The lengths of two components will be measured several times. The uncertainty in each

Chapter 3, Problem 20

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The lengths of two components will be measured several times. The uncertainty in each measurement of the length of the first component is \(\sigma_{1}=0.02 \mathrm{~cm}\), and the uncertainty in each measurement of the length of the second component is \(\sigma_{2}=0.08 \mathrm{~cm}\). Let \(\bar{X}\) denote the average of the measurements of the first component, and let \(\bar{Y}\) denote the average of the measurements of the second component. The total length of the two components will be estimated with the quantity \(\bar{X}+\bar{Y}\).

a. Find the uncertainty in the total length if the first component is measured 4 times and the second component is measured 12 times.

b. Find the uncertainty in the total length in terms of n if the first component is measured n times and the second component is measured 16 − n times.

c. Determine the best way to allocate 16 measurements between the components by determining the value of n that minimizes the uncertainty.

Equation Transcription:

   

   

   

   

Text Transcription:

\sigma_{1}=0.02 cm

\sigma_{2}=0.08 cm

\bar{X}

\bar{Y}

\bar{X}+\bar{Y}