Lab 1: Measurements

Physics I Lab

Section 2054-002 Wednesday 12 PM - 2 PM

Lab Instructor: Dr. Michael Zelin

September 2, 2016

By Abbigale Holloway

1. Objective: Introduction to measurement and error analysis 2. Introduction: Real-world measurements always involve some error (uncertainty [1]). Even simple measurements of an object’s dimensions have some systematic (particularly, instrument) errors and random errors [1]. The instrument error can be estimated as a fraction (1/10, 1/5, 1/2, etc.) of the least count of an instrument, the smallest non-zero measurement that it can reliably make [2]. For example,

considering that on average a resolution limit of a human eye is ~200 m, the instrument error of a meter stick with the smallest divisions in millimeters will be 0.2 mm. Random errors are caused by unpredictable factors, such as electrical noise in electrical circuits and sample variation (sampling error). This is why it is a good practice to take multiple measurements and use multiple samples to find an average (mean) value and standard deviation characterizing scatter of data around the mean [2]. For example, samples used for measuring a material density can have different structure, for instance, different volume fraction of pores in cork, that will cause different measured values. Proper statistical procedures are to be used for interpreting measurement results to infer findings made on a limited number of samples on the entire class of studied objects. In relative terms, errors can be expressed using a percent error calculation. For example, if the true weight of an object is 100 kg, but the result of measurement was 110 kg, the percent error would be: 100% ( 110 – 100 ) /100 = 10%. Percent instrument errors are added together when making multiple

measurements for calculating a certain quantity [1]. For example, in calculating density, =m/V, the total percent error will be equal to sum of percent errors of measuring mass, m, and volume, V. 3. Apparatus and Materials: Vernier, dial, and digital calipers, micrometer, meter stick, digital scale, a set of samples (Fig. 1)

Fig. 1 (a) – Vernier, digital, and dial calipers, (b) - materials used

4. Procedure and Results: This lab consists of two parts: - Part I: use micrometers to measure the thickness of coffee filters - Part II: use Vernier and digital calipers to measure dimensions of objects made of various materials and a digital scale to determine their mass to find the density from the measurements.

Vernier

Vernier

caliper

Digital caliper

Dial caliper

Table 1: Least Counts

Instrument Least Count

Meter stick 1 mm Micrometer 0.5 mm

Vernier caliper 0.1 cm

Dial caliper 0.1 cm

Digital caliper 0.01 mm

Digital scale 0.01 g

Part I. The following diagrams illustrate micrometer readings – go here to view a video [3].

Figure 2. Illustrations of reading a micrometer: a - [4], b -[5]

1. Measure thickness of 1 filter. Find an average value, from N- readings, xi, and a standard deviation,

s, as [2, 6, 7]:

You can use Excel spreadsheet or an online statistical calculator, for example, MathPortal.org [7] - see Appendix I for details.

2. Measure thickness of 10 filters, and then find the average filter thickness. How do the average values obtained in steps 1 and 2 compare: are they close, or there is a statistically significant difference? To answer this simple question, one should use statistical procedures for

comparing two means to determine if the difference between them is statistically significant. Without going too deep in statistics theory [2], you can use graphic calculators that have special functions for hypothesis testing or online utilities. For example, if you chose to use MathPortal.org calculator [7[, you will need to provide information on the type of the test and input required values to obtain the answer – see Appendix I for details. 3. Based on the thickness of a stack of 10 filters, find the average filter thickness; use that and the total thickness of the stack of all your filters, to estimate the number of filters in that stack. Show your calculations, and compare the value you get to the actual filter count (find the percent error.)

Table 2: Micrometer practice

Reading Thickness of 1 Filter ()

Thickness of 10 Filters ( )

Average Filter Thickness ( )

Thickness of Stack of all Filters ( )

Student 1 0.3 0.63 0.063 0.73

Student 2 0.4 0.82 0.082 1.2

Student 3 0.6 0.77 0.077 1.35

Average 0.43 0.74 0.074 1.09

Standard deviation 0.1528 0.0985 0.01 0.3235

Estimated # of filters:

15 Actual # of filters: 21 % Error: 28.57%

Part II. Figures 3a and 3b illustrate Vernier caliper's parts and readings, respectively – view a video [8].

Figure 3. A schematic diagram illustrating (a) – parts [9] and (b) – reading [10] of a Vernier caliper. Figure 4 illustrates reading of a dial caliper – see details here [11]

Table 3 Vernier calipers practice

Object Mass ( g ) Volume (cm) Experimental Density (g/cm3)

Accepted Density (g/cm3)

Percent Error (%)

Aluminum Rod 21.01 7.89 2.66 2.7

Cork 1.31 1.89 0.69 0.24

Steel 3.55 0.41 9.187 7.8

Acrylic 1.46 1.18

Graphite 1.87 2.266

Wood 0.55

Brass 7.09 2.197 8.55

Copper 7.40 8.9

Iron 6.61 6.98

Rubber 1.46 1.52

Plastic 1.25 1.0

Aluminum Cube 2.28 2.7

To assess statistical error in density calculations, fill the table below based on results obtained by other groups for aluminum cylinder and cork samples:

Figure 4 Illustration of reading of a dial caliper [11]

Table 4 Density of an aluminum cylinder and a cork samples

Group 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Al

Density,

( )

2.01 2.60 2.66 2.82 N/A 2.70 2.66 2.81 2.30 2.60 2.78 2.61 2.72

Cork

Density,

( )

0.62 0.69 0.69 0.38 N/A 0.67 0.66 0.40 0.3 0.69 0.42 0.62 0.69

Calculate the mean and standard deviation values and construct a confidence interval for the 95%

confidence level. You can use a graphical calculator or a Web-based statistical package, for example,

on one of these sites: WolframAlpha.com [12], EasyCalculation.com [13], Sample-Size.net [14].

Table 5 Mean, Standard Deviation, and Confidence Interval Values of the Class Density Data

Calculated on WolframAlpha.com and EasyCalculation.com

Mean Standard Deviation Confidence Interval

Aluminum Rod Density 2.60583 0.23244 2.458 to 2.754

Cork Density 0.56917 0.14805 0.4751 to 0.6632

5. Analysis & Summary

Comment on:

More automated measuring devices might have been more precise than manual measuring devices.

By using the Vernier caliper in each group to calculate the volume, there should be little variance

between the measured values in each group. However, some students might not have known how to

properly use the measuring devices, and could have recorded inaccurate values for the

measurements used in their density calculations. One of the main sources of error in the

measurements was that each student measured a different item, and each students’ interpretation of

the measured values was different. Also, each group had different sets of materials. Slight differences

in the cork or aluminum rods could have caused different measurements for each group. In addition,

some of the students were more comfortable with the measuring devices than other students, and

might have gotten more accurate readings of the values than those recorded by students

unaccustomed to the measuring devices that might not have watched the videos provided on how to

use each device. This could cause great variances in the data recorded by each group, which could

affect the statistical analysis. The deviation from the mean value was lower for the cork than for the

aluminum rod. The cork was in a cube shape, and might have been easier to measure, resulting in

more accurate records of the values used to calculate volume.

6. Questions:

1. A professor's error in grading students' Labs is +/- 5 points. What is the percent error of his

grading if an average (mean) class grade is 80?

</Insert answer here.>

2. An Engineering student who synthesized a new

revolutionary nanocrystalline material with a record

high strength came to you for an advice because he had

heard that you had completed a special lab on

measurements and are a guru in reporting measurement

data. The student measured an average grain (crystal)

size of 10 nm. The instrument error is 0.1 nm, and the

margin of error (statistics used to determine the

confidence interval) is 0.25 nm. What is your advice on

how to report the result?

He should report his value as being 10 nm within a margin of error of 0.25 nm.

3. Two Biology students A and B came to you with their measurements of spacing of the same

manatee's ribs performed on two pictures taken under different angles: 90 degrees and 30 degrees

with respect to the object, respectively. They have heard that you had helped an Engineering student

to solve a problem related to measurement results and were hoping you could help them. Would you

expect identical results? Will their measurements have a systematic error and a sampling error?

Whose measurements will you believe more? If there are wrong measurements, can they be fixed?

How?

Fig 6. Pictures of a manatee's skeleton taken under (a) 90 degrees and (b) 30 degrees with respect to

Fig. 5 An atomic resolution electron microscopy

micrograph showing structure of a nano-palladium [14]

the skeleton exhibited in a local museum

I would not expect their measurements to be the same. The image with the 90 degree angle would

have more space shown between the ribs, and would be more accurate. There would be a sampling

error in their data, because one of the students sampled the data from an image that was turned to

display less space between each rib. These measurements can be fixed by retaking the image at a

more horizontal angle so the student can correctly view the

space between the ribs.

4. A Marine Biology student needed precise measurements of

the pearl beads diameters. The student have heard that you had

got 180 points (including creativity points) for your

Measurement Lab and have helped an Engineering student and

two Biology students and wanted your advise with respect to i.

the right instrument and ii. expected instrument

error. What are your answers?

The correct instrument to use would be the Vernier caliper, and the expected error would be 0.1cm.

5. Three Bio-Engineering students needed to conduct measurements to assess regularity in leafs'

spacing (Fig 8a) and veins' spacing (Fig 8b). They had an argument about the procedure:

a. Student A suggested to use a meter stick on pictures carefully taken from the plants

b. Student B was going to use a meter stick on real plants

c. Student C insisted on using a digital caliper on real plants to ensure a high precision measurements

They came to you for an advice because of your solid reputation based on your help to a bunch of

other students. Which procedure would you recommend?

The procedure suggested by Student C because it would be the most accurate.

Fig. 7 Pearl beads

Figure 8 Pictures of plants for measurements of spacing of (a) leafs and (b) leafs' veins

7. References

1. College Physics, https://cnx.org/contents/Ax2o07Ul@9.35:S7pqHKDm@7/Accuracy-Precision-

and-Signifi

2. Introductory Statistics, https://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de

3. How to Read Micrometers, https://www.youtube.com/watch?v=oiAutI0i5YE

4. The Virtual Machine Shop,

http://www.jjjtrain.com/vms/measure_mic_reading/measure_mic_reading_07.html

5. Measurement, MIT Tutorial,

http://web.mit.edu/2.670/www/Tutorials/Machining/measure/Description.html

6. MathsIsFun.com, https://www.mathsisfun.com/data/standard-deviation-formulas.html

7. MathPortal.org, http://www.mathportal.org/calculators/statistics-calculator/t-test-

calculator.php

8. How to read a Venear Caliper, https://www.youtube.com/watch?v=4hlNi0jdoeQ

9. http://www.hasorc.com/hasoha/physic1/1_measurement_of_length.html

10. Vernier Caliper Use, http://hyperphysics.phy-astr.gsu.edu/hbase/class/phscilab/vernier.html

11. Precision Measuring, Project Lead The Way, http://slideplayer.com/slide/8756832

12. WolframAlpha.com

13. EasyCalculation.com

14. Sample-Size.net

15. Karlsruhe Institute of Technology, Germany, https://www.knmf.kit.edu

16. Online gallery for this class, http://www.sponsorschoose.org/colleges/asuphysics

18. Aqua-calc Conversions and Calculations http://www.aqua-calc.com/

9. Acknowledgments: This Lab is based on original materials developed by Dr. Bruce Johnson in

1999 and updated by Steven Hoke in 2014. Dr. Koushik Biswas, Dr. Bin Zhang, and Dr. Ross Carroll

contributed to the discussion.

10. Credits: Numerous materials developed at various schools and organizations, including MIT,

Rochester Institute of Technology, etc. were used to leverage the best resources on the topic. Students

are strongly encouraged to submit their materials and get a credit:

Samuel Schratz, Environmental Science, - see here

Appendix I. Statistical Calculators

There are numerous utilities for statistical processing of experimental measurements. A few examples

are given below: you can find better resources and include in your Lab report and share with other

students (replace text given in orange). It is strongly suggested that you try to find the formulas used

in the calculators to make sure that the calculations are based on a solid theoretical foundation.

1. Mean and standard deviation. If you want to go beyond the Excel spreadsheet, you can use

MathPortal.org [7]: screenshots provided below should help you to check right options.

Check Show me an explanation box: provided explanations are very helpful for understanding basics

of statistics as illustrated by the following example :

Just keep in mind that by convention, symbol is used for the population standard deviation: they

should have used s for the sample standard deviation. Also, calculated number has too many

significant digits: it should not be more than the accuracy of

measurements.

2. Confidence interval. Confidence interval is determined as a

mean value +/- margin of error. You can use

WolframAlpha.com [11] calculator:

You can also use (especially, if you are a

biology or a medical student) a calculator, from

the Clinical and Translational Sciences

Institute, UCSF [14] (note, that it provides

appropriate used formulas in contrast to some

other calculators):

3. Comparing two means. You can use

MathPortal.org [7] calculator:

Appendix II. Supplemental Materials

You can use these images or your own images and any changes in the text to perfect the Lab report. It

is our ambition to come up with the best Lab report ever. How? We hope to challenge and reward

your creativity and critical thinking and leverage value of your work by using a new crowdsourcing-

crowdfunding system [16]. Here are some supplemental images you can use in your reports: go here

for more [16].

Materials

Micrometer

Meter

Stick