Physics 140L
Laboratory Manual
by
H. Butner, A. Fovargue, K.Giovanetti,
L. Lucatorto, G. Niculescu, T. O’Neill, B. Utter
James Madison University
Harrisonburg, VA 22807
2009
c
2009-2010
Department of Physics and Astronomy
James Madison University
All rights Reserved
Contents
1 A Mean Lab (Introduction to PHYS140L)
1.1 The value of a Measurement
(In Class activity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
8
2 Picturing Motion
(In Class activity)
17
3 Spreading the Data
(At Home activity)
27
4 Gauging the Force
(In Class activity)
37
5 Dropping the Ball
(At Home activity)
45
6 Atwood’s Machine
(In Class activity)
51
7 Sliding along
(At Home activity)
63
8 Crashing Carts
(In Class activity)
71
9 Happy and Sad Balls
(At Home activity)
81
10 Poe’s Pendulum
(In Class activity)
87
11 Functions/Air Drag
(At Home activity)
95
12 Comedy of Errors
(Final Lab Part I)
99
3
13 Tale of Woe
(Final Lab Part II)
113
14 Appendix 1: Curve Fitting
120
15 Appendix 2: Excel Spreadsheet
124
16 Appendix 3: Establishing Uncertainty
128
17 Appendix 4: Suggestions for Data Handling
138
4
List of Figures
1.1
1.2
1.3
1.4
Vernier Caliper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reading: 2.64 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Micrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Metric micrometer reading equals 23.15 mm. 23 whole divisions (= 23 mm);.
0 mm divisions are uncovered (= 0.0 mm);15 0.01 mm divisions line up on
the thimble (= 0.15 mm). . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1
2.2
2.3
Position versus time plots for four situations. . . . . . . . . . . . . . . . .
Velocity versus time plots for the situations described in Fig. 2.1. . . . .
Position, velocity, and acceleration versus time (blank) plots. . . . . . . .
17
18
25
3.1
3.2
Sample Plot with Labels . . . . . . . . . . . . . . . . . . . . . . . . .
Sample Plot with Trendlines . . . . . . . . . . . . . . . . . . . . . .
32
33
5.1
Experimental setup for the Bounce Procedure . . . . . . . . . . . . . . .
49
6.1
6.2
Atwood machine: A sliding mass is connected to a falling mass via a pulley 53
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.1
Friction and Normal Forces . . . . . . . . . . . . . . . . . . . . . . . . .
64
8.1
Case 1: Both carts at rest initially (Note: your setup may be the mirror
image of this figure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Case 2: Inelastic collision with objects moving with the same final velocity.)
Case 3: Elastic collision. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cart positions after elastic collision, Case 3A. . . . . . . . . . . . . . . .
Cart positions after elastic collision, Case 3B. . . . . . . . . . . . . . . .
Cart positions after elastic collision, Case 3C. . . . . . . . . . . . . . . .
74
76
77
78
78
79
8.2
8.3
8.4
8.5
8.6
10.1 Pendulum cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 A warning about plots – pay attention to the scales on your axes! . . . .
10.3 Experimental setup for the pendulum experiment. . . . . . . . . . . . . .
8
9
9
87
88
93
12.1 Calorimeter Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
12.2 Calorimeter with boiler . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5
6
List of Tables
1.1
1.2
1.3
1.4
Measurements of a Cylindrical mass with
Measurements of a Cylindrical mass with
Calculated volume of a cylindrical mass .
. . . . . . . . . . . . . . . . . . . . . . .
a
a
.
.
vernier caliper. . . .
micrometer caliper. .
. . . . . . . . . . . .
. . . . . . . . . . . .
.
.
.
.
13
14
14
15
3.1
3.2
Sample Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
34
4.1
4.2
4.3
4.4
4.5
4.6
Force Calibration . . . . . . . . . . . . . . . . . . .
Setup and Performance of Force Probe Experiment
Analysis of the First Data (10N Scale) . . . . . . .
Analysis of the Second Data (10N Scale) . . . . . .
Analysis of the Third Data (50N Scale) . . . . . . .
Analysis of the Fourth Data (50N Scale) . . . . . .
.
.
.
.
.
.
40
43
43
43
44
44
5.1
Dropping the ball: Sample table for the raw experimental data. . . . . .
48
6.1
6.2
6.3
Example Cart Mass Table - YOUR NUMBERS WILL BE DIFFERENT
Example Case A Table - Remember Units! . . . . . . . . . . . . . . . . .
List of Required Items . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
60
61
7.1
7.2
Experiment: Sliding Along. Data table. If needed, feel free to make copies
of this table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experiment: Sliding Along. Results table. . . . . . . . . . . . . . . . . .
68
69
8.1
Instructor check off table for “Crashing Carts” experiment.
. . . . . . .
80
9.1
9.2
CR Measurements using the happy and sad balls. . . . . . . . . . . . . .
CR Averages and SD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
85
11.1 Description of activities and assignments for functions/air drag . . . . . .
97
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
12.1 Example - Mass of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
12.2 Barometric Pressure vs Boiling Temperature of Water . . . . . . . . . . . 111
1
2
Chapter 1
A Mean Lab (Introduction to
PHYS140L)
Welcome to Physics 140L
In this laboratory course, you will explore how to perform experiments, and learn how to
account for experimental uncertainties. Along the way, you will be exposed to material
covered in lectures, and hopefully have a better sense of how the physics works.
Look around. In the first class you may see that the total number of students is more
than 16. Ideally we want to have only two students per lab station. To achieve that, we
have broken the labs into different types - those that are done in the lab, and those that
are done at home.
• There are two lab sections - called Group A and Group B. Each group will alternate
doing labs in the laboratory (every other week). On weeks that your group is not
meeting in the laboratory, you will be doing your laboratory “at home” with your
lab partner. This means that you and your lab partner will meet somewhere other
than P&C 2286 to do the lab.
• To make the lab sections balanced, your instructor will identify which group (A or
B) that you are in. If you are in A group, you will start off with the laboratory
the second week. If you are in B group, you come in for the lab in the third week.
Group A will finish the labs before Thanksgiving, Group B will finish the week
after Thanksgiving. See Table 1 for the breakdown of the schedule.
• You and your lab partner will be getting a lab kit which will be used for “at home”
labs. Keep track of it and its materials. If you fail to turn it in, or turn it in
missing materials, you will charged a fee. Your course grade will be marked
as incomplete until you and your lab partner turn the lab kit in or pay
the replacement fee.
• When doing at home labs, you can meet anywhere or anytime, so long as you turn
the lab in by the due date. By the way, you and your lab partner have the block
3
of time assigned to the lab period free that week - so you ALWAYS have at least
one meeting time free.
• Before each lab, you will be expected to read the lab. Since the time in the laboratory is short, you can’t waste time coming in without reading the lab. To
encourage reading before the lab, there will be a reading quiz on that lab - usually
administered through blackboard. Check with your instructor for details. It will
be due before the lab starts - and it will count toward your grade. Note that it is
open-book. You are encouraged to have the lab manual with you when you take it.
Table 1 shows the lab schedule for each Group.
Activity
Week
Week of
Group A
A Lab
Group B
B Lab
1
Aug 24
In lab
A Mean Lab
In class
A Mean Lab
2
Aug 31
In lab
Picturing Motion
3
Sep 07
At home
Spreading the Data
In lab
Picturing Motion
4
Sep 14
In lab
Gauging the Force
At home
Spreading the Data
5
Sep 21
At home
Dropping the Ball
In lab
Gauging the Force
6
Sep 28
In lab
Atwood’s Machine
At home
Dropping the Ball
7
Oct 05
At home
Sliding Along
In lab
Atwood’s Machine
8
Oct 12
In lab
Crashing Carts
At home
Sliding Along
9
Oct 19
At home
Sad or Happy
In lab
Crashing Carts
10
Oct 26
In lab
Poe’s Pendulum
At home
Sad or Happy
11
Nov 02
At home
Flight of the Filter
In lab
Poe’s Pendulum
12
Nov 09
In lab
Comedy of Errors
At home
Flight of the Fillter
13
Nov 16
At home
Tale of Woe
In lab
Comedy of Errors
14
Nov 23
No class
Thanksgiving
No class
15
Nov 30
At home
Tale of Woe
Grading
The grading for the labs will be broken down as follows:
• 15% - Online Quizzes - These will be “reading quizzes” that will be due before each
lab (both in-lab and at home labs).
• 30% - In-class Labs - The lab generally will be complete by the end of the lab.
• 30% - At-home Labs - The lab will be due by the end of the lab period the week
you are scheduled for the lab, or whatever other time your lab instructor requires.
4
Your instructor will let you know if they prefer hard-copies or electronic copies of
your reports.
• 25% - Final lab report (includes the work done on the final two labs - Comedy of
Errors and Tale of Woe). While you will work with your partner on the experiment,
the actual write-up will be your own.
As with any course, if you are having trouble getting the work done, talk to your
instructor!
The instructors will let you know their grading requirements, and also what to do
if you miss a class or have an excused emergency. Just skipping a lab without a valid
excuse will get you a zero for the lab, so always check with your instructor as soon as you
can. Failure to do assigned quizzes will also lower your grade. In the event of a conflict
or problem with a scheduled lab, the student must make prior arrangements with the
instructor. Otherwise a documented medical excuse is required.
Purpose
For these laboratory experiments, there are three main goals:
1. Become familiar with experimental procedures, including how to identify and solve
problems that arise with real measurements.
2. Become familiar with how to include uncertainties in the analysis of an experiment
and how to estimate the overall uncertainty in your experimental results.
3. Become familiar with how to present your results in a form that others can understand.
Note: While we will be reinforcing concepts you will learn in your introductory physics
courses, our focus will be on developing your experimental skills - not trying to demonstrate every equation you might see in the course. Although we will see topics in parallel
to the Phys 140 an Phys 240 lecture courses, there will be times when we will explore
areas you might not have seen in your lectures.
Introduction to the Mean Lab
Experiments are often portrayed in movies and TV as requiring that the scientist be
brilliant, wear white lab coats, and/or be as obscure as possible when talking to mere
mortals. Actually most good science starts off with common sense methods and simple
questions about how the technique can be improved at every step of the process. If
you are cooking a soup and carefully adding in various ingredients and monitoring the
cooking, and then at the last moment pour it on the floor, you will not get a very good
tasting soup. Experiments require that a scientist pay attention every step along the way
until the experiment (and analysis) is completed, and identify if the results are reasonable
as you proceed.
5
The phrase “results are reasonable” is one that students often misinterpret.
Getting reasonable results does not mean that your experiment agrees with the ”expected result” to as many digits on your calculator as possible.
In many of the experiments you will be doing, it is very likely that you will not agree
with the “expected result”. That doesn’t imply that your experiment is wrong - but it
does mean that you need to identify and account for any possible sources of uncertainty
(or errors) in your experiment. If you were to repeat the experiment, you would work to
reduce the sources of uncertainty that you identified.
Uncertainty and error have different meanings:
• Error
In physics, we use the term “error” to refer to the difference between
a value and its correct or true value. The true value, of course, is often not known.
• Uncertainty - In physics, we use the term “uncertainty” to estimate the difference between a calculated or measured value and the true value.
We measure some physical quantity with an instrument. The values reported by the
instrument are in error by a certain amount. Since we do not actually know the exact
true value of a quantity, we do not know the instrumental error. Instead, we realize
that the instrumental value is an estimate of the true value and the uncertainty in our
measurement is an estimate for the error. Thus the uncertainty is based on the technique
that you are using for the lab. If you use an instrument with poor resolution, then you
will have larger uncertainties. For example, if you measured length of a field with a meter
stick that only meters and decimeters marked on it, you would have more uncertainty
in your measurement than if you used a meter stick with millimeters marked on it. You
would have “more resolution” with your measuring device in the second case.
A key point is that every measurement has associated with it an uncertainty. That
uncertainty needs to be recorded as you are taking measurements. In addition, that
uncertainty is quantified. You are not allowed to just say “I think the uncertainty is...”.
You have to have a way of estimating the uncertainty. This lab will illustrate how an
experimenter might do this for a simple measurement.
No uncertainty can be introduced into any discussion unless you can define
a quantitative estimate of its size. All such estimates need to be justified.
Appendix 3 goes into a more in-depth discussion of the difference between error and
uncertainty.
Formulas
Mean, Standard Deviation, and Standard Deviation of the Mean
Statistics are a way for an experimentalist to estimate quantities of interest from the
experimental data.
The mean (average of the data points) usually is a good estimate of the quantity
measured. If the data has a random component, then averaging several samples together
should act to cancel out that random fluctuation. That results in the mean being a better
6
estimator of the experimental result than any single data point would. Here the bar over
the x indicates the mean of x.
x1 + x2 + x3 + x4 + ... + xN )
= mean(M)
(1.1)
N
Standard deviation (SD) is one statistic measure that can be used to estimate the uncertainty of an experiment. It is an estimate of the error for any one of the measurements
averaged.
x̄ =
σ=
s
(x1 − x̄)2 + (x2 − x̄)2 + .... + (xN − x̄)2
= standard deviation(SD)
N −1
(1.2)
The standard deviation σ is usually a property of the measurement technique. It
describes how spread out the data points are around the mean. As you collect more
data points, σ tends to approach a value that is roughly the width of the spread in
measurements. It seems reasonable that the measurement should become more reliable
as the number of trials N increases. The standard deviation of the mean (SDM) can be
thought of as the statistical uncertainty in x. We can therefore equate the experimental
uncertainty ∆x with the SDM
σ
σx = √ − standard deviation of the mean(SDM)
N
(1.3)
We can adopt the following:
1. The best estimate of x, is x̄
2. The statistical uncertainty of x is ∆x = σx .
3. We can then write x = x̄ ± ∆x
In the case where we only take one measure, then the resolution of our observation can
be used to define the uncertainty. I.e. it is a educated guess based on the measurement
technique.
Again - a more detailed discussion of these concepts is presented in Appendix 3 and
Appendix 4.
Let us see how to apply these ideas in practice.
7
1.1
The value of a Measurement
(In Class activity)
Introduction
This lab serves as an introduction to measurement taking and experimental statistics
through the use of calipers.
Accurate measurement requires appropriate tools. When measuring a tabletop, we
could use a meter stick to produce a suitable measurement. The meter stick has graduations small enough to attain a measurement to within a millimeter. One can make a
measurement accurate to within a thousandth of a meter. This is good accuracy if the
table is roughly a meter or longer.
To use a meter stick to measure the thickness of a pencil would be inappropriate.
Assuming a pencil is roughly 5 mm in diameter; one would want a tool that could give
measurements accurate to a fraction of a millimeter. The vernier and micrometer calipers
were developed to perform such measurements.
The vernier caliper (Fig. 1.1) is a fairly simple measurement tool. It has two parts:
a stem with the fixed main scale (cm) and the vernier, a secondary scale. Each part of
the caliper forms a jaw to grasp the item being measured. Ten vernier scale divisions fit
within nine stem divisions (remember the stem is the fixed part), so each vernier division
is 9/10 as long as a stem division (refer to Fig. 1.2). When the jaws of the caliper are
closed, the first line of the vernier, the zero line, coincides with the zero line of the main
scale.
Figure 1.1: Vernier Caliper
To make a measurement with the vernier caliper, the jaws must be tightly closed
around an object. Wherever the zero line of the vernier falls on the main scale indicates
8
the number in the tenths place of measurement. The next line on the vernier that aligns
with the main scale indicates the hundredths place, as shown in Fig. 1.2.
Figure 1.2: Reading: 2.64 cm
The micrometer caliper (Fig. 1.3) is another tool for measuring short lengths. It is
more precise than the vernier caliper because it can measure within thousandths of a
millimeter.
Figure 1.3: Micrometer
To use the micrometer caliper, an object must be placed between the screw and the
frame. The thimble is then turned to advance the screw until the object is touched.
The ratchet may click to let one know enough force has been applied and to prevent
9
over tightening. Like the vernier caliper, there are two scales on the micrometer caliper,
a circular scale on the thimble and a longitudinal scale along the barrel containing the
screw. The longitudinal scale is divided into half millimeter increments, and the circular
scale has fifty divisions. Rotating the circular scale through one full revolution advances
the screw by 0.5 mm (the distance between two marks on the longitudinal scale). Rotating
the thimble through one scale division (the distance between marks on the circular scale)
advances the screw 1/50th of 0.5 mm or 0.01 mm.
To read the micrometer, first observe the position of the circular scale on the longitudinal scale. This yields the number of millimeters to the nearest 0.5 mm. Next, note
which line on the circular scale aligns with the axial line on the longitudinal scale. This
gives the fractional portion of the millimeter reading.
Figure 1.4: Metric micrometer reading equals 23.15 mm. 23 whole divisions (= 23 mm);.
0 mm divisions are uncovered (= 0.0 mm);15 0.01 mm divisions line up on the thimble
(= 0.15 mm).
Formulas
For this experiment, in addition to the statistical quantities discussed above it would be
good to remember the following definition/formula:
Volume of a cylinder:
V = πr 2 h
(1.4)
Where r is the radius and h is the height of the cylinder.
Equipment/Materials
For this experiment you will need the following: vernier caliper, micrometer, penny or
slug, magnifying lens
10
• a vernier caliper
• a micrometer
• a penny or slug to be measured
• a magnifying lens
Experimental Procedure
1 Draw a 4 inch line using a ruler on a piece of paper.
2 Measure the line in centimeters to the greatest precision the ruler will allow.
3 Record the number of centimeters.
4 Calculate the conversion factor between inches and centimeters (divide the two
numbers).
Use the golden rule for reporting measurements: Report all of the digits that
you know with certainty, plus the first digit that you must estimate.
in
Length of line:
How many significant figures are in your measurement?
(this is determined by your ruler).
Which is the uncertain digit?
Length of line:
cm
How many significant figures are in your measurement?
(this is determined by your ruler).
Which is the uncertain digit?
Calculation of the conversion factor:
(take the ratio of your measurements and include units)
5 Now calculate (see eq. ??) the percent error between the actual value (look it up)
and the value you came up with:
6 Take a cylindrical mass (penny) and measure its diameter and height with the
vernier caliper. Record this in Table 1.1.
7 Repeat step 6 at least five more times. Be sure to take the caliper off the mass
between measurements.
8 Repeat steps 6 and 7 using the micrometer caliper. Record your results in Table
1.2
9 Compute the mean and the standard deviation (eq. ?? ??) of your measurements
and record them in Tables 1.1 and 1.2
11
10 Find the volume of the cylindrical mass using your two sets of measurements. Enter
your results in Table 1.3. Remember to use the mean values in your calculations
and use the appropriate number of significant digits.
11 Now that you have the volume, estimate the error in your figures by propagating
the uncertainty. Record these values in the section below.
Questions:
• Does percent error pertain to accuracy or precision? Explain.
• How could error be improved in this experiment?
• Why are several observations better than one in an experiment?
Main points to remember!!
• All measurements have an associated uncertainty, which should be quantified.
• A calculated result has an associated uncertainty based upon its dependent values.
• The design of an experiment and the skill of conducting an experiment affect the
uncertainty in the measurement.
• Uncertainty is used to compare results and draw conclusions.
12
No.
Height Diameter
[mm]
[mm]
Radius
[mm]
1
2
3
4
5
6
Standard
Deviation
Uncertainty
of device
Table 1.1: Measurements of a Cylindrical mass with a vernier caliper.
Data Analysis and Results
Vernier
[mm3 ]±
Vol=
Micrometer
Vol=
[mm3 ]±
[mm3 ]
[mm3 ]
13
No.
Height Diameter
[mm]
[mm]
Radius
[mm]
1
2
3
4
5
6
Standard
Deviation
Uncertainty
of device
Table 1.2: Measurements of a Cylindrical mass with a micrometer caliper.
Vernier Micrometer
Volume
Table 1.3: Calculated volume of a cylindrical mass
14
Lab kit returned by
(your name)
on
(Date)
Instructor’s Signature
or initials
Table 1.4:
Lab kit Return Page
When you return your lab kit, your instructor will sign below if you so desire:
15
16
Chapter 2
Picturing Motion
(In Class activity)
Motion Match Pre-Lab
For the following scenarios, use the coordinate system to sketch a position versus time
(x vs. t) graph for the conditions indicated:
x
x
t
t
Object moving in positive
direction at constant speed.
Object at rest.
x
x
t
t
Object moving in negative
direction at constant speed.
Object accelerating in positive
direction starting from rest.
Figure 2.1: Position versus time plots for four situations.
17
Notice that the initial position (the x position at t=0) is not specified — only the rate
of change of position (velocity) or how the velocity changes (acceleration) are indicated.
Any of the curves above can be shifted up or down and still be correct.
Now, for the same situations, sketch a velocity versus time plot (v vs. t) using the
axes below.
v
v
t
t
Object moving in positive
direction at constant speed.
Object at rest.
v
v
t
t
Object moving in negative
direction at constant speed.
Object accelerating in positive
direction starting from rest.
Figure 2.2: Velocity versus time plots for the situations described in Fig. 2.1.
Motion Match
One of the most effective methods for describing motion is to plot graphs of distance,
velocity, and acceleration vs. time. From such a graphical representation, it is possible to
determine in what direction an object is going, how fast it is moving, how far it traveled,
and whether it is speeding up or slowing down. In this experiment, you will use a Motion
Detector to determine this information by plotting a real time graph of your motion as
you move across the classroom.
The Motion Detector measures the time it takes for a high frequency sound pulse to
travel from the detector to an object and back. Using this round-trip time and the speed
of sound, you can determine the distance to the object; that is, its position relative to the
detector. Logger Pro will perform this calculation for you. It can then use the change in
position to calculate the object’s velocity and acceleration. All of this information can
18
be displayed either as a table or a graph. A qualitative analysis of the graphs of your
motion will help you develop an understanding of the concepts of kinematics.
An object’s velocity is determined the rate of change of position:
v=
∆x
dx
=
dt
∆t
(2.1)
A positive velocity indicates a position that is moving in the positive direction. In this
case, that means away from the Motion Detector. A negative velocity indicates an object
moving in the opposite direction.
Similarly, the acceleration is the rate of change of velocity:
a=
dv
∆v
=
dt
∆t
(2.2)
These definitions lead to a couple useful consequences for velocity and position plots.
The slope of a position versus time graph is the velocity of the object. The slope of a
velocity versus time plot is the acceleration. (We won’t focus on two additional important
relationships: The integral, or area under the curve, for an acceleration versus time
plot is equal to the change in velocity. The integral of a velocity versus time plot is a
displacement, or change in position.)
Since the positions are measured, there are experimental errors associated with them.
Since the velocity is calculated by subtracting two positions at two different times, you
will find that the experimental velocities will typically have larger errors than the position
measurements. That is to be expected.
Equipment/Materials
For this experiment you will need the following:
• Logger Pro
• Vernier Motion Detector
• Meter stick
• Masking tape
• Cardboard tube
• Racquetball
Experimental Procedure
Part I. Preliminary Experiments
1. Connect the Motion Detector to DIG/SONIC 1 port on the Lab Pro Interface.
19
2. Place the Motion Detector so that it points toward an open space at least 4 m long.
Use short strips of masking tape on the floor to mark the 1 m, 2 m, 3 m, and 4 m
distances from the Motion Detector. Be sure to remove the tape when you are done
with the lab.
3. Prepare the computer for data collection by opening Exp 01A from Intro Physics
folder (see icon on the desktop). One graph will appear on the screen. The vertical
axis has distance scaled from 0 to 5 meters. The horizontal axis has time scaled
from 0 to 10 seconds.
4. Using Logger Pro, produce a graph of your motion when you walk away from the
detector with constant velocity. To do this, stand about 1 m from the Motion
Detector and have your lab partner click “Collect”. Walk slowly away from the
Motion Detector when you hear it begin to click. Carefully examine the graph to
insure you understand the measurement. Choose “Experiment Menu” then “Store
Latest Run” to save a good run. Repeat the motion, if it’s better then “Store
Latest Run” if it’s not better, try again.
5. Be prepared to explain what the distance vs. time graph will look like if you walk
faster. Check your prediction with the Motion Detector.
6. Check the distance vs. time graphs that you sketched in the Preliminary Questions
section (Fig. 2.1) by walking in front of the Motion Detector. Once you get a
nice graph save the data so you can show it to your instructor (4 graphs total).
Discuss with your instructor how well your pre-lab sketches match your Motion
Detector graphs. Explain any differences. Now, click on the vertical axis and
change “position” to “velocity”. Compare the resulting plots to what you answered
in Fig. 2.2. Again, comment on the results and explain any differences.
Similarities/differences in position plots:
Similarities/differences in velocity plots:
20
Part II. Distance vs. Time Graph Matching
7. Prepare the computer for data collection by opening “Exp 01B”. A distance vs.
time graph will appear.
8. Describe how you would walk to produce this target graph:
9. To test your prediction, choose a starting position and stand at that point. Start
data collection by clicking Collect. When you hear the Motion Detector begin to
click, walk in such a way that the graph of your motion matches the target graph
on the computer screen.
10. If you were not successful, repeat the process until your motion closely matches
the graph on the screen. Use the “Store Latest Run” command to save your best
attempt. Show your instructor when you have a close fit.
11. Prepare the computer for data collection by opening “Exp 01C” and repeat Steps
8 – 10, using a new target graph.
12. Answer the questions for Analyzing Part II on the next page before proceeding to
Part III.
Part IIl. Velocity vs. Time Graph Matching
13. Prepare the computer for data collection by opening “Exp 01D”. You will see a
velocity vs. time graph.
14. Describe how you would walk to produce this target graph:
15. To test your prediction, choose a starting position and stand at that point. Start
Logger Pro by clicking Collect. When you hear the Motion Detector begin to click,
walk in such a way that the graph of your motion matches the target graph on the
21
screen. It will be more difficult to match the velocity graph than it was for the
distance graph. Have your instructor initial your graph when you get a good fit.
16. Prepare the computer for data collection by opening “Exp 01E”. Repeat Steps 14
– 15 to match this graph. Match the graph and answer the questions for Analyzing
Part III below.
17. Remove the masking tape strips from the floor.
Data Analysis and Results
Analyzing Part II. Distance vs. Time Graph Matching
1. Explain the significance of the slope of a distance vs. time graph. Include a
discussion of positive and negative slope.
2. What type of motion is occurring when the slope of a distance vs. time graph is
zero?
3. What type of motion is occurring when the slope of a distance vs. time graph is
constant?
4. What type of motion is occurring when the slope of a distance vs. time graph is
changing? Test your answer to this question using the Motion Detector.
22
Analyzing Part IIl. Velocity vs. Time Graph Matching
Return to the procedure and complete Part III.
5. Using the velocity vs. time graphs from Part III, sketch the distance vs. time graph
for each of the graphs that you matched. In Logger Pro, switch the vertical axis to
a position vs. time graph to check your answer. Do this by clicking on the y-axis
label and unchecking velocity; then check distance. Click to see the distance graph.
6. What does the area under a velocity vs. time graph represent? Test your answer
to this question using the Motion Detector.
7. What type of motion is occurring when the slope of a velocity vs. time graph is
zero?
8. What type of motion is occurring when the slope of a velocity vs. time graph is
not zero? Test your answer using the Motion Detector.
A Final Experiment: Motion Under Constant Acceleration
In kinematics, one special case that we frequently see is the motion of an object in free
fall. For instance, if we drop a ball that bounces up and down, the object is accelerating
due to gravity (except for the short intervals when it collides with the ground). Below,
23
sketch a plot of the height, velocity, and acceleration versus time, when a ball is dropped
and allowed to bounce a few times:
Now, construct an experiment to test your predictions. Attach the Motion Detector
to a ring stand, placed on the table, such that it points directly downward into the large
cardboard tube resting on the ground. (The tube merely restricts the ball from bouncing
out of view of the detector.) Drop a racquetball down the tube, recording with the
Motion Detector. Again, change the vertical axis from position to velocity and then to
acceleration to compare with your prediction. (Remember that the position is relative
to the detector, so that it will increase as the ball falls to the ground.)
Comparison of data with prediction:
24
position
velocity
t
acceleration
t
t
Figure 2.3: Position, velocity, and acceleration versus time (blank) plots.
25
26
Chapter 3
Spreading the Data
(At Home activity)
Purpose
In this lab, we will work with Excel as a way of displaying and processing data. Many
of you are familiar with Excel spreadsheets. For you, this lab might be primarily review.
For others, you know only a few excel commands, so much of this will be new.
• You will learn how to set up a basic Excel Spreadsheet (with labels)
• You will learn how to add data into individual cells
• You will learn how to add multiple data points into columns or rows
• You will learn how to name cells
• You will learn how to use named cells and simple Excel functions to calculate new
entries
• You will learn how to plot data from the spreadsheet
Why the emphasis on Excel as a way of recording, analyzing , and plotting the data?
It provides a relatively quick way to process even large amounts of data. In addition, it
is possible to define relationships such that we can estimate new parameters based on
the experimental data as well as estimate uncertainties.
What is a spreadsheet?
A spreadsheet is a way of storing data in tables. In addition, it is possible to use values
in the tables to calculate new values automatically as the tables are updated.
A typical spreadsheet might start off as follows:
Excel calls a spreadsheet (or table) a Worksheet. Open a new worksheet in Excel.
Along the bottom of the worksheet you will notice a number of tabs. You can change
between different worksheets by clicking on a different sheet.
27
Table 3.1: Sample Spreadsheet
A
1
2
3
4
...
B
C
D
...
X
12
In Excel, the spreadsheet typically has columns labeled with letters, and rows labeled
with numbers. To identify a particular table entry, otherwise known as a cell, you simply
give a letter and a number. For example, in the spreadsheet located above, the cell C2
contains the letter X. The cell D4 contains the number 12.
Entering Data
If you want to pick a particular cell, you can move your mouse to that cell and click.
In your new worksheet, go to sheet 1, click on A1. It is now highlighted. If you enter
a number such as “34”, it will be recorded in the cell. Click on a different cell, say B2.
Here you can enter a phrase. Enter “Test Phrase”. If you hit return, then you will see
that the mouse (highlighted cell) moves down one to B3. You are ready to enter more
data. The cell ID (the column and row) are also present at the top of the tool bar.
Cells can contain data of all types: Numbers, Dates, Labels, Formulas, and Functions.
That data can be displayed in a number of different formats, including percent, dollars,
integer, among many others.
If you left-click with your mouse, you can select a cell. If you hold the left button
down, you can select a range of cells. In contrast, the right button will usually reveal
advanced features in a pull-down format. For example, if you are working with a graph,
you can use the mouse to select a region of a plot (for example the title or data) and then
pull-up options for that region (to change the title format, or the source of the data).
Naming Cells
One of the great advantages of spreadsheets is that you can name cells, which allows
much greater flexibility in their use in formulas.
In Excel, choose “Name” under the “Insert” menu. Choose “Define...” in the list of
options. A dialog box will appear. Enter “chosen name” at the top of the dialog box.
Be aware that if there is a name in an adjacent cell, Excel will use that name by default.
The actual cell address that you are naming appears at the bottom of the box - it is
also highlighted on the spreadsheet. In Excel, the cell name will include the worksheet
name, and have $ characters added to refer to the column and row entries. If you want
to edit the cell address, then you can highlight the entry in the dialog box and alter it.
28
When you type return (or hit ok in the dialog box), the cell will now have the new name
associated with it.
Another feature is that you can paste this name onto other cells. Do do that, name
a cell. Choose a new cell and select “Name” under the “Insert” menu. Choose “Paste”.
A different dialog box appears - listing all the names of the named cells. You select one,
and then hit OK. The new cell now will have a formula that refers to the named cell.
Whatever the contents of the named cell are, they are now also part of the new cell as
well. If you change the value of the named cell, the second cell’s contents also change.
Try making a spreadsheet that contains named cells in A2 (named as distance),
A4 (named velocity) and A6 (named as acceleration). Put in values of 2.0, 4.0, 8.0
respectively. Paste the names into B2, B4, and B6 respectively. You should see the same
values as the named cells. Now change the values of A2, A4, A6 to 8.0, 4, 0, 2.0. If the
values of B2, B4, and B6 are not 8.0, 4.0, 2.0, then you have named one or more them
incorrectly. Make sure you can do handle naming cells before proceeding.
Formating Cells
You can control the format of the cells (how many digits are displayed. Select the cells
you want to change. Go to “View” and click on “Formating Palette”. A box will appear
to the side. Under the category “Number”, you will see various options for how you
want your numbers displayed. Usually it defaults to “General”. However, most of the
time, you will want to control the number of digits displayed. To do that, choose the
“Number” option, and then click on the buttons below to shift the digits left or right. It
starts off with two digits past the decimal point. Set it so three digits are displayed, i.e.
0.000).
Note that you can also alter the display format of a single cell by clicking on it and
then changing its format using the “Formating Palette’. You are NOT changing the
underlying
Entering Formulas
Now that we know how to create named cells, we can create formulas easily. You can
create formulas just using cell locations (such as E3) but it is easier to check your work
if you use a name (such as distance or acceleration).
The contents of a cell in Excel will be considered a formula if the first character is an
equal sign. Two examples:
• =B3
Set the cell’s contents to whatever is in cell B3
• =vo Set the cell’s content to the cell named v0. If you have not yet named a cell
v0, then the error message “#NAME?” will appear in the cell.
Excel has a large number of functions defined for you to use. These include common
math functions like:
• ∗ multiplication
29
• − subtraction
• / divide
• ˆ raise to the power (i.e. 10ˆ 4 === 10,000)
• () set the order of operations
• For example: =36*B3+vo+7 Multiply the contents of cell B3 by 36, and add the
contents of vo and the value of 7 to the total.
• For example: =36*(B3+vo+7)
Multiply the total by 36.
Add the contents of cells B3, v0, and the value 7.
In addition, there are many special functions that you can use in formulas. To see
what is possible, choose a cell in your spreadsheet and “insert” ”function” (on the menu).
The dialog box that pops up will list many possible functions. Choose one that looks
familiar and hit OK. The next dialog box that pops up will help you select the arguments
(i.e. cells) that you need. You can either enter the cell numbers or click on the arrow
- which allows you to go back to the spreadsheet and select the cells using your mouse.
Hitting Enter will then complete the selection.
Depending on the function you select, you may have to select a cell that contains an
angle (for something like sin() or a list of cells for a function like sum() that requires
several cell entries.
You also can type a function directly into the cell. You can use the mouse to choose
the cells you want as arguments for functions directly.
Create a list of cells, containing say 5 numbers. Find the functions (AVERAGE,
STDEV, SQRT, and COUNT. We want to find the mean (i.e. average), standard deviation (i.e. STDEV), and standard deviation of the mean. The first two are easy, as
Excel has those functions defined. To find the standard deviation of the mean is a little
trickier. Recall from our first lab that:
σ
(3.1)
σx = √ − standard deviation of the mean(SDM)
N
So, we will want to define a function that takes the result of the standard deviation
(the cell containing STDEV) and divides it by the square root (which is SQRT) of the
number of cells in our list. We could count and enter the number 5. However, there are
times when it is useful to have Excel keep track of the number of cells. To do that, we
use a function called COUNT, which will see how many cell entries are in our list. For
example, if our list spanned C4 to C8, we would have 5 entries.
The formula in that case would be: “ =STDEV(C4:C8)/SQRT(COUNT(C4:C8))”
You can replace STDEV(C4:C8) by the cell that contains STDEV(C4:C8).
So, create the three cells containing the mean, standard deviation, and standard
deviation of the mean for your list of 5 numbers.
If you change your numbers in your original list to 12, 10, 9, 8, 11, you should find
that you get 10.000, 1.581, and 0.707. If you don’t, check your formula entries.
30
Plotting
Often the best way to analyze data is to plot it. We will use plots frequently to examine
our data.
To plot data, decide which columns should be plotted.
1. Choose “Insert” in the menu and choose “Chart”
2. Chose “XY (Scatter)” on the first dialog window. Click “Next”
3. Choose the data to plot by switching from “Data Range” to “Series” using the
folder tab near the top of the dialog window.
• (a) Click “Add” to get your first data series.
• (b) Click the button at the right of the “X Values:” entry window. The dialog
box disappears and you highlight the cells in the appropriate column. Hit
Enter after the box shows all the desired column entries have been chosen.
4. Now use the “Y Values:” entry window and chose your Y data.
5. Click Finish
As you might expect there is a lot of refinements that you can apply to your data.
Multiple data sets can be plotted. You can also add labels or colors. Experiment by
clicking with either the left or right mouse button on various portions of a graph and see
what you change.
To play with this, let us create a simple spreadsheet. X numbers = 1, 2, 3, 4, 5, 6,
7 , 8 Y numbers = 1, 4, 9, 16, 25, 36, 49, 64. Enter these numbers into the spreadsheet
into two columns X and Y. Create a plot, and label it.
A sample chart can be found on Blackboard or in the desktop folder (Intro Physics
Lab/excel worksheets/2CurvesOn1Chart.xls). The worksheet delves further into chart
(plot) making.
You should end up with something like Figure3.1:
Note that we did not actually put a chart title - so we got a default chart title name.
Trendlines
A nice feature of Excel allows you to plot a trendline on a plot. To do that, you click on
the graph you have made on a specific data point. Right clicking will bring up a menu.
Depending on where you are on the graph, different menus might appear. Choose “add
trendline”. You can add the equation to the graph by setting the appropriate option on
the options page.
For the sample plot, select a polynomial of order two. You will get something like
Figure 3.2.
Note that some trendline options might be blanked out. That usually means that
you have a zero in your x data, which cause some functions to be ignored by Excel.
31
Figure 3.1: Sample Plot with Labels
32
Figure 3.2: Sample Plot with Trendlines
33
Saving Data
For any spreadsheet that you wish to keep, you will need to save a copy of the file. You
might wish to save as you go along, so that a computer glitch at the final entry does not
wipe out an hour or more or work. To do that, just go to “File” and use the “Save As”
entry. Be sure to keep a copy somewhere (like a flash-drive or your own account) where
the file will not be removed. On the computers in the lab, the files are removed every
Sunday. Your instructor can set up storage areas for you if needed.
Text Box
To add a textbox, open the “View” menu. Select Toolbars and Click “Drawing”. To
insert a Text Box, which is the one with the little A on it. Use your mouse to drag the
box to the size you want. Start typing. When you are done, move the mouse outside the
Text Box.
Some Sample Data
To illustrate what you have learned in this lab, you and your partner will each create a
new worksheet. The worksheets will be turned into your instructor when the lab is due.
Below in Table 3.2, you will find 4 columns of numbers. Choose one column. Your
lab partner and you should choose DIFFERENT columns. While you and your partner
can help each other, the spreadsheet you create should be your own work.
Table 3.2: Test Data
Column
Beep!
Beep!
Wiley E Coyote
(m/s)
10
10
10
10
10
2
1
1
0
0
Chases
(m/s)
10
9
8
7
6
5
4
3
2
1
The RoadRunner
(m/s)
10
10
10
10
15
20
50
100
100
100
All Around
(m/s)
10
10
2
2
1
1
-1
-2
-10
-20
For your spreadsheet, do the following:
1. Put your name, your lab partner’s name, and your lab section/group into a Text
Box.
2. Label one cell as Column
34
3. In the next cell to the right - enter the label from the column you chose.
4. Label one cell as Time (seconds)
5. Enter the numbers 1 through 10 in the 10 cells below the Time label
6. Label one cell as “Velocity (meters/second)”
7. Enter in the 10 cells below that column the data entries from the column you chose.
8. Label one cell as Mean (meters)
9. In the column to the right - calculate the mean of the ten data entries
10. Label one cell as SD (meters/sec)
11. In the column to the right - calculate the standard deviation of the entries
12. Label on cell as SDM (meters/sec)
13. In the column to the right - calculate the standard deviation of the mean
14. In the cell beside the one labeled as Velocity, label that cell as “Distance traveled
(meters)”
15. In the cells beside the velocity data, enter a formula (distance traveled) =velocity
* 1.0 where the velocity is the value of the velocity cell, and time is one second.
16. Calculate the mean, SD, and SDM of the distance traveled.
17. Plot velocity vs time and label the axes (remember units)
18. Plot distance traveled vs time and label the axes (remember units)
19. Save your work and turn it in.
35
36
Chapter 4
Gauging the Force
(In Class activity)
Purpose
Computers can be used for a host of applications. In this laboratory, the computer will
serve to record the data, analyze the data, and display the data. You will be introduced to
the lab’s equipment and methods. Thus, this lab should be seen as a training exercise for
future labs. One key thing to note is where important information about the equipment
or the software can be found. That will be useful for trouble-shooting later. As you go
along, if you have questions or are unclear on something, consult with your instructor.
• To learn the basics of data collection with Logger Pro software, LabPro (interface)
and measurement probes hardware.
• To learn to manipulate and analyze data
• To write out a brief summary of the experiment
Materials
• Logger Pro software
• LabPro interface
• force probe
• ring stand
• weights
37
Introduction
In these labs, the instructor will serve the role of mentor, which means that they will offer
suggestions on how you can solve your problem - rather than just telling you what to do.
For you to benefit, you need to be thinking about the lab and raising specific questions.
If your lab is not working, then you will need to be able to say identify where the problem
is. A common example is: check to make sure that the power is on (otherwise LabPro
will not work).
At each lab station there is a colored binder that contains notes on the equipment and
on the software you are using. The binder will contain diagrams and explanations that
aid in the use and understanding of both the hardware and the software. Most equipment will come with a manual. A manual will describe the installation, maintenance,
and calibration procedures. It has diagrams illustrating proper connections and how to
properly use the equipment. For this lab, the colored binder is your laboratory station
manual. If you have a problem, then the solution can be found in this binder or your
notes. When you solve a problem, it is important to make notes of what the problem
was and how you solved it. You might encounter a similar problem in the future!
You should become familiar with the contents of the binder, since you will be expected
to refer to the binder from time to time when conducting your experiments.
As you proceed through the labs, if you have suggestions on how the binder can be
improved, please let the instructor know.
Note that as the labs become more advanced, you will take on more responsibility
about how to conduct the experiment. The labs will discuss a relationship or quantity
of interest and let you work out the details of the procedures and the analysis.
In this lab, we will:
1. Hook up a probe through the LabPro to the computer.
2. Set up the software to record the data from the probe.
3. Record a set of data.
4. Analyze the data to verify the relationship between measured quantities.
5. Repeat the measurement several times
6. Explore the meaning of mean, standard deviation, and standard deviation of the
mean.
If you have trouble, call on your classmates or the instructor for help.
However, after you receive help, be sure that you repeat the process on your
own in the future.
38
Making A Measurement
At the end of the lab, there is section for the instructor record their evaluations as you
proceed. As you complete each step, have your instructor enter a comment in this table.
The Diagram
FInd in your manual the diagram that shows the basic setup. Examine your connections
and see that they agree with the diagram. In this lab, your sensor is a device called
a ”Dual-Range Force Sensor”. Look at the manual for the sensor (located later in the
manual) and read through its description. Record the range and resolution of the sensor
in the table at the back.
Set the force probe to the 10 N scale using the switch on the force probe. Find the
LabPro and note where various probes can be attached. At this point you may have to
make an educated guess as to where your probe will attach. Plug the force probe into
CH1. The white connector should slide in easily. Do not force it. You are trying to
develop an overview.
Running the Software
Start up the Logger Pro program. Logger Pro will set up the probes and the interface
for LabPro. You can then record data according to your specifications. Be sue that the
LabPro is on and is connected to the computer (via the USB port). Check the status of
the sensor connection to the LabPro interface. On the top toolbar of Logger Pro, click
the LabPro button on the left. A dialog box should open showing the interface and the
attached probes. If the correct sensor is not automatically shown in the correct input
window, click the “Identify” button. If you apply a force to the probe by gently pulling
on the hook, the value of the input window and the top toolbar changes appropriately.
If you have a problem, first try and solve it yourself. If you can’t solve the
problem, ask your instructor for help.
Tutorials for Logger Pro
You need to be able to use the Logger Pro program and you will need to come comfortable
with using the mouse and menu bar. A complete guide to the LabPro interface is in the
station manual (green binder) along with a Quick Reference Manual for Logger Pro.
There is also a short description of the force probe (Dual-Range Force Sensor). Scanning
these will be useful for an overview of how the lab equipment will work.
The help files for Logger Pro are complete and easy to use. Use the “file” menu
and the “open” command to start the first tutorial in the tutorial folder (01 Getting
Started.xmbl). Since the tutorial will be displaying temperature data, you will need
to ignore sensors. Take a few minutes to explore this tutorial. You can return to the
tutorials, help files and manuals as you go through the lab. When you would like to
continue with the force probe, use the “new button” on the “file” menu. This will close
39
the tutorial and the Logger Pro software should automatically sense the force probe and
return you to your previous configuration.
Calibrating the Force Probe
Calibrate the force probe. Either reach the calibration dialog box by clicking on the probe
picture in the LabPro setup dialog window (same window as before) or use calibrate in
the “experiment” pull-down menu.
To calibrate, apply a known force by hanging a weight from the probe.
Enter the value (type it). Record the point by clicking the KEEP point. Apply a
different known force. Enter its value. Record this calibration point. The probe should
now be calibrated.
Remember that force and mass are different quantities. The force that we are measuring is the result of gravity. This is illustrated by the following calculation:
Mass
(gm) (kg)
200
0.2
Acceleration
(m/s2 )
9.8
Force
Comment
(N)
1.96 F=mass x acceleration
Table 4.1: Force Calibration
Configure the collection mode as Events with Entry. Use the data collection button
on the toolbar (to the left of the Lab Pro button). Set the entered value to be the mass
in grams. To perform a measurement, hang a mass from the probe, click the collection
button and enter the mass value.
Once the probe is configured, a good experimentalist will check the setup by performing test measurements. Add some weight and check that the results agree with your
expectations.
One nice feature of calibrating the probe is that each experiment can fix the zero
value. You may recalibrate so that the force on the probe with holder only is centered
as zero. Now, record a second point where the force is calculated using only the mass
that was added without including the holder mass. This calibration allows you to have
the mass holder in place but you don’t need to include it as part of the recorded mass.
40
Performing A Measurement and Analysing the Results
Perform an experiment using 10 measurements that vary between 0 and a few (less than
10) Newtons.
Copy the data to an Excel spreadsheet using cut and paste. Label the columns x and
y. Add one more column “y uncertainty”. Take a guess at the value for the uncertainty
in each y value and enter it in the column. Add a text box to your spreadsheet and
describe your reasoning for the error you assigned to the y values. You should mention
that the errors for the x values are assumed to be very small and therefore have been
neglected for this experiment.
All numbers used or measured in any laboratory experiment must have
an assigned uncertainty. Develop the habit now of always including an error.
Plot the data using Excel. (see the appendix or the previous lab). Add two trendlines
to the plot. For the first, use the function y = AeBx ; for the second, use y=mx+b, where
A, B, m and b are the parameters found in the fit, and y and x represent the variables.
You will need to set the trendline to display the equation on the chart. If necessary,
ask your instructor or colleagues for help. Note that if you have a zero value for x or y,
exclude that data point - some trendlines will not be able to be used.
The trendline function represents an attempt to mathematically describe the relationship between the force and the hanging mass. Enter these parameter values into a table
below the graph. Be sure to include a column for the errors in the parameters. For the
moment, just enter “unknown” for the errors of the fitted parameters. Add comments
and labels where necessary. Add a text box and enter a brief comment on the fits to the
data. Have your name and your partner’s name on the spreadsheet. Save the spreadsheet. Show the instructor your completed spreadsheet. Be prepared to discuss what you
think it means. Feel free to discuss procedures with other students in the lab.
Repeating Measurements
Record ten or more measurements using the same mass.
Copy this data into another worksheet in your spreadsheet. Find the mass, standard
deviation (SD), and the standard deviation of the mean (SDM). Based on your work
in previous labs, interpret what these quantities mean. You may wish to consult the
Appendix and your notes from the earlier labs.
To highlight these important results, calculate and label these values in the spread
sheet. Be sure you understand the correct results. Put a textbox in your spreadsheet
that clarifies the meaning of the mean, SD, and SDM for this experiment. Have your
instructor review your work.
41
Changing Resolutions
If you change the scale on the sensor from 0-10N to 0-50N, you have poorer resolution in
your measurements.
First put a mass on - does it give the same value you expect?
Change the calibration - does it give the value you expect for the test mass?
Repeat the previous two sections and see how your results change. Use the identical
masses that you did in the previous sections to minimize uncertainties due to different
masses.
Record your results on two new worksheets and show your instructor your results.
42
Table 4.2: Setup and Performance of Force Probe Experiment
Step Performed
Instructor’s Evaluation
Computer and Interface Connected
Probe Attached and Configures
Calibration Performed and understood
Verified that measurement sensible
Data Recorded
Table 4.3: Analysis of the First Data (10N Scale)
Step Performed
Instructor’s Evaluation
Data and Errors Entered
Data plotted and Fit
Fit Parameters Tabulated
Table 4.4: Analysis of the Second Data (10N Scale)
Step Performed
Instructor’s Evaluation
Data Recorded
Statistics Calculated
43
Table 4.5: Analysis of the Third Data (50N Scale)
Step Performed
Instructor’s Evaluation
Data and Errors Entered
Data plotted and Fit
Fit Parameters Tabulated
Table 4.6: Analysis of the Fourth Data (50N Scale)
Step Performed
Instructor’s Evaluation
Data Recorded
Statistics Calculated
44
Chapter 5
Dropping the Ball
(At Home activity)
Introduction
In this experiment you will:
• determine the local acceleration due to gravity, g.
• determine the factors that influence the precision of the experiment.
• determine the accuracy of the measured quantity, g.
Formulas
An object that is moving in a linear fashion under constant acceleration can be modeled
by the equation of linear motion:
x(t) = x0 + v0 t + 1/2at2
(5.1)
Where x(t) is the position of the object at time, t; x0 is the position of the object
at time t = 0; v0 is the velocity of the object at time t = 0; a is (the magnitude of) the
acceleration of the object. The acceleration of the object can be found from the sum of
the forces acting on the object using Newton’s Second Law:
Fnet = ma
(5.2)
In Eq. 5.2, Fnet is the net force1 acting on the object and m is the mass of the object.
For this experiment we will assume that the force on the object due to air resistance is so
small that it can be neglected. Thus, the net force on the object is only the gravitation
1
Note that force, acceleration, velocity, and displacement are all vectors. Because the motion studied
in this experiment is one dimensional these equations only deal with the magnitudes of these quantities.
45
attraction between the ball and the Earth. This force is given by Newton’s Law of
Universal Gravitation:
mM
(5.3)
F = G 2 = ma = mg
r
Here G is the universal gravitation constant (≃ 6.6730 × 10 −11 Nm2 kg −2 ), m is
the mass of the object; M is the mass of the Earth (≃ 5.9742 × 1024 kg); and r is the
distance between the center of the Earth and the center of mass of the object. For small
objects at the Earth’s surface this distance is the radius of the Earth at the point of the
experiment (≃ 6.371 × 106 m). If the Earth were perfectly spherical and all experiments
were conducted at the surface of the Earth, then the local acceleration due to gravity
(symbolized by g) would be
GM
g= 2
(5.4)
r
The “g” of Eq. 5.4 is the acceleration that an object feels at the surface of the Earth.
If the object starts with zero initial velocity (v0 = 0) with the initial position is some
height (above the ground), h, and the final position = 0 then Eq. 5.1 becomes:
0=h−
gt2
2
(5.5)
Solving this last equation for “g” yields:
2h
(5.6)
t2
Here, we’ve been careful with our signs since we are considering up to be positive,
the downward acceleration due to gravity is a = −g. Solving for t yields:
g=
2h
(5.7)
g
Now consider what happens if the object bounces. By symmetry, the time from the
first bounce to the second bounce is twice the time for the object to fall from the bounce
height to the ground. Using the new bounce height, hb and the bounce time, tb yields
the following equation:
2hb
tb = 2sqrt
(5.8)
g
The local acceleration due to gravity, g can be found from the bounce height and
bounce time by the following equation:
t = sqrt
g=
8hb
t2b
Equipment/Materials
For this experiment you will need the following:
• stopwatch
46
(5.9)
• Measuring tape
• Hi-Bounce ball (35 mm diameter).
• Hi-Bounce ball (45 mm diameter).
Experimental Procedure
1 Find a suitable site for the experiment where the bouncing ball will not hit people
or equipment.
2 Measure the height of the drop to the nearest cm.
3 Place the smaller ball so that the bottom of the ball is at the measured height.
Using the stopwatch, determine the time from the release of the ball to the impact
with the floor.
• Suggestion: Have one lab partner hold the ball at a fixed height while the second
lab partner is operating the timer. The first partner should initiate a brief countdown (3-2-1-drop) which the second partner should use to start the timer coincident
with the release of the ball. The second partner should listen and stop the timer
when the ball bounces.
4 Repeat 1–3 at least fifteen times.
5 Organize the data in a suitable Excel spreadsheet with appropriate labels. An
example data table might look like Table 5.1 below.
6 Repeat the sixteen drops from the same height using the larger 45 mm diameter
ball. Organize the data in a suitable Excel spreadsheet.
7 For each of the quantities measured in the experiment, determine the uncertainty
in the measurement. Use/Add two new columns to the table(s) and label them
appropriately, as shown in Table 5.1.
Bounce Procedure
The experiment will be re-run using a slightly different procedure:
1 Place the smaller ball so that the bottom of the ball is at the given height. Drop
the ball from the given height and measure the height of the first bounce to the
nearest cm (bounce height).
3 Using the stopwatch, determine the time from the first bounce on the floor to the
second bounce (bounce time).
• Suggestion: Have one lab partner hold the ball at a fixed height while the second
lab partner is operating the timer. The first partner should the release the ball. The
second partner should listen, start the timer when the ball bounces the first time
and then stop the timer when the ball bounces the second time. The first partner
should mark the height of the first rebound so that it can be reliably measured.
47
Ball
Trial No. Height
[m]
1
Diameter =
Height
Time
Uncertainty [m] [s]
mm
Time
Uncertainty [s]
...
16
Table 5.1: Dropping the ball: Sample table for the raw experimental data.
3 Repeat 1–2 at least fifteen times.
4 Organize the data in a suitable Excel spreadsheet with appropriate labels.
5 Repeat the sixteen drops from the same height using the larger 45 mm diameter
ball. Organize the data in a suitable Excel spreadsheet.
6 For each of the quantities measured in the experiment, determine the uncertainty
in the measurement.
Data Analysis and Results
1 Add a column to the Excel spreadsheet and label it “Experimental g”. Include the
appropriate units. Determine the local acceleration due to gravity g, for each of the
measurements. Be sure to round off the reported value of “g” to the appropriate
number of significant figures. At the bottom of each of the four tables, include three
new lines labeled “Average g”, “Standard Deviation (SD) of g” and “Standard
Deviation of the Mean (SDM) of g”. Calculate the average, standard deviation,
and standard deviation of the mean for each table of data.
2 Determine a value of the local acceleration due to gravity, g, by using Eq. 5.4.
Report this value in the spreadsheet as “Average Earth surface g”. Note that this
value is NOT the textbook value of ‘9.81 m/s2 .
3 Using the data from Google Earth and the National Geodetic Survey, the value
of the local acceleration due to gravity for the second floor JMU Physics lab is
9.79888(2) m/s2 The appropriate applet can be found at
http://www.ngs.noaa.gov/cgi-bin/grav_pdx.prl
.
48
Height
Bounce height
Time
Bounce Time
Figure 5.1: Experimental setup for the Bounce Procedure
• The reason for the discrepancy in the values of g is that the Earth is not perfectly
round and James Madison University is not at sea level. Report this value in the
spreadsheet as “Accepted JMU g”.
4 Determine the uncertainty in the experimental value of g using the standard deviation of the mean (SDM). Compare your measurement to the generally accepted
JMU measurement by determining how well the accepted value lies within your
value ± your uncertainty (your Standard deviation of the Mean). See Eq. 5.10.
5 If the number calculated above is less than 3, then your value did not exclude
the accepted value with at least 95% certainty. In other words, your value is
consistent with the accepted JMU value. Using a text box, add a statement to
your spreadsheet about whether each of your four values is consistent with the
“Accepted JMU g”.
6 Did you have systematic error in your data compared with the accepted JMU value
of g? How do you know? How do you account for this error? Using a text box,
49
add a statement to your spreadsheet answering these questions.
7 Were your results consistent across all four experiments? Is the drop or bounce
method better at determining g? How do you know? Using a text box, add a
statement to your spreadsheet answering these questions.
8 Does the size of the ball appear to change the value of g? How do you know? Using
a text box, add a statement to your spreadsheet answering these questions.
|
Accepted JMU g − Y our value
|
Y our Uncertainty
(5.10)
Lab Report
The spreadsheet should have the following clearly labeled items:
A Four data tables with trial, height, uncertainty in height, time, and uncertainty in
time.
B Four experimental averages, experimental standard deviations and experimental
standard deviations of the mean.
C Accepted Earth surface g and Accepted JMU g.
D Four comparisons of experimental value to the accepted JMU g.
E Four statements about the consistency of the experimental g with the accepted
JMU g.
F A statement about the possibility of systematic error in the experiment.
G A statement comparing the bounce versus the drop method of determining g.
H A statement comparing the size of the ball and the determination of g.
50
Chapter 6
Atwood’s Machine
(In Class activity)
Newton’s second law (F = ma) is a cornerstone of physics. Given that, how can one test
or verify the law. Given that you are in an lab, the answer is that it can be tested - as
all laws of physics can be. How one tests physical laws is one of the things this lab will
help you explore.
To do that, we will use a device known as ’“Atwood’s machine” - which involves a
pulley. Our machine is modified to use a cart on a low-friction surface to help isolate
the physical processes. The need to minimize friction is one of the major experimental
concerns when trying to verify Newton’s second law.
Purpose
1. To use a cart track as a system for minimizing the effects of friction.
2. To develop sound methods for insuring that experiments are working.
3. To examine analysis techniques.
4. To test Newton’s second law.
This lab will illustrate for you many of the processes involved in performing an experiment. One key process is the act of questioning. A good experimentalist is continually
examining their experiment and questioning what can be done to improve the experiment and remove or reduce sources of uncertainty. To help you see how an experimenter
thinks, there will be questions posed in the lab manual that are designed to alert you to
important issues. In future experiments, students will be expected to raise (and answer)
these sorts of questions on their own.
51
Materials
For this experiment we will need the following:
1. Photogate with LabPro
2. LoggerPro software
3. cart tracks
4. carts
5. weight hanger
6. balance
7. assorted masses
Background Theory
In Figure 6.1 you see a very simple system, consisting of two masses (m1 , m2 ). The
masses are connected by a massless, inextensible string, which passes over a massless,
frictionless pulley. When you apply Newton’s second law to the system, if you define T
as the tension on the string, you find the following equations:
F orces on mass 1 :
m1 a = T
(6.1)
F orces on mass 2 :
m2 a = m2 g − T
(6.2)
If you add Equations (6.1) and (6.2), then the final equation of motion is given by:
(m1 + m2 )a = m2 g
(6.3)
This last equation can be tested experimentally. However, to do so will require the
aid of an (almost) frictionless cart track. The “almost” will be something that you as an
experimenter will have to keep in mind.
Constant Accelerating Force
We can explore (6.3) in several ways. The first would be to keep the acceleration force
Fa = m2 g constant. We can then check and see how the acceleration a depends on the
total mass m1 + m2 . To see how, rewrite Equation (6.3) as follows:
Let
Fa = m2 g
(6.4)
then, we can write:
a = Fa
1
m1 + m2
52
(6.5)
Figure 6.1: Atwood machine: A sliding mass is connected to a falling mass via a pulley
53
When written this way, we see that if we plot acceleration vs the reciprocal of the
1
)), it should be a straight line that runs through the origin with a
total mass (i.e. ( m1 +m
2
slope equal to Fa .
Constant Total Mass
Another way to check the 2nd law using Equation (6.3) is to keep the total mass (m1 +m2 )
constant and check the dependence of the acceleration a on the acceleration force Fa =
m2 g. We will write Equation (6.3) as follows:
Let
Fa = m2 g
(6.6)
then, we can write:
1
Fa
a=
m1 + m2
(6.7)
At first glance the equations are the same, but they have been written in a way to
help you recognize what is being changed. In this case, if we plot a graph of acceleration
versus the accelerating force, we should find a straight line through the origin with a slope
equal to the reciprocal of the fixed total mass.
1
slope =
m1 + m2
(6.8)
Of Slopes and Intercepts
To summarize: In the cases above, Equation (6.5) and (6.7) express the acceleration in
terms of y = mx + b, where y = acceleration, m = the slope of the fit, b is the intercept
at x =0 (set to 0) , and x is the thing you are changing.
In the case of the constant acceleratingh force,i the slope is given by Fa since you are
1
).
varying the total mass (or more precisely m1 +m
2
h
i
1
In the case of the constant total mass, the slope is given by m1 +m
, since you are
2
varying the accelerating force Fa .
Thus, we have two cases (constant accelerating force, constant total mass) that we
can use to test if Newton’s second law.
Experimental Techniques
To do this experiment, you will use a cart track. The cart track is a device that provides
an approximately frictionless system for mechanics experiments. The cart rides on wheels
with minimal contact with the track, resulting in very low friction.
To measure the acceleration of the cart, you will use LoggerPro software, the LabPro
interface, and a photogate. The spokes of the pulley create blocked and unblocked states
in the photogate, triggering the Logger Pro software.
54
Figure 6.2: Experimental Setup
55
To set up the Logger Pro software, open the experimental setup file “Atwood’s Machine”. You will find the file in the Experimental Setup Files / Intro Physics folder on
your laboratory desktop. This file will configure the photogate and set up the collection
mode. The setup uses the photogate to measure the time interval between the arrival
of adjacent pulley spokes at the photogate. From this time and the distance between
the spokes (which is already initialized in Logger Pro) the average velocity and average
acceleration are calculated and recorded. Examine the experimental setup, including the
software, before proceeding. Now is a good time to ask your instructor if you have any
questions about how the data is to be taken.
Measurements and Analysis
A word about good experimental techniques. When faced with a new experiment, one
should think about what is necessary to successfully perform the measurement. The first
step is to understand what will be recorded and why are those quantities being measured
and not other ones?
In this experiment, for example, we want to explore the motion of objects under the
influence of applied forces. To do that, the experiment is designed to measure time intervals. If you know (or can determine) the distance moved by the cart over a measured
time interval, then you can calculate velocity and acceleration. In a more detailed experiment, the methods used to measure time and distance should be explored to verify
their accuracy. For this lab, in the interest of time, you can assume that the procedures
are adequate.
Understanding the Experimental Setup
The goal of this experiment is compare the acceleration measured to the acceleration
predicted by Newton’s second law from a measured force and a measured mass. List
these three quantities on your data sheet.
Some questions that should be answered by an experimenter are:
• What forces are being applied ? Did you consider all of them?
• Which forces are relevant according to the ideal design?
• Which forces may influence the experiment because the experiment is not ideal?
• What object is actually experiencing the force and moving? Think about this
question before you answer.
• How does one measure the applied force?
Create an Excel spreadsheet, and briefly summarize your response to the
above questions in a paragraph or series of short answers in a text box in
your spreadsheet. If you understand the experiment, you are more likely to perform
the experiment correctly and identify your primary sources of uncertainty.
56
Testing Your Equipment
A good experimentalist will also test their equipment. Since the photogate is designed
to directly measure time intervals, you should test it. Move the cart by hand very slowly
so that the pulley spokes initiate photogate state changes. If you manipulate the motion
in simple ways, you can determine what is actually being measured.
First try some movements and note what is recorded by Logger Pro. Then perform
controlled motions where you predict what the results should be. With a watch you can
roughly measure time intervals and compare what Logger Pro measures with what you
expect. From your observations of the photogate, describe what starts the data recording
and what is measured. Logger Pro produces a table of values of t, x, v, and a. You should
be able to ascertain how the time and position data is measured. (Logger Pro uses those
data points to derive the velocity and acceleration data. You don’t need to discover the
algorithm used to find v and a).
Summarize in a text box in Excel what and how Logger Pro measures the
t and x data.
Testing Your Experimental Procedure
Here we will use the constant accelerating force to explore our experimental procedure.
Measure the mass of the cart and record this value, mc on your spreadsheet. Remember to clear label your units. Also estimate the uncertainty in your measurement. One
example is recorded in 6.1
mass of cart
uncertainty in mass
510 grams
16 grams based on reproducibility and ability to read scale
Table 6.1: Example Cart Mass Table - YOUR NUMBERS WILL BE DIFFERENT
In this exercise, you will want to verify that the track is level. There are leveling
knobs on both supports. Now place one cart somewhere in the middle of the track so
that it is not moving. If it starts moving, that is a good hint that the track needs some
serious adjustment! Be sure that the cart does not have a tendency to roll one way or
the other before you continue.
After the track is level, attach the mass hanger and string to the cart as shown in
Figure 6.2. Start with an accelerating mass of m2 = 50 grams. Hold the cart at the end
of the track and start the LogPro program. After releasing the cart, the event timer will
be triggered by the pulley spokes passing through the photogate.
57
Verify and demonstrate that your measurement works
Once you have acquired your data, you need to examine it.
• Copy the columns of t, x, v, and a into your Excel spreadsheet.
• Label the columns appropriately.
• Plot the data for a trial (x vs t and v vs t).
• Using your Excel spreadsheet, calculate the mean, SD, and SDM for the acceleration
column.
• Fit the velocity versus time data to a straight line
• Compare the slope from the fit to the average of the accelerations.
• Repeat the measurements a few times and compare your results.
As you take the data, consider the following questions:
• Are the values reasonable?
• Does the fitted curve pass close enough to all the data points?
• Do similar measurements (trials) give similar results? How close should results be
to each other? (Hint—they are not going to be identical.)
• What did you expect? If you predicted incorrectly, do you understand why?
• How do you expect acceleration to depend on time?
• How can you examine the data to verify the expected behavior?
• If verified, what analysis method do you suggest for estimating the acceleration
from the measurements of acceleration?
• How do you expect the velocity to depend on time?
• How can you examine the data to verify the expected behavior?
• If verified, what method do you suggest for estimating the acceleration from the
measurement of a velocity?
58
To complete the above analysis, you will need to include the following in your spreadsheet:
• 1 sample data table from Logger Pro (do not submit all your data).
• a plot and fit of the velocity versus time
• a plot and comments on the position versus time
• the analysis of the acceleration column (including the mean, SD, and SDM)
• a statement with your overall analysis of these measurements
In your statement, make sure it is clear what conclusions should be drawn
and why various tables and plots are included. Imagine the reader is familiar
with the experiment but may not have read the manual. Note: Instructors
will not do your analysis.
At this point, you are ready to record data efficiently. You will remain alert to prevent
problems from ruining your data but you should be able to quickly record the necessary
data for the parts A and B below. Look at how much time is remaining. Cut, paste, and
save the data before leaving. For this lab, if necessary, the analysis can be done outside
the lab. Your instructor will tell you how to turn in the lab (electronic or hardcopy) and
the due date.
59
Case A - Constant Accelerating Force
You will now proceed to study how the acceleration changes as the mass of the cart
changes. Analyze each measurement by fitting the velocity versus time and by averaging
the acceleration data (i.e. taking the mean, SD, and SDM).
Repeat the experiment in 50 g increments by adding 50 g to the cart for each new
trial. For all measurements, there are 50 grams on the string - providing the constant
accelerating force.
Do the following mass values:
• m1 =mc = mass of cart
• m1 =mc + 50 g
(attach 50 grams to the cart)
• m1 =mc + 100 g
(cart plus 100 grams attached to the cart)
• m1 =mc + 150 g
(cart plus 150 grams attached to the cart)
• m1 =mc + 200 g
(cart plus 200 grams attached to the cart)
At this stage, you will need to summarize this part of the experiment. Put your
results into a table. You might wish to use different worksheets to organize your data.
m2
50 grams
Fa =m2 g
m1
uncertainty
in mass
uncertainty = 1 gram
acceleration
(velocity data)
uncertainty
acceleration
uncertainty
(acceleration data)
Table 6.2: Example Case A Table - Remember Units!
Once you have obtained an acceleration value for each mass, plot a graph of the
acceleration a versus the reciprocal of the total mass ([1/(m1 + m2 )]. Include comments
with your result.
Compare the slope of the graph with the theoretical value of Fa = m2 g.
Your answer to this part requires a table, a plot, and comments.
Continue on to the next section once you have your data - do the analysis later.
60
Case B - Constant Total Mass
Start with 200 grams attached to the cart and m2 = 50 g. The constant total mass for
this part is m1 + m2 = mc + 250g.
Remove 20 grams from the cart and add the same 20 grams to the accelerating mass
m2 , and find the acceleration for this system as before. Repeat this step until you have
measured the acceleration for the following values of m2 : 70g, 90g, 110g, and 130g.
Graph the acceleration a versus the accelerating force Fa = m2 g. Determine the slope
of the graph. Comment on the result.
Compare the slope of the graph with the theoretical value of ([1/(m1 + m2 )].
Your answer to this part requires a table, a plot, and comments
Summary of Required Write-ups
Item
Instructor’s Evaluation
Summary: Basic measurement====physics
Summary: Equipment operation
Measurement of Mass
Evaluation of the Experiment:
Analyze first measurement, repeat, summarize
Table of data recorded for mass added to cart
Plot of a vs ([1/(m1 + m2 )],
comments on validity of result
Comparison of expected vs observed values
for the applied force
Table of data recorded when moving mass to hanging
position
Plot of a vs applied force,
comments on validity of result
Comparison of expected vs observed values
for the total mass
Table 6.3: List of Required Items
61
62
Chapter 7
Sliding along
(At Home activity)
Introduction
Any good shoe store (be it brick–and–mortar or on-line) sells a bewildering array, perhaps
one hundred or more different sports shoes. Besides the all–important “coolness” factor,
one of the more important characteristics that differentiates between these shoes is the
amount of friction (“grip”, “traction”, etc.) that their soles provide. While friction forces
are often view as unwanted, performance limiting side effects (overheating of motors,
bearings, tires, speed–limiting air drag forces, etc.), they are beneficial in numerous
human activities: imagine a car stuck in mud, with its wheels spinning helplessly, or a
person with smooth-soled shoes trying to walk across ice.
A convenient quantity that gauges the amount of friction between two surfaces is
called the coefficient of friction, µ (Greek letter “mu”, pronounce myoo or moo). In this
experiment you will investigate the effect of different surfaces, and different weights on
the coefficient of friction.
Formulas
The coefficient of friction1 between two surfaces is defined as:
Ff
(7.1)
N
Where Ff is the magnitude of the friction force between the surfaces and N is the magnitude of the normal force on the surface. The index s or k denotes the “static”/”kinetic”
coefficient of friction. Note that µs can be larger than one while µk is always smaller
than one. Because it is a ratio of like quantities, the friction coefficient is dimensionless.
Note that eq. 7.1 merely states that the friction force is proportional with the normal
µs/k =
1
If the two surfaces move with respect to each other the coefficient of friction is called “kinetic
coefficient of friction”. If the two surfaces do not move with respect to each other the coefficient of
friction is called static.
63
force. If, as shown in Fig. 7.1 the surface is horizontal and one has a (relatively) small
object moving across a massive, immovable object (i.e. floor), the normal force will be
equal to the weight of the object, N = W eight.
N
v
Fapplied
Ff
Weight
Figure 7.1: Friction and Normal Forces
While one cannot directly measure the friction force, we can measure the force needed
to overcome the friction force, as shown in Fig. 7.1. If the applied force Fapplied equals
the friction force Ff then the object will be in equilibrium, therefore it will experience
uniform motion (the small velocity “v” shown).
Equipment/Materials
For this experiment you will need the following:
• Spring Scale (included in your PHYS140L kit)
• String
• A shoe (anything except very low profile shoes or flip–flops will do)
• A way to attach the string to the shoe (tying to shoelace, using a small bit of tape,
a bent paper clip, etc)
• Access to both “smooth” and “rough” horizontal surfaces to test the shoe on.
Smooth surface examples: most tiled floors, hardwood floors, floor in the JMU
Physics and Chemistry building (and in most other JMU buildings). Rough surface
examples: carpet, concrete (not painted), asphalt.
Experimental Procedure
0. Calibrate the spring scale. Without any weight attached to it, hold the spring scale
vertically and make sure that the spring scale reads 0 (zero). If it does not move
64
the metal ruler (the one that has marked pounds and newtons on it) up or down
as required.
1. Attach shoe to spring scale using a ∼one foot piece of string (and tape, bent paper
clip, etc).
2. Weigh the shoe using the spring scale (Gently lift the shoe off and having it hang
on the spring scale). Record this weight in Table 7.1. Pay attention to units.
3. Place the shoe on a horizontal surface of your choosing (either “rough” or “smooth”).
4. Record in the “Observations” area a brief description of the surface and of the shoe
(brand, how new/worn out it is), etc.
5. using the spring scale drag, very gently, the shoe across your test surface. Be sure
to pull parallel to the ground. Your laboratory partner, positioned a distance away
might be able to better asses if you are keeping the pulling force parallel to the
ground. Adjust as necessary.
6. As you pull the shoe across the surface note the value of the pulling force as measured by the spring scale. If the shoe is moving with uniform motion (i.e. very
slowly, in a straight line, not speeding up, not slowing down) then the force recorded
by the spring scale will be equal in magnitude with the friction force between the
shoe and the surface. Record this force in the data table.
7. Repeat steps 5–6 at least four more times (you should consider switching places
with your lab. partner mid-way through this process.
8. Repeat steps 5–7 for at least another surface. Record your results in Table 7.1.
9. Repeat steps 5–8 for a different shoe model. Record your results in Table 7.1.
Data Analysis and Results
1. Determine the average, standard deviation, and standard deviation of the mean of
the weight for each shoe that you measured. Record these values in Table 7.2.
2. Determine the average, standard deviation, and standard deviation of the mean of
the pulling force (needed to achieve uniform motion) for each shoe–surface combination that you measured. Record these values in Table 7.2.
3. Using Eq. 7.1 compute the kinetic coefficient of friction µ for each shoe–surface
combination. Record these values in Table 7.2.
4. Comment (in no more than 3-5 paragraphs) your coefficient of friction results? Do
the number make sense for each shoe–surface pair? Ordering the pair of surfaces
according to their µs do they line up as you would expect or there were some
surprising results?
65
5. Comment (in no more than 3-5 paragraphs and using formulas and estimates as
appropriate) on the precision of your µ measurement, given the SD, SDMs you
computed for both the weight and the pulling force averages.
6. As the head(s) of the advertising department of the company that markets this
shoe (pick one of those tested) design a one–page add that will help market this
product, incorporating (some) of the results of your measurement. Print/send in
electronic format to your instructor this add.
66
• Do not forget that an add “marketing” one of the shoes
tested must be produced/turned–in!
• Do the µ results make sense? Comment below.
• How precise are your µ results? Comment below.
67
No. Weight
[N]
Pulling Force
[N]
Description
Table 7.1: Experiment: Sliding Along. Data table. If needed, feel free to make copies of
this table.
68
Shoe
–Surface
Average
Weight
[N]
SD
SDM
Weight Weight
[N]
[N]
Average SD
Pull
Pull
[N]
[N]
SDM
Pull
[N]
Table 7.2: Experiment: Sliding Along. Results table.
69
Friction
Coefficient
µ
70
Chapter 8
Crashing Carts
(In Class activity)
Introduction
In this lab, you will investigate the conservation of momentum in three different cases:
1. inelastic event with both carts initially at rest
2. inelastic collision with both carts having the same final velocity
3. elastic collision
In each case, with no net force acting on the system, the total momentum of both
carts is conserved. This means that the total momentum after the event (e.g., a collision)
is the same as before the event, that is, initial momentum equals final momentum1 .
In this process you will also learn how to:
• set up & become familiar with using two sensors connected to the LabPro interface.
• make predictions of experiment and then test
• verify conservation of momentum laws by observation and analysis
• see the effects of error propagation on the final results
Formulas
Momentum conservation can be written as:
p~i = p~f
1
(8.1)
It would be helpful to remember that momentum, like velocity and acceleration is a vector quantity.
71
Here the indexes i and f denote the initial and final value of the momentum of the
system. Recall that the momentum of an object is simply defined as its mass times its
velocity:
~p = m~v
(8.2)
For the particular case of a system of two objects (as you will study in this lab.) eq. 8.1
becomes:
m1~v1i + m2~v2i = m1~v1f + m2~v2f
(8.3)
Where the indexes 1 & 2 denote the two objects/carts.
Equipment/Materials
For this experiment you will need the following:
• a cart track
• two carts (one with spring plunger)
• a cart launcher
• two photogates
• LabPro interface
• two 500g masses
• detection sails for carts
• balance
• meter stick
Experimental Procedure
This experiment has many similarities to the Atwood’s Machine lab you did previously.
Recalling the techniques you used in that lab, how you set up your spreadsheet and your
uncertainty analysis (See section on uncertainties below) will greatly assist in this lab.
It might be a good idea to start a spreadsheet now.
Measure the mass of each cart with the detection sails installed. Record these values
in your spreadsheet. Include uncertainty estimates. Have your instructor check this off.
As in the Atwood’s Machine lab, it is critical that the track be level and stable. There
should be a level tool available for your use to check this. As a check you can place one
cart somewhere near the middle of the track so that it is not moving. Be sure the cart
does not have a tendency to roll one way more than the other.
In this experiment cart velocities are required in addition to the cart masses. The velocities are measured using photogate sensors, attached to the LabPro interface units and
72
utilizing the LoggerPro software. The velocity is found when the “sail” passes through
the photogate and blocks the infrared beam. It is determined by LoggerPro dividing the
length of the sail by the time that the photogate beam is blocked. This of course is an
average velocity over the time interval.
This experiment requires the use of two sensors connected to the LabPro interface.
Use the following procedure to set them up for reading:
1 Open LoggerPro. (sensor operation can be checked at this point by blocking the
infrared beam of the photogate and observing if the red LED on top of the sensor
illuminates)
2 Under Experiment, click on “Set Up Sensors”, then “LabPro: 1”, or click on the
small LabPro icon in the upper left-hand corner of the screen.
3 Click on the photogate icon in the “DIG/SONIC1” window. If this icon does not
appear, select “Photogate” from the drop-down menu in the “DIG/SONIC1” window. block the photogate beam now and verify the “Gate State” where indicated
above the table in the LoggerPro screen)
4 Click on the photogate icon in the “DIG/SONIC1” window, select “Gate Timing”
under Current Calibration.
5 Again click on the photogate icon, click on “Set Distance or Length”
6 Measure the length of the sail on the cart that will be passing through the photogate
connected to the “DIG/SONIC1” port and enter that value in meters. Remember
to also enter this value (and its uncertainty) in your spreadsheet.
7 Repeat steps 3–6 for the second photogate connected to “DIG/SONIC2”. Then
close the LabPro pop-up window.
8 Under Experiments, click on “Data Collection”.
9 Under Mode, select “Time Based”, enter 10 s for length of data collection and for
sampling rate enter 2000 samples/second. Click Done.
You may want to adjust the width of the table in the LoggerPro screen so that the
velocity columns for both photogates are shown.
For all of your data collection, one of your team members should be designated to
catch the cart(s) after passing through the photogate. By “softly” catching the carts, it
will keep the carts from being involved in secondary collisions, either with track bumpers
or the floor, and therefore preventing damage to the carts. This is more critical in the
higher velocity runs.
Set up the photogates so that the sail on the carts pass through the beam. Position
the photogate stands along the track about 40 to 50 cm apart. Click Collect and push
one of the carts slowly so that it passes through both photogates to make sure you set up
the data collection correctly and so that you can get a feel for what velocity is produced
73
by a given force (push). Try different levels of “push”. Have your instructor witness this
after the data collection works correctly.
Case 1 (Carts initially at rest):
In this experiment, both carts will start initially at rest between the photogates.
Momentum is imparted to the carts by the release of a spring plunger in one of the carts.
Because the carts are initially at rest, both the initial momentum and the final momentum of the system is zero:
m1~v1f + m2~v2f = ~0
(8.4)
From eq. 8.4 it is clear that regardless on how you choose (left–to–right or right–to–left)
your coordinate system one of the final velocities will point in the positive direction of
the axis and one final velocity will point in the opposite direction of the axis.
You will run experiments for three cart/mass arrangements:
• Case 1A no extra masses on either cart
• Case 1B a 500g mass on Cart 2
• Case 1C two 500g masses on Cart 2 (note, measure the masses of these weights
and record with their uncertainties)
For these three sub-cases; assuming that the final velocities can be written as: ~v1f = k~v2f ,
with k a scalar constant, predict the value of k for these three sub-cases based on eq.
8.4. Record these three values in your spreadsheet including their uncertainty. Have your
instructor check this off.
Set up the two carts between the photogates as in Fig. 8.1 below:
Photogate 1
Cart 1
Cart 2
Photogate 2
Cart Launcher
plunger
Figure 8.1: Case 1: Both carts at rest initially (Note: your setup may be the mirror
image of this figure)
The cart labeled Cart 1 has a built-in spring plunger with 3 set positions. To set the
spring plunger, push the plunger in, and then push the plunger upward slightly to allow
74
one of the notches on the plunger bar to “catch” on the edge of the small metal bar at
the top of the hole. After setting the plunger, it is released by tapping the trigger button
on top of the front end cap. To ensure that you do not give the cart an initial velocity,
other than that supplied by the spring plunger, release the trigger by tapping it with a
rod or stick using a flat edge. Practice this until you are confident that your releasing
technique is not affecting the cart velocity.
Set the plunger to the middle position. Position the carts (Case 1A, no extra masses
on either cart) so that the end of the spring plunger is touching Cart 2, click Collect in
LoggerPro, and then release the plunger to propel the two carts through the photogates.
• Does the initial position of the carts relative to the photogates affect the results?
• What is the optimum position of the photogates with respect to the starting position
of the carts?
• Adjust the positions of the photogates as necessary.
Record the measured velocities in your spreadsheet. Repeat for at least three trials
so that you obtain velocities that are about the same. Calculate k for each trial and
find its average value. Compare this to your predicted value for Case 1A considering
uncertainties.
Repeat the above for Cases 1B and 1C. Do the results make sense? Explain in a text
box in your spreadsheet. Review with your instructor.
Case 2 (Inelastic collision):
In the inelastic collision, the carts will stick together. This is accomplished with
velcro pads on the ends of the carts. For this experiment, Cart 2 will be initially at rest
and since the carts stick together, the final velocity of both carts will be the same (note,
this will simplify eq. 8.3).
To start this experiment, position Cart 2 initially at rest between the photogates
while the other cart will start outside the photogates. Cart 1 will be propelled to collide
with Cart 2 in an inelastic collision. The cart launcher attachment will be used to propel
Cart 1.
• Before performing the collision experiment, you should understand the operation
of the cart launcher.
• Loosen the thumbscrew on the adjustable latching clamp on the plunger and move
it into a position so that when the trigger lever is set on the latching clamp, the
indicator is at about 3.5 cm.
• Set the launcher by compressing the spring and hooking the trigger lever on the
latching clamp. Place Cart 1 up against the rubber tip on the end of the plunger.
• Position the first photogate about 10 cm away from the cart and the second photogate another 40 cm away.
• Launch the cart by releasing the trigger lever while collecting data in LoggerPro.
75
• You may find it helpful to have one of the members your team hold down the track
when the cart is launched in order to keep the relative position of the track and
photogates the same.
• Repeat this so you are comfortable with this operation.
• Record the velocity obtained at this plunger setting.
Repeat the steps above for spring compression levels of about 2.5 cm and 1.5 cm. Have
your instructor check your velocity measurements.
Now for this set of experiments with inelastic collisions, you will run the following
three sub-cases:
• Case 2A: no extra masses on either cart
• Case 2B: a 500g mass on Cart 2, no extra mass on Cart 1
• Case 2C: a 500g mass on Cart 1, no extra mass on Cart 2
Adjust the cart launcher spring compression to propel Cart 1 (without added mass)
at a velocity of about 0.5 m/s. This may require some trial and error; adjust as necessary.
Set up the two carts relative to the photogates and cart launcher as in Fig. 8.2 below.
Make sure the spring plunger in Cart 1 is well secured in its fully compressed position;
you want it to stay there.
Cart 1
Cart 2
Cart Launcher
Figure 8.2: Case 2: Inelastic collision with objects moving with the same final velocity.)
Launch Cart 1, recording the velocities with LoggerPro. When the “attached” carts
pass through the second photogate, is it better to use the velocity value from Cart 1 or
Cart 2? When you are satisfied with this operation, show your instructor. Again, consider
how the location of the photogates may improve your data and adjust accordingly.
Do this for sub-cases 2A, 2B and 2C, recording all your data. You may want to
repeat each trial to be sure your velocities are generally repeatable and you are not
getting spurious data.
76
For all trials, calculate the initial and final system momentum values with uncertainties. Evaluate your trials by comparing the initial and final momentum. Is the absolute
value of the difference between the initial and final momenta less than the sum of the
uncertainties of the initial and final momenta? If not, how does it compare to two times
the sum of the uncertainties of the initial and final momenta? Summarize in a text box
in your spreadsheet. Review your results with your instructor.
Case 3 (Elastic collision):
An elastic collision is one in which the two carts bounce off each other and in which
both momentum and kinetic energy are conserved.
In this experiment, the setup will be similar to Case 2 with Cart 1 being propelled by
the cart launcher and Cart 2 initially at rest. However, since it is an elastic collision, the
carts will be turned around so that their magnet ends face one another. Use the same
cart launcher setting so that Cart 1 has an initial velocity of about 0.5 m/s.
For this set of experiments with elastic collisions, you will run the following three
sub-cases:
• Case 3A: no extra masses on either cart
• Case 3B: a 500g mass on Cart 2, no extra mass on Cart 1
• Case 3C: a 500g mass on Cart 1, no extra mass on Cart 2
Set up the two carts relative to the photogates and cart launcher as in Figure 3 below.
Figure 8.3: Case 3: Elastic collision.
Before you start collecting data, predict and sketch the positions of the carts after
the collision on the track in Fig. 8.4 through Fig. 8.6 below. Use arrows to indicate the
cart directions and indicate the positive velocity direction. Show your instructor.
After your instructor has checked your sketches and setup, do a trial run of case 3A
to check the data collection, again considering how the location of the photogates may
improve your data.
Do this for sub-cases 3A, 3B and 3C, recording all your data. Record data as you did
in Case 2.
77
Figure 8.4: Cart positions after elastic collision, Case 3A.
Figure 8.5: Cart positions after elastic collision, Case 3B.
For all runs, calculate the initial and final system momentum values with uncertainties. Compare the initial and final momentum, as you did in Case 2.
Did your results match your cart position predictions? If not, what part was different?
Describe in a text box on your spreadsheet. Review these results with your instructor.
Data Analysis and Results
(Computing the uncertainty in velocity)
Recall how the velocity is calculated with the photogate sensor. The length of what
blocks the infrared beam (the sail in this lab) is divided by the time the beam is blocked.
Hence, the photogate actually measures a time differential, ∆t, vs. directly measuring a
velocity. Now, you have measured the length of the sail and estimated the uncertainty for
that measurement, but how can you determine an uncertainty for the time measurement?
During the sensor setup you told LoggerPro to collect data at a rate of 2000 samples/s.
That means the photogate is checking every 1/2000 of a second, that is, every 0.0005 s,
to see if the infrared beam is blocked or unblocked. Now consider a situation in a run
78
Figure 8.6: Cart positions after elastic collision, Case 3C.
where, for instance, the sensor determines at t = 0.1000 s the beam is unblocked and
then at the next sampling, t = 0.1005 s, it is blocked. Furthermore, lets assume somehow
we know that the beam actually becomes blocked at t = 0.1001 s. This would mean that
the time LoggerPro records as when the beam becomes blocked is in error by 0.0004 s.
How much could be the worst case error here? Since the photogate is measuring a ∆t,
what will happen to the error if a similar sort of event happens when the beam becomes
unblocked? Use the sum of these potential, worst case errors, as your uncertainty.
Since there are numerous runs and velocity measurements, and each velocity has a
separate uncertainty calculation, the result would be that you are spending a lot of time
calculating velocity uncertainties. To save time, it is suggested that you calculate the
uncertainty for one velocity. This may be a typical value for velocity, or the minimum or
maximum velocity observed (consider which would give you the worst case uncertainty).
In addition, for this uncertainty, and the velocity it is based on, calculate a relative
uncertainty. Use this relative uncertainty throughout this lab for all velocities. State on
your spreadsheet how you determined your uncertainty value.
79
No. Topic
1
Mass of carts & Uncertainties
2
Sensor & data collection
“push” vs velocity
3
Case1: k predictions
4
Case1: k results
5
Cart Launcher
velocity measurements
6
Case2: Setup & operation
7
Case2: Inelastic collision
Results
8
Case3: Elastic collision
prediction sketches
9
Elastic collision results
Instructor Evaluation
Table 8.1: Instructor check off table for “Crashing Carts” experiment.
80
Chapter 9
Happy and Sad Balls
(At Home activity)
Introduction
In this experiment we will explore the properties of the “sad/happy” balls found in your
PHYS140L take home kit. To begin with, locate these two objects: the two seemingly
identical black balls, about an inch in diameter. To convince youseleves that these two
are only apparently identical do the following quick test: drop (do not throw) each ball
in turn from a height of a couple of feet on a hard surface (tile, hardwood floor, etc.).
This little test should be enough to convince you that the two balls are not identical.
Formulas
For this experiment you will need the following formulas/concepts:
Density is a scalar quantity that measures how compact an object is:
m
(9.1)
V
With m the mass and V the volume of the object.
Archimedes’ Principle states that for every object imersed in a fluid there is an upward
force equal to the weight of the fluid displaced.
In order to “float” an object’s density needs to be lower than that of the fluid in
which it is imersed.
In a collission between two objects (let’s label them “1” and “2”) the restitution
coefficient is defined as:
v2f − v1f
CR =
(9.2)
v1i − v2i
Here the index i/f denotes the initial/final state (i.e. before and after collission). v is of
course the magnitude of the velocity for objects “1” and “2”. Note that as it is a ratio of
like quantities, CR is dimensionless. For an elastic collission CR would be equal to 1; for
a perfectly inelastic collission CR would be equal to 0 (zero); most/all collission between
real objects will have CR s somewhere in between zero and one.
ρ=
81
For the particular case in which one of the objects is massive and static (like in the
collission between a ball and a fixed floor) eq. 9.2 becomes:
CR =
v1f
v1i
(9.3)
Due to limitations in the PHYS140L take home kit you will not be able to measure
(reliably) either of the velocities in eq. 9.3. Neglecting air drag one can, however, equate
the kinetic energy of the object before/after the collission with the potential energy of
that object: K.E. = P.E.. Given the definitions for kinetic and potential energy it is
straightforward to show that:
CR =
s
h1f
h2i
(9.4)
Here hi denote the height from which the object is dropped while hf is the height to
which the object will bounce.
Equipment/Materials
For this experiment you will need the following:
• Happy and Sad balls
• Meter stick/tape
• Access to different types of floors (a hard surface floor and a carpeted floor)
• Plastic cup/container and table salt.
Experimental Procedure
Determination of the density of the “happy” and “sad” balls.
The formula (eq. 9.1) for density calls for the measurement of both the mass and the
volume of an object. The spring scale in your PHYS140L kit is not sensitive enough to
provide a good measurement for the masses of either the “happy” or the “sad” balls.
In principle one could weigh a bunch of identical “happy” balls using the spring scale
provided, divide by the number of balls to get the mass of a single ball, then repeat for
the “sad” ball. That is a good procedure when large numbers of identical objects are
available. Unfortunately that is not the case in this experiment.
Alternatively one can estimate directly the density of the two balls using the following
procedure:
• Take a container (plastic cup, drinking glass) large enough to contain the two balls
(do not use too big a vessel!) and fill it to a height of ∼1.5–2 inches with water
(you need the liquid level to exceed the diameter of the balls but not by much).
82
• Drop the two balls in your container. What do you observe? Do the balls sink or
float?
• Start adding regular table salt 1 , one teaspoon at the time to the water. Stir a
little to allow the salt to disolve. Are the balls floating or sinking?
• After a few teaspoons of salt one of the balls will start floating. Make a note of
which of the two balls is the first one to float (if in doubt dry it out on a paper
towel and drop it against a hard floor)
• This simple experiment should allow you to order the two balls according to their
density. In principle one could get an actual measurement of the density by keeping
track of the amount of salt added. For this assignment just being able to tell which
ball is more/less dense is enough.
Determination of the coefficient of restitution CR
1 Drop (from a previously measured height) one of the balls against a hard floor
(tile, hardwood floor, cement, etc.) and measure the height to which the ball will
bounce.
2 Record both the starting height and the bounce height in Table 9.1.
3 Repeat the procedure at least five more times (as long as you record it appropriately
the starting height need not be exactly the same from step to step), record the
results in Table 9.1
4 Repeat steps 1–3 for the other ball. Record the results.
5 Repeat steps 1–4 this time dropping the balls against a carpeted floor (if you have
access to one, a Persian rug would be an acceptable substitute). Do you notice
anything different with respect to the previous set of throws?
6 For all trials above compute CR using eq. 9.4.
7 Average your CR values for each ball–type – floor combination. Record these averages and the spread (SD) of your CR measurements in Table 9.2.
Data Analysis and Results
The ball that floats first is the
ball. That means that this ball has
of higher/lower) density than the other ball.
Questions:
(chose one
• Is CR a property of the ball?
1
Table salt will cloud the water somewhat; that is OK. If you have it, you can substitute pickling salt
or Kosher salt - the resulting solution might be less cloudy.
83
No. hi
[m]
hf
[m]
CR
Ball & Surface
Table 9.1: CR Measurements using the happy and sad balls.
84
No. Ball & Surface
CR
SD for CR
Table 9.2: CR Averages and SD.
• If one would freeze the two balls, will their respective CR s be larger or smaller?
Explain.
• Same question if one would be heat (by putting them in boiling water for instance).
Do not try either of these!
• Imagine that the “happy” and “sad” balls, having the same initial velocity - for
instance by rolling each one down the same incline (an open book, propped at one
end, your PHYS140/240 book comes to mind, would make a suitable incline; just
roll the ball along the binding), collide (in turn) with a third ball, initially at rest
on a table/floor. Which would send this third ball farthest? Would it be the happy
ball? Or the sad ball? Explain.
85
86
Chapter 10
Poe’s Pendulum
(In Class activity)
Period of a Pendulum
A swinging pendulum keeps a very regular beat. It is so regular, in fact, that for many
years the pendulum was the heart of clocks used in astronomical measurements at the
Greenwich Observatory. What determines how quickly the pendulum moves back and
forth?
Angle (degrees)
1 cycle
1
0. 5
0
0
10
20
30
40
50
60
70
80
-0. 5
-1
Time (s)
Figure 10.1: Pendulum cycle.
Each back-and-forth motion is called a cycle. The time (usually measured in seconds)
that it takes for one cycle to occur is called the period, and is given the symbol T . The
frequency, or the number of cycles per second, is inversely related to the period: f = 1/T .
A simple pendulum consists of a mass hanging from a pivot on a string. In an ideal
pendulum, we imagine that the mass is concentrated at a point, the string has no mass,
87
and there is no friction or air resistance.
There are at least three things you could change about a simple pendulum that might
affect its period:
• the mass of the pendulum bob
• the length of the pendulum, measured from the center of the bob to the point of
support
• the amplitude of the pendulum swing (or how far from vertical, in degrees, the
pendulum is initially pulled back when it’s released)
To investigate the pendulum, you need to do a controlled experiment; that is, you
need to make measurements, changing one variable at a time, while keeping all other
variables constant. Conducting controlled experiments is a basic principle of scientific
investigation.
In this experiment, you will use a Photogate capable of microsecond precision to measure the period of one complete swing of a pendulum. By conducting a series of controlled
experiments with the pendulum, you can determine how each of these quantities affects
the period.
In this lab, you will plot your data and draw relationships from the plot. One word
of caution: Examine the plots in Fig. 10.2. What do you think might be the function
f (x) plotted on the left? It is apparently linear. However, the plot in the center shows
exactly the same data with the vertical scale starting at zero. The variation is only a
couple percent, so maybe the function is a constant. The plot on the right shows the
same points along with three additional data points. In this case, there seems to be a
clear (and non-linear) trend. Maybe we didn’t have enough data to see how f(x) depends
on x.
10.2
12
12
10
10
8
8
6
6
4
4
2
2
f(x)
10
9.8
9.6
9.4
0
0
0
1
2
3
4
0
1
2
x
x
3
4
0
1
2
3
4
5
x
Figure 10.2: A warning about plots – pay attention to the scales on your axes!
When determining whether a parameter is relevant in affecting the period of the
pendulum, you must both vary the parameter(x-axis) by a sufficient amount to
in order to potentially observe a measurable change and make sure that the
scales on your plots are not so small that a slight change appears significant.
There is no “correct scale” for your plots, but you should be conscious of this as you
draw conclusions from your data.
88
Equipment/Materials
For this experiment you will need the following:
• Vernier photogate
• Protractor
• Strings with 4 different masses
• Ring stands (2) and pendulum clamp
• Meter stick
Experimental Procedure
1. Attach the string and mass to the pendulum clamp. Attach the Photogate to the
second ring stand. Position it so that the center of the mass blocks the Photogate
while hanging straight down, as shown in Fig. 10.3. Use care when releasing
the mass that it doesn’t strike the Photogate. The length of the pendulum
is the distance from the pivot point (bottom of the clamp) to the center of mass
of the pendulum bob. Connect the Photogate to the DIG 1 port on the LabPro
Interface.
2. Prepare the computer for data collection by opening “Exp 14” from the Intro to
Physics folder. A graph of period vs. time is displayed.
3. Temporarily move the mass out of the center of the Photogate. Notice the reading
in the status bar of Logger Pro at the top of the screen, which shows when the
Photogate is blocked. Block the Photogate with your hand; note that the Photogate is shown as “blocked.” Remove your hand, and the display should change to
unblocked. Click “Collect” and move your hand through the Photogate repeatedly.
After the first blocking, Logger Pro reports the time interval between every other
block as the period, since the mass blocks the photogate twice during one complete
cycle. Verify that this is so.
4. Design and conduct an experiment to determine how the period depends on the
mass of the pendulum bob (m).
5. Design and conduct an experiment to determine how the period depends on length
of the pendulum (L).
6. Design and conduct an experiment to determine how the period depends on the
initial amplitude (θ0 ).
89
Data Analysis and Results
• For each of your three experiments, construct a data table and a graph (or graphs)
that represents your data.
• Determine the empirical formula for the relationships that exist between the variables tested and the period of a pendulum. That is, try to write a relationship
T ∝ ... which shows how the period depends on the three experimental variables
(“∝” means “is proportional to”).
As an example, from Newton’s 2nd Law (F = ma), we know that given a fixed
mass, a ∝ F and given a fixed force, a ∝ m1 .
Ask questions as necessary, but also use experience and techniques from the previous
labs.
• Use appropriate estimates of uncertainty to qualify how well you know these relationships and formulas. In the lab next week, we will focus more on how to
determine experimental relationships, including determining the errors in the parameters.
• Briefly write up your findings and be prepared to justify your conclusions.
Your lab report should include:
• Clearly labeled data and plots for each of the three cases.
• Uncertainties for all measured quantities listed and explained.
• Analysis tabulated and clarifying comments included.
• Results clearly stated and demonstrated.
A final measurement and a look ahead
As mentioned above, the lab next week will enable us to quantitatively determine a
functional dependence, including the errors of the fitting parameters. In order to have
reliable data for this at-home activity and to save time next week, we will collect the
data now. It should take about 15 minutes to go through the following steps. Briefly,
here is the goal of the analysis that you will perform next week:
When you solve physics problems involving free fall, often you are told to ignore
air resistance and to assume the acceleration is constant and unending. In the real
world, because of air resistance, objects do not fall indefinitely with constant acceleration.
Instead, their acceleration is decreased by air resistance – if you drop a feather, for
instance, it will quickly reach an almost constant velocity referred to as terminal velocity,
vT . (Even objects that aren’t as clearly affected by air drag, like a baseball or a skydiver,
will reach terminal velocity if allowed to fall far enough!)
90
Air resistance is sometimes referred to as a drag force. Experiments have been done
with a variety of objects falling in air. These sometimes show that the drag force is
proportional to the velocity and sometimes that the drag force is proportional to the
square of the velocity. In either case, the direction of the drag force is opposite to the
direction of motion.
Mathematically, the drag force can be described using Fdrag = −cv or Fdrag = −cv 2 .
The constant c is called the drag coefficient and depends on the size and shape of the
object. When falling, there are two forces acting on an object: the weight, mg, and air
resistance, −cv or −cv 2 . At terminal velocity, the downward force is equal to the upward
force, so mg = −cv or mg = −cv 2 , depending on whether the drag force follows the first
or second relationship. That is, mg = −cv n where n is equal to 1 for the linear drag
force and 2 for the squared drag force.
How can we tell if the right drag rule for our coffee filters is the linear or the squared
type? Mathematically, we can see if we take the log of both sides:
ln(mg) = ln(kvTn )
(10.1)
ln(m) + ln(g) = ln(k) + n ln(vT )
(10.2)
ln(m) = n ln(vT ) + ln(k/g)
(10.3)
You might notice that if we call ln(m) “y” and ln(vT ) “x”, this is in the form of
y = mx + b. In order to determine which power is more appropriate, you will take your
data for mass and velocity and make a plot of ln m vs. ln vT . In fitting this plot to a
straight line, you will find that the slope n will be equal to the power.
For today, we just need to measure the terminal velocity for a few different masses.
• Disconnect the photogate used for the pendulum and connect the motion detector.
Start Logger Pro by opening ”‘Exp 10, Air Resistance”’ in the Intro Physics Lab
Folder to take data with the motion detector.
• Position the motion detector approximately 2 meters off the ground, facing down.
Place a coffee filter in the palm of your hand and hold it about 0.5 m under the
Motion Detector. Do not hold the filter closer than 0.4 m to the motion detector.
• Click collect to begin data collection. When the Motion Detector begins to click,
release the coffee filter directly below the Motion Detector so that it falls toward
the floor. Move your hand out of the beam of the Motion Detector as quickly as
possible so that only the motion of the filter is recorded on the graph.
• If the motion of the filter was too erratic to get a smooth graph, repeat the measurement. With practice, the filter will fall almost straight down with little sideways
motion.
• The velocity of the coffee filter can be determined from the slope of the distance vs.
time graph. At the start of the graph, there should be a region of increasing slope
(increasing velocity), and then it should become linear. Since the slope of this line
is velocity, the linear portion indicates that the filter was falling with a constant
91
or terminal velocity (vT ) during that time. Drag your mouse pointer to select the
portion of the graph that appears the most linear. Use Logger Pro to find the slope
of the straight line.
• Record the slope in a data table (a velocity in m/s). Repeat the measurement twice
more to verify that the results are typical.
• Repeat these steps for two, three, four, and five coffee filters. Record the data to
be used at home next week.
92
clamp
L
m
photogate
Figure 10.3: Experimental setup for the pendulum experiment.
93
94
Chapter 11
Functions/Air Drag
(At Home activity)
Introduction
This lab is designed to introduce and review concepts important to the analysis of data.
The delivery of this lab is through a special web based tool called LonCapa. This is
similar to Blackboard but with features more inline with solving scientific problems.
Step–by–step Guide
First connect to the web application:
http://lc.cit.jmu.edu/adm/login?domain=jmu
Second login using you standard JMU username and password.
Third choose the course PHYS140L. You should have only a few options. Any one
of your courses can deliver material via the LonCapa system and you will automatically
be given access to those courses but no others.
Fourth be sure you are on the Navigation page (Button near top of the window)
Click on Navigate Contents
Fifth there is a folder called OnlineLab.sequence. Click on this folder to start the
lab.
The material provided consists of
1 basic information,
2 data to be analyzed in Excel,
3 short quizzes to test your knowledge as you read the material.
You must
1 Complete the quizzes
95
2 Hand in an Excel spread sheet based on the assignments requested in the lab.
Other information:
• quizzes will allow multiple tries so you can change your answer if you realize later
in the lab that you made a mistake in your previous attempt(s)
• you can navigate to different sections by returning to the navigation page clicking
on the folder, followed by clicking on the part of interest.
There are 9 parts to the lab and the student should visit each part at least once:
• 4 of the parts require answers either based on material you should master or results
from the requested analysis
• 5 informational pages with assignments
• One Excel spread sheet must be built as you work through the material. It will
contain the following work sheets:
– straight line
– in fit
– non linear
– non linear fit
– logarithms
Lab parts
The lab is designed to introduce concepts briefly. Some might be straightforward others
might be complex. Students are encouraged to supplement the material with other
sources. There is web a page where some additional material can be found:
http://csma31.csm.jmu.edu/physics/Courses/P140L/index.htm
[The link “SUMMARY and GLOSSARY for Fitting lab” is an outline of the lab]
and there are very good references on the web for all topics covered. You can keep
multiple windows open, cut and paste ideas from these sources and from LonCapa into
a file with your accumulated notes. You can share reference material with colleagues.
(Students are expected to submit their work for the assignments and may
not simply copy another student’s work.)
Students are often confused as to what data analysis means and entails. This lab
explores how a set of data can be examined to learn or test an idea. The first step is to
review functional relationships (straight line, polynomials, exponentials). There is a short
discussion as to the ways that the data to be analyzed can be measured. Then we see
what we mean by comparing data to a model (expected behavior). Usually the student
has some basic idea as to how to analyze data that follows a straight line. This is reviewed
96
No
Description
Assignments
1
2
3
info. & Assignments
info. & Assignments
Answers to be
submitted
Answers to be
submitted
info. & Assignments
info. & Assignments
7
Introduction
EXCEL line plot
Find slope and intercept
(2 parts)
Fit line
(2 parts)
Fit line
non linear
plot
Which function
8
non linear fit
9
logarithms
4
5
6
Answers to be
submitted
Answers to be
submitted
info. & Assignments
Table 11.1: Description of activities and assignments for functions/air drag
with some emphasis on thinking about how to judge when a line best matches the data.
A broader option is to compare data to more complicated functions. In making these
comparisons the student is asked to consider how the uncertainty should enter in this
judgment. Also there is freedom to allow the some aspects of what is usually considered
to be the function type to be changed and therefore investigated. On can ask does the
data follow a linear function, quadratic or cubic function? Finally t his question can be
asked using only the straight line analysis tools if one uses logarithms to first transform
the data. Hopefully, the final exercise highlights this powerful technique.
97
98
Chapter 12
Comedy of Errors
(Final Lab Part I)
This lab is the first part of a two part lab. The data obtained in this lab will be used by
you to estimate physical quantities. The actual estimates and write-up of the results are
the focus of the second lab. For this portion of the lab, you will investigate new physical
phenomena that we have not covered yet in class. We will use the techniques we have
been developing to investigate the new area.
Purpose
1. To become familiar with the temperature probe as means of measuring temperature
2. To measure errors carefully
3. To obtain all the data necessary to determine the latent heat values of water for
the processes of fusion and vaporization
4. To investigate new physical phenomena in the laboratory
Materials
For this experiment we will need the following:
1. Calorimeter with outside insulating container and stirrer
2. Boiler with hot plate
3. Balance
4. Warm water
5. Ice
6. Thermometer or Thermometer sensor (Lab Pro, Logger Pro-software)
99
Figure 12.1: Calorimeter Setup
100
Background Theory
When a substance changes state, a certain amount of heat is exchanged between a substance and its surroundings. The amount of heat needed to melt a substance (or to be
removed to freeze the substance) depends both on how much stuff there is (mass) and
as well as what the substance is. Heat of fusion (Lf ) is the term applied to the ratio of
exchanged heat (Qf ) per unit mass m, when a specific substance melts or freezes, or:
Qf
(12.1)
m
Lf has units of calories/gram. If melting occurs, heat is absorbed by the substance
from the surroundings. Freezing on the other hand, implies a reverse process where heat
flows from substance to surroundings.
A similar expression, Lv , heat of vaporization is associated with the liquid to vapor
process, or its opposite; again, the same rationale applies:
Lf =
Lv =
Qv
m
Qv is the exchanged heat.
101
(12.2)
Experimental Determination of Lf
To experimentally determine the heat of fusion of water, one uses a calorimeter container
of a given mass, mc , and specific heat, C. Water of mass mw is poured into an insulating
container. The initial Temperature To of water and container is then measured. Ice with
mass mi is then added to the container where melting takes place. You may assume that
the ice temperature during the melting process remains at 0o C.
The law of conservation of energy may be applied during the above mixing/melting
process. The result is:
Heat gained by mi = Heat lost by mw and mc
(12.3)
This can be expressed as:
mi (Lf + Cw (Tf − 0)) = mw Cw (T0 − Tf ) + mc Cc (To − Tf )
(12.4)
The two terms on the left represent the heat gained by (1) the ice in melting and
(2) the melted ice water going from 0o C to Tf . The two terms on the right side are,
respectively, heat loss by (1) water originally in the calorimeter and (2) the aluminum
calorimeter. The symbols Cw and Cc stand for the specific heats of water and the
aluminum calorimeter, respectively, for which the numerical values are 1.00 and 0.22
cal / (o C-g).
Equation 12.4 readily yields a value for Lf when all other quantities appearing in the
expression have been measured or are given.
Derive an equation for Lf and show your instructor the result before proceeding:
Formula for Lf
Instructor’s checkoff
102
Experimental Determination of Lv
A similar approach to that above is followed in determining the heat of vaporization. In
this instance steam of mass ms from a boiler is directed into the calorimeter where it
mixes with the water already in the container and brings the temperature of the system
from an initial temperature To to a final temperature of Tf . A parallel statement to 12.3
reads:
Heat lost by ms = Heat gained by mw and mc
(12.5)
This can be expressed as:
ms (Lv + Cw (Tbp − Tf )) = mw Cw (Tf − To ) + mc Cc (Tf − To )
(12.6)
The two left hand terms represent, respectively, heat loss by (1) steam in condesation
and (2) condensed steam (as liquid) in changing temperature from Tbp to Tf . It will be
noted that Tbp denotes the boiling point temperature. The above equation yields a value
of Lv where all other quantities are measured or given.
One complication is that the boiling point of water (Tbp ) is actually a function of the
barometric pressure. As the atmospheric pressure varies, then the boiling point of water
will change as well. Table 12.2 gives values of barometric pressures and the corresponding
Tbp .
Derive an equation for Lv and show your instructor the result before proceeding:
Formula for Lv
Instructor’s checkoff
103
Experimental Measurements
Proper Setup of Formulas
As shown above several measurements must be plugged into a complicated formula in
order to compute the final result. There will be a more complete discussion of these
calculations in the Appendices and the next lab. To properly estimate your final error,
you will be required to combine uncertainties. In this lab, we will take the data you will
need to estimate values for Lf and Lv properly including errors.
Take time to setup the entry of your data into a spreadsheet so that you will be able
to calculate uncertainties. The best way is to proceed in steps, not to try to obtain the
final values in a single calculation.
For example, as can be see above, one of the bits of information you will need is
the temperature difference between the final temperature and the original temperature
(Tf - To ) (often written as ∆T). Perform this single calculation and also calculate the
uncertainty in ∆T. Each temperature has an uncertainty associated with it, so how would
you figure out the error in ∆T? Record how you would solve for ∆T below.
In a similar fashion, 12.4 requires you to calculate the error in the product mc Cc ∆T.
Each of the terms (mc , Cc , and ∆T) has an uncertainty associated with them. However,
since they are combined in a product rather than a sum, you use a different rule to
combine them. Record what formula you would use to solve for the error in mc Cc ∆T
below and have your instructor check it. Your instructor will tell you what value to assign
to Cc and what uncertainty you can assign to it.
Because some of the rules for combining uncertainties require fractional uncertainties
and some require absolute uncertainties, it will be convenient to provide columns for both
types of uncertainties. It will also be useful to name the cells so that you can enter the
formulas easily and understand which terms belong to which types of error.
Show your instructor your formulas for ∆T and mc Cc ∆T before proceeding.
Formula to calculate uncertainty in ∆T
Formula to calculate uncertainty in mc Cc ∆T
104
Instructor’s checkoff
Instructor’s checkoff
Let us consider how you might set up the cells for the mass of the water you use in
your experiment - and the error in that mass estimate.
We might set up a table as follows. We have named some of the cells. We also have
provided the excel column name to make them easier to see.
Table 12.1: Example - Mass of Water
excel col A
Comment
Total Mass
Mass of Calorimeter
Mass of Water
B
name
MWC
MC
MW
C
value
55
4
MWC - MC
D
units
g
g
g
E
abs. unc.
2
2
?
F
fractional unc.
=E2/MWC
=E3/MC
=?
G
comment
unc based on measure error
unc based on measure error
unc calculated by formula
The “?” in column E and F above would be the formulas that you would use to
estimate the uncertainty in the mass of the water. In column E, it would be the term
that you would associate with the sum rule for uncertainties. In column F, you might use
the product rule. This is similar to the exercise you did for the ∆T and mc Cc ∆Tabove.
Check with your instructor if you have questions.
The key point - take some time to set up your tables to make sure that you record
the appropriate data and uncertainties!
105
Proper Setup of Experiment
There are two good calibration points for checking and calibrating thermometers for
today’s lab.
• 0 o C - freezing point, or the ice-water equilibrium temperature at atmospheric
pressure of 760 mm of Hg.
• 100 o C - boiling point, or the water-steam equilibrium temperature at atmospheric
pressure of 760 mm of Hg. Try not to let the bottom of the thermometer
touch the bottom of the boiler. Be careful - you can easily burn your
hand on the boiler or other hot pieces of equipment.
To calibrate the temperature sensor, you will need to do the following:
• Startup Logger Pro
• Load setup heat file
• Check the setup to ensure the configuration is sensible (rate, duration)
• Calibrate
To calibrate the thermometer, you will need to do the following:
• Use care as thermometers break easily
• Examine the scales and make sure that you can read them properly
• Check that the thermometer reads correctly at 0 o C and 100 o C.
In addition to the thermometer, we will use a device called a calorimeter. A calorimeter is an insulated container designed to minimize thermal transfer between the experiment and the outside world. A diagram of a calorimeter and its components is given in
12.1.
106
Watching water boil
Check with your instructor about these two steps. It is often a good idea to start the
water in the boiler going so that the experiment will be ready to go.
Step 1 - Look at 12.2 and make sure that you know where the tube from the boiler will
plug into the calorimeter. What will happen is that you will heat the boiler, generating
steam. The steam will be conducted by the hose into the calorimeter, condensing into
water, and heating the water existing in the container. To make the experiment as
simple as possible, we will not attach the hose until the boiler is producing steam. Before
proceeding, check with your instructor if you have questions.
Step 2 - Heat the water in the boiler, allowing it to come to a boil. While waiting
for the water to boil, place the open end of the tube into a beaker (so that no one is hit
by steam). Make sure that the tube is positioned as in Figure 2 so that steam does not
condense inside the tube. We are starting this step early to make sure that the water
in the boiler is ready when you start your heat of vaporization experiment. While the
water is heating up, you can proceed to the Heat of Fusion experiment below.
107
Figure 12.2: Calorimeter with boiler
108
Heat of Fusion Procedure
You may find it useful to take a picture or two of the equipment (or make a sketch) of
your experiments, which will be helpful for next week’s lab.
Step 1- Measure the mass of the empty inner calorimeter and stirrer. Record this
value on the data table as mc . Estimate the uncertainty in this measurement.
Step 2 - Record the room temperature as Tr . Add water at about 10 o C above room
temperature to your inner calorimeter container so that the container is filled to about
60% of capacity. You can easily decrease or increase the water temperature by adding a
little bit of cold or hot water. Be sure to stir well so that all the water is at the same
temperature.
Step 3 - Measure the mass of the inner container (with stirrer) with the water. Record
this as mc+w . Record an uncertainty with that measurement. This can be used with the
results of step 1 to estimate the mass of the water you are using.
Step 4 - Place the inner container within the outer insulated container as shown in
Figure 1. Place the lid on the container so that the stirrer handle is sticking out and
place the stopper and temperature probe in the large hole in the top of the container.
The probe should be positioned so that the tip of the probe (thermometer) is between 1
and 2 cm below the surface of the water surface. Record the temperature of the water
(and of course an error estimate). Wait until you get a few stable readings.
Step 5 - Remove an ice cube from the insulated container on the side table, holding
it in a paper towel. Place the cube in the inner calorimeter container, replace the lid of
the calorimeter and gently start stirring the water until the ice has completely melted.
Monitor the temperature of the water until it reaches its lowest value. Record this as
Tf (along with an error estimate).
Step 6 - Measure the mass of inner container (including the stirrer) with the water.
Record this as mc+w+i. Record an uncertainty with this measurement. You can use this
value of the total mass and the values from previous steps to find the mass of the (now
melted) ice, mi .
At this point, you have all the experimental data and uncertainties that you will need
to estimate Lf . We will do that in the next lab, where you will do a complete lab write-up
of this experiment.
109
Heat of Vaporization
Note - it is assumed that for this part you will record all the uncertainties as you are
going along. Again, a sketch or photo of your setup will be useful as you proceed.
Step 1 - Refill your container with cool water and record the mass as in the previous
experiment.
Step 2 - As before, measure the starting temperature of the water (and record as To ).
It should be at most 10 o C above room temperature. It can be cooler.
Step 3 - If you have not already done so, look at Figure 2 and make sure that you
know where the tube from the boiler will plug into the calorimeter. What will happen
is that you will heat the boiler, generating steam. The steam will be conducted by the
hose into the calorimeter, condensing into water, and heating the water existing in the
container. To make the experiment as simple as possible, we will not attach the hose
until the boiler is producing steam. Before proceeding, check with your instructor if you
have questions.
Step 4 - If you have not already done so, heat the water in the boiler, allowing it to
come to a boil. While waiting for the water to boil, place the open end of the tube into a
beaker (so that no one is hit by steam). Wait at least one minute after the steam starts
coming out of the tube before plugging it into the 1 cm hole in the calorimeter. Make
sure that the tube is positioned as in Figure 2 so that steam does not condense inside
the tube.
Step 5 - As the steam is entering the calorimeter, gently stir the water and monitor the
temperature. When the temperature reads about 15 o C above room temperature (and
several o C above your starting temperature, remove the tube and continue to monitor
the temperature until it peaks. Record this value as Tf .
Step 6 - Remove and weigh the inner container. Record this value as mc+w+s . This
value, and your previous measurements can be used to calcuate the mass of the (nowcondensed) steam.
Step 7 - Read the current barometric pressure from the room barometer and record
this in your data table. With this value use Table 12.2 to find the boiling point of water
in the lab and record this as Tbp .
110
Table 12.2: Barometric Pressure vs Boiling Temperature of Water
P
Temp
o
C
mm of Hg
682
97.0
692
97.6
707
98.0
718
98.4
728
98.8
739
99.2
749
99.6
760
100.0
771
100.4
782
100.8
793
101.2
805
101.6
P
Temp
o
C
mm of Hg
687
97.2
702
97.8
712
98.2
723
98.6
733
99.0
744
99.4
755
99.8
765
100.2
776
100.6
788
101.0
799
101.4
810
101.8
Before Leaving
• Make sure that you show your formulas for Lf and Lv to the instructor.
• Make sure that you show your uncertainty formulas to the instructor. You will be
expanding those formulas in the next lab, so any questions you have, ask now!
• Make sure that you show your lab data tables to the instructor.
• Make sure that you and your lab partner EACH have a copy of the lab data tables.
• Make sure that you each have a copy of any photos or sketches that you might have
made.
111
112
Chapter 13
Tale of Woe
(Final Lab Part II)
This lab is the second part of a two part lab. The data obtained in the first part of
the lab will be used by you to estimate physical quantities. The actual estimates and
write-up of the results are the focus of this lab. In this lab, you will draw together the
error analysis techniques you have practiced to estimate the latent heat of vaporization
and fusion for water. Each student will write up their own lab report to hand in.
Purpose
1. To use our experimental data to estimate latent heat of fusion and vaporization
2. To estimate our uncertainty in the various physical quantities
3. To write up a report describing our experiment and analysis
4. To investigate new physical phenomena in the laboratory
Materials
For this experiment we will need the following:
1. Copy of our data tables from 13.
Review from Last Lab
The term latent heat refers to the energy exchanged between a unit mass and its surroundings as it changes state (solid to liquid or gas to liquid). Different substances
will have different latent heats, and the amount of latent heat will be different when a
substance melts/freezes compared to when it vaporizes/condenses.
In the last lab, you explored what happens as material is melted (ice turning to water)
or condensed (steam turning to water) though the use of a calorimeter. You carefully
113
measured a number of quantities (and estimated associated uncertainties) for the two
experiments you did. One experiment investigated the latent heat of fusion (ice and
water) and the other explored the latent heat of vaporization (steam and water).
At the end of the last lab, we had all the information we needed to actually estimate
the latent heat of fusion Lf and latent heat of vaporization Lv .
Estimating Lf and Lv and estimating their Uncertainties
The expressions for Lf and Lv trial are given in chapter 12. As part of that lab, you
solved for Lf and Lv . Use those formulas to set up expressions and solve for Lf and Lv .
Remember that you had two or more trials, so solve for Lf and Lv for each trial you did.
In addition to the value of Lf and Lv , you also need to evaluate the uncertainty in
the estimate. As you realized in the previous lab, the uncertainty is not quite as straightforward as some earlier labs. The expression for Lf (and Lv ) involves both sums and
products. To properly evaluate the uncertainties, you will need to break your expression
down into smaller parts.
For example, suppose we were interested in the errors associated with an expression
such as
GM1 M2 GM1 M3
+
(13.1)
F =
r2
d2
where we knew G, M1 , M2 , M3 , r and d, as well as the uncertainties associated with each
of the terms. To find the uncertainty in F, we would have to find the uncertainties with
term1 and term 2 where they are defined as:
term1 =
GM1 M2
r2
(13.2)
and
GM1 M3
(13.3)
d2
Finally there is the total uncertainty attributed to summing the two term’s uncertainties together. Only then do you have an estimate of the uncertainty in F.
So, you will need to write out your expressions for the uncertainties in Lf and Lv
broken down into smaller terms, and then sum those terms together if you are to estimate
the uncertainty in Lf and Lv properly. If you need an example of how to get started,
look at the example for Lv in Appendix 3. Note that for your error analysis, you should
use the techniques discussed by your instructor.
Record in your spreadsheet the final uncertainty estimate for each trial of Lv and Lf .
term2 =
Comparing Lf and Lv with previously measured values
This laboratory experiment is tricky, as there are a lot of ways that your data might
have been affected. It is common for many experimenters to find values of Lf and Lv
significantly different from the ”accepted” values of Lf = 80 cal/g and Lv = 539 cal/g.
114
As experimenters we are not interested in “matching” the “accepted” result. Instead, we
are interested in understanding if our results are consistent within our estimates of the
uncertainties.
So, take your data and the corresponding uncertainties and start comparing...
Step One
Are your values for Lf (remember you had at least two trials in the last lab) consistent
with each other within the errors? What about Lv ?
For example, suppose I had measured g twice in an experiment. The first time I got
9.01 m/s2 with an uncertainty of ±0.62 m/s2 . The second trial gave a result of 8.72 m/s2
with an uncertainty of ±0.53 m/s2 . The two trials are consistent within the estimated
uncertainties.
Step Two
Are your values for Lf and Lv consistent with the established values?
In the case above, my two values of g (9.01±0.62 m/s2 and 8.72±0.53m/s2 are more
than one ”sigma” away from the accepted value of 9.80 m/s2 . This does not mean that
my trials were wrong! It probably means that there were sources of uncertainty in my
experiment that I did not account for properly or completely. If one can identify those,
one can improve the experiment for the next time.
A more detailed discussion of how you can interpret “sigma” values is presented in
Appendix 3 in the section comparing theoretical and experimental values. As that section
illustrates, a useful way to present your data when comparing with other experimental
values or a theoretical estimate is to plot your data (with 1 sigma error bars) against the
other estimates. That allows you and your reader to quickly assess how well the values
are in agreement.
Under no circumstances should you ”fix” the numbers so that your values of Lf and
Lv match the established values. Your results are valid for your experiment. Nor should
you ”bump” up the uncertainties to make your values consistent within the uncertainties.
Your estimates of the uncertainties in your experiment were what you estimated at the
time to be reasonable. In the future, you can try and think of ways to improve your
techniques or run tests of your assumptions.
Step Three
Rather than focus on how well your experimental result matches (or misses), take a look
at your various error terms. Identify the three largest sources of uncertainties in your
experiment. The uncertainty might be in terms of absolute
For example, in my gravity case, maybe the measurement of time had a large uncertainty percentage wise (i.e. I had errors of 6% on my time estimates). Or perhaps I had
an uncertainty of 0.1m on my distance measurements. In some cases, a large uncertainty
in absolute terms will contribute a lot. In other cases, the uncertainty will be dominated
by the relative amount of uncertainty (i.e. fractional or percentage uncertainty). Since
115
the estimates of Lf and Lv involve several different quantities, it is important to look and
see if you can identify which terms contribute the most to your overall uncertainty.
The Lab Write-up - what makes up a lab report?
This lab, in addition, to providing you a chance to work through a more complex case
of error analysis than you have done before, will also serve as way to communicate your
results to other people. Each lab partner will write their own version of the lab report.
Your audience should be another PHYS 140L student who has not seen the lab
before. Therefore you will have to describe things clearly, so that the student could go
and reproduce your lab if they needed to.
A good lab report has the following parts:
• Title Page - Includes a title, your name, names of your lab partner, section
number, and your instructor’s name. It should also include a brief (one or two
sentence) statement that describes the purpose of the experiment. You can consider
the statement an abstract of what your lab (done in 12).
• Introduction - Include a discussion of the physics and the formulas that you are
studying in the lab. Be sure you state what aspects of physics you were investigating
and how this physics will clarify your overall purpose. (Maximum 1 page of text).
• Procedure - Use a bullet or list style to write this section. There must be a diagram
for all your apparatus. This can be a simple block pencil sketch or something
more elaborate. If you took a photo of your setup during the lab, you could use
that. Label important pieces of equipment in your diagram. If you carefully show
important aspects of the setup, you can avoid extensive discussion in the report.
You can not just copy and paste the lab description from the previous lab. It has
to be in your own words - because it was your experiment.
Remember to include a description of how you calibrated your instruments. Describe briefly how you checked that your measurements were being done correctly.
(Maximum 2 pages of text).
• Data - You should present a table containing some of the measured quantities and
the associated uncertainties in the report. You should also make available the excel
spreadsheet with the complete data set so that the instructor can examine the data.
(No more than one page of text).
• Analysis and Results - Here you will provide a brief description of your analysis.
The discussion should include sample calculations. (No more than two pages). In
addition, you will present a summary of your results of the analysis. You might
include a table containing your final values for Lf and Lv . This is where you
can discuss how various uncertainties impacted your overall result for Lf and Lv .
Remember to keep proper track of units and of significant figures. (No more that
two pages of text). Note that you can split the analysis and results into two subsections if you wish.
116
• Conclusion - You present a brief summary that informs the reader of what you
investigated and how well the experiment succeeded in performing the investigation. Note that does not mean how well you matched the “established values” but
whether you identified and kept track of uncertainties. You can compare your results with the established values and suggest areas that would be worth exploring
in future experiments. This section should be short and to the point - the reader
should be clear on what you accomplished. (No more than one page of text).
You can review with your lab partner the details of your experiment and procedures.
However, your write-up and analysis should be your own. This is why you should have
copies of all the sketches, photos, and data that you and your lab partner took the
previous week.
You will write up a lab report that will include proper accounting for uncertainties
and will include all of the sections listed above. It will be due on your laboratory meeting
time. Your instructor will let you know if they prefer hardcopy or electronic submission.
117
General Advice
When writing, remember that what might be crystal clear to you at the conclusion of an
experiment is not clear to your reader. In general, it should be written so that a good
student or an instructor from any of the other lab sections could read and understand your
report. A classmate would make a good proofreader. Be sure that you draw attention in
your write up to all the important points and link all the pieces of the lab.
Some general comments:
• Don’t get things wrong - proofread any formulas, read what you are saying. For
example, if you start talking about boiling water and adding the resultant ice to
the calorimeter, you can expect to lose points.
• If you include tables, graphs, calculations, etc., then explain why you are including
them. Also explain what conclusions the reader should take away from the included
material.
• Plots and diagrams are very effective ways to communicate information but if you
expect your plot to illustrate a certain relationship, don’t assume the reader will
make the connection. Tell the reader what conclusion (or information) they should
draw from the plot.
• Don’t include all the raw data - just a sample is fine. But you will need to make
your overall data available to the instructor.
• Don’t make complicated arguments you don’t understand. I.e. Re-read the argument and if it is confusing to you, it will be confusing for your reader as well!
• Avoid hand-waving arguments.
118
Appendices
1.
2.
3.
4.
Curve Fitting
Excel Spreadsheet
Establishing Uncertainty
Suggestions for Data Handling
119
APPENDIX 1: FITTING DATA
There are several methods that one can use to find a function that passes through a set of data
points thereby revealing a mathematical relationship. To perform a fit the experimenter must
choose a functional form.
Functional form defines the mathematical relationship between the dependent variable (y) and
independent variable (x). The form usually contains parameters whose values must be chosen to
fix the relationship. Examples of some functional forms follow.
A
Line:
Exponential:
y = mx + b
Polynomial
y = ax 3 + bx 2 + cx + d
y = Ao e
Parameters: m, b
Parameters: Ao, λ
λx
Parameters: a, b, c, d
computer program usually changes the parameters of interest in some pattern that is designed to
find the best values for the parameters in an efficient way. The y-values calculated with the fit
function are compared to the data y-values. The quality of the fit is judged by the difference.
The method employed to search for the best parameters is unimportant as long as a good fit is
found. Consider the following function
F (t ) = A e − B t + C t + D
There are four parameters, A, B, C, and D. Choosing different values for these parameters
results in a different lines as shown in the graph below. Both lines represent the same functional
form F(t) but with different values for the parameters.
F(t)=Aexp(Bt) +Ct + D
700
600
F(t)
500
400
300
200
100
0
0
20
40
60
80
100
120
time t
120
Neither function passes through the experimental values, shown as triangles with error bars. A
fitting program keeps changing the parameter values and testing if the new line passes through
the data points. When the line passes sufficiently close to the data (y-values from fit are close to
the y-values of the data) the fitting program returns the values of the parameters. A good fitting
routine can vary the parameters again to see how much a parameter can change while still
passing through the error bars. This allows the routine to establish an uncertainty (range of
possible values) for each parameter. In the laboratory, data analysis almost always requires
both a value and an uncertainty. The fitting routines DATAFIT, Logger Pro and GRAPHICAL
ANALYSIS provide values and uncertainties. Student therefore need to be able to pass data to
one of these fitting programs, run the fit and retrieve parameters and uncertainties. If one uses a
routine that doesn’t provide parameter uncertainties then an alternativemethod to determine
these uncertainties is required.
Trendline: Excel provides trend lines for charts. These lines are made to pass through the data.
The parameters can be viewed by displaying the trend line function on the chart. The
disadvantage of this method is that it doesn't indicate the uncertainty in the parameters.
Finding Uncertainty (repeated trials method): An uncertainty can be determined (for a trend
line analysis) by measuring more than one data set. Trend lines can be placed on each of the
different data sets and the parameter values from each dataset (e.g. slope of a straight line) can
be put into a table and compared using the SD to estimate the uncertainty in the fitted parameter
(e.g. slope). This method requires the experimenter to repeat the experiment so that
independent datasets are compared. This method can be used to estimate an uncertainty for any
fit method. As mentioned above, routines such as DataFit provide uncertainties based on one
data set. The two methods should agree.
DataFit: A separate program, DataFit, is one of the best tools for general fitting.
ƒ Start program using the DataFit icon.
ƒ Enter the number of independent variables (usually 1).
ƒ Decide if you want to have a column for y uncertainties (standard deviation column) or no
column (usually no column).
ƒ Hit OK.
ƒ Paste the data to the data window.
ƒ Choose regression under the solve menu.
ƒ Choose nonlinear.
ƒ Choose the functional form from among the options or provide a custom function.
ƒ If the fit is successful the results can be obtained by choosing - detailed…- in the results
menu. Scroll down until you find the table Regression Variable Results use Value and
Standard Error for each parameter.
The results contain the parameters and their uncertainties (standard error) as well as a host of
plots and other indicators.
Ask your instructor to show you the procedure. While your instructor is demonstrating, add
your own comments to the above procedure so that you can perform a fit on your own.
121
Graphical Analysis: This package is supplied as part of the data collection and analysis tools
from Vernier Software. This package allows the experimenter to enter or import data, to plot,
calculate, graph and fit data. It has a complete set of tools so that a full analysis can be
performed. It provides text boxes for comments, and graphs with sophisticated display options.
Graphical analysis is a fairly complete, additional spread sheet which is available for student
use. Copies of the software are available for installation on your home computer. Ask your
instructor.
Start Graphical Analysis by double clicking on the GA icon. Open a new analysis by choosing
new under the file menu. Import or cut & paste data to the table window. Use the toolbar
“Curve Fit” button or choose curve fit from the analyze menu. Choose the data to fit in the
window that appears (y-column). Choose the functional form and click the “try fit” button.
Complete the process by using the “OK” button. A window should appear on the graph showing
the parameters and the associated uncertainties. The root mean square error, RMSE, is also
given.
Vernier Tech Info Library TIL # 1014 (from website)
MSE: Mean Square Error, for every data point, you take the distance vertically from the
point to the corresponding point on the curve fit and square the value. Then you add up
all those values for all data points, and divide by the number of points. The squaring is
done so negative values do not cancel positive values. The smaller the Mean Squared
Error, the closer the fit is to the data.
RMSE: Root Mean Squared Error is just the square root of the mean square error. That
is probably the most easily interpreted statistic, since it has the same units as the quantity
plotted on the y axis. The RMSE is thus the distance, on average, of a data point from
the fitted line, measured along a vertical line.
LoggerPro: Logger Pro, also provided by Vernier, has fitting functions available. These come
in handy when recorded data needs to be fit quickly. Logger Pro does provide an estimate of the
uncertainty for the fitted parameters.
• Highlight a section of data with the mouse.
• From “Analyze” menu choose “Curve fit”.
• Highlight a function.
• Hit “Try Fit” button.
• Hit “OK”
• The results appear on the graph.
If you do not see the parameter uncertainties in the fitting summary dialog box then right click
on the dialog box to obtain get the options and check the appropriate boxes.
There are several interesting options available. You can vary the parameters and see how the
function changes. You can define additional functions. Also see Logger Pro help files.
122
The following functions are useful for the restricted case of a straight-line relationship
between the dependent and the independent variable.
Linest: The function LINEST returns the slope and intercept data from a straight line LSQ fit.
Since there are several values returned you must:
• Enter the function LINEST(y range, x range, 1, 1) into a cell.
• Select a range of cells (2 cells across, 5 cells down) that include this formula in the
upper left corner.
• Type F2 function key followed by “Cntl-Shift-Enter”. (this is Excel’s array entry)
• The slope and intercept values are in the first row of this 2 x 5 array
• The uncertainties for the slope and intercept are in the second row of this array.
Regression Analysis: This is a just a fancy name for straight-line fitting. It assumes that the
relationship between the variables is linear. It therefore can be used to find the best straight line
that passes through as set of (x,y) pairs. The regression function returns a full set of quantities
that can be used to describe the quality of the fit. It also provides estimates of the uncertainty in
the slope and intercept. To perform a regression in excel:
ƒ Use Data Analysis item in the tools menu.
ƒ Choose regression.
ƒ Enter the y and x values.
ƒ Hit OK.
ƒ View the sheet with the results.
ƒ The intercept and uncertainty are tabulated.
ƒ A value for the slope (x variable 1) and an associated uncertainty for this value are also
tabulated.
123
Appendix 2: Working with Excel
This appendix will cover ways that Excel can be used to display and analyze data. Since this is a
major component of the lab, the student is encouraged to take notes and document in his/her
own words how a method or tool can be used. Example spreadsheets can also be stored on the
network for future reference. This is a brief guide. Excel has many features and a host of
methods to accomplish the same result. If you know a method that differs from the one
contained in this guide then you may use it and share it with your colleagues. As with most
applications the student should explore beyond the specific lab instructions until the method is
well understood. Your instructor should be able to help clarify. Once you understand the
concept or method add details and summarize in your own words for future reference.
WHAT IS A SPREADSHEET
Spreadsheets store data in tables. Excel refers to each table as a WORKSHEET. One can
change to a different worksheet using the tabs along the bottom (sheet1, sheet2). A specific
location in a table (worksheet) is called a CELL. A letter and a number, for example A3,
identify a cell. “A” identifies the column. “3” identifies the row. To choose a cell, move the
cursor to the cell location and click the mouse. The contents of the cell are shown in the cell
and in more detail in a space at the top (part of a toolbar). The cell may be empty. The cell ID
(e.g. A3) also appears at top in the tool bar. Cells contain data of all types: numbers, dates,
labels, formulas, and functions. The data in a cell can be displayed in many formats: date
formats, percent, dollars, integer, and others.
ENTERING DATA
Choose a cell. You may then enter numbers or text from the keyboard. You can edit the cell’s
data either in the cell or in the location provided as part of the toolbars. Use your mouse to
choose the cell and the data entry or edit point for the cell chosen.
MOUSE
The mouse is a powerful tool in the Excel environment. The normal left-click is used to choose
cells and locations (data entry windows). The left-click normally is also used to hit buttons
(cancel, ok). Holding the left button down allows you to select a range of cells. The right button
often reveals advanced features in a pull-down menu format. This is very useful when working
with plots and graphs. Often the original plot needs updates and the mouse can be used to select
a region of a plot (e.g. title or data) and then pull-up options for that region (format title, change
the data source).
SAVING DATA
If the work will be important for future experiments you should save the file to the network or a
floppy disk. If you want to safeguard current work you can save the file on the local computer
(Data erased every Sunday). Your instructor will help students create a temporary area to save
files and show you how to save the spreadsheet in this area using the "Save As" menu item
under the "File" menu. Once you have saved the work in a file with a unique name you can
periodically use the "Save" item in the "File" menu to update your work. It is good practice to
periodically store your work. This enables you to recover from mistakes made while using the
spreadsheet and computer problems.
124
NAMING CELLS
Names will simplify the use of Excel equations. Choose "Name" under the "Insert" menu.
Choose "Define..." in the list. A dialog box appears. Enter "chosen name" at the top of the
dialog box. (Note: If there is a name in an adjacent cell Excel will use this name by default.)
The cell that is being named appears at the bottom of the dialog box. The cell name will include
the worksheet name and have $ characters added. If this is the correct cell simply click the OKbutton. (Note: Excel defaults to the current cell location.) If you want to chose a different cell
then click the button in the bottom right-hand corner (RED arrow button) of the dialog box.
This lets you select which cell or range of cells will be named. You will need to complete the
cell selection with the ENTER key. The dialog box will disappear while you are choosing the
cells and reappear when ENTER is hit.
Test the process. Name a cell. Choose a new cell. Choose "Name" under the "Insert" menu.
Now choose "Paste". A dialog box appears with the names of all the named cells. Choose one of
your named cells and then hit OK. The new cell now has a formula that refers to the named cell.
Hit enter. The cell contents should now be the same as the named cell. Change the value in the
named cell and the new value appears in both cells.
ENTERING FORMULAS
A cell’s contents are interpreted as a formula if the first character is an equal sign. A formula
can refer to another cell by call name or by naming the cell as described above.
=B3
=vo
set the cell’s contents to whatever is in cell B3
set the cell’s content to the cell named vo. (If no cell has been named vo the error
message #NAME? appears.)
Common math operations can be used in formulas
*
multiplication
subtraction
^
raise to the power
/
divide
()
sets order of operations
=36*B3+B4+7
multiply the contents of cell B3 by 36 and add the contents of B4
and the value 7.
There are a many special functions that can be used within a formula. Choose a cell and “insert”
(on toolbar) “function” (on this menu). Choose a familiar function from the dialog box. When
you have chosen your function hit OK. A new dialog box appears. This will aid in getting the
arguments needed. The dialog box is similar to the one used in naming cells. The RED arrow
button returns you to the spreadsheet so you can choose cells. The ENTER key completes the
selection. If you chose the function =sum( ) then you need to supply a list of cells to sum. If
you chose =sin( ) then you need to supply an angle. The arguments of functions can be other
cells.
=sin(B3)
takes the sine of the value in cell B3.
=sum(B4:B8) sums the range of cells from B4->B8.
125
Functions can be typed directly from the keyboard and the arguments for formulas and
functions can be supplied by choosing cells with the mouse.
PLOTTING
Graphs or plots are powerful ways to visualize and analyze data. These tools will be used
frequently in the lab.
To plot data decide which columns should be plotted.
1. Choose "Insert" in the menu and then "Chart".
2. Choose "XY (Scatter)" on the first dialog window. Click "Next >".
3. Choose the data to plot by switching from "Data Range" to "Series" using the folder tab near
the top of the dialog window.
a. Click "Add" to get your first data series.
b. Click the button at the right of the "X Values:" entry window. The dialog box disappears
and you highlight the cells in the time column. Hit Enter after the box shows that all the
desired times are selected.
4. Now use the "Y Values:" entry window and choose your position data.
5. Click finish.
There are a number of refinements available for improving the graph. Labels, colors, or
additional data can be added or changed. Experiment by clicking with either the left or right
mouse button on various portions of the graph and seeing what you can change. You will need
to explore the various options to become proficient at plotting data.
A sample plot is shown below. The excel datasheet that generated this plot can be found in the
desktop folder
==
Intro Physics Lab/excel worksheets/ 2CurvesOn1Chart.xls
The worksheet describes some methods for creating charts (plots). Students may open up this
folder and experiment with plotting.
126
time (s)
1
2
0
0
-20
-1
v (m/s)
1
x (m)
2
20
1
-1
-2
1.5
40
0.5
position (m)
velocity (m/s)
-40
0
TRENDLINE
To add a trendline you click on the graph on a data point. Right click to bring up a menu.
(Choosing different sections of the graph will cause different menus to appear.) Choose "add
trendline". Put the equation on the graph by setting the appropriate option on the trendline
options page.
127
APPENDIX 3: Establishing Uncertainty
Every number used in the laboratory must be recorded with an uncertainty.
This appendix will discuss methods for obtaining that uncertainty. This section should be used
in conjunction with the Error Analysis section, which contains the definitions, formulas and
some additional examples.
Error and uncertainty have different meanings.
• Error is the difference between a value and its correct value or true value. The true
value, of course, is not known.
• Uncertainty is an estimate of the difference between a calculated or measured value and
the true value.
An instrument measures values that are in error by a certain amount. Since the exact true value
of a quantity is unknown, instrumental error is also unknown. The experimenter is forced to
estimate and to judge the estimation. The measured value is an estimate for the true value and
the uncertainty is an estimate for the error.
It will be important to understand what is an acceptable difference between two results. How
much does one allow results to differ before the experiment is judged to have a serious flaw?
The uncertainty is used to compare two values. It provides insight, when comparing two
consecutive measurements, when comparing results with the theory, and when comparing two
independent experiments. Experimenters know that their uncertainty is a safety net. Because
you have an uncertainty, your results cover a range of possible values. Large uncertainties make
an experiment very defensible. The result cannot be called into question if the uncertainty is so
large that all reasonable results are included. On the other hand, an extremely small uncertainty
is a sign of a high quality experiment, the smaller the uncertainty the better the experiment.
This means that the optimal uncertainty has to balance these two opposing goals. The
uncertainty should not be so small as to guarantee failure, nor so large that the experiment has
no merit. It is, of course, unethical to arbitrarily increase or decrease an uncertainty without
justification. Knowing that the uncertainty you claim determines the correctness and quality of
the result under all future scrutiny, many experimenters spend considerable effort searching for
unknown sources of error (increasing uncertainty) and pushing the limits of a technique
(minimizing uncertainty ).
COMMON SENSE UNCERTAINTY
There are different approaches for establishing uncertainty but in this lab the focus will be on
common sense rather than rigorous mathematical analysis. With this laboratory course focusing
on different aspects of the experimental process, some of the rigor that is required for a real
experiment will be relaxed. Often a student can use a simple straightforward method for
guessing an uncertainty. Be sure to check with you lab instructor if you are not sure of your
method.
• The scale on the ruler can be read to about 0.5mm. The uncertainty can be estimated
based on this limitation to be 0.5 mm.
128
•
•
•
•
•
A few independent measurements (3 trials) can be used to calculate SD and SDM (see
below). One of these two values could serve as the uncertainty. The discussion below
should help you decide which one to use in any given situation.
Some instruments may show no indication of error in their measurement. In an actual
experiment a separate measurement could be used to determine the uncertainty or the
experimenter could consult the instrument manual. In this lab the student may conclude
that the instrument contributes a negligible error. This is true, for example for some
voltage measurements.
Instrument uncertainty is sometimes given in a manual.
Tex book constants are typically good to 3 significant figures.
Your instructor may prefer to provide the uncertainty for some quantities.
Human Error, Hand waving arguments– Student is not allowed to introduce an uncertainty
unless the student has developed a method to measure that uncertainty. There will be no hand
waving arguments permitted. If you believe that your experiment may be subject to errors that
have not been included then you either develop a method to measure the uncertainty or you
ignore it.
Absolutely no error can be introduced into any discussion unless some quantitative
estimate can be made for its size and all estimates need to be justified.
The correct approach is to find a way to estimate these additional uncertainties, include them
and reevaluate or to simply state that the experiment does not agree with theory within the
uncertainty.
Not allowed:
• The results are in agreement with predictions because in addition to the uncertainty
measured there were some effects due to wind resistance.
• We suspect that human error and an uneven table are the source of the difference
between our result and the theoretical result.
Allowed:
• By measuring for a longer time period we were able to see a loss in energy over many
oscillations due to friction. As shown in the figure this resulted in a 2% change in the
energy for one period. …
• Comparing the measurements of different students we were able to see an average
deviation of 3 mm, which we attribute to a differences in reaction times. We therefore
are factoring in a 3mm uncertainty in our analysis. …
• This experiment includes all of the measurable uncertainties that we found. The result
for g, however, differs significantly from the accepted value (4 times the uncertainty).
Re-examining possible pitfalls and carefully re-measuring g did not change our result.
You will be expected to include all the important sources of errors but you cannot merely state
that something might be a source of error. You need to provide a justifiable guess as to how
large it is. If you decide wind resistance might have influenced your measurement then, in order
to mention it, you must think of a way to figure out how large an influence it is. If you cannot
129
find all the errors in your experiment, you may have to conclude that your experiment failed to
demonstrate the principle the lab was exploring. It is okay to have a failed experiment as long
as another experimenter using the same equipment would get the same result. There are some
sources of error that are to complex or subtle to be discovered in an introductory lab. There may
be instruments that are not calibrated and cannot be tested by the student. Materials and
components may be flawed in ways that are undetectable. It is advisable to do a dry run and
perform calculations immediately to see that things are going as planned but sometimes even
well designed experiments fail. If your experiment is unsuccessful and there is no obvious flaw
then you should receive a good grade. Naturally your instructor will try and see why you failed.
Your report may therefore require a more complete description and may be more difficult to
write.
Unknown – There will be times when estimating the uncertainty is not critical to the particular
laboratory procedure. The student should still include the uncertainty as unknown. You should
check with your instructor to see if this appropriate.
Constants – Values from the textbook are usually given to 3 significant figures. You can use
this rule for most of the constants used in the lab. The value of g, 9.80 m/s2, should be assigned
an uncertainty of 0.01 m/s2.
MEAN - STANDARD DEVIATION -STANDARD DEVIATION OF THE MEAN
These are called statistics. They are functions of the data points that can be used by the
experimentalist as an estimate for a quantity of interest. Usually the mean is a good estimate of
the quantity measured. Most students already understand that averages can be superior to a
single measurement. The underlying assumption is that the measurements are random. Some
data points are greater and some less than the true value. Averaging tends to cancel these
fluctuations. Be sure that you are comfortable with the notion that the mean of these data
qualifies as a good estimator for the result.
The standard deviation will be used as one estimate of uncertainty. The interpretation of the SD
is a more subtle point. Since the SD is the average deviation of the measurements from their
central value, one expects the SD could be used to estimate the uncertainty in a typical
measurement. If 10 measurements of the same mass are on average 6 gm from their central
value then assigning an uncertainty of 6 gm to each measurement is reasonable. The SD is then
assigned as the error for each of the 10 measurements. Those close and those far from the mean
are given the same value for their uncertainty (6 gm).
To summarize:
MEAN is an estimate for the true value of the quantity being measured.
STANDARD DEVIATION is an estimate for the error in any ONE of the measurements
averaged.
The averaging process that provides the mean value often reduces the actual error. As
mentioned above the mean is superior to a single measurement because of the cancellation due
130
to the averaging. One can go further and state that averages are better when more values are
used. Let N be the number of individual measurements. As N increases the average value
improves as an estimate of the true value. If this point doesn’t seem obvious accept it for now
and we will explore it later. This leads to the conclusion that the mean is closer to the true value
than the standard deviation may suggest and that the uncertainty of the mean should depend on
N. In fact another statistic, the standard deviation of the mean, SDM, is usually used to estimate
the error if the mean is used as the estimate of the true value rather than SD the uncertainty in
one of the individual measurements.
STANDARD DEVIATION OF THE MEAN is an estimate of the error associated with
using the mean as an estimate for the true value.
A discussion of the mean, SD and SDM must include the limitations of these statistics as
estimators. It is probably apparent that one cannot improve a measurement by simply recording
and averaging more and more data. The limits arise due to a second type of error. These errors
are called systematic errors. They are different from random errors because they influence
each measurement in the same manner. A ruler that is too short is an example of a systematic
error and such a ruler will measure all values to be short. Averaging cannot correct for this
error.
SYSTEMATIC errors cannot be reduced by averaging and they limit the extent to which
averaging data can be used to reduce experimental error.
When an experimenter judges, based on an evaluation of the experiment, that systematic errors
could be a significant then the SD should be used so that one doesn’t underestimate the error.
SD can be used as an overall estimate of the error (uncertainty) when the student suspects
that there are systematic limits.
An actual experiment must explore the extent of both types of error and develop methods to
evaluate both of these errors. The introductory physics labs do not always require this level of
thoroughness.
More on Measurement Differences and Uncertainty
An expression can be constructed for the likelihood of obtaining a certain result given the true
value TV and an uncertainty σ. This expression is often a gaussian function. For illustration let
us assume that you know the length of a field is exactly 50m (TV) and when measuring the
length of the field the average uncertainty is 10m (σ). The gaussian function would be
⎛ 1 ⎞
Pr = ⎜
⎟e
⎝ σ 2π ⎠
⎛ x − TV ⎞
−1 ⎜
2 ⎝ σ ⎟⎠
2
A plot of this function that describes this situation is shown below. The lines show those x
values that are one σ (40, 60) away from TV (50).
131
The likelihood of a measurement falling somewhere in this region is 68.3% (1 σ, 40 to 60). If
we increase the range of values (2 σ, 30 to 70) there is a 95.4% probability that a measurement
will fall somewhere within this range. If we increase the range of values (3 σ, 20 to 80) there is
a 99.7% probability that a measurement will fall somewhere within this range. The probability
of getting a measured value outside these ranges is 31.7%, 4.6% and 0.3%, respectively. You
can conclude that finding two measured values of the same quantity that are 1 σ apart in not that
unlikely but finding two measured values 3 σ apart is very unlikely and probably indicates one
of the measurements is bad.
When do my measurements agree with another experiment or with a theoretical value ?
The graph above shows 6 measurements that were performed and compared to a theoretical prediction of
g (circles, 9.8 m/s-s). The first thing to note is that both the theoretical (circles) and the experimental
(squares) results have an associated uncertainty. Also note that measurement A, B, and C have a
theoretical uncertainty that is very small compared to the experimental uncertainty. For measurements E,
F, and G the experimenter is using a prediction for g that has a comparable uncertainty. A rigorous
evaluation of the agreement or disagreement involves probability statements. However, our first goal is
to get a sense of what the measurements mean and in this lab this will be the only requirement. Here are
three rules of thumb one may use. The experiments are labeled as 1, 2 and 3 sigma. Sigma denotes the
uncertainty chosen by the experimenter and is reflected in the size of error bars drawn.
132
10
B
A
9.9
E
D
C
F
9.8
9.7
9.6
1
S
I
G
M
A
9.5
9.4
9.3
2
S
I
G
M
A
9.2
9.1
1
S
I
G
M
A
3
S
I
G
M
A
2
S
I
G
M
A
3
S
I
G
M
A
9
0
2
4
6
8
10
12
1. Measurements and/or theoretical results that agree to within one sigma are in agreement.
a. Shown as case A and D.
2. Measurements and/or theoretical results that agree to within 2 sigma are in agreement but
suggest there may be problems. The experimenter needs to review his/her data and methods.
The experimenter might duplicate the experiment.
a. Shown as case B and E.
3. Measurements that agree only at the 3 sigma level are in disagreement.
a. Shown as case C and F.
SIGNIFICANT FIGURES
WHEN YOU OBTAIN YOUR FINAL RESULT AND YOUR FINAL UNCERTAINTY
BE SURE TO STATE THESE RESULTS WITH THE CORRECT NUMBER OF
SIGNIFICANT FIGURES.
The Error Analysis Section of this lab (front of manual) has a detailed description of significant
figures with numerous examples. In general, numbers presented in spreadsheet tables do not
need to given with the correct number of significant figures. On the other hand, showing an
excessive number of digits can clutter a spreadsheet and make it difficult to read. Intermediate
results do not need to be given with the correct number of significant figures. Summary tables
that are providing results for a lab section or a final result should be listed with the correct
number of significant figures. If you are unsure ask your instructor.
133
COMPOUNDING UNCERTAINTIES.
When a number of measured quantities are combined to get a result, the uncertainty associated
with the result is a function of the uncertainties of measured quantities used to calculate the
result. Examples:
• distances and times to find g,
• temperatures and masses to measure the heat capacity
This type of analysis can be quite complicated. The correct way to add independent
uncertainties is to add uncertainties in quadrature. The formulas in Section 3 of Appendix 4 in
this Manual follow this rule and there one sees that many of the formulas sum squares and then
take the square root.
The following analysis will use a simpler, less rigorous, approximation. This approximation
will overestimate the uncertainties. (always double check with your instructor as to the level of
rigor expected in an analysis):
ƒ Absolute uncertainty refers to actual uncertainty.
ƒ Relative uncertainty involves the ratio of the uncertainty to another quantity. The relative
uncertainty can be expressed as:
a) fractional uncertainty - the ratio of the absolute uncertainty to the measurement
b) percent uncertainty – the fractional uncertainty times 100
Since dealing with percent uncertainties involves multiplying by 100 and then later dividing
by 100 to get back to an absolute uncertainty, it is suggested to use fractional uncertainty to
avoid this step, as is used in the discussion below.
ƒ When quantities add or subtract, add absolute uncertainties.
ƒ When quantities are multiplied or divided, the fractional uncertainty in the result is the sum
of the fractional uncertainties in the quantities used in the calculation.
ƒ When a quantity A is raised to the power j, B=Aj. The fractional uncertainty is j times the
fractional uncertainty in A. (fractional uncertainty B) = j (fractional uncertainty A)
ƒ For formulas that consist of several different operations, combine the uncertainties as you
perform the calculation. A spreadsheet is ideal for this type of calculation.
ƒ Uncertainties (absolute & relative) are always positive. If when calculating uncertainties,
the measurement or calculated value is negative, then use the absolute value.
ƒ One can find uncertainties by plugging in values +/- the uncertainty into a formula and see
how the result changes. This can be misleading when some of your values should be
combined as smaller values with others as larger values to get the largest fluctuation.
Students can typically ignore this effect.
ƒ More complicated formulas will require more complicated relationship. Discuss these with
your instructor.
Example of Calculating Uncertainties
Equations 4 and 6 shown below are extracted from a lab designed to measure Lf and Lv.
mi(Lf + Cw(Tf-0)) = mwCw(To - Tf) + mcCc(To - Tf)
(4)
134
msLv + msCw(Tbp-Tf) = mwCw(Tf - To) + mcCc(Tf - To)
(6)
How do we find the uncertainty, for example, in equation 6 for L assuming that we measure the
following quantities:
quantity
mass of water
final temperature
initial temperature
boiling point of water
mass of the steam added
mass of the container
known sp. heat of water
known sp. heat of
copper
mw
Tf
To
Tbp
ms
mc
Cw
413.6
44.7
22.6
99.1
15.5
60.3
1.000
gm
o
C
o
C
o
C
gm
gm
cal/gm oC
uncertainty
0.14
Δmw
0.3
Δ Tf
0.3
Δ To
0.2
Δ Tbp
0.14
Δms
0.07
Δmc
0.0
ΔCw
fractional uncert.
0.0003
Δmw/ mw
0.0067
Δ Tf / Tf
0.0133
Δ To / To
Δ Tbp / Tbp 0.0020
0.0090
Δ ms /ms
0.0012
Δmc/mc
0.0000
ΔCw/Cw
Cc
0.0924
cal/gm oC
ΔCc
ΔCc/Cc
0.0
0.0000
The quantities Cw and Cc are assumed to be known exactly. The experimenter measures several
quantities and determines the uncertainties in each quantity measured. There are various
techniques for finding the uncertainty including statistical analysis and consulting instrument
specifications.
Let us solve equation 6 for Lv.
msLv = mwCw(Tf - To) + mcCc(Tf - To) - msCw(Tbp-Tf)
Lv =
mw Cw ( T f − To ) + mc Cc ( T f − To ) − ms Cw ( Tbp − T f )
ms
Substituting the values above we find Lv = 543.
There are three terms added together
term1 =
mw C w ( T f − To )
ms
135
term2 =
mc Cc ( T f − To )
ms
term3 = − Cw ( Tbp − T f )
To determine the uncertainty we first note that each term includes a difference in temperatures.
The uncertainty associated with these differences (add absolute uncertainties) is
Term
Tf - To
Tbf - Tf
Value
22.1
54.4
uncertainty
0.3 + 0.3 = 0.6
0.3 + 0.2 = 0.5
fractional uncertainty
0.0271
0.0092
Once the temperature difference and its uncertainty have been determined, each term becomes a
product (or quotient) of numbers. Therefore we add fractional uncertainties to get the
uncertainty in each term. For example, to determine the uncertainty for term 1 we sum the
fractional uncertainty in the mass of water, the fractional uncertainty for specific heat, the
fractional uncertainty for the temperature difference (above table) and the fractional uncertainty
for the mass of steam.
Term
1
2
3
Value
589.7135
7.944194
54.4
fractional uncertainty
0.0003 + 0 + 0.0271 + 0.0090 = 0.0365
0.0012 + 0 + 0.0271 + 0.0090 = 0.0373
0 + 0.0092 = 0.0092
uncertainty
21.536
0.297
0.500
Since you add the terms together we must sum up absolute uncertainty of each term in the sum.
ΔLv = 21.536 +0.297 + 0.5 = 22.3
ΔLv /Lv =.041 or 4.1%
Final result
Lv = 543 ± 22
cal/gm
136
ERROR ANALYSIS CHECK SHEET
List of all directly measured quantities:
Name of quantity
Value
Uncertainty Method for estimating error
Some measured quantities are indirect and must be calculated from a set of direct
measurements.
Indirect measurements Quantity:
Formula or relationship used to calculate this quantity:
List of Direct quantities
absolute uncertainty
fractional uncertainty
TOTAL ERROR
FINAL RESULTS - USE THE CORRECT NUMBER OF
SIGNIFICANT FIGURES.
137
Appendix 4
SUGGESTIONS FOR HANDLING DATA
Significant Figures
Experimental Errors (uncertainty)
Statistical Treatment of Errors (uncertainty)
Error (uncertainty) Analysis Cookbook
by
Dr. D. Chodrow
Definition of terms:
Error is the difference between a value and its correct value or true value.
Uncertainty is an estimate of the difference between a calculated or
measured value and the true value.
138
1. SIGNIFICANT FIGURES
Concepts and Definitions
No measurement is exact. Consequently, whenever we measure any quantity it is necessary
to state both the measured value and some estimate of the precision. The number of
SIGNIFICANT FIGURES used in stating a measured value indicates the precision. The number
of significant figures in a number is defined as follows:
1) The leftmost nonzero digit is the most significant digit.
2) If there is no decimal point, the rightmost nonzero digit is the least significant digit.
3) If there is a decimal point, the rightmost digit is the least significant digit, even if it is
a zero.
4) The number of significant figures is the number of digits from the least significant
digit to the most significant digit, inclusive.
Examples:
1)
4630
4630.
4630.000
0.000 20
0.000 200 0
has
has
has
has
has
3
4
7
2
4
significant figures.
significant figures
significant figures.
significant figures.
significant figures.
2) Those zeroes whose only function is to locate the decimal point in a decimal fraction such
as 0.000 456 or a large integer such as 6 789 000 000 are not significant. Such numbers are
best expressed in scientific notation with only the significant figures given. The numbers in
this example would be given as
4.56 x 10-4 which has 3 significant figures, and
6.789 x 109 which has 4 significant figures.
When a measured value is written down, the POSITION OF THE LEAST SIGNIFICANT
FIGURE indicates the magnitude of the precision.
Examples:
3) If you state that the length of a rod is 34.76 cm, you are implying that the leftmost three
figures are certain and that the least significant figure is uncertain to some degree. In other
words, you are stating that the length of the rod is probably not less than 34.7 cm and not
more than 34.6 cm, and that you are reasonably confident that the length is 34.76 cm.
139
4) You use a balance which is known to be accurate only to within 0.2 gm to make a
single measurement of the mass of a machine screw. Even if the balance pointer indicates
a value of 2.637 gm, you may only state the mass of the screw as 2.6 gm. This is because
the second and third decimal places are meaningless here since the first place is already
uncertain.
Arithmetic with Significant Figures
SUMS AND DIFFERENCES: Suppose that we use three different methods to measure the
lengths of the sides of a triangle. The resulting lengths, each given with the proper number of
significant figures, are 27.113 cm, 8.63 cm and 19.2 cm. We wish to determine the perimeter
(the sum of the lengths of the sides of the triangle. Proceeding without regard to significant
figures, we add the lengths of the sides to get
27.113 cm
8.63 cm
+ 19.2 cm
54.943 cm.
We must now interpret this result. In any number obtained by measurement, all digits following
the least significant digit are UNKNOWN. Therefore, the lengths of each side, to the nearest
thousandth of a cm, are 27.113 cm, 8.63X cm and 19.2YZ cm, where X,Y and Z stand for
COMPLETELY UNKNOWN digits. The correct result for the perimeter is
27.113 cm
8.63X cm
+ 19.2Y2 cm
______________
54.9AB cm
where A and B are also unknown digits. The sum therefore has only three significant figures
and is correctly given as 54.9 cm.
From this example, we see that the rule for determining the number of significant figures is a
sum or a difference is:
THE LEAST SIGNIFICANT DIGIT OF THE RESULT IS IN THE SAME COLUMN
RELATIVE TO THE DECIMAL POINT AS THE LEAST SIGNIFICANT DIGIT OF
THE NUMBER ENTERING INTO THE SUM OR DIFFERENCE WHICH HAS ITS
LEAST SIGNIFICANT DIGIT FARTHEST TO THE LEFT.
Example 5: Let x = 4.231, y = 32.6, z = 29, and w = 31.7
A) If p = x+y, y is the number whose least significant digit, 6, is farthest to the left, so
140
p = 4.231 + 32.6 = 36.831 = 36.8 Thus p = 36.8 to the proper number of significant
figures
B) If q = x + y - z, z is the number whose least significant digit, 9, is farthest to the left, so
a = 4.231 + 32.6 - 29 = 7.831 = 8 Thus q = 8 to the proper number of significant figures.
C) If r = z - w, then r = -3 to the proper number of significant figures.
PRODUCTS AND QUOTIENTS: Suppose that we use different methods to measure the
lengths of the adjacent sides of a rectangle, and that the results given to the proper number of
significant figures are a = 3.24 cm and b = 4.112 cm. We wish to determine the area of the
rectangle. Proceeding without regard to significant figures, we get
A = ab = 3.24 cm x 4.112 cm = 13.32288 cm2
We must now interpret this result. The least significant digits of a and b are uncertain to
some degree. Let us assume an uncertainty of 2 in the least significant digits. Then a could have
any value between 3.22 cm and 3.26 cm, while b could have any value between 4.110 cm and
4.114 cm. Therefore A = ab could have any value between
Amin = 3.22 cm x 4.110 cm = 13.2342 cm2
and
Amax = 3.26 cm x 4.114 cm = 13.41164 cm2
Therefore the first decimal place is the first uncertain figure, and the area is properly reported as
A = 13.3 cm2
which has THREE significant figures.
If a had been determined to two significant figures, a = 3. 2 cm, while b = 4.112 cm, we
would have found
A = 3.2 cm x 4.112 cm = 13.1584 cm2
Now, since a could have any value in the range from 3.0 cm to 3.4 cm while b could have any
value in the range from 4.110 cm to 4.114 cm, we would have
Amin = 3.0 cm x 4.110 cm = 12.33 cm2
and
Amax = 3.4 cm x 4.114 cm = 13.9876 cm2
Now the first digit to the left of the decimal point is uncertain, so the area is properly reported as
141
A = 13 cm2
which has TWO significant figures.
From these examples we see that the rule for determining the number of significant
figures in a product (or a quotient) is:
THE NUMBER OF SIGNIFICANT FIGURES IN A PRODUCT OR QUOTIENT IS
THE SAME AS THE NUMBER OF SIGNIFICANT FIGURES IN THE FACTOR
WITH THE FEWEST SIGNIFICANT FIGURES.
Example 6: Let x = 6.63 x 1.0-4, y = 9.0346, z = 47320 and t = 4.2 Then
A) f = xy/t = 6.63 x 10-4, y = 9.83A6/ 4.2 = 1.552 x 10-3. This must be rounded off to two
significant figures, so
f = 1.6 x 10-3
B) g = 3x2z/y = 3x(6.63 x 10-4)2 x 47320 / 9.8346 = 6.345068 x 10-3 or, to three significant
figures,
g = 6.35 x 10-3
C) h = z/y2 = 47320 / (9.8346)2 = 489.2505636, or to four significant figures (z has four
significant figures), h = 489.3
ROUNDOFF ERRORS, A WARNING EXAMPLE: It is often advisable to ignore the
preceding rules for arithmetic with significant figures during intermediate stages in a
calculation, although the final result must be given with the correct number of significant
figures. This is because rounding off intermediate values in a long calculation may lead to
arithmetic errors. The following example demonstrates this point. We wish to find the value of
q = x4y3z
where x = 1.36, y = 1.26 and z = 5.2. According to the rule for products and Quotients the value
of q should have two significant figures. Before the days of hand-held electronic calculators, a
common labor-saving technique was to round all data and the results of all intermediate steps to
the lowest number of significant figures. In this case, x would be rounded to x1 = 1.4 while y
would be rounded to y1 = 1.3. Then, to two significant figures,
x14 = 1.44 = 3.8416 = 3.8
142
y13 = 1.33 = 2.197 = 2.2
and
q1 = x14y13z = 3.8 x 2.2 x 5.2 = 43.472 = 43
The result a is incorrect because both x4 and y3 have been
overestimated. We now calculate a, keeping all the figures provided by a ten-digit calculator:
x4 = 1.364 = 3.42102016
y3 = 1.263 = 2.000376
and
q = 1.364 x 1.263 x 5.2
= 3.42102016 x 2.000376 x 5.2 = 35.58529844
To two significant figures, q = 36. The value q1 = 43 is 19% too large.
In order to avoid arithmetic errors arising from premature rounding off:
DO NOT ROUND OFF THE INTERMEDIATE STAGES OF A LONG
CALCULATION. INSTEAD, DO THE ARITHMETIC AS IF ALL THE DATA
CONSISTED OF EXACTLY KNOWN VALUES, USING YOUR CALCULATORS
MEMORY TO STORE ANY INTERMEDIATE RESULTS. THE FINAL RESULT
SHOULD THEN BE ROUNDED OFF TO THE CORRECT NUMBER OF
SIGNIFICANT FIGURES.
143
2. EXPERIMENTAL ERRORS (uncertainty)-- AN INTRODUCTION
All measured quantities contain inaccuracies. These inaccuracies complicate the
problem of determining the "true" value of a quantity. Therefore, the object of
experimental work must be to determine the best estimate of the "true" value of the
quantity being measured, together with an indication of the reliability of the
measurement.
There are two main sources of experimental error systematic errors and statistical
errors. SYSTEMATIC ERRORS are associated with the particular instruments or
technique used. They can result when an improperly calibrated instrument is used or
when some unrealized influence perturbs the system in some definite way, thereby
biasing the result of the measurement. An example of such an influence is the small
amount of friction due to air resistance, which acts on a dropped object in such a way as
to reduce its acceleration by a small unknown amount.
Sometimes it is possible to correct for systematic errors. If, for example, we know
that a voltmeter is calibrated so that it always reads 10% too low, it is a simple matter to
compute the correct voltage by multiplying the meter reading by 10/9. Most of the time,
however, the task of discovering and compensating for systematic errors is very difficult,
requiring great familiarity with the experimental techniques and equipment used. There
are no general methods for dealing with systematic errors.
No matter how carefully a measurement is made, it will possess some degree of
variability. The errors which result from the lack of precise repeatability of a
measurement are called STATISTICAL ERRORS or RANDOM ERRORS. It is often
possible to minimize statistical errors by judicious choice of measuring equipment and
technique, but they can never be eliminated completely. We must therefore learn how to
determine the statistical error associated both with a single directly measured quantity
and with a result which is calculated from several measured quantities.
The terms ACCURACY and PRECISION are often used to describe the reliability
of a measurement. Although these terms are commonly used interchangeably, they have
very different meanings in scientific work. A quantity is determined with great
ACCURACY if the result of the measurement is close to the "true" value. In other words,
great accuracy is equivalent to small systematic errors. A quantity is determined with
great PRECISION if the measurements are closely repeatable. In other words, great
precision is equivalent to small statistical errors.
It is possible for a measurement to be precise without being accurate or vice versa.
The aim of scientific work is to achieve both accuracy and precision.
There is a third type of error, which is due entirely to poor experimental technique
and carelessness. These errors are called BLUNDERS or MISTAKES and are totally
unacceptable in scientific work. They can be eliminated completely with a reasonable
amount of care. Be sure that you understand what you are supposed to do in the
144
laboratory before you start any experiment. Read the instructions. If you do not
understand how to use a piece of equipment or how to analyze your data, reread the
instructions. If you are still confused, ask your laboratory instructor for help.
LABORATORY REPORTS CONTAINING BLUNDERS WILL SUFFER A SEVERE
GRADE PENALTY.
3. STATISTICAL TREATMENT OF EXPERIMENTAL DATA
Introduction
The variability inherent in any repeated measurement makes it impossible to
determine with absolute certainty the "true" value of a physical quantity. However, it is
possible to make several measurements of such a quantity and to use them to estimate
both the value of the quantity and the statistical uncertainty of the estimate. (Uncertainty
will be used to signify an estimate of the error.) It is also possible to estimate the value
and uncertainty of a result which is calculated from other quantities whose values and
uncertainties have been determined.
In the rest of this section we will assume that all sources of systematic error have
been eliminated or compensated for so that only the statistical uncertainties remain to be
dealt with.
Estimating the Best Value and Uncertainty of a Measured Quantity
Let us assume that we have made N INDEPENDENT measurements of a quantity
x, with the resulting values
x1, x2, x3, .......... xn.
The problem before us is to make the best guess of the "true" value of x and of the
statistical uncertainty or error in x.
We first define the MEAN or AVERAGE of the measurements to be
X=
( x1 + x 2 + x 3 +... x n )
N
or
x=
1 N
∑ xk
N k=1
In general, a bar over a quantity indicates the mean of that quantity.
145
If systematic errors have been eliminated and each of the N measurements is equally
reliable, then the best estimate of the "true" value of x is the mean x :
Best value = Mean value
Now we must determine the reliability of this estimate. To do this, we must find
the UNCERTAINTY (estimate of error) in x, Δx, which is defined by saying that it is
very likely that the "true" value of x lies in the range from x - Δx to x + Δx.
The precise meaning of "very likely" depends on the particular method used to
compute Δx. We will use a method for which the probability of the "true" value of x
lying in the range from is about 2/3. We then present the result as
x = x ± Δx
The uncertainty Δx depends both on the number N of measured values and on the
dispersion, or scatter of the individual measurements about their mean. A useful measure
of the dispersion is called the STANDARD DEVIATION. It is approximately the same
as the r.m.s. (root-mean-square) deviation. The RMS and SD are defined by the following
expressions
( x1 - x )2 + ( x 2 - x )2 + ... + ( x N - x )2
rms = (x - x ) =
N
2
( x1 - x )2 + ( x 2 - x )2 + ...+ ( x N - x )2
σ=
N −1
σ=
1 N
( x k - x )2
∑
N −1 k=1
Roughly 2/3 of the measured values of x should lie in the range from x - σ to x + σ.
The equations above define the standard deviation σ but do not provide an easy
way to compute it. Many hand-held calculators have pre-programmed algorithms for
calculating x and σ for a set of data. If such a calculator is not available and N is large so
that N-1 can be replaced by N, the following equation provides an easier calculation for
σ:
σ ≅ ( x2 ) - ( x )2
146
The standard deviation σ is usually a property of the measuring technique or equipment,
and can often be thought of as a measure of the "free play" in the equipment. It seems
intuitively reasonable that the reliability of a measurement should increase as the number
N of data values increases. We should therefore expect that the experimental error Δx
should decrease as N increases. This involves a statistical quantity called the standard
deviations of the mean, σx .
σx is the statistical uncertainty in x.
DEVIATION OF THE MEAN is
It can be shown that the STANDARD
σx=
σ
N
Therefore
1)
The best estimate of x is x .
2)
The statistical uncertainty of x is Δx = σx.
This means that the probability that the "true" value of x lies between x - Δx and x + Δx
is roughly 2/3. We then write
x = x ± Δx
Notice that the denominator in the equation for x vanishes when N = 1. This is because
the concept of a statistical uncertainty based on the dispersion of the data becomes
meaningless when there is only a single data value. Occasionally it is necessary or
convenient to make only one measurement of a quantity. In that case, the statistical
uncertainty in that quantity should be taken to be the RESOLUTION of the measuring
device, which is the smallest increment in the quantity which can be distinguished. For
example, if the thickness of a rod is measured once using a caliper which can be read to
within 0.02 cm, then the uncertainty in t is Δt = 0.02 cm.
Example 1: Ten measurements of the length of a stretched spring yield the values (all in
cm)
7.2, 6.9, 7.1, 7.0, 7.1, 7.2, 6.9, 7.0, 6.9, 7.1
Solution -- In this case, N = 10. The best estimate of x is the mean
x=
1 N
∑ xm
N k=1
147
= (7.2 + 6.9 + 7.1 + 7.0 + 7.1 + 7.2 + 6.9 + 7.0 + 6.9 + 7.1) / 10
x = 7.04cm
Note that we do not round x off to 2 significant figures. In fact, we do not round
anything off until all the calculations are finished.
To find the standard deviation use a calculator which is pre-programmed or make the
following calculations
2
(x ) =
1 N 2
∑ xm
N k=1
= (7.22 + 6.92 + 7.12 + 7.02 + 7.12 + 7.22 + 6.92 + 7.02 + 6.92 + 7.12) /10
( x 2 ) = 49.574 cm2
Then (using the approximation that N-1=9 is about the same as 10).
σ = ( x 2 ) - ( x )2 = 49.574 - (7.04 )2 = 0.111cm
which we have rounded to three significant figures. The standard deviation of the mean
is
σx=
σ
N
=
0.111cm
10
= 0.035cm
148
This is the uncertainty in x, Δx = 0.035 cm. Then x = (7.04 ± 0.035) cm.
We can now determine the correct number of significant figures in x. It is highly
probable that the "true" value of x lies between x - Δx = 7.005 cm and x + Δx = 7.075
cm. We see that the tenth' place is certain but the hundredths' place is not. Therefore, x
should be stated with three significant figures. Since it does not make any sense to give a
numerical value for the uncertainty in a figure which is completely uncertain, we round
the uncertainty off to one significant figure (or two at the most) and quote the value of x,
with its uncertainty, as
x = (7.04 ± 0.04) cm.
Avoid serious blunders by rounding off only at the end of the calculation.
Example 2: In the previous example we saw how the number of significant figures
quoted in an experimental result is determined by the uncertainty. Usually we only give
the uncertainty to one figure. For example, if our calculations yield v = 43.2684 m/s and
Δv = 0.02162 m/s we should quote the result as
v = (43.27 ± 0.02) m/s
Sometimes however, we will quote the uncertainty to two significant figures and keep an
extra figure in the mean. This is done only when it is necessary to prevent rounding off
in such a way as to affect two figures in the result. For example, if T = 8.9631 s and ΔT
= 0.3421 s, we may quote T as either
T = (9.0 ± 0.3) s
or
T = (8.96 ± 0.34) s
Strictly speaking, the first equation is correct, but the second equation is more
convenient. However, it would be totally incorrect to state that T = (8.9631 ± 0.3421) s.
Sometimes instead of stating the ABSOLUTE uncertainty Δx, one states the RELATIVE
uncertainty in a quantity x. There are two ways to do this. The FRACTIONAL
uncertainty in x is
FractionalError =
Δx
x
while the PERCENT UNCERTAINTY in x is simply 100 times the fractional uncertainty
149
PercentError = 100
Δx
x
Example 3: If M = 4.79 kg and ΔM = 0.08 kg then the fractional uncertainty in M is
ΔM 0.08kg
=
= 0.017
M
4.79kg
and the percent uncertainty is 1.7%, which could equally well have been rounded off to 2
%. We may quote M as either
M = (4.79 ± 0.08) kg
or
M = 4.79 kg ± 2 %
Note that we never give a percent uncertainty to more than two significant figures.
Propagation of Uncertainties (errors): Estimating the Best Value and Uncertainty of a
Result Calculated from Several Independently Measured Quantities
Let the quantity q depend on the quantities x, y, z, .... through the equation
q = f(x, y, z, ...)
If x, y, z, ... are measured independently with the results
x = x ± Δx, y = y ± ΔY,z = z ± Δz
we must find the best value for q together with its uncertainty.
We will assume that the relative uncertainties in x, y, z, ... are small. Then the best
estimate for q is found by substituting the best estimates for x, y, z, ... into the equation
which defines q:
q = f( x, y,z,... )
150
There are two very important special cases for which the statistical uncertainty in q can
be calculated from easy to remember formulas:
Special Case 1 -- q is a SUM
q = Ax + By + Cz + ....
where A, B, C, ... are constants. In this case, the uncertainty in q is
2
2
2
Δq = A2 ( Δx ) + B2 ( Δy ) + C2 ( Δz ) +...
Special Case 2 -- q is a PRODUCT
B
q = K x A y zC
where K, A, B, C, . . . . . are constants. In this case, the fractional uncertainty in q is
2
2
2
⎛ Δy ⎞
Δq
⎛ Δx ⎞
⎛ Δz ⎞
= A2 ⎜ ⎟ + B2 ⎜ ⎟ + C2 ⎜ ⎟ +...
⎝ x ⎠
⎝ z ⎠
q
⎝ y⎠
and the uncertainty is
⎛ Δq ⎞
⎟q
Δq = ⎜
⎝ q ⎠
Examples:
4) If q = 7y2 and y = 26.3 + 0.8, find q and Δq.
2
Solution: Here y = 26.3 and Δy = 0.8. Then q = 7( y ) = 4841.83. We find Δq by first
finding the fractional uncertainty
151
Δq
Δy 2x0.8
=2
=
.3 = 0.0608
y
26
q
Then the uncertainty is Δq = 0.0608 q = 0.0608 x 4841.83 = 294. Then, since q = 4.8 x
103 and Δq = 0.3 x 103, q = (4.8 ± 0.3) x 103.
5)
If w = 18z -1/3 and z = (7.24 ± 0.06), find w and its uncertainty.
Solution: Here z = 7.24 so w = 18(7.24)-1/3 = 18 x 0.5169 = 9.304.
Now
Δw
1 Δz
== 0.00276
3 z
w
so Δw = 0.00276 x 9.304 = 0.0257 = 0.03. Then w = 9.30 ± 0.03.
6) A rectangle of sides x and y has perimeter L = 2x + 2y and area A = xy. x and y are
measured and found to be x = (3.0 ± 0.1) m and y = (2.65 ± 0.02) m. Find the perimeter
and area of the rectangle together with their uncertainties.
Solution: Here x = 3.0 m, y = 2.65 m, Δx = 0.1 m and Δy = 0.02 m.
Perimeter: L = 2 x + 2 y = 2(3.0) + 2(2.65) = 11.3 m. Since L is a sum,
2
2
ΔL = 22 ( Δx ) + 22 ( Δy ) = 0.204 m
Then L = (11.3 ± 0.2) m.
Area: A = xy = (3.0) (2.65) = 7.95 m2 . Since A is a product,
2
2
⎛ Δy ⎞
ΔA
⎛ Δx ⎞
= 12 ⎜ ⎟ + 22 ⎜⎜ ⎟⎟ = 0.034
A
⎝ x ⎠
⎝ y ⎠
152
Then ΔA = 0.034 A = 0.034 x 7.95 m2 = 0.27 m2, and A = (7.95 ± 0.27) m2 or
A = (8.0 ± 0.3) m2
7)
A cylinder of radius r and height h has volume V = πr2 h and surface area
S = π r2 + 2 πrh. If the radius and height of the cylinder have been measured with the
results r = (4.60 ± 0.05) cm and h = (6.0 ± 0.1) cm, find the volume and surface area of
the cylinder together with their uncertainties.
Solution: Here r = 4.60 cm, h = 6.0 cm, Δr = 0.05 cm and Δh = 0.1 cm.
Volume:
V = π ( r ) ( h ) = π (4.60 ) (6.0) = 398.9 cm3
2
2
Since V is a product,
2
2
ΔV
⎛ Δr ⎞
⎛ Δh ⎞
= 22 ⎜ ⎟ + 12 ⎜ ⎟ = 0.027
⎝ r ⎠
⎝ h⎠
V
then V = 0.027 V = 0.027 x 398.9 cm3 = 10.8 cm3 = 11 cm3, and V = (399±11) cm3 or
V = (4.0 ± 0.1) x 102 cm3.
8)
Great care is needed when two experimentally determined quantities whose values
are close are to be subtracted. For example, if x = (123 ± 4) cm and y = (129 ± 3) cm and
if the quantity of interest is w = y - x, then
w = y - x = 129 - 123 = 7cm
while
2
2
2
2
2
2
Δw = (-1 ) ( Δx ) + (1 ) ( Δy ) = ( Δx ) + ( Δy ) = 5cm
so w = (7 ± 5) cm. Even though x and y are determined with precision of 3.3% and 2.3%,
w has a percent uncertainty of 71%.
153
4. ERROR (uncertainty) ANALYSIS COOKBOOK
Measurements
Suppose N values are recorded
x1, x2, x3, ...xN
Mean or Average: (Best estimate = Mean value)
x=
Standard Deviation: σ =
x1 + x 2 + x 3 + K x N
N
1 N
( x k - x )2 ≅ ( x 2 ) - ( x )2
∑
N − 1 k=1
Standard Deviation of the Mean: (Statistical Uncertainty = Standard Deviation of the
Mean)
σx=
σ
N
Propagation of Uncertainties
Suppose q = f(x,y,z,....)
Best Estimate:
q = f( x, y,z,K )
Statistical Uncertainty: (General Case)
Δq = (
∂q 2
∂q
∂q
) ( Δx )2 + ( )2 ( Δy )2 + ( )2 ( Δz )2 +K
∂x
∂y
∂z
Statistical Uncertainty: (Special Case where q is a sum q = Ax + By + Cz ...)
2
2
2
Δq = A2 ( Δx ) + B2 ( Δy ) + C2 ( Δz ) +K
Statistical Uncertainty: (Special Case where q is a product q = KxAyBzC....)
|
Δq
Δx 2
Δy 2
Δz 2
) + B2 (
) + C2 (
) +K
|= A2 (
q
x
y
z
154
155