In a simple pendulum the period of oscillation `(T)` is related to the length of the pendulum `(L)` as

Options

A. `(l)/(T)` =constant
B. `(l^(2))/(T)` =constant
C. `(l)/(T^(2))` =constant
D. `(l^(2))/(T^(2))` =constant

Explanation

To solve the problem regarding the relationship between the period of oscillation \( T \) of a simple pendulum and its length \( L \), we can follow these steps:

### Step 1: Understand the Formula for the Period of a Simple Pendulum
The period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where:
- \( T \) is the period of oscillation,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity (a constant).

### Step 2: Rearrange the Formula
We can rearrange the formula to express the relationship between \( T \) and \( L \). First, we can isolate \( T \) on one side:
\[ T = 2\pi \sqrt{L} \cdot \frac{1}{\sqrt{g}} \]
This shows that \( T \) is proportional to \( \sqrt{L} \).

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ T^2 = (2\pi)^2 \cdot \frac{L}{g} \]
This simplifies to:
\[ T^2 = \frac{4\pi^2}{g} \cdot L \]

### Step 4: Identify the Constant
From the equation \( T^2 = \frac{4\pi^2}{g} \cdot L \), we can see that if we rearrange it, we get:
\[ \frac{T^2}{L} = \frac{4\pi^2}{g} \]
Here, \( \frac{4\pi^2}{g} \) is a constant. Thus, we can conclude:
\[ \frac{T^2}{L} = \text{constant} \]

### Step 5: Determine the Correct Option
From the relationship \( \frac{T^2}{L} = \text{constant} \), we can deduce that:
\[ L \propto T^2 \]
This implies that if we express it in terms of \( L \) and \( T \), we can also say:
\[ \frac{L}{T^2} = \text{constant} \]
Thus, the correct answer is:
\[ \frac{L}{T^2} = \text{constant} \]

### Conclusion
The correct option is:
\[ \text{Option 3: } \frac{L}{T^2} = \text{constant} \]

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