Question

Four persons measure the length of a rod as 20.00 cm, 19.75 cm, 17.01 cm and 18.25 cm. The relative error in the measurement of average length of the rod is :

• A. 0.24
• B. 0.06
• C. 0.18
• D. 0.08

Hint:

Relative error is a dimensionless quantity. If you are asked for percentage error, simply multiply the relative error by 100.

Answer 1

The Correct Option is B

To find the relative error in the measurement of the average length of the rod, follow these steps:

1. Calculate the average length of the rod:
   • The individual measurements given are: 20.00 cm, 19.75 cm, 17.01 cm, and 18.25 cm.
   • The average length, \(L_{\text{avg}}\), is calculated as follows: \(L_{\text{avg}} = \frac{20.00 + 19.75 + 17.01 + 18.25}{4}\)\(= \frac{75.01}{4} = 18.7525 \, \text{cm}\)
2. Calculate the absolute errors: Calculate the error for each measurement with respect to the average length.
   • The absolute error for each measurement is the difference between the measurement and the average:
     • Error for 20.00 cm: \(|20.00 - 18.7525| = 1.2475\)
     • Error for 19.75 cm: \(|19.75 - 18.7525| = 0.9975\)
     • Error for 17.01 cm: \(|17.01 - 18.7525| = 1.7425\)
     • Error for 18.25 cm: \(|18.25 - 18.7525| = 0.5025\)
3. Calculate the mean absolute error:
   • The mean absolute error is:\(\frac{1.2475 + 0.9975 + 1.7425 + 0.5025}{4} = \frac{4.49}{4} = 1.1225\)
4. Calculate the relative error:
   • The relative error is calculated by dividing the mean absolute error by the average length: \(\text{Relative Error} = \frac{1.1225}{18.7525} \approx 0.0598\)
   • Rounded to two decimal places, the relative error is approximately 0.06.
5. Conclusion:
   • The relative error in the measurement of the average length of the rod is 0.06.