lab 1: measurements physics i lab section 2054-002 ...€¦ · lab 1: measurements ... micrometer...
TRANSCRIPT
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/[email protected]: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