This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ENGR 201 Evaluation and Presentations of Experimental Data I
Fall 2011 – 201115 Basic Measurements
R. Carr, R. Quinn, Introduction to the Art of Engineering1
Edited by K. Scoles, T. Chmielewski, D. Miller 1 Summary
All scientific and engineering knowledge about the physical world and its governing
principles has been gained by observation and experimentation. The numbers used to
describe physical phenomena and properties are called physical quantities. In order to
be consistent each physical quantity must be expressed in some accepted units whose
values are referred to some accepted standards. In any measurement of a physical
quantity, there is always some experimental error. There are a variety of methods used
to identify, control and minimize these errors. This experiment will provide an opportunity
to measure length, a basic physical quantity, and develop skill in using a variety of
instruments designed for this purpose. It will also provide an opportunity to learn and
apply concepts, practices and procedures fundamental to all types of scientific and
engineering experimentation. Educational Objectives
After performing this experiment, students should be able to:
1. Determine the accuracy and precision of instruments.
2. Measure length using a linear scale (ruler), a vernier caliper and a micrometer.
3. Properly acquire and record data using these instruments.
4. Analyze data to identify and/or minimize error.
5. Select an optimum method of measurement for a given length measurement
application.
6. Construct a histogram. Background Information
All analog measurements have error and a consequent uncertainty. Errors are
classified as systematic or random. Systematic errors are usually categorized as
instrumental, personal, or extraneous. An instrumental error is due to faults or limitations
of the measuring device. This includes improper calibration as well as broken devices.
Personal errors vary from one observer to the next and indicate any bias the observer
may have. Extraneous errors are introduced by the environment in which measurements
are taken. For example, air currents from a fan or window may alter the readings of
mass obtained on a mass scale.
Hysteresis is another phenomenon that may contribute to error. An instrument is said
to have hysteresis when it shows a different reading for the same measured quantity
depending on whether the quantity is approached from above or below.
Some of the systematic errors may be corrected using a calibration curve. A plot of
the instrument reading against the standard being measured is called a calibration curve. 1 Carr, R. and R. Quinn, Introduction to the Art of Engineering, 7th ed., Wiley Custom Services, 2005. Page 1 of 12 We can imagine an ideal instrument for which each measurement exactly equals the
quantity being measured. Thus the calibration curve fpr an ideal instrument is a line of
slope one through the origin. Figure 1 depicts calibration curves for an ideal instrument,
a nonideal instrument and an instrument with hysteresis. Figure 1: Calibration curves for (1) an ideal instrument, (2) a nonideal linear instrument, (3) a nonideal, nonlinear instrument with hysteresis.
Random error is statistical in nature. These errors change with time and/or position,
and have an associated probability. An increase in the number of measurements taken
will reduce the effect of these errors because they tend to cancel out. Many times it is
impossible to eliminate the errors in a method of measurement. In these cases it is
important to be able to reproduce the same readings. In other words, the errors should
be consistent in all measurements.
All errors affect the results to varying degrees. As measurements are used to
compute other physical quantities, the errors are carried throughout in the computation.
This compounding of error as it is carried at each consecutive step is called propagation
of error. Examples
Example 1: Uncertainty
1. The diameter of a rod is given as 32.41 ± 0.02 mm. Thus the actual diameter
may be anywhere between:
1.1. a maximum of: 32.41 + 0.02 = 32.43 mm.
1.2. a minimum of: 32.41 – 0.02 = 32.39 mm.
2. The mass of a rod is given as 10 grams with a 20% error. Thus the actual mass
of the rod may be anywhere between:
2.1. a maximum of: 10 + 10 • (0.2) = 12 grams
2.2. a minimum of and: 10 – 10 • (0.2) = 8 grams Page 2 of 12 Example 2: Accuracy The accuracy of a measurement is its deviation from the actual
value of the quantity being measured. If, for example, a certain balance measures a
100 grams standard mass as 110 grams, its accuracy is only 10%. Similarly, the
accuracy of an instrument measures the deviations of its readings from known inputs. Of
course the accuracy depends on the input, so one arbitrarily deﬁnes the accuracy of an
instrument as a percentage of its fullscale reading. If a voltmeter with a 100 V range has
an accuracy of 2%, its reading over this range would be accurate within ±2 volts.
Example 3: Precision
The precision of an instrument has to do with the repeatability of its readings. If the
balance from the previous example gives ﬁve different readings (99.0 g, 101.0 g, 100.0 g,
99.5 g and 100.5 g) for the same standard mass of 100 grams, then its precision would
be ± 1.0 g since the individual measurements deviate from the average (100.0 g) by at
most ±1.0 g.
Example 4: Propagation of Error in a Volume Calculation
The linear dimensions of a metal bar are measured within an uncertainty of ±0.1 inch as
illustrated in Fig. 2. Find the maximum and minimum values for the volume V of the
metal bar.
If the measurements were exact, the volume V would be given by the product: V =
Length x Width x Height = 2.7" • 2.7" • 11.5" = 83.8 in3. 2. 7 ± 0.1”
Figure 2 Metal Bar But the measurements are not exact and the actual volume of the bar could lie between:
1. a maximum of 2.8" • 2.8" • 11.6" = 90.9 in3 .
2. a minimum of 2.6" • 2.6" • 11.4" = 77.1 in3.
Notice how a seemingly small error in the original measurements is magniﬁed in the
volume calculation. Page 3 of 12 Finally, we would like to review two related concepts: least count and sensitivity.
Least count is the smallest increment of the measurement unit that can be detected with
the instrument. Sensitivity is defined by the equation: Sensitivity = ΔOutput
ΔInput In approaching a given experimental problem, various criteria can determine which
method of measurement is optimum or "best". For example, high priority may be given to
€
the errors a method will introduce and the effect of such errors on the end result. Clearly
an uncertainty of ±1 tsp. salt in a large pot of soup prepared for 20 people is not as
significant as ±1 tsp. salt in an individual serving. In another application an engineer
might have to give primary consideration to the practicality of each method. An engineer
working in the field will find it inconvenient to carry an analytic balance. A less precise
trip balance may be the best choice for reasons of convenience alone. Therefore, the
purpose of each measurement must be clearly defined. In this experiment, our purpose
is to learn about experimentation and we will explore different devices and concepts. For
our purposes, all equipment will be assumed to be equally practical.
Mean & Standard Deviation
Suppose a measurement is performed on N objects giving the data:
{x1, x2 , ... , xN}. The average or arithmetic mean is given as:
N 1
x = ∑ xi
N i =1 (1) where xi is an individual measurement and N is the total number of measurements.
The absolute deviation di of an individual reading xi from the mean value x is defined
as: € di = x i − x (2) The standard deviation of a set of numbers is denoted by the Greek letter sigma. For
a complete set of data, the standard deviation is defined by: € σ= 1N
∑( x − x)
N i =1 i 2 (3) However, if the set of points xi represents only a sample of the possible readings then
we must use the sample standard deviation formula: € σ= 1N
∑( x − x)
N − 1 i =1 i 2 (4) The standard deviation tells how much a typical measurement will deviate from the
mean. That is, it is a measure of the dispersion of the readings from the mean value.
The most precise method of measurement is the one that yields the smallest standard € Page 4 of 12 deviation. In other words, the smaller the deviation from the mean is, the more
repeatable the reading is. The most accurate method may not be the most precise.
Based on the data and these observations it is possible to select the instruments that
comprise the optimum or "best" method of measurement. Using the standard deviation
as the uncertainty, the range of most likely values should be specified in the report.
More on Histograms
A histogram is a convenient pictorial representation of the distribution of a set of
collected data. A histogram is a graph composed of rectangles. The rectangles
composing the histogram lie over nonover lapping intervals called class intervals or bins.
The area of these rectangles is proportional to the frequency of each interval, which is
the number of observations that fall in each class interval. Usually a histogram is
constructed using equal length intervals, so that the frequencies are proportional to the
heights of the rectangles.
The issue of how many rectangles or bins (K) are appropriate for a particular size of
data set2 is addressed by the relationship in equation 5. An estimate of K ≈ N1/2 works
well for large N. K = 1.87( N − 1) 0.40 +1 (5) We would like to introduce at this point some further properties of histograms.
Histograms can be unimodal, bimodal, and multimodal. A histogram that increases to a
€
peak and then decreases is a unimodal histogram. A bimodal histogram is one with two
different peaks and similarly, a histogram with more than two peaks is called multimodal.
A unimodal histogram may be further classified as either symmetric, positively
skewed or negatively skewed. A histogram is said to be symmetric if both the left and
right halves are mirror images of each other. A positively skewed histogram is one with
its right side more stretched out than the left. When the stretching is mostly toward the
left then it is said to be a negatively skewed histogram. Figure 3 shows examples of
these various types of histograms. (Note that a smooth curve has been drawn to
represent the tops of each rectangle.) Figure 3. Symmetry and Modality of Histograms 2 Kendal, M.G. and A. Stuart, Advanced Theory of Statistics, Vol. 2, Griffin, London, 1961 Page 5 of 12 How to Use the Calipers
Each caliper consists of jaws for holding the object to be measured and two bars
with scales the main scale and the vernier scale. Calipers are useful for measuring
outside diameters with the large ﬂat jaws (number 1 in Fig. 4), inside diameters with the
inside jaws (number 2 in Fig. 4), and hole depths (number 3 in Fig. 4). Both scales are
marked in inches and in millimeters. For the purposes of this lab, take all measurements
in inches. 1 Outside jaws: used to take external measures of objects
2 Inside jaws: used to take internal measures of objects
3 Depth probe: used to measure the depth of objects
4 Main scale (cm) 5 Main scale (inch)
6 Vernier (cm)
7 Vernier (inch)
8 Retainer: used to block movable part Figure 4. Caliper showing two sets of measuring jaws and the protruding
depth probe.3
The object to be measured is ﬁrst placed between the jaws of the calipers and then the
jaws are adjusted to obtain a snug ﬁt. How to Measure in Inches
Each division on the standard scale corresponds to 0.025 inches. Each division on the
sliding scale corresponds to 0.001 inches or one thousandths of an inch.
1 Find the division mark on the standard scale that lies just before the zero mark
on the sliding scale. This division gives the length to the nearest 0.025 inch which
does not exceed the true length. Thus the true length is always a little larger. 2 Next look for the division mark on the sliding bar which exactly lines up with a
division mark on the standard bar. Read this number using the scale on the
sliding bar and add it to the previous number. 3 http://upload.wikimedia.org/wikipedia/commons/9/96/Vernier_caliper_new.png Page 6 of 12 Example
In Fig. 5, the division mark on the standard bar that lies just before the zero mark on the
sliding bar is 1.375 inches. The division marks on the two scales line up at the division
corresponding to 0.005inches so that the total measurement is: 1.375" + 0.005" = 1.380" Figure 5. Example of combining the results at the caliper zero mark and point
where division marks on sliding and standard bars line up Page 7 of 12 Using the Micrometer A B Figure 6. Micrometer showing measuring bar (A) and adjustor (B). Measuring Bar A
•The lines on the top half of the measurement bar are in increments of 0.05 inches.
•The lines on the bottom half of the measurement
bar are in increments of 0.025 inches Adjustor B
•The numbers on the adjustor increase from 0 to 25 (i.e. 0.000 in. to 0.025 in.). Each full rotation of the
adjustor moves the column 0.025 inches or one
increment on the bottom half of the measurement bar. Page 8 of 12 Example
Measurement reading
on measurement bar
(rounded down to
nearest
line). + Measurement
reading on
adjustor = Measurement reading 0.25 in .+ 0.0179 in. = 0.2679 in. •N.B. The “9” in 0.0179 is the
estimated value between the
lines. This is the precision of
the instrument. You might want to take photographs of the caliper and micrometer that you actually
use for your report since they may look different from the ones in this handout. Page 9 of 12 Procedures
In this experiment you will be provided with test specimens (washers and ball
bearings) having different shapes and sizes as well as various test instruments for
measuring length. Each sample lot should contain eight to ten specimens of the same
apparent size. The linear dimensions of each specimen will be measured using a Vernier
caliper, a micrometer, and a ruler. Ideally, the specimens in each given lot are exactly
the same. Your experimental observations may determine otherwise.
In this experiment, errors could include poor alignment of the caliper or micrometer.
Personal errors arise from changes in perspective, angle of sight or even the lighting in
the room. Part 1. Measuring Lengths
You are to measure the linear dimensions of 10 washers of the same apparent size
and 10 ball bearings of the same apparent size using three measurement devices  a
vernier caliper, a micrometer and a ruler. The dimensions to be measured are given in
Table 1. Each student in the group should measure all objects and record their own data
(a sample spreadsheet for data logging is posted on the website and must be included in
the lab report).
As you take the data, consider the sources of error in each measurement, and how
they may change from instrument to instrument. For example, gently close the jaws of
your micrometer. Does the instrument indicate 0.0000, i.e. does it have a zero offset? Is
the offset positive or negative? What type of error would an offset introduce? There are
methods to adjust the instrument to remove the offset, but we will not do this at this time.
Table 1. Measurement Set (ID = inside diameter, OD = outside diameter) Object
Washer
Ball Bearing Instrument
Ruler
Caliper Measurement
ID, OD
ID, OD Caliper
Micrometer Diameter
Diameter 1. Create a table in your lab notebook for your first set of measurements (or use the
spreadsheet provided in the laboratory folder). For example, the first table may
be for the two ruler measurements of each washer.
1.1. Record the date, time, the names and emails of your partners and any other
important details. Normally this would include the calibration info and
manufacturer, model, and serial numbers of the instruments used.
1.2. Make enough columns and rows to record each dimension measured as
well as it’s uncertainty.
1.3. Label the table clearly as to what data this represents
1.4. Take your data. The order of measurements does not matter, so one student
can use the caliper while another uses the micrometer or ruler.
1.4.1. If you would like, you can add each data point to both your notebook
and spreadsheet as you go (see items 34 below)
2. Repeat for each set of measurements
3. Create a spreadsheet for all your data, graphs and histograms. Any spreadsheet Page 10 of 12 4.
5.
6.
7. program is acceptable, just so there is a common data file format for exchanging
data with your partners.
Transfer your data into the spreadsheet that you just created. Note the smallest
unit on each scale.
Use your spreadsheet to calculate the mean and standard deviation for each set
of measurements.
Exchange measurement data with your partners, combine them with yours, and
recalculate the means and standard deviations.
The TA will collect one set of measurements from each group for the caliper
measurement of washer diameter and post the values so you can compare
measurements across the entire section. Part 2. Creating Histograms
1. Compile your team’s data for each measurement in a separate column in your
spreadsheet. For example, list all of the washer thicknesses measured with
calipers in one column and the washer thicknesses measured with the
micrometer in another. For a team of three you should have approximately 30
values in each column. Be sure to label each column with a title and unit.
2. Create histograms for each of your data sets – including yours, your lab groups,
and the laboratory section. Report
The report format should follow that in the “LabReportGradingRubric” within the
Laboratory Items folder on Bb Vista site for the course. At no time should you cut and
paste text from your lab handout into your report. You are going to be generating quite a
histograms for your report. Only put the few that you want to discuss in detail into the
body of your report. The others can go into an appendix.
Here are some questions to consider in your discussion of results and conclusions,
but do not limit yourself to these:
• If you needed a measurement to the nearest 0.1 inch, what instrument would you
choose? How about 0.01 in, 0.001 in, or 0.0001 in?
• How did your individual measurements compare to your team results?
• Comment on the accuracy and precision of the measurements. The washers you
have are stamped parts, and you would not expect them to have high precision.
What size bolt do you think they are designed for? On the other hand, ball
bearings need to be high precision to prevent wear and friction. How do the sets
of bearing measurements compare?
• Which instrument produces the narrowest distribution of results? What shape do
the distributions have?
• Which standard deviation function did you use in your calculations? Why?
Each student should upload a copy of the report in PDF format to the Bb Vista site.
One (identical) hardcopy per group should be turned in to your TA. Reports are due one
week after your lab meeting. Further Information
•
• How to use and read a vernier caliper,
http://www.tresnainstrument.com/how_to_read_a_vernier_caliper.html
Vernier caliper tutorial,
Page 11 of 12 •
•
• http://www.physics.smu.edu/~scalise/apparatus/caliper/tutorial/
How to use a micrometer, http://www.youtube.com/watch?v=oHqaLMEHlnE
How to read an outside micrometer, http://www.pgiinc.com/howtoreoumi.html
How do they get the balls in ball bearings so perfectly round and smooth?,
http://www.howstuffworks.com/question513.htm Page 12 of 12 ...
View
Full
Document
 Fall '08
 Miller

Click to edit the document details