While measuring the acceleration due to gravity by a simple pendulum, a student makes a positive error of 1% in the length of the pendulum and a negative error of 3% in the value and a of time period. His percentage error in the measurement of g by the relation `g = 4pi^(2)(l//T^(2))` will be

Options

A. A) 0.02
B. B) 0.04
C. C) 0.07
D. D) 0.1

Answer

To find the percentage error in the measurement of acceleration due to gravity \( g \) using the formula

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

we need to analyze the errors in the measurements of length \( l \) and time period \( T \).

### Step 1: Identify the errors
- The student makes a **positive error of 1%** in the length \( l \).
- The student makes a **negative error of 3%** in the time period \( T \).

### Step 2: Determine the relationship of errors
Using the formula for \( g \), we can express the percentage error in \( g \) in terms of the percentage errors in \( l \) and \( T \).

The formula for \( g \) can be rewritten as:

\[ g \propto \frac{l}{T^2} \]

### Step 3: Calculate the percentage error in \( g \)
The percentage error in \( g \) can be calculated using the formula for the propagation of errors:

\[ \text{Percentage error in } g = \text{Percentage error in } l + 2 \times \text{Percentage error in } T \]

Substituting the values:

- Percentage error in \( l = +1\% \)
- Percentage error in \( T = -3\% \)

Now substituting these into the equation:

\[ \text{Percentage error in } g = 1\% + 2 \times (-3\%) \]

Calculating this gives:

\[ \text{Percentage error in } g = 1\% - 6\% \]
\[ \text{Percentage error in } g = -5\% \]

### Step 4: Conclusion
The percentage error in the measurement of \( g \) is **-5%**. This indicates that the measured value of \( g \) is lower than the actual value by 5%.

### Final Answer
The percentage error in the measurement of \( g \) is **-5%**.

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