In case of a simple pendulum, time period versus length is depicted by

Options

A.
B.
C.
D.

Explanation

To find the relationship between the time period (T) and the length (L) of a simple pendulum, we start with the formula for the time period of a simple pendulum:

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

Where:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

### Step 1: Rearranging the Formula
We can express the time period \( T \) in terms of \( L \):

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

### Step 2: Identifying Variables
In this equation:
- \( T \) is the dependent variable (y-axis),
- \( L \) is the independent variable (x-axis).

### Step 3: Expressing T in Terms of L
We can rewrite the equation to highlight the relationship between \( T \) and \( L \):

\[ T = 2\pi \cdot \frac{1}{\sqrt{g}} \cdot \sqrt{L} \]

### Step 4: Identifying the Form of the Equation
This equation can be expressed in the form of \( y = mx^{1/2} \), where:
- \( y = T \),
- \( x = L \),
- \( m = 2\pi \cdot \frac{1}{\sqrt{g}} \).

### Step 5: Squaring Both Sides
To find a linear relationship, we can square both sides:

\[ T^2 = \left(2\pi \cdot \frac{1}{\sqrt{g}}\right)^2 \cdot L \]

This gives us:

\[ T^2 = \frac{4\pi^2}{g} \cdot L \]

### Step 6: Identifying the Graph
This equation \( T^2 = kL \) (where \( k = \frac{4\pi^2}{g} \)) is a linear equation in terms of \( T^2 \) and \( L \). However, if we plot \( T \) versus \( L \), it will yield a parabolic shape because \( T \) is proportional to the square root of \( L \).

### Conclusion
Thus, the graph of the time period \( T \) versus the length \( L \) of a simple pendulum is a parabola opening upwards in the first quadrant.

The correct answer is option B, which depicts a parabolic relationship.

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