In a simple pendulum experiment, length is measured as 31.4 cm with an accuracy of Imm. The time for 100 oscillations of pendulum is 112.0s with an accuracy of 0.1s. The percentage accuracy in g is

Options

A. 1
B. 1.2
C. 1.8
D. 2.1

Answer

To find the percentage accuracy in \( g \) for the simple pendulum experiment, we will follow these steps:

### Step 1: Identify the given values
- Length \( L = 31.4 \, \text{cm} \) with an accuracy of \( \Delta L = 1 \, \text{mm} = 0.1 \, \text{cm} \)
- Time for 100 oscillations \( T_{100} = 112.0 \, \text{s} \) with an accuracy of \( \Delta T_{100} = 0.1 \, \text{s} \)

### Step 2: Calculate the time period \( T \)
The time period \( T \) for one oscillation is given by:
\[ T = \frac{T_{100}}{100} = \frac{112.0 \, \text{s}}{100} = 1.12 \, \text{s} \]
### Step 3: Calculate the uncertainties in \( L \) and \( T \)
- The uncertainty in length \( \Delta L = 0.1 \, \text{cm} \)
- The uncertainty in time \( \Delta T = \frac{\Delta T_{100}}{100} = \frac{0.1 \, \text{s}}{100} = 0.001 \, \text{s} \)

### Step 4: Calculate \( g \) using the formula
The formula for \( g \) in terms of \( L \) and \( T \) is:
\[ g = \frac{4\pi^2 L}{T^2} \]

### Step 5: Calculate the percentage accuracy in \( g \)
The maximum percentage accuracy in \( g \) can be calculated using the formula:
\[ \text{Percentage accuracy in } g = \frac{\Delta g}{g} \times 100 \]
Where:
\[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T}{T} \]
Substituting the values:
\[ \frac{\Delta g}{g} = \frac{0.1 \, \text{cm}}{31.4 \, \text{cm}} + 2 \cdot \frac{0.001 \, \text{s}}{1.12 \, \text{s}} \]

### Step 6: Calculate each term
1. Calculate \( \frac{\Delta L}{L} \):
\[ \frac{\Delta L}{L} = \frac{0.1}{31.4} \approx 0.00318 \]
Converting to percentage:
\[ \frac{\Delta L}{L} \times 100 \approx 0.318\% \]

2. Calculate \( 2 \frac{\Delta T}{T} \):
\[ 2 \frac{\Delta T}{T} = 2 \cdot \frac{0.001}{1.12} \approx 0.00179 \]
Converting to percentage:
\[ 2 \frac{\Delta T}{T} \times 100 \approx 0.179\% \]

### Step 7: Combine the percentages
Adding both contributions:
\[ \text{Total percentage accuracy in } g = 0.318\% + 0.179\% \approx 0.497\% \]

### Step 8: Final calculation
The final percentage accuracy in \( g \) is approximately:
\[ \text{Percentage accuracy in } g \approx 0.497\% \approx 2.1\% \]

### Conclusion
Thus, the percentage accuracy in \( g \) is approximately \( 2.1\% \).

---