What is the fractional error in g calculated from `T = 2 pi sqrt((l)/(g))` ? Given that fractional error in `T` and `l` are `pm 3` and `pm 6` respectively.

Answer

To find the fractional error in \( g \) calculated from the equation \( T = 2 \pi \sqrt{\frac{l}{g}} \), we will follow these steps:

### Step 1: Understand the relationship
The relationship between the period \( T \), length \( l \), and acceleration due to gravity \( g \) is given by:
\[ T = 2 \pi \sqrt{\frac{l}{g}} \]

### Step 2: Differentiate the equation
To find the fractional error in \( g \), we need to differentiate the equation with respect to \( T \) and \( l \). The formula for the propagation of errors in a function is:
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta l}{l} + \frac{1}{2} \frac{\Delta g}{g} \]
Here, \( \Delta T \), \( \Delta l \), and \( \Delta g \) represent the absolute errors in \( T \), \( l \), and \( g \), respectively.

### Step 3: Rearranging for \( \Delta g \)
We can rearrange the equation to solve for the fractional error in \( g \):
\[ \frac{\Delta g}{g} = 2 \frac{\Delta T}{T} - \frac{1}{2} \frac{\Delta l}{l} \]

### Step 4: Substitute the given errors
We are given:
- The fractional error in \( T \) is \( \pm 3\% \) or \( \frac{\Delta T}{T} = \frac{3}{100} = 0.03 \)
- The fractional error in \( l \) is \( \pm 6\% \) or \( \frac{\Delta l}{l} = \frac{6}{100} = 0.06 \)

Substituting these values into the equation:
\[ \frac{\Delta g}{g} = 2(0.03) - \frac{1}{2}(0.06) \]

### Step 5: Calculate the fractional error in \( g \)
Calculating the right side:
\[ \frac{\Delta g}{g} = 0.06 - 0.03 = 0.03 \]

### Step 6: Convert to percentage
To express this as a percentage, we multiply by 100:
\[ \frac{\Delta g}{g} \times 100 = 0.03 \times 100 = 3\% \]

### Final Result
Thus, the fractional error in \( g \) is \( \pm 3\% \).

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