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 2` and `pm 3` respectively.

Explanation

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 given equation is:
\[ T = 2\pi \sqrt{\frac{l}{g}} \]
We need to find the fractional error in \( g \).

### Step 2: Rearranging the equation
We can rearrange the equation to express \( g \) in terms of \( T \) and \( l \):
\[ g = \frac{4\pi^2 l}{T^2} \]

### Step 3: Apply the error propagation formula
To find the fractional error in \( g \), we will use the formula for error propagation. The fractional error in a function of multiple variables can be expressed as:
\[ \frac{\Delta g}{g} = \frac{\Delta T}{T} + 2 \frac{\Delta l}{l} \]
where \( \Delta T \) and \( \Delta l \) are the absolute errors in \( T \) and \( l \) respectively.

### Step 4: Substitute the given fractional errors
We are given:
- Fractional error in \( T \) is \( \pm 2\% \) or \( \frac{\Delta T}{T} = \frac{2}{100} = 0.02 \)
- Fractional error in \( l \) is \( \pm 3\% \) or \( \frac{\Delta l}{l} = \frac{3}{100} = 0.03 \)

### Step 5: Substitute into the error formula
Substituting these values into the error propagation formula:
\[ \frac{\Delta g}{g} = 0.02 + 2 \times 0.03 \]

### Step 6: Calculate the result
Calculating the right side:
\[ \frac{\Delta g}{g} = 0.02 + 0.06 = 0.08 \]

### Step 7: Convert to percentage
To express this as a percentage:
\[ \Delta g \text{ (percentage)} = 0.08 \times 100 = 8\% \]

### Final Answer
The fractional error in \( g \) is \( \pm 8\% \).

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