The lengths of two components will be measured several times. The uncertainty in each measurement of the length of the first component is σ1= 0.02 cm, and the uncertainty in each measurement of the length of the second component is σ2 = 0.08 cm. Let denote the average of the measurements of the first component, and let denote the average of the measurements of the second component. The total length of the two components will be estimated with the quantity

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.

Answer

Step 1 of 4</p>

Uncertainty means standard deviation of the measurements

The uncertainty in the length of the first component is

The uncertainty in the length of the second component is

Let denote the average of the measurements of the first component

Let denote the average of the measurements of the second component

The total length of the components will be estimated with the quantity

Step 2 of 4</p>

a) We need to find the uncertainty if the first component is measured 4 times and

The second component is measured 12 times

=

=

=0.0251