The length of a pendulum is measuredas 1,01 m and time for 30 oscillation is measured as one minute 3 s. Error length is 0.01 m and error in the 3 s. The percentage error in the measurement of acceleration due to gravity id,

Options

A. 1
B. 5
C. 10
D. 15

Answer

To find the percentage error in the measurement of acceleration due to gravity (g), we will follow these steps:

### Step 1: Identify the given values
- Length of the pendulum, \( l = 1.01 \, \text{m} \)
- Error in length, \( \Delta l = 0.01 \, \text{m} \)
- Time for 30 oscillations, \( T_{30} = 1 \, \text{min} \, 3 \, \text{s} = 63 \, \text{s} \)
- Error in time, \( \Delta t = 3 \, \text{s} \)

### Step 2: Calculate the time period for one oscillation
The time period \( T \) for one oscillation is given by:
\[ T = \frac{T_{30}}{30} = \frac{63 \, \text{s}}{30} = 2.1 \, \text{s} \]

### Step 3: Use the formula for acceleration due to gravity
The formula relating the time period, length, and acceleration due to gravity is:
\[ T^2 = \frac{4\pi^2 l}{g} \]
From this, we can express \( g \) as:
\[ g = \frac{4\pi^2 l}{T^2} \]

### Step 4: Calculate the percentage error in \( g \)
The percentage error in \( g \) can be calculated using the formula:
\[ \frac{\Delta g}{g} \times 100 = \frac{\Delta l}{l} \times 100 + 2 \times \frac{\Delta t}{T} \times 100 \]

### Step 5: Calculate each component of the percentage error
1. **Calculate the percentage error due to length \( l \)**:
\[ \frac{\Delta l}{l} \times 100 = \frac{0.01}{1.01} \times 100 \approx 0.9901\% \]

2. **Calculate the percentage error due to time \( T \)**:
\[ \frac{\Delta t}{T} \times 100 = \frac{3}{2.1} \times 100 \approx 142.857\% \]
Since this is multiplied by 2 in the formula:
\[ 2 \times \frac{\Delta t}{T} \times 100 \approx 2 \times 142.857\% \approx 285.714\% \]

### Step 6: Combine the errors
Now, we can combine these errors:
\[ \text{Total percentage error in } g = 0.9901\% + 285.714\% \approx 286.7041\% \]

### Conclusion
Thus, the percentage error in the measurement of acceleration due to gravity \( g \) is approximately **286.70%**.

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