A student performs an experiment for determination of ` g [ = ( 4 pi^(2) L)/(T^(2)) ] , L ~~ 1m` , and he commits an error of `Delta L`. For `T` he takes the time of `n` oscillations with the stop watch of least count ` Delta T` . For which of the following data , the measurement of `g` will be most accurate ?

Options

A. `Deltal=0.5, Deltat=0.1, n=20`
B. `Deltal=0.5, Delta=0.1,n=50`
C. `Delta=0.5, Deltat=0.01,n=20`
D. `Delta=0.1, Deltat=0.05,n=50`

Answer

To determine the most accurate measurement of \( g \) using the formula

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

where \( L \) is the length and \( T \) is the time period of oscillation, we need to analyze how errors in the measurements of \( L \) and \( T \) affect the calculation of \( g \).

### Step-by-Step Solution:

1. **Understand the Formula**:
The formula for \( g \) is given as:
\[ g = \frac{4 \pi^2 L}{T^2} \]
Here, \( L \) is the length of the pendulum (approximately 1 m), and \( T \) is the time period for \( n \) oscillations.

2. **Identify the Errors**:
- The error in \( L \) is denoted as \( \Delta L \).
- The time \( T \) is measured for \( n \) oscillations, and the error in time measurement is denoted as \( \Delta T \).

3. **Calculate the Relative Error in \( g \)**:
The relative error in \( g \) can be expressed as:
\[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T}{T} \]
Since \( T \) is the time for \( n \) oscillations, we can express \( T \) as:
\[ T = \frac{t}{n} \]
where \( t \) is the total time for \( n \) oscillations. The error in \( T \) can therefore be expressed as:
\[ \Delta T = \frac{\Delta t}{n} \]
Thus, the relative error in \( T \) becomes:
\[ \frac{\Delta T}{T} = \frac{\Delta t}{t} \cdot \frac{1}{n} \]

4. **Substituting Back into the Error Formula**:
Substitute \( \Delta T \) back into the error formula:
\[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \cdot \frac{\Delta t}{t} \cdot \frac{1}{n} \]

5. **Minimizing the Error**:
To minimize the error in \( g \), we need to:
- Minimize \( \Delta L \) (the error in length).
- Maximize \( n \) (the number of oscillations measured).

6. **Analyzing the Options**:
Given different options for \( \Delta L \) and \( n \):
- Identify which option has the smallest \( \Delta L \) and the largest \( n \).
- If two options have the same \( n \), choose the one with the smaller \( \Delta L \).

7. **Conclusion**:
After analyzing the options, the most accurate measurement of \( g \) will be the one with the maximum \( n \) and minimum \( \Delta L \).