Choose the correct length (L) versus square of time period `(T^2 )` graph for a simple pendulum executing simple harmonic motion.

Options

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Answer

To determine the correct graph of length (L) versus the square of the time period (T²) for a simple pendulum executing simple harmonic motion, we can follow these steps:

### Step 1: Understand the formula for the time period (T) 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: Square the time period formula
To find the relationship between \( T^2 \) and \( L \), we square both sides of the equation:
\[ T^2 = (2\pi)^2 \frac{L}{g} \]
This simplifies to:
\[ T^2 = \frac{4\pi^2}{g} L \]

### Step 3: Identify the relationship between T² and L
From the equation \( T^2 = \frac{4\pi^2}{g} L \), we can see that \( T^2 \) is directly proportional to \( L \). This means that if we plot \( T^2 \) on the y-axis and \( L \) on the x-axis, the relationship will be linear.

### Step 4: Determine the characteristics of the graph
Since \( T^2 \) is directly proportional to \( L \), the graph will be a straight line that passes through the origin (0,0). This indicates that as the length \( L \) increases, the square of the time period \( T^2 \) also increases linearly.

### Step 5: Choose the correct graph
Based on the analysis, the correct graph will be a straight line that starts from the origin and has a positive slope. Any graph that does not meet these criteria (e.g., curves or does not pass through the origin) would be incorrect.

### Conclusion
Thus, the correct length (L) versus square of time period (T²) graph for a simple pendulum executing simple harmonic motion is a straight line passing through the origin.

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