Sure, let's go through the steps needed to determine the error in the estimated value of [tex]\( g \)[/tex].
1. Understanding the Problem:
- We have a formula for [tex]\( g \)[/tex] given by:
[tex]\[
g = 4\pi^2 \frac{L}{T^2}
\][/tex]
- The errors in the measurements are:
- Error in [tex]\( L \)[/tex] ([tex]\( \delta L \)[/tex]) is [tex]\( \pm 2\% \)[/tex]
- Error in [tex]\( T \)[/tex] ([tex]\( \delta T \)[/tex]) is [tex]\( \pm 3\% \)[/tex]
2. Error Propagation for the Given Formula:
- To find the error in [tex]\( g \)[/tex] ([tex]\( \delta g \)[/tex]), we need to use the rules of error propagation for multiplication and division.
- The relative error in a product or quotient [tex]\( Q = \frac{A}{B} \)[/tex] is the sum of the relative errors of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. If [tex]\( Q = A \cdot B \)[/tex], again the relative error is the sum of the relative errors.
- For the formula:
[tex]\[
g = 4\pi^2 \frac{L}{T^2}
\][/tex]
we need to consider:
- The coefficient [tex]\( 4\pi^2 \)[/tex] is a constant and has no error.
- [tex]\( L \)[/tex] has an error of [tex]\( \pm 2\% \)[/tex]
- [tex]\( T \)[/tex] has an error of [tex]\( \pm 3\% \)[/tex] but since it is [tex]\( T^2 \)[/tex] in the denominator, the error in [tex]\( T^2 \)[/tex] will be twice the error in [tex]\( T \)[/tex], so [tex]\( \pm 6\% \)[/tex].
3. Calculating the Total Error:
- The total relative error in [tex]\( g \)[/tex] is then the sum of the relative errors in [tex]\( L \)[/tex] and [tex]\( T^2 \)[/tex]:
[tex]\[
\text{Total error in } g = \text{Error in } L + 2 \cdot \text{Error in } T
\][/tex]
[tex]\[
\delta g = 2\% + 2 \cdot 3\%
\][/tex]
[tex]\[
\delta g = 2\% + 6\%
\][/tex]
[tex]\[
\delta g = 8\%
\][/tex]
4. Choosing the Correct Option:
- From the options provided:
- (1) [tex]\( \pm 8\% \)[/tex]
- (2) [tex]\( \pm 6\% \)[/tex]
- (3) [tex]\( \pm 3\% \)[/tex]
- (4) [tex]\( \pm 5\% \)[/tex]
The correct answer is [tex]\( \pm 8\% \)[/tex].
Therefore, the error in the estimated value of [tex]\( g \)[/tex] is [tex]\( \pm 8\% \)[/tex].
Answer: (1) [tex]\( \pm 8\% \)[/tex]
