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 a 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: MTMotivation, 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 for 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

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.

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

Instructions on caliper use -> explain in lecture?!?

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 -> Muttern oder 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 Pick one of the cylinders and measure its length and diameter.

Submit a complete measurement of the length and the diameter. Use at least 10 measurements. (repetition)

Now measure all 4 cylinders (reproducibility)

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)!

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, value for 1.1 ruler
3. the thicker object (A or B)
4. value for the complete measurement of one cylinder, length and diameter