Experiment 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

Necessary further reading: how to use a caliper!, error propagation, rounding to significant figures!:

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

[2] Puente León, Messtechnik. Berlin, Heidelberg: Springer Berlin Heidelberg, 2019.

[3] Fornasini, The Uncertainty in Physical Measurements. New York, NY: Springer New York, 2008.

[4] Kameke, Messwerte und Messunsicherheit. Hamburg, TUHH, 2019.

[5] Lectures: MT-Motivation, Measurement Theory

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]

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)

(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 three types of measurements that can be made with a vernier caliper gauge: Inside measurements, outside measurements and depth.
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. Leading zeros are always non significant!

Some examples:

The result of an addition/subtraction has as many decimal places as the number with the fewest decimal places. [3]

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. [3]

Examples:  3.123*4.54=14.2     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
  • 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!

1.1 O-ring measurements:

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

1.1.2 Which value is closer to the real value?

1.2 Find out, which of the two flat objects has a higher thickness t, object A or B.

Fig. 1: object A (left) and object B (right)

1.3 Cylinder measurement:

   1.3.1 Pick the cylinder 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 cylinder 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 Take the measurements of the cylinder again to find the volume of the cylinder by carrying out the following steps:

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

1.4.2 Calculate the propagated uncertainty (using the accuracy as variable uncertainty a_x) and supply a complete measurement of the volume (rounded to significant figures and including the uncertainty)! Use the equation given in the explanation 4.2.

1.5 Measure 10 beans from the central container.

1.5.1 Measure the length, width and depth of the beans and make a table with all values.

1.5.2 Assume that a bean's volume V equals its depth multiplied with its length and its width: . Calculate the propagated uncertainty and supply a complete measurement of the volume (rounded to significant figures)!

For this, first calculate the standard deviation for each measured variable (d, l and w) and then use this as the uncertainty of the variable a_xj for the calculation of the propagated uncertainty.

1.5.3 Calculate the amount of beans that fit into the container (47 x 39 x 55 mm) you took them from.  Assume that there is no air in between the beans. Just consider the absolute volume of the container and the beans.

1.6 (creative) Find a way to measure the volume of a very irregular shape (e.g. your hand) and describe your steps.

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 (VIPS)

The following values should be entered to vips in Stud.ip. Please note that only 30 minutes are available for entering values in vips, after which the vips entry is automatically terminated. The test is only to hand in the final values, so be prepared and have all values ready and at hand!

1. Matriculation number
   
2. Which step was for testing the reproducibility of the experiment?
   select from:
   - 1.3.1
   - 1.3.2
   - 1.3.3
   relevant section: 6.1.3.
   
   
3. Which step was for testing the repeatability of the experiment?
   select from:
   - 1.3.1
   - 1.3.2
   - 1.3.3
   relevant section: 6.1.3.
   
   
4. Please specify the accuracy of the measurement instrument you are using for task 1.4.1
   unit: [mm]
   relevant section: 6.1.4.1
   
   
5. Please enter the complete measurement result for the cylinder volume of task 1.4.2
   unit: [mm³]
   enter your aswer like this:  (measurand +/- propagated uncertainty)mm³
   relevant section: 6.1.4.2
   
   
6. Please type your table values for task 1.5.1
   unit: [mm]
   Use the structure proposed in this example:
   
   [3.5 4 2; 5 6 3]
   relevant section: 6.1.5.1
   
   
7. Please enter the complete measurement result for the beans volume of task 1.5.2
   unit: [mm³]
   enter your aswer like this:  (measurand +/- propagated uncertainty)mm³
   relevant section: 6.1.5.2
   
   
8. Please explain the steps necessary to measure the volume of your hand briefly. Include your resulting value in mm³
   (free text)
   relevant section: 6.1.6
   
   
9. Please enter the total calculated number of beans in the container
   relevant section: 6.1.5.3