The period of oscillation of a simple pendulum is `T = 2pi sqrt(L/g)` . Measured value of 'L' is 1.0 m from meter scale having a minimum division of 1 mm and time of one complete oscillation is 1.95 s measured from stopwatch of 0.01 s resolution. The percentage error in the determination of 'g' will be :

Options

A. `1.13%`
B. `1.03%`
C. `1.33%`
D. `1.30%`

Answer

To determine the percentage error in the value of 'g' based on the given measurements, we can follow these steps:

### Step 1: Identify the formula for 'g'
The formula for the acceleration due to gravity 'g' in terms of the period of oscillation 'T' and the length 'L' of the pendulum is given by:

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

Rearranging this formula to solve for 'g', we have:

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

### Step 2: Identify the measured values and their uncertainties
- Length \( L = 1.0 \, \text{m} \) with a minimum division of 1 mm, which gives an uncertainty:
\[ \Delta L = 0.001 \, \text{m} \]

- Time period \( T = 1.95 \, \text{s} \) with a resolution of 0.01 s, which gives an uncertainty:
\[ \Delta T = 0.01 \, \text{s} \]

### Step 3: Calculate the relative errors
To find the percentage error in 'g', we need to calculate the relative errors in 'L' and 'T':

1. **Relative error in L**:
\[ \frac{\Delta L}{L} = \frac{0.001}{1.0} = 0.001 \]

2. **Relative error in T**:
\[ \frac{\Delta T}{T} = \frac{0.01}{1.95} \approx 0.005128 \]

### Step 4: Use the formula for the error in 'g'
The formula for the relative error in 'g' based on the contributions from 'L' and 'T' is:

\[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \cdot \frac{\Delta T}{T} \]

Substituting the values we calculated:

\[ \frac{\Delta g}{g} = 0.001 + 2 \cdot 0.005128 \approx 0.001 + 0.010256 = 0.011256 \]

### Step 5: Calculate the percentage error in 'g'
To find the percentage error, we multiply the relative error by 100:

\[ \text{Percentage error} = \frac{\Delta g}{g} \times 100 \approx 0.011256 \times 100 \approx 1.1256\% \]

Rounding this to two decimal places, we get:

\[ \text{Percentage error} \approx 1.13\% \]

### Final Answer
The percentage error in the determination of 'g' is approximately **1.13%**.

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