To determine the most accurate measurement of \( g \) from the given options, we will analyze the formula and the errors involved in the measurements.
### Step-by-Step Solution:
1. **Understanding the Formula**:
The formula for calculating \( g \) is given by:
\[
g = \frac{4 \pi^2 L}{T^2}
\]
where \( L \) is the length of the pendulum and \( T \) is the time period of oscillation.
2. **Identifying Errors**:
- The student commits an error in measuring \( L \) denoted as \( \Delta L \).
- The time for \( n \) oscillations is measured, and the stopwatch has a least count error \( \Delta T \) along with a human error of 0.1 seconds.
3. **Calculating the Time Period**:
The time period \( T \) for one oscillation is given by:
\[
T = \frac{t}{n}
\]
where \( t \) is the total time for \( n \) oscillations.
4. **Error in Time Period**:
The error in the time period \( \Delta T' \) can be expressed as:
\[
\Delta T' = \frac{\Delta T}{n}
\]
This shows that the error in the time period decreases as \( n \) increases.
5. **Relative Error in \( g \)**:
The relative error in \( g \) can be expressed as:
\[
\frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T'}{T}
\]
Substituting \( \Delta T' \):
\[
\frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T}{nT}
\]
6. **Minimizing Errors**:
To achieve the most accurate measurement of \( g \), both \( \Delta L \) and \( \Delta T' \) should be minimized. This can be done by:
- Choosing the smallest \( \Delta L \).
- Choosing the largest \( n \) to minimize \( \Delta T' \).
- Ensuring \( \Delta T \) (least count error) is also minimized.
7. **Evaluating Options**:
We need to evaluate the given options based on the values of \( \Delta L \), \( n \), and \( \Delta T \) to find which combination yields the smallest relative error in \( g \).
- **Option A**: \( \Delta L = 5 \, \text{mm}, n = 10, \Delta T = 0.2 \, \text{s} \)
- **Option B**: \( \Delta L = 5 \, \text{mm}, n = 15, \Delta T = 0.15 \, \text{s} \)
- **Option C**: \( \Delta L = 5 \, \text{mm}, n = 20, \Delta T = 0.1 \, \text{s} \)
- **Option D**: \( \Delta L = 1 \, \text{mm}, n = 20, \Delta T = 0.1 \, \text{s} \)
- From the options, **Option D** has the smallest \( \Delta L \) (1 mm), the largest \( n \) (20), and the smallest \( \Delta T \) (0.1 s).
8. **Conclusion**:
Therefore, the most accurate measurement of \( g \) will be obtained from **Option D**.
