In an experiment to determine the acceleration due to gravity `g`, the formula used for the time period of a periodic motion is `T = 2pisqrt((7(R - r)/(5g)`. The values of `R and r` are measured to be `(60 +- 1)mm and (10 +- 1)mm`, repectively. In five successive measurment, the time period is found to be `0.52s, 0.56s,0.57s,0.54s and 0.59s`. the least count of the watch used for the measurement of time period is `0.01s`. Which of the following satement `(s)` is `(are)` true?

Options

A. The error in the measurement of `r` is `10%`
B. The error in the measurement of `T` is `3.57%`
C. The error in the measurement of `T` is `2%`
D. The error in the determined value of `g` is `11%`

Answer

To solve the problem, we need to analyze the given information step by step.

### Step 1: Understand the Formula
The formula for the time period \( T \) is given as:
\[ T = 2\pi \sqrt{\frac{7(R - r)}{5g}} \]
where \( R \) and \( r \) are measured values.

### Step 2: Identify the Values and Errors
The values of \( R \) and \( r \) are given as:
- \( R = 60 \pm 1 \) mm
- \( r = 10 \pm 1 \) mm

### Step 3: Calculate the Mean Time Period
The time periods measured are:
- \( T_1 = 0.52 \) s
- \( T_2 = 0.56 \) s
- \( T_3 = 0.57 \) s
- \( T_4 = 0.54 \) s
- \( T_5 = 0.59 \) s

To find the mean time period \( T_{mean} \):
\[ T_{mean} = \frac{T_1 + T_2 + T_3 + T_4 + T_5}{5} \]
Calculating:
\[ T_{mean} = \frac{0.52 + 0.56 + 0.57 + 0.54 + 0.59}{5} = \frac{2.78}{5} = 0.556 \text{ s} \approx 0.56 \text{ s} \]

### Step 4: Calculate the Uncertainty in Time Period
The least count of the watch is \( 0.01 \) s. The uncertainty in the mean time period can be calculated as:
\[ \Delta T = \frac{\text{max} - \text{min}}{2} = \frac{0.59 - 0.52}{2} = 0.035 \text{ s} \approx 0.02 \text{ s} \]

### Step 5: Calculate Percentage Error in Time Period
The percentage error in the time period is given by:
\[ \text{Percentage Error} = \frac{\Delta T}{T_{mean}} \times 100 = \frac{0.02}{0.56} \times 100 \approx 3.57\% \]

### Step 6: Calculate the Error in R and r
For \( R \):
\[ \text{Percentage Error in } R = \frac{1}{60} \times 100 \approx 1.67\% \]
For \( r \):
\[ \text{Percentage Error in } r = \frac{1}{10} \times 100 = 10\% \]

### Step 7: Calculate the Total Percentage Error in g
Using the formula for the propagation of error:
\[ \text{Total Percentage Error} = \text{Percentage Error in } R + \text{Percentage Error in } r + 2 \times \text{Percentage Error in } T \]
Substituting the values:
\[ \text{Total Percentage Error} = 1.67 + 10 + 2 \times 3.57 \approx 1.67 + 10 + 7.14 = 18.81\% \]

### Conclusion
Based on the calculations:
1. The percentage error in \( R \) is approximately \( 1.67\% \).
2. The percentage error in \( r \) is \( 10\% \).
3. The percentage error in \( T \) is \( 3.57\% \).
4. The total percentage error in \( g \) is approximately \( 18.81\% \).