Lab1(Experimental Uncertainty)

1Experimental Uncertainty Calculation

Experimental Uncertainty Calculation

Lecturer: Mrs. A. Baker

Student Name: Kimani Lawrence

Course Title: Engineering Mechanics Lab 1

Course Code: MEE2018

Institution: University of Technology, Jamaica

Course Code: MEE2018 Student ID#: 1104173


AbstractThe lab report was on experimental uncertainty calculation. The aim was to determine the uncertainty

associated with the lack of accuracy of measuring equipment, the random variation in the measurands and

lack of repeatability in the output of the measuring system. The experiment was done in groups. It

implemented the use of triple beam balances, vernier calipers and four 3.8 cm internal diameter washers.

Each member of the group measured the mass, internal diameter, outer diameter and the thickness of each

washer. The results were tabulated and used for calculations to obtain the major results volume, density,

mean, standard deviation, bias error, precision error and the different uncertainties. In the analysis of this

data it could be assumed that the manufacture was very consistent. This was shown when the

experimenter calculated the density to be 0.33 g/cm3 less than that of which was published by the

manufacture. This confirmed the objectives of the experiment. The experiment showed that sources of

errors are in the environment and is very unpredictable; therefore one should be very careful when taking

measurements in order to be accurate. Conclusions could then be made.



Table of ContentsNomenclature..............................................................................................................................................4

Objectives:...................................................................................................................................................5

Theory:........................................................................................................................................................5

Apparatus:...................................................................................................................................................6

Procedure:...................................................................................................................................................6

Results:........................................................................................................................................................8

Calculations:................................................................................................................................................9

Discussion:................................................................................................................................................10

Conclusion:...............................................................................................................................................10

Recommendations.....................................................................................................................................11

References.................................................................................................................................................11

Appendix C (Sample Calculations)...........................................................................................................12

Appendix B (Raw Data)………………………………………………………………………………………………………………………..15



Nomenclature

Symbol DefinitionOD Outer DiameterID Inner Diameterl Thicknessρ DensityV VolumeB Bias ErrorP Precision Errorr0 Radius of Outer Ring

r1 Radius of Inner Ringm masst Student-t DistributionSẍ Standard Deviation of the Mean

Sx Standard Deviation of the Measurement

W UncertaintyWV Uncertainty of the Volume

Wρ Uncertainty of the Densityπ Constant [pi] = 22/7

∂ Partial DifferentialWl Uncertainty of the ThicknessWm Uncertainty of the mass

Wr0 Uncertainty of the Radius of the Outer Ring



Objectives: To determine the uncertainty associated with the lack of accuracy of measuring equipment

To determine the uncertainty associated with random variation in the measurands

To determine the uncertainty associated with the lack of repeatability in the output of the

measuring system

Theory:A measurement error, in the lab, is defined as the difference between the measured value and the

true value of the measurand. There are two main types of errors namely precision errors and bias errors.

Precision errors are errors caused by the lack of repeatability in the output of the measuring system. Bias

errors are consistent repeatable errors, caused by factors associated with the measuring instrument, such

as incorrect calibration or weak batteries in digital instruments. The main difference between bias error

and precision error is how the calculation is done. Precision error is the difference between the true value

and the average reading. Bias error is the difference between the average reading and the true value.

There are three major types of bias errors. These are calibration errors, loading errors and spatial

errors. It is very important to determine the errors prior to the start of the experiment. This will allow the

experimenter to correct the measurement at the end of the experiment or decide whether to use a different

measuring instrument for the experiment. All experimental uncertainties estimates should be made at the

same confidence level. It is normally made at 95%. This implies that 95% of the time an uncertainty is

made, the actual error will be less than the estimated uncertainty.



Apparatus: OHAUS 600 Series Triple Beam Balance of least count 0.1gram. The subsystems associated with

this instrument are the metal plate which is the sensing element, the sliding weights which is the

modification sub system and the lever arm which is the indicator and recorder.

Aerospace Vernier Caliper of least count 0.002cm. The subsystems associated with this

instrument are the jaws which are the sensing element, the slide assembly of the jaws which is the

modification sub system and the vernier scale which is indicator or recorder.

Four 3.8 cm (ID) flat steel washers of density 7.85 g/cm3

Procedure:The experiment was done in groups of four.

The measuring instruments were inspected.

The manufacture, model, least count and the three subsystems of the instruments were recorded.

The measuring instruments were calibrated.

The washer was placed on the plate on the triple beam balance. This displaced the lever arm.

The weights on the lever arm were adjusted until the arm returned to its initial condition.

The positions of the weights on the lever arm were observed and recorded.

The readings were summed. This revealed the weight of the washer.

Each member of the group took individual turns to measure the weights of each washer.

The upper jaws of the caliper were placed in the center of the washer and then slide down until it

touched the ring of the washer.

The experimenter observed the reading on the caliper and then recorded it. This gave the internal

diameter of the washer.

The lower jaws of the caliper was opened to about ¾ the size of the washer.



The washer was placed perpendicular between the jaws and the experimenter slowly pulled it

through the jaws.

The experimenter observed the reading on the caliper and then recorded it. This gave the outer

diameter of the washer.

The washer was then placed parallel between the jaws of the caliper and the experimenter closed

the jaws on the washer.

The experimenter observed the reading on the caliper and then recorded it. This gave the

thickness of the washer.

All readings were collected and recorded.



Results:Washer # Mass [g] ID [cm] OD [cm]

1 102.7 4.17 7.65 0.431 13.93 7.372 110.6 4.17 7.636 0.46 14.79 7.483 111.6 4.144 7.64 0.464 15.02 7.434 111.3 4.17 7.64 0.46 14.81 7.515 104.5 4.142 7.654 0.42 13.67 7.646 105.7 4.14 7.628 0.428 13.80 7.667 107.9 4.142 7.644 0.462 14.98 7.208 110 4.15 7.648 0.432 14.01 7.859 111.9 4.168 7.644 0.468 15.10 7.41

10 114.1 4.17 7.616 0.48 15.32 7.4511 122.3 4.162 7.618 0.5 15.99 7.6512 113 4.166 7.61 0.466 14.85 7.6113 102.7 4.16 7.638 0.438 14.12 7.2714 105.1 4.174 7.638 0.432 13.89 7.5715 104.7 4.7 7.678 0.46 13.32 7.8616 108.6 4.154 7.644 0.46 14.88 7.30

n 16 16 16 16 16 16Mean 109.17 4.19 7.64 0.45 14.53 7.52

5.086285 0.135821 0.016264 0.02164014 0.7228022 0.19079Bias Error [B]

0.05 0.001 0.001 0.001

2.709718 0.072359 0.008665 0.01152879

2.710179 0.072365 0.008722 0.01157208 0.3738011 0.28645

Thickness, l [cm]

Volume [cm3]

Density [g/cm3]

Standard Deviation

Precision Error [P]

Uncertainty [W]



Calculations:Volume [V] = πl(ro

2-r12)

Density [ρ] = m/V

Bias Error [B] = ½ * least count

Precision Error [P] = tSẍ

Sẍ = Sx/√n

:. P = t(Sx/√n)

Uncertainty [W] = (B2 + P2)1/2



Discussion:In carrying out the experiment various problems were encountered. Firstly it was found that the triple

beam balance did not retain its calibration after each use. This was observed when the experimenter

measured the same washer more than once. It was realized that the reading deviated further and

further away in that same direction at each instance. As a result, the experimenter had to recalibrate it

after each use. Continuing the experiment, the measurement of the outer diameter of the washer was

very tedious. Each time a measurement was made, the experimenter got different readings. The most

accurate procedure found was to slowly allow the washer to open the jaws of the vernier caliper to its

fullest. At this point, the experimenter would be certain that it was the true outer diameter of the

caliper that was measured. The calculated density of the material used to make the washer is 7.52

g/cm3 and the published density is 7.85 g/cm3. It can be said that the method used by the manufacture

was consistent. This is evident when a density of 7.52 g/cm3 was calculated, which was under the

published density by 0.33 g/cm3. The cause of this would most likely be some minor experimental

errors.

Conclusion:The aim was to determine the uncertainty associated with the lack of accuracy of measuring

equipment, the random variation in the measurands and lack of repeatability in the output of the

measuring system. Now in the experiment, it was found that there was a contrast between the

measured value and the true value. There was a very small deviation from the true value which the

experimenter could neglect. It can therefore be concluded that the uncertainty associated with the lack

of accuracy of measuring equipment, the random variation in the measurands and lack of the

repeatability in the output of the measuring system can be determined. This should also be done

within the limits of experimental errors.



Recommendations To better the output of future experiments, the experimenter recommends that experiments be

done more than once. This will decrease the chances of having an error in the output in the measured value.

ReferencesGlenn Elert (1998 January 20). Density of Steel, University of Wisconsin-Stout Physics Department,

Retrieved September 21, 2011 from http://hypertextbook.com/facts/2004/KarenSutherland.shtml


http://hypertextbook.com/facts/2004/KarenSutherland.shtml


Appendix C (Sample Calculations)






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Lab1(Experimental Uncertainty)