Experiment 1 - Mechanical Measurements

For this experiment, it is not necessary to use your own laptop (other than to take notes).

If you want to print this description, the pdf export might not work properly, so you can download one here: (version Oct 22, 2025 ).

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 either 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