1 Mechanical Measurements

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

• Puente León, Messtechnik. Berlin, Heidelberg: Springer Berlin Heidelberg, 2019.
• Fornasini, The Uncertainty in Physical Measurements. New York, NY: Springer New York, 2008.
• Kameke, Messwerte und Messunsicherheit. Hamburg, TUHH, 2019.
• Lectures: MT-Motivation, Measurement Theory

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.

Example systemic error: a scale always shows a few grams, although nothing is on the scale.

Example statistic error: when reading a scale with an indicator needle, the reader makes small mistakes
(Serious/major errors: errors due to the experimenter, e.g. improper instrument handling/arrangement, can be easily and generally avoided with some care and preparation. E.g. reading the ammeter as a voltage)

It can therefore be concluded, that a single measurement doesn’t have any implications as no knowledge about the statistic error exists!

(1)

(2)

(3)

A complete measurement result for a statistically determined quantity x is therefore with the measurand M_x and the uncertainty u_x.

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.

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.

Some examples:

The result of an addition/subtraction has as many decimal places as the number with the fewest decimal places.

Examples:  11.234+0.0007=11.235     11+3.432=14     11.00+3.432=14.43

The result of a multiplication/division has as many significant figures as the number with the fewest significant figures.

Examples:  3.123*4.54=14.18     3.123*0.1=0.3     3.123*0.0012=0.0037

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
  • 10 cylinders of type A
  • 2 flat objects -> Unterlegscheiben!
  • 1 O-ring

6.      Experiment Procedure

1.1 Measure the inner diameter of the o-ring using the caliper and the ruler. Note both values.

1.2 Find out, which of the two flat objects is thicker, A or B.

1.3.1 Pick one of the cylinders and measure its length and diameter.

   1.3.2 Put the cylinder down. Measure the same part again, using the same caliper.

   1.3.3 Now ask your neighbor to measure your wooden part with their caliper!

• Did 1.3.1, 1.3.2 and 1.3.3 yield the same result? Which of the steps was for testing reproducibility (A) and which for repeatability (B)?

1.4 Measure the volume of one of the cylinders by carrying out the following steps:

Measure every size of the cylinder that is needed to find the volume. What accuracy does your measuring instrument have?

Calculate the propagated error and supply a complete measurement of the volume (rounded to significant figures and including the uncertainty)!

1.5 Submit a complete measurement of the length and the diameter. Use at least 10 measurements.

1.6 (creative) Find a way to measure the weight of your hand (only the hand, not the arm) at home and describe your steps.???

7.      Evaluation of Experiment Results

The following values should be entered to vips in Stud.ip:

1. Matriculation number
2. Value for 1.1 caliper
3. value for 1.1 ruler
4. the thicker object (A or B)
5. value for cylinder - length (1.3.1)
6. value for cylinder - diameter: measurement  (1.3.1)
7. value for cylinder - length (1.3.2)
8. value for cylinder - diameter: measurement  (1.3.2)
9. value for cylinder - length (1.3.3)
10. value for cylinder - diameter: measurement  (1.3.3)
11. reproducibility
12. repeatability
13. complete measurement of the volume of the cylinder 1.4
14. complete measurement result for 1.5 - length and diameter
15. (free text) explain your measurement steps and the final result of 1.6

Resources:

[1] DIN 1319-4. Grundlagen der Meßtechnik. Teil 3: Auswertung von Messungen - Meßunsicherheit (9.2). Ausgabe 2, Februar 1999.