The error in the measurement of the length of a simple pendulum is `0.1%` and the error in the time period is `2%`. What is the possible percentage of error in the physical quantity having the dimensional formula `LT^(-2)`?

Options

A. `4.1%`
B. `8.2%`
C. `2.2%`
D. `1.1%`

Explanation

To find the possible percentage of error in the physical quantity with the dimensional formula \( LT^{-2} \), we can follow these steps:

### Step 1: Identify the given errors
We are given:
- The error in the measurement of length (\( L \)) is \( 0.1\% \).
- The error in the measurement of time period (\( T \)) is \( 2\% \).

### Step 2: Understand the relationship of the physical quantity
The physical quantity with the dimensional formula \( LT^{-2} \) can be expressed as:
\[ X = L \cdot T^{-2} \]

### Step 3: Use the formula for error propagation
The percentage error in a product or quotient can be found using the following formula:
\[ \frac{\Delta X}{X} = \frac{\Delta L}{L} + n \cdot \frac{\Delta T}{T} \]
where \( n \) is the power of \( T \) in the expression. In our case, since \( T \) is in the denominator and raised to the power of 2, we have:
- \( n = -2 \)

Thus, the equation becomes:
\[ \frac{\Delta X}{X} = \frac{\Delta L}{L} - 2 \cdot \frac{\Delta T}{T} \]

### Step 4: Substitute the values of errors
Now, substituting the values of the errors:
- \( \frac{\Delta L}{L} = 0.1\% = \frac{0.1}{100} \)
- \( \frac{\Delta T}{T} = 2\% = \frac{2}{100} \)

Substituting these into the error propagation formula:
\[ \frac{\Delta X}{X} = \frac{0.1}{100} - 2 \cdot \frac{2}{100} \]

### Step 5: Calculate the total error
Now, calculate the right-hand side:
\[ \frac{\Delta X}{X} = \frac{0.1}{100} - \frac{4}{100} \]
\[ \frac{\Delta X}{X} = \frac{0.1 - 4}{100} \]
\[ \frac{\Delta X}{X} = \frac{-3.9}{100} \]

### Step 6: Convert to percentage
To express this as a percentage:
\[ \Delta X = -3.9\% \]

Since we are interested in the magnitude of the error:
\[ \text{Possible percentage of error in } X = 3.9\% \]

### Final Answer
The possible percentage of error in the physical quantity having the dimensional formula \( LT^{-2} \) is \( 3.9\% \).

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