In measuring 'g' using a simple pendulum a student makes a positive error of 1% in length of pendulum and a negative error of 3% in the value of time period. Percentage error in measurement of value of g is

Options

A. `2%`
B. `4%`
C. `7%`
D. `10%`

Answer

To find the percentage error in the measurement of the acceleration due to gravity \( g \) using a simple pendulum, we can follow these steps:

### Step 1: Understand the relationship between \( g \), length \( L \), and time period \( T \)
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
From this equation, we can derive that:
\[ g = \frac{4\pi^2 L}{T^2} \]

### Step 2: Determine the percentage errors
The problem states that:
- There is a positive error of 1% in the length \( L \).
- There is a negative error of 3% in the time period \( T \).

### Step 3: Calculate the relative errors
The percentage error in \( g \) can be calculated using the formula for propagation of errors. The percentage error in \( g \) is given by:
\[ \text{Percentage error in } g = \text{Percentage error in } L + 2 \times \text{Percentage error in } T \]

### Step 4: Substitute the values
Substituting the given values into the equation:
- Percentage error in \( L = +1\% \)
- Percentage error in \( T = -3\% \)

Thus, we have:
\[ \text{Percentage error in } g = 1\% + 2 \times (-3\%) \]
\[ = 1\% - 6\% \]
\[ = -5\% \]

### Step 5: Interpret the result
The negative sign indicates that the measurement of \( g \) is underestimated due to the errors in measurement.

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

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