1. Context of the Experiment
“Nothing reveals the lack of mathematical education more, than an excessively precise calculation.” - Carl Friedrich Gauß
Measurements are supposed to give certain information about a measurand, but they can never be exact due to influencing factors as the measuring procedure, environmental influences, the skills of the person measuring etc. Therefore, when measuring physical quantities, the measured values always scatter around the “true value”. Each measurement yields a different result, although the measured quantity doesn’t change. This inaccuracy is called “error” or more precise: uncertainty. To determine this uncertainty, multiple measurements can be conducted, while all other influences should be kept the same.
2. Learning Goals of this Experiment
- Knowing: uncertainty, complete measurement, errors, steps in handling calipers
- Abilities: calculate uncertainties and absolute/relative errors, handling calipers, choosing the correct measurement instrument
- Understand: measurement accuracy and being able to assess it, review of measurement method
3. Literature
[1] DIN 1319-4. Grundlagen der Meßtechnik. Teil 3: Auswertung von Messungen - Meßunsicherheit (9.2). Ausgabe 2, Februar 1999.
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Necessary further reading: how to use a caliper!, error propagation, rounding to significant figures!
4. Basics/Fundamentals
4.1 uncertainty
A complete measurement can only be specified by taking the uncertainty into account, it can be displayed as where M is the average or the one measurement and u is the uncertainty. The uncertainty can also be specified as a relative value, e.g. in percent of the measured value M. A measured value that has multiple decimal digits can show more digits than the uncertainty allows, therefore rounding to the last decimal digit of the uncertainty is common.
4.2 errors
Errors that can influence a measurement can be either systemic or statistic. Systemic errors are dependent e.g. on the instrument and always have the same amount and sign and are reproducible, so they appear every time the measurement is conducted. Some kind of offset can be used to compensate for systemic errors, but there could also be unknown systemic errors that therefore can’t be compensated. Statistic errors cannot be corrected because they are random by definition, but they can be described by average values and standard deviation, if multiple measurements were conducted. The standard deviation then is a measure of the scattering of the values around the average, for a big standard deviation, the values scatter very far from the average. It can therefore be concluded, that a single measurement doesn’t have any implications as no knowledge about the statistic error exists! [2]
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A complete measurement result for a statistically determined quantity x is therefore
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4.3 vernier caliper gauge
There are four types of measurements that can be made with a vernier caliper gauge: Inside measurements, outside measurements, depth and steps.
When the vernier caliper is closed, the scale should read exactly 0. To take a measurement of an inside dimension, the jaws of the caliper are now opened until they exactly cover the area to be measured. To measure an outside dimension, the caliper jaws are opened and then closed again until they touch the object. For measuring depths (e.g. drilled holes), the depth gauge at the end of the caliper can be used - the main element of the caliper is pressed against the higher surface and the depth gauge is positioned at the deepest point of the hole.
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For reading, the main scale in mm is now used first to read the digit before the decimal point. This can be read at the position in front of the dash of zero on the slider. In order to read the decimal place, the position in the vernier scale (german: Nonius-Skala) must be found where the line of the large scale exactly coincides with one of the smaller scale.
4.4 significant figures
Significant figures are figures in a number, that carry meaning. To avoid misunderstandings, only significant ending zeroes are written, non significant zeros are omitted by using the scientific notation with power of ten. Leading zeros are always non significant!
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A result can never be given with more accuracy than the biggest decimal number of the uncertainty, therefore, rounded numerical uncertainty values should be reported with two (or, if required, three) significant digits and rounded up. The measurement result should be rounded at the same decimal place as the associated uncertainty, e.g. M=123.4561V to M=123.46V if u(M)=0.7819V is rounded up to u(M)=0.79V. [1]
5. Technical Basics & Preparations
Before measuring, these aspects of measuring should be considered and thought trough:
- What is supposed to be measured? What values?
- Which accuracy? Is it even possible to measure this with this accuracy?
- What measuring device can be used for this? Does it provide the necessary accuracy?
- What are the possible (systemic) errors and external limitations that exist in the setup?
- Then: find out about the complete measurement
General instructions
- Always try to pick the measurement instrument that alters the measurement the least and is the most accurate – there might be a trade-off! Use the instrument you deem most useful!
- Try to minimize systemic errors where possible!
Preparations:
- Gather all the necessary measurement objects and instruments:
- Caliper
- Ruler
- 1 cylinder
- 2 flat objects
- 1 O-ring
- 10 beans out of the container
- Prepare a table to record all your measurements.
- Familiarize yourself with the caliper and its function.
6. Experiment Procedure
1.0 Take your caliper and find the offset value (the value you can still read, even though the caliper is fully closed). This offset value will later need to be subtracted from every measured value. Remark: This step is usually not necessary for high quality calipers!
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1.7 Go to one of the teststands and open the Matlab App for Experiment 1. Enter your matriculation number and the number of the Experiment Kit you are working with. Then enter your values for 1.0, 1.1, 1.2 and 1.3. All values and calculations for 1.4-1.6 will need to be supplied to vips later.
7. Evaluation of Experiment Results
The following values should be entered to vips in Stud.ip:
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