The acceleration due to gravity is measured on the surface of earth by using a simple pendulum. If `alphaandbeta` are relative errors in the measurement of length and time period respectively, then percentage error in the measurement of acceleration due to gravity is

Options

A. `(alpha+1/2beta)xx100`
B. `(alpha-2beta)`
C. `(2alpha+beta)xx100`
D. `(alpha+2beta)xx100`

Answer

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

### Step-by-Step Solution:

1. **Understand the Formula for the Time Period of a Simple Pendulum**:
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

2. **Rearranging the Formula**:
From the formula, we can express \( g \) in terms of \( T \) and \( L \):
\[ g = \frac{4\pi^2 L}{T^2} \]

3. **Identify Relative Errors**:
Let \( \alpha \) be the relative error in the measurement of length \( L \):
\[ \alpha = \frac{\Delta L}{L} \]
Let \( \beta \) be the relative error in the measurement of the time period \( T \):
\[ \beta = \frac{\Delta T}{T} \]

4. **Differentiate the Equation for g**:
To find the error in \( g \), we can differentiate the equation \( g = \frac{4\pi^2 L}{T^2} \):
\[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T}{T} \]
This can be rewritten using the relative errors:
\[ \frac{\Delta g}{g} = \alpha + 2\beta \]

5. **Convert to Percentage Error**:
To convert the relative error into percentage error, we multiply by 100:
\[ \text{Percentage Error in } g = \left(\alpha + 2\beta\right) \times 100 \]

6. **Final Expression**:
Thus, the percentage error in the measurement of acceleration due to gravity \( g \) is:
\[ \text{Percentage Error in } g = (\alpha + 2\beta) \times 100\% \]

### Final Answer:
The percentage error in the measurement of acceleration due to gravity is \( (\alpha + 2\beta) \times 100\% \).

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