis, we will get a straight line passing through the origin.

5. **Conclusion on the Shape of the Graph**: Although \( T \) is not directly proportional to \( l \), the graph of \( T^2 \) versus \( l \) is a straight line, indicating that \( T \) increases with the square root of \( l \). Therefore, the graph between \( T \) and \( l \) is not linear but rather a parabolic curve when plotted as \( T \) against \( l \).

### Final Answer:
The graph between the time period \( T \) and length \( l \) of a simple pendulum is a parabolic curve opening upwards, indicating that \( T^2 \) is proportional to \( l \).