The periodic time T of a simple pendulum are observed , for different length L if a graph of 'log T' against 'log L' is plotted , the slope of the graph at T = 2 s is

Options

A. 2
B. `1//2`
C. `sqrt(2)`
D. `sqrt(2)`

Answer

To solve the problem, we need to analyze the relationship between the periodic time \( T \) of a simple pendulum and its length \( L \). The formula for the period of a simple pendulum is given by:

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

where:
- \( T \) is the period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

### Step 1: Rearranging the Formula
First, we can rearrange the formula to express \( T^2 \) in terms of \( L \):

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

### Step 2: Taking Logarithms
Next, we take the logarithm of both sides to facilitate plotting \( \log T \) against \( \log L \):

\[ \log T^2 = \log \left( 4\pi^2 \frac{L}{g} \right) \]

Using the properties of logarithms, we can separate the terms:

\[ \log T^2 = \log(4\pi^2) + \log L - \log g \]

### Step 3: Simplifying the Logarithmic Equation
Since \( \log T^2 = 2 \log T \), we can rewrite the equation as:

\[ 2 \log T = \log(4\pi^2) - \log g + \log L \]

Let’s denote \( \log(4\pi^2) - \log g \) as a constant \( C \):

\[ 2 \log T = C + \log L \]

### Step 4: Rearranging to the Linear Form
Now we can rearrange this equation to express it in the form \( y = mx + b \):

\[ \log L = 2 \log T - C \]

This can be rewritten as:

\[ \log L = 2 \log T - C \]

### Step 5: Identifying the Slope
From the equation \( \log L = 2 \log T - C \), we can see that the slope \( m \) of the line when plotting \( \log T \) (on the y-axis) against \( \log L \) (on the x-axis) is \( \frac{1}{2} \).

### Conclusion
Thus, the slope of the graph of \( \log T \) against \( \log L \) at \( T = 2 \) seconds is:

\[ \text{slope} = \frac{1}{2} \]