Modulus (E) is to be found from a force (F), width (w), thickness (t), and strai
ID: 3119196 • Letter: M
Question
Modulus (E) is to be found from a force (F), width (w), thickness (t), and strain (e) reading from
E=F/twe
The following data was obtained experimentally and from product calibration records.
a) Build a table of bias and precision values for each of the four measurements below.
b) Determine the modulus and its uncertainty from the propagation of error for this experiment.
Modulus Test
Standard Deviation (n>30)
Accuracy
Force (lbs)
8000
5
100
Thickness (in)
0.125
0
0.001
Width (in)
1
0
0.001
Strain (me)
2100
10
100
Modulus Test
Standard Deviation (n>30)
Accuracy
Force (lbs)
8000
5
100
Thickness (in)
0.125
0
0.001
Width (in)
1
0
0.001
Strain (me)
2100
10
100
Explanation / Answer
Relative error = |Experimental value - Accepted value|/ Accepted value X100%
Relative error =+
Consider the following results of velocity measurements: 0.38, 0.38, 0.35, 0.44, 0.43, 0.42 m/s. The average value of these six velocity measurements is equal to: v = (0.38 + 0.38 + 0.35 + 0.44 + 0.43 + 0.42) / 6 = 0.40 m/s. Next, one needs to calculate the deviations from the average velocity: 0.38 - 0.40 = 0.02 m/s; 0.38 - 0.40 = 0.02 m/s; 0.40 - 0.35 = 0.05 m/s; 0.40 - 0.44 = -0.04 m/s; 0.40 - 0.43 = -0.03 m/s; 0.40 - 0.42 = -0.02 m/s. The general formula for calculation of the average value xAV (sometimes also called mean value) is as follows: ! xAV = 1 n x1 + x2 + x3 + ...+ x ( n ) where n is the number of repeated measurements (for this example n = 6). The values of the deviation from the average value are used to calculate the experimental error. The quantity that is used to estimate these deviations is known as the standard deviation ! sx and is defined as: ! sx = 1 n "1 x1 " x ( AV ) 2 + x2 " x ( AV ) 2 + ...+ xn " x ( AV ) 2 [ ] The standard deviation squared - ! sx 2 is the sum of squares of deviations from the average value divided by (n - 1). The subscript usually indicates the quantity that the standard deviation is calculated for, e.g., sv stands for the standard deviation of velocity measurements, whereas sa is the standard deviation for acceleration data. For the previously discussed example of velocity measurements we have: ! sv = 1 6 "1 (0.02 m/s) 2 + (0.02 m/s) 2 + (0.05 m/s) 2 + (0.04 m/s) 2 + (0.01 m/s) 2 + (0.02 m/s)2 [ ] = = 1 5 #0.0062(m2 /s2 ) = 0.0352 m/s $ 0.04 m/s We use the standard deviation as the value of the experimental error. The final result of measurements and error analysis should be written as: v = vAV ± sv = 0.40 ± 0.04 (m/s) (do not forget to write the appropriate units!) The general format for presenting experimental results with experimental error is given by one of the following expressions: “final result” = “average value” ± “standard deviation” x = xAV ± sx units or x = xAV ± sx (units) or x = (xAV ± sx) units Obviously, the average value and the standard deviation must have the same units.
Experimental value - Theoretical value Theoretical value X 100%
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