Acceleration due to gravity is to be measured using simple pendulum. If a and b are percentage errors in the measurement of length and time period respectively, then percentage error in the measurement of acceleration due to gravity is

Options

A. `a-2b`
B. `2a-b`
C. `a+2b`
D. `b+2a`

Explanation

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

### Step 1: Understand the formula for acceleration due to gravity
The formula for the acceleration due to gravity using a simple pendulum is given by:
\[ g = \frac{4\pi^2 L}{T^2} \]
where \( L \) is the length of the pendulum and \( T \) is the time period of the pendulum.

### Step 2: Identify the errors in measurements
Let:
- \( A \) be the percentage error in the measurement of length \( L \).
- \( B \) be the percentage error in the measurement of time period \( T \).

### Step 3: Determine the relationship of errors in \( g \)
To find the percentage error in \( g \), we need to consider how the errors in \( L \) and \( T \) affect \( g \).

Using the formula for \( g \):
\[ g = 4\pi^2 L T^{-2} \]

### Step 4: Apply the error propagation formula
The relative error in \( g \) can be derived from the formula:
\[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T}{T} \]
where:
- \( \Delta g \) is the absolute error in \( g \),
- \( \Delta L \) is the absolute error in \( L \),
- \( \Delta T \) is the absolute error in \( T \).

### Step 5: Convert to percentage errors
Now, converting to percentage errors:
\[ \frac{\Delta g}{g} \times 100 = \left( \frac{\Delta L}{L} \times 100 \right) + 2 \left( \frac{\Delta T}{T} \times 100 \right) \]

Substituting the percentage errors \( A \) and \( B \):
\[ \text{Percentage error in } g = A + 2B \]

### Step 6: Conclusion
Thus, the percentage error in the measurement of acceleration due to gravity \( g \) is:
\[ \text{Percentage error in } g = A + 2B \]