Download Physics 1000 Lab Manual Spring 2012

Department of Physics
and Astronomy
Physics 1000
Lab Manual
Spring 2012
Contents
Lab Reports and Marks
i
Sample Lab Outline
iv
Sample Lab Report
vii
1 Statistics
1
2 Free Fall
12
3 Projectile Motion
16
4 Centripetal Acceleration
23
5 Principles of Mechanics
28
6 Conservation of Mechanical Energy
32
7 Conservation of Momentum
38
8 Impulse and Momentum
43
9 Motion of Centre of Mass
51
10 Angular Acceleration of a Disk
56
11 Nuclear Half-life
62
A Reading Vernier Scales
67
B Error Analysis
69
Lab Reports and Marks
This section includes an outline for your lab reports and a sample lab report based on an
experiment called "Conservation of Energy on an Inclined Plane" (also included).
Since your marks for the lab are derived from your written reports it is important to understand what is expected as a report. Read the sample experimental outline "Conservation
of Energy on an Inclined Plane" first, then the lab report outline, and conclude by reading
the sample report.
Lab Report Outline:
Lab reports must have the following format:
Title Page: Includes the title of experiment, the date the experiment was performed, your
name, your partner’s name(s), and lab section.
Results: List data in tables whenever possible. Each table must have a title and carry
units for each column. Show one sample calculation for each step in the analysis,
carrying the units through the calculations. Consult the sample lab report to see how
sample calculations should be done. Any other measurements associated with the
data (parameters, constants, etc.) should be clearly labelled within this section.
Analysis: Complete all steps and questions in the analysis section of the experimental
outline. The analysis section contains primarily equations and numerical work that
must be carried out in order to obtain usable results. When asked to plot a graph
be sure to label each axis including units. Each graph must have a descriptive title
placed above the graph (note: y vs x is not a descriptive title).
Discussion: A few written paragraphs outlining the important facets and details of your
experimental results. Examples of what questions you should be asking yourself are
provided in each lab outline, but do not feel limited by these questions. If you have
other observations or conclusions that you would like to discuss, then do so. This
section is the most important, yet most difficult.
i
Conclusion: A final sentence(s) stating the outcome(s) of the experiment written in the
context of the objective of the experiment. Your conclusions should be supported by
arguments made in your discussion.
If you glance through a scientific journal you will observe that the outline listed above is a
simplified version of most journal formats. Items such as introductions and procedures have
been provided for you within the manual. It is important to note here that presentation
matters; the sections of your report should appear in the same order listed above.
The difference between a "good" lab write-up and a poor one is often attention to detail. As
you write your report you must assume that the person who will read it knows little about
your experiment. You are presenting a case upon which you will base your conclusion. Guide
the reader through the report by clearly labeling tables and graphs with descriptive titles,
succinctly presenting your discussion of the experiment and results, and unambiguously
stating your conclusion(s).
Marking scheme
Each group is required to submit one lab report for each experiment unless directed otherwise. Some labs are designed to be done individually. Groups can be no more than 2
students. When experiments are done in groups one lab report for the group is sufficient,
although individual group members may submit their own report if they prefer. When
group reports are marked each member of the group receives the same mark.
In some cases, several groups will team up on a given apparatus to obtain their results.
Each group will then share the same data, but ALL analysis must be done by each group
individually. This includes tabulating the original data; you are not allowed to ‘share’
spreadsheets.
Lab reports are typically due one week after each experiment is completed, unless instructed
otherwise.
Lab reports will generally be marked out of 10 (longer labs may be marked out of 15). Your
lab reports will be marked on the following criteria:
Completeness:
Correctness:
Presentation:
Reason:
-
Have you completed all the necessary tables, graphs, and calculations?
Are all the required sections present, i.e. title page, conclusion, etc.?
Do you have the proper labels and units for all graphs and tables?
Are your numerical calculations done correctly?
Are the sections of the report in correct order?
Are the results presented in a clear and concise manner?
Are the spelling, grammar, and punctuation correct?
Do your conclusions and arguments seem reasonable, based on your results?
ii
Note that the list of comments for each section given above is not exhaustive. There are
many different aspects to a good lab report which are not given here. Refer to the Lab
Report Outline section above to obtain more guidance on each section.
Your final lab mark is calculated as:
Total Marks Received / Total Marks Possible
Absence from Lab sessions
If you must miss a lab section, contact your lab instructor as soon as possible to let them
know you will be absent, preferably before the lab, if possible. You will need to make
arrangements to do the lab at some other time. If you are able to attend a different lab
section within the same week, then let your lab instructor know. He or she can make the
necessary arrangements with the instructor of this other lab session. Under no circumstances
are you allowed to put your name on a report for a lab that you did not participate in.
iii
Sample Lab Outline
Objective
The objective of this experiment is to investigate the principle of conservation of energy
using an inclined plane.
Apparatus
• Air Track
• Sonar System
• Carpenter’s Level
• Meter Sticks
Background
The near frictionless surface of an air track represents a close approximation to something
called an isolated system. Such a system would be expected to demonstrate the principle
of conservation of mechanical energy.
By definition an isolated system does not exchange mass or energy with its surroundings.
Any mechanical energy put into the system will remain there and we should be able to
account for it as either kinetic or potential energy. If the principle of conservation of
mechanical energy is correct then the total energy in the system should remain constant.
Tipping the air track at an angle creates what physicists call an inclined plane. In the
absence of friction an object placed on an inclined plane will begin to move in the downhill
direction.
iv
Energy is introduced into the inclined plane system by placing the glider at the top of
the incline. Until released all the energy in the system is in the form of potential energy.
Once moving some of the potential energy is transformed into kinetic energy. As the glider
progresses down the incline the potential energy continues to decrease while the kinetic
energy increases. At any instant in time, however, the sum of the two should be constant
and equal the initial amount of potential energy in the system.
Procedure
Set the air track at a slight angle to the horizontal (about 2 degrees). Place the glider at
the high end of the air track and allow it slide to the low end. Track the position of the
glider during its travel using the sonar equipment. Save your data on a disk. Be sure to
record the mass of the glider and measure the angle of incline with two meter sticks and
the carpenter’s level.
Analysis
Construct a data table like the one below and make the appropriate calculations.
Time
(s)
Position of
Glider (cm)
Velocity of
Glider (cm/s)
Kinetic Energy
of Glider (J)
Potential Energy
of Glider (J)
Total Energy
(J)
From your calculations plot on a single graph the potential, kinetic and total energies as a
function of time.
v
Discussion
From your results, can you conclude that energy was conserved? If not, where might it
been lost?
How accurately were you able to measure the angle of incline? What impact(s) might an
error in the measurement of the angle of incline have on the results of the experiment?
vi
Sample Lab Report
CONSERVATION OF ENERGY ON AN INCLINED PLANE
Larry Brown
Curly Jones
Moe Smith
Thursday, Jan. 8, 2010
Lab Section #2
vii
Table 1. Inclined Plane Data Table
Mass of glider (g)
∆t (s)
Angle of incline (°)
Time
(s)
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.05
2.10
2.15
2.20
2.25
2.30
2.35
2.40
Position
(cm)
198.43
196.85
195.20
193.06
191.05
189.29
186.75
184.50
182.29
179.70
176.75
173.91
170.92
167.76
164.50
161.28
157.83
153.97
150.49
146.43
142.54
138.51
134.38
130.25
125.77
121.37
116.81
220 ± 0.5
0.0500 ± 0.00025
2.7 ± 0.5
Velocity
(m/sec)
0.316
0.330
0.428
0.402
0.352
0.508
0.450
0.442
0.518
0.590
0.568
0.598
0.632
0.652
0.644
0.690
0.772
0.696
0.812
0.778
0.806
0.826
0.826
0.896
0.880
0.912
0.932
Potential
Energy (J)
0.2015
0.1999
0.1982
0.1961
0.1940
0.1922
0.1897
0.1874
0.1851
0.1825
0.1795
0.1766
0.1736
0.1704
0.1671
0.1638
0.1603
0.1564
0.1528
0.1487
0.1448
0.1407
0.1365
0.1323
0.1277
0.1233
0.1186
Kinetic
Energy (J)
0.0110
0.0120
0.0202
0.0178
0.0136
0.0284
0.0223
0.0215
0.0295
0.0383
0.0355
0.0393
0.0439
0.0468
0.0456
0.0524
0.0656
0.0533
0.0725
0.0666
0.0715
0.0751
0.0751
0.0883
0.0852
0.0915
0.0955
Total
Energy (J)
0.213
0.212
0.218
0.214
0.208
0.221
0.212
0.209
0.215
0.221
0.215
0.216
0.218
0.217
0.213
0.216
0.226
0.210
0.225
0.215
0.216
0.216
0.212
0.221
0.213
0.215
0.214
Sample Calculations
Notes:
1.) The position data column represents the distance from the sonar unit to the glider.
2.) The frequency of the sonar unit was 20.0 ± 0.1 Hz, which translates into a ∆t of .0500
±0.00025 seconds.
viii
Velocity Calculation:
Velocity is defined as:
∆Position
∆time
Each entry in the velocity column was calculated as:
Sample calculation:
Pn − Pn+1
∆t
198.43cm − 196.85cm
= 31.6 cm/s = 0.316 m/s
0.05s
Potential Energy:
Potential energy is defined as U = mgh.
Given that the data can be thought of as the hypotenuse of a right-hand triangle, the height
can be calculated as,
h = (position) sin θ
∴ U = mg(position) sin θ
Sample calculation: U = (.220 kg) (9.80 m/s2 ) (1.9843 m) (sin (2.7°)) = 0.2015 J
Kinetic Energy:
Kinetic energy is defined as K = 1/2mv 2
Sample calculation: K = (0.5) (0.220 kg) (.316 m/s)2 = 0.0110 J
Total Energy:
Total energy is the sum of the potential and kinetic energies, E = U + K.
Sample calculation: E = 0.2015 J + 0.0110 J = 0.2125 J = 0.213 J
ix
Figure 1: Potential, kinetic, and total energy of a single glider on an inclined plane.
x
Figure 2: The velocity of the glider on the track as a function of time.
Discussion
It can be seen from Fig. 1 that energy was conserved as the glider descended the air track.
There are some slight deviations from a constant energy value that appear to be due to
similar deviations in the kinetic energy values. This is probably due to small errors in the
velocity values, most likely caused by timing errors in the motion sensor.
We checked to see if air resistance was a problem. We found that the velocity of the glider
(see Fig. 2) was linear and therefore air resistance did not appear to affect our results. The
fact that the velocity was linear also tells us that acceleration on the air track is constant.
Measurement of the angle of the air track is a concern. If the angle is improperly measured the potential energy calculations will be inaccurate. The result could be an apparent
increase or decrease in the total energy when in reality energy is conserved.
We measured the angle several times and used the average value as the actual angle of
incline for the air track. We found the accuracy of the measurement to be about one half
of one degree. The limiting factor in measuring the angle is the carpenter’s level. The
experimental calculations are quite sensitive to the value used for the angle of incline. This
is because this angle is used to calculate the potential energy. Using the same data, and
varying the angle over a range of 1.5 degrees, we were able to get a variety of results. Over
xi
this range of angles we could show a loss of energy or a gain of energy.
Conclusion
Energy was conserved on the inclined plane.
xii
LAB
1
Statistics
Objective
The objective of this exercise is to familiarize the student with the use of Gaussian distributions, linear regression, and Chauvenet’s criterion as they relate to data analysis for
experiments in introductory physics.
Background
Statistical analysis plays a very important role in experimental work. In this exercise we are
concerned with the connotations of terms such as average, standard deviation, and normal
distribution, as these terms have particular meanings when associated with experimental
work. In addition, we will develop quantitative tools for judging the validity of data.
Throughout the semester we will associate the term average or mean with the "best estimate
of the actual value". Standard deviation will represent the "average value of uncertainty in
the average" or probable error, and a normal distribution will be viewed as a probability
distribution. We will use techniques such as linear regression and Chauvenet’s criterion to
provide methods for assessing the validity of data gathered during experiments.
The following sections are brief overviews of the important points as they relate to statistical
analysis.
Gaussian Distributions
The Gaussian curve, named after Karl Friedrich Gauss, was devised in 1795. Gauss found
that his experimental data would not repeat itself exactly from trial to trial during exper-
1
2
LAB 1. STATISTICS
iments. After plotting the results of a particular experiment he developed his now famous
curve.
Gaussian distributions are also called normal distributions or bell curves. The term bell
curve obviously arose from the symmetrical curved shape of the Gaussian distribution while
the name normal distribution is related to one of the mathematical requirements of a Gaussian distribution; that it be normalized. In practice all three terms are used interchangeably
and refer to a symmetrical distribution of values caused by random error. There are other
types of distributions used in statistical analysis which differ from Gaussian distributions
(e.g. Poisson distributions or binomial distributions). We will confine our interest solely to
Gaussian distributions as they are applicable to the type of experimental work we will do
this semester.
Any time measurements are made they are subject to some degree of uncertainty. This
is known as experimental error. When the error is small and random (as opposed to systematic) then it can be shown that the repeated measurements of the same quantity will
cluster around the actual or true value of the measurement. Further, it can be shown that
distribution of values around the true value will be symmetrical and have a curved shaped
(the bell curve).
The "actual", "best", or "true" value of a set of measurements is, as you probably already
know, called the average or mean. It can be shown mathematically that this is the case,
however, we will be content at this time to simply accept this fact.
If we denote a measurement as x, then the average of a set of x’s can be calculated as:
Average = x̄ =
N
X
xi
i=1
N
(1.1)
BACKGROUND
3
The width of the distribution curve about the average is also important. If it is narrow
then most values are close to the average. If it is wide then the values are not as close
to the average. A narrow distribution indicates higher precision (lower error) than a wide
distribution.
The width of a Gaussian distribution is denoted by its standard deviation. Standard deviation represents the average difference of a given measurement from the value of the mean.
If you look closely at the equation for calculating standard deviation you can see that you
are taking the average of the square of the differences between the data points in the set and
the average of the set. If the standard deviation is numerically large then the distribution is
wide. If it is numerically small then the distribution is narrow. Naturally then the standard
deviation is a measure of experimental error.
A Gaussian distribution (shape of the curve) is expressed mathematically as
"
#
−(x − x̄)2
N = Nmax exp
,
2σ
where N
Nmax
x̄
σ
=
=
=
=
the number of times a given measurement is observed,
the greatest number of times a single measurement is observed,
average value of all measurements,
standard deviation, defined below.
σx2 = lim
N →∞
where N
xi
(1.2)
N
1 X
(x̄ − xi )2 ,
N i=1
= total number of measurements,
= results of the ith measurement.
(1.3)
4
LAB 1. STATISTICS
There is an interesting and useful relationship between a Gaussian distribution and its
standard deviation. In any Gaussian distribution 68% of the total area under the curve
falls in the region from x̄ − σ to x̄ + σ. 95% of the total area falls between x̄ − 2σ to x̄ + 2σ.
This tells us that if an experiment is done to measure some quantity, although the result is
not duplicated exactly each time, there will be a definite pattern in the results. First there
will be a tendency toward some central value (x̄). Second, 68% of all values will fall within
one standard deviation of x̄ and 95% of all values will fall between 2 standard deviations of
x̄. Thus, it is reasonable to associate σ with the uncertainty in a given set of measurements
and to view the Gaussian distribution as a probability curve.
For experiments done in the lab, use the following definition of standard deviation σx . (On
your calculator this function is usually denoted as σn−1 .)
σx2 =
N
1 X
(x̄ − xi )2
N − 1 i=1
(1.4)
It can be shown that the standard deviation of a data set is related to the probable error
in the value of the average for the data set. The derivation of this is too complex to be
considered at this time, but the result is worth knowing. The standard deviation of the
mean σm is given by
σx
σm = √
(1.5)
N
√
Note the factor of 1/ N present. This tells us that the more measurements we take, the
lower the standard deviation of the mean, and therefore the more confident we are in that
mean. This leads us to a very simple conclusion: More measurements = More accurate
results. All of your results should be presented with uncertainties (where possible), with
the uncertainty being represented by the standard deviation of the mean. Thus
x̄ ± σm
(1.6)
BACKGROUND
5
An example: Consider the measurement of a steel rod using a micrometer (accurate to .001
cm). The results are listed below.
length xi (cm)
10.351
10.354
10.350
10.351
10.350
10.352
10.352
10.349
10.352
10.349
(x̄ − xi ) × 10−3 (cm)
0
-3
1
0
1
-1
-1
2
-1
2
10
P
xi
10 = 10.351 cm
(x̄ − xi )2 × 10−6 (cm2 )
0
9
1
0
1
1
1
4
1
4
10
P
i=1
(x̄ − xi )2 = 22 × 10−6 cm2
i=1
−5 1
−3
2
The standard deviation is: σx = [ 2.2×10
10−1 ] = 1.6 × 10 cm
The standard deviation of the mean is: σm =
σx
√
N
=
0.002
√
10
= 5 × 10−4
Therefore, the length of the rod is: 10.3510 ± .0005 cm.
Chauvenet’s Criterion
Another important consideration in experimental work is judging the validity of data. Inevitably when doing experiments data is generated which upon inspection seems to be
incorrect. If mistakes have been made during the experiment then it is easy to discard the
data, however, if the experimental process is found to be valid then one must treat the data
very carefully. More than one Nobel prize in physics has been awarded because someone
would not throw away what was considered to be "bad data" at the time.
We will use Gaussian distributions as probability distributions in assessing the validity of
data. As an example consider the following set of measurements of some quantity:
3.9, 3.5, 3.7, 4.0, 3.4, 4.1, 1.9
At first glance the value 1.9 appears to be incorrect when compared to the other values.
Can the value of 1.9 simply be disregarded because it differs from the rest? If no evidence of
malfunction in equipment and/or mistake in procedure can be found then one must provide
a justification for the rejection of data. One school of thought suggests that rejection of
data is never justified. A somewhat more practical solution is to assess the probability that
the suspect data is correct (Chauvenet’s criterion).
6
LAB 1. STATISTICS
In order to use Chauvenet’s criterion the student must be comfortable with the concepts
of average and standard deviation and have realized that a Gaussian distribution is in fact
a probability curve. The first step in assessing the data is to calculate the average and
standard deviation of the data (including the suspect value of 1.9).
x̄ = 3.50 σx = 0.75
Knowing the magnitude of the standard deviation makes it possible to calculate the difference between the suspect value (xsus ) and the average value in terms of the number of
standard deviations (z).
z=
|xsus − x̄|
|1.9 − 3.5|
=
= 2.1
σx
0.75
Recall that in a Gaussian distribution 68% of all measurements fall within 1 standard
deviation of the mean, 95% within 2 standard deviations of the mean and so on. Another
way of saying this is that 32% of all values will be outside one standard deviation from
the average value and only 5% of all values will be farther away from the average than two
standard deviations. In other words the farther away a value is from the average value, the
less probable it is to be present in the data set.
It is possible to calculate the exact probability if one has a table or a computer to provide
the values. Table 1 lists the area under a Gaussian curve as a function of the width measured
in standard deviations. Let’s call the probability of being part of this area, Prob(inside).
Since all of the area under the curve represents 100% of the possibilities then the probability of being outside a certain portion of the curve (Prob(outside)) would simply be 1 Prob(inside). For our example we get 96.43% or .9643 from Table 1, meaning that 96.43%
of all our data should be within 2.1 standard deviations of the average.
Prob(outside 2.1σx ) = 1 − Prob (inside 2.1σx )
= 1 − .9643 = .0357 ≈ .036.
Therefore only 3.6% of all measurements would be farther away from the average.
Now the important part. In experimental work there is no such thing as part of a measurement. You either make a measurement or you don’t make a measurement. How many
measurements do you need to make in order that one measurement represents 3.6% of the
total number of measurements?
The number .036 suggests that if there were 28 (1/.036) measurements we should expect
to see a value of 1.9, however, in a set of 7 measurements we would expect only .25 of a
measurement (.036 × 7) to have a value around 1.9. We must therefore decide whether
.25 is sufficient to indicate that 1.9 is an "improbable" value for this set of measurements.
Chauvenet’s criterion sets the value of "improbability" at .5. In other words if the probability
of a value falling outside a given number of standard deviations (Prob (outside)) multiplied
BACKGROUND
7
by the total number of measurements is less than .5 then the data may be rejected.
The process for using Chauvenet’s criterion is summarized as:
1. Calculate x̄, σx using all the data.
2. Find the difference d between your suspect point xsus and x̄.
d = |xsus − x̄|
3. Express this difference d as a multiple z of the standard deviation σx .
d = zσ, z = d/σ
4. Find z on the table provided, and record the percentage shown. Call this percentage
P(inside).
5. Calculate P(outside), where P(outside) = 1 - P(inside).
6. Multiply P(outside) by the number of data points in the set (N ).
If this number is ≥ .5 the data point(s) is good.
If this number is < .5 reject the data point(s).
NOTE: If data is rejected a new x̄, σx must be calculated based on the remaining data.
An Alternate Approach......
Suppose you have 500 data points of which 35 are suspicious. Do you need to use Chauvenet’s Criterion on each point? No, Chauvenet’s Criterion is based on the number of data
points in the set. Begin by calculating P(outside) from
P(outside) =
.5
number of data points
and then work backwards to calculate the bounds of σx for valid data. The steps for this
procedure are summarised below.
1. Calculate x̄, σx using all the data.
2. Calculate P(outside), where P(outside) =
0.5
N .
3. Calculate P(inside) = 1 - P(outside).
4. Find P(inside) on the table provided, and record the corresponding z. P(inside) will
typically lie between 2 values on the table. In these cases, choose the value that
corresponds to the higher z score.
5. Find the range of acceptable values of x:
Minimum x: x̄ − zσx
Maximum x: x̄ + zσx
Any measurements outside this range of acceptable values can be rejected.
8
LAB 1. STATISTICS
Table 1.1: Gaussian Probability: Area under the Gaussian Curve
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.00
7.97
15.85
23.58
31.08
38.29
45.15
51.61
57.63
63.19
0.80
8.76
16.63
24.34
31.82
38.99
45.81
52.23
58.21
63.72
1.60
9.55
17.41
25.10
32.55
39.69
46.47
52.85
58.78
64.24
2.39
10.34
18.19
25.86
33.28
40.39
47.13
53.46
59.35
64.76
3.19
11.13
18.97
26.61
34.01
41.08
47.78
54.07
59.91
65.28
3.99
11.92
19.74
27.37
34.73
41.77
48.43
54.67
60.47
65.79
4.78
12.71
20.51
28.12
35.45
42.45
49.07
55.27
61.02
66.29
5.58
13.50
21.28
28.86
36.16
43.13
49.71
55.87
61.57
66.80
6.38
14.28
22.05
29.61
36.88
43.81
50.35
56.46
62.11
67.29
7.17
15.07
22.82
30.35
37.59
44.48
50.98
57.05
62.65
67.78
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
68.27
72.87
76.99
80.64
83.85
86.64
89.04
91.09
92.81
94.26
68.75
73.30
77.37
80.98
84.15
86.90
89.26
91.27
92.97
94.39
69.23
73.73
77.75
81.32
84.44
87.15
89.48
91.46
93.12
94.51
69.70
74.15
78.13
81.65
84.73
87.40
89.69
91.64
93.28
94.64
70.17
74.57
78.50
81.98
85.01
87.64
89.90
91.81
93.42
94.76
70.63
74.99
78.87
82.30
85.29
87.89
90.11
91.99
93.57
94.88
71.09
75.40
79.23
82.62
85.57
88.12
90.31
92.16
93.71
95.00
71.54
75.80
79.59
82.93
85.84
88.36
90.51
92.33
93.85
95.12
71.99
76.20
79.95
83.24
86.11
88.59
90.70
92.49
93.99
95.23
72.43
76.60
80.29
83.55
86.38
88.82
90.90
92.65
94.12
95.34
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
95.45
96.43
97.22
97.86
98.36
98.76
99.07
99.31
99.49
99.63
95.56
96.51
97.29
97.91
98.40
98.79
99.09
99.33
99.50
99.64
95.66
96.60
97.36
97.97
98.45
98.83
99.12
99.35
99.52
99.65
95.76
96.68
97.43
98.02
98.49
98.86
99.15
99.37
99.53
99.66
95.86
96.76
97.49
98.07
98.53
98.89
99.17
99.39
99.55
99.67
95.96
96.84
97.56
98.12
98.57
98.92
99.20
99.40
99.56
99.68
96.06
96.92
97.62
98.17
98.61
98.95
99.22
99.42
99.58
99.69
96.15
97.00
97.68
98.22
98.65
98.98
99.24
99.44
99.59
99.70
96.25
97.07
97.74
98.27
98.69
99.01
99.26
99.46
99.60
99.71
96.34
97.15
97.80
98.32
98.72
99.04
99.29
99.47
99.61
99.72
3.0
3.5
4.0
4.5
5.0
99.73
99.95
99.994
99.999
99.999
-
-
-
-
-
-
-
-
-
BACKGROUND
9
Linear Regression
Linear regression or least linear squares fitting provides a method by which the "best"
straight line fit for a set of data may be calculated. Linear regression does not determine if
the relationships in a data set are linear. That must be established by some other means
(usually from theory). For example the speed of an object undergoing constant acceleration
will change linearly with respect to time. A plot of speed vs. time would therefore be a
straight line. Linear regression could be used to determine the slope and intercept of the
plot. The fact that the plot is linear is a consequence of the physics of the situation, not the
use of linear regression. If the plot where a curve rather than linear it would be meaningless
to find a slope and intercept using linear regression.
If we wish to fit a straight line through a set of data in which x is the independent variable
and y the dependent variable the following must be true:
• each y value must be governed by a Gaussian distribution with standard deviation of
σy .
• the uncertainty in x must be small.
• the uncertainties in y must be uniform.
• the relationship between x and y must be linear, i.e. y = mx + b.
Deriving the equations for linear regression is beyond the scope of Physics 1000. They are
developed by applying probability techniques to Gaussian distributions. The results of the
derivation, called the normal equations, allow the calculation of the best fit slope and the
intercept of the plot.
INTERCEPT =
(
P
xi 2 )(
P
SLOPE =
N(
P
yi ) − (
∆
P
xi yi ) − (
∆
P
xi )(
P
xi )(
P
yi )
x i yi )
(1.7)
(1.8)
where
∆=N
X
xi 2 − (
X
xi )2
You will use these equations in part II of this exercise.
(1.9)
10
LAB 1. STATISTICS
Procedure: Part I
This exercise requires two sets of data, which are the results of measurement. Each set
must be measured or estimated to the indicated tolerance.
Set #1. The height of each student in Physics 1050 (to the nearest cm).
Set #2. The diameter of the metal plate located at the front of the lab (to the nearest
mm).
In the first case you will be provided with additional data from previous classes.
Analysis: Part I
1. For each set of data calculate the average, standard deviation, and standard deviation
of the mean. Plot a distribution (histogram) of the number of repetitions of a value
vs the range of values.
2. Use Chauvenet’s criterion to determine the "validity" of the extreme points in the
distributions.
3. Answer the following questions:
(a) How is the meaning of "average" different for these two sets of data? What does
the average actually represent in each case?
(b) Should the distribution for set #1 appear Gaussian? Think of the different
"types" of people measured, and if their heights are the same. How should the
distribution appear if these different types are considered? Is this represented in
your results?
(c) Locate the definitions of random error and systematic error (see Appendix B).
Describe the effects of these two types of errors on a Gaussian distribution.
(d) When you apply Chauvenet’s Criterion to set #1 are you assessing the "validity"
of the extreme points in the distribution (think about your answer to part (a))?
Procedure: Part II - Linear Regression
The data set below represents the position of a moving object with respect to a fixed point.
Beneath each position measurement is the corresponding time of the measurement. This
relationship is known to be linear (y = mx + b).
Time (s)
Position (m)
0.00
6.50
1.00
10.7
2.00
19.9
3.00
21.3
4.00
26.9
5.00
32.7
6.00
38.2
7.00
44.1
8.00
49.5
ANALYSIS: PART II
11
Using a spreadsheet program, plot the data (position vs time) using a scatter plot, making
sure to NOT join the points. Print this plot, and draw on your best estimation of the
best-fit line using a ruler and pen.
Analysis: Part II
Determine the slope and intercept of your hand-drawn line, and clearly label them in your
report as the estimated slope and intercept. To do this, simply pick two points on your line
for which you can easily determine the position and time coordinates. Use m = rise / run
to find the slope, and use this m value in conjunction with one of your points to find the
intercept b using the equation y = mx + b.
Now using the equations contained in the section on linear regression (equations 1.8 and
1.7), calculate the slope and intercept of the actual best-fit line. You will need to construct
a suitable table to find all the necessary sums, which is much easier if a spreadsheet program
is used.
Compare your estimated slope and intercept with those calculated by linear regression and
comment.
What are the physical interpretations of slope and intercept in this case?
LAB
2
Free Fall
Objective
The objectives of this experiment are to show that a freely falling body has constant acceleration, g, and to determine a value for g.
Background
The term free fall implies a one dimensional motion for which the motivating force is solely
that of gravity. An object falling freely under the influence of gravity in vacuum has
an acceleration that is constant, therefore, as the object falls its velocity increases at a
constant rate. Any object that moves in a straight line, with equal changes in velocity in
equal intervals of time, is said to be moving with uniform, or constant, acceleration.
It should be noted that the terms velocity and speed are not exactly interchangeable. Velocity is a vector, meaning that values for both magnitude and direction are required to
fully describe the motion. Speed is by definition the magnitude of the velocity vector. For
this experiment we will use the term velocity since the object will experience changes in
both speed and direction.
From an experimental point of view the easiest way to determine if the acceleration of an
object is constant is to plot its velocity as a function of time. If the plot is linear then
the equation which governs the plot is velocity = (acceleration)(time) + initial velocity
(recall the equation of a straight line y = mx + b). The slope of the plot is the value of the
acceleration while the intercept represents the velocity of the object when the experiment
began.
12
PROCEDURE
13
The definition for average velocity is
vave =
Displacement
∆~x
=
Time
∆t
(2.1)
where ∆ means "change in". Equation 2.1 defines "average velocity over an interval of
time". A useful point to remember in situations where the acceleration is constant is that
the average velocity over some interval of time is equal to the instantaneous velocity at the
mid point (in time) of that interval.
In this experiment we will toss a large rubber ball vertically into the air and track its
position with a sonar system. The ball rises about two metres and then falls. The speeds
associated with these displacements are low enough that air resistance should not be a
significant factor.
Regardless of the fact that the ball first travels upward and then downward the only force
acting on the ball after it is tossed is the force of gravity. If the acceleration of the ball
is constant then we should see a linear plot of velocity versus time despite the change in
direction.
The motion of the ball is tracked by a sonar system that can accurately measure the location
of the ball by measuring the time of flight of an ultra sonic sound pulse reflected off the
ball. Since the sound pulses are sent at regular intervals of time, ∆t is a constant.
Procedure
The sonar apparatus will be set-up at the front of the lab. Practice tossing the ball vertically
upward such that it rises and falls in a straight line above the sonar device. When you are
ready, start the DataStudio program and proceed as follows:
1. Click the "Setup" button, and ensure that the motion sensor is displayed. If not, click
"Add Sensor" and find the motion sensor in the list.
2. Click the "Start" button to begin collecting data. You should hear a ticking sound
while the sonar system is collecting data.
3. Position the ball over the sensor and drop it from rest. Make sure you are prepared
to catch the ball before it hits the sensor. It may help to have one lab partner drop
the ball while the other catches the ball.
4. Toss the ball into the air, allowing it to rise and fall, catching it before it hits the
sonar system (keep your arms to the side so as not to interfere with the sonar beam).
Repeat this step several times to get many trials in your data file.
5. Once you are satisfied with the data use the "Export Data" entry in the "File" menu
to save your data to a suitable medium.
14
LAB 2. FREE FALL
Open your file in a spreadsheet program, plot all the data, and select the portions of data
which correspond to the flight of the ball. The portion you use should appear as a smooth
parabola, with few erroneous points. Ultimately, the data that you keep will be decided by
the velocity graph, so don’t be too picky about where the data starts and ends. Transfer
the points you select by copying and pasting the appropriate entries into a new spreadsheet
file or worksheet within the same file.
Analysis
We wish to construct a graph of velocity vs. time for each case. The first step is to construct
a data table like the one below:
Time
0
t1
t2
..
.
Position
y1
y2
y3
..
.
Velocity
v1 = (y2 − y1 )/∆t
v2 = (y3 − y2 )/∆t
v3 = (y4 − y3 )/∆t
..
.
Acceleration
a1 = (v2 − v1 )/∆t
a2 = (v3 − v2 )/∆t
a3 = (v4 − v3 )/∆t
..
.
tn−2
tn−1
tn
yn−2
yn−1
yn
vn−2 = (yn−1 − yn−2 )/∆t
vn−1 = (yn − yn−1 )/∆t
an−2 = (vn−1 − vn−2 )/∆t
The position and elapsed time columns in this data table are the data you selected for the
flight of the ball. Reset the time to start at zero. Each successive time value will then be ∆t
greater than the previous one, since the change in time will be constant in this experiment.
∆t is determined by the frequency of the sensor, and should be easy to determine from your
initial data. Calculate the velocity as indicated above by calculating the displacement of the
ball as the difference in two successive positions of the ball and dividing by the corresponding
change in time. The acceleration column will be calculated in a similar manner. Be sure
to include appropriate units for all columns, and sample calculations for your velocity and
acceleration columns. Also note that there are blank spaces in the bottom right cells of this
table; you should attempt to understand why this is the case.
Create one table for each case done (2 total), and clearly label which table corresponds to
the dropped ball, and which corresponds to the tossed ball. Each table is created in an
identical manner, using different data.
Plot the velocities against elapsed time and determine the acceleration due to gravity from
the slope of the graph (use linear regression). Your graph should appear as having a constant
downward slope. If there are portions of your graph which have an upward slope, these do
not correspond to free fall; simply remove the necessary points from your table until there
are no upward sloping portions (the graph will automatically update when removing these
points). Your graph should have 2 series (lines), one for each case (dropped and tossed).
DISCUSSION
15
Make sure each case is clearly labelled on your graph. If you need help adding 2 series to a
graph, ask your instructor.
Discussion
Compare your experimental values for g with the accepted value and compute the percentage difference in your results. The percentage difference is defined as %difference =
|actual - measured|/actual, where actual and measured refer of course to the actual value
and the measured value. Comment on the disparity in your result and the actual value;
what do you feel may have affected your results? Which case gave better results? The
answer to this question may help you identify any errors in the procedure.
Without repeating the experiment, can you determine from your data if air resistance is a
problem in this experiment? Think of how air resistance would affect your graph, noting
that the acceleration due to gravity is towards the ground, but air resistance always opposes
the direction of motion.
LAB
3
Projectile Motion
Objective
The objectives of this experiment are to show that motion in two dimensions can be treated
independently in each dimension and to determine a value for g, the acceleration due to
gravity.
Background
Projectile motion, or trajectory motion as it is sometimes called, refers to the two dimensional motion of an object projected into the air at an angle where gravity is the only force
(e.g., throwing a baseball or basketball). Analysis of this motion is based on the premise
that the vertical and horizontal motions are independent of each other. In order to verify
this premise, the trajectory of a steel ball will be analyzed in both the vertical (y) and the
horizontal (x) directions. By way of illustration we consider the following example (that is
different from the experiment that is actually performed).
Figure 3.1: Independent Motion
A ball is thrown into the air (Fig. 3.1, point A) having velocity in both the horizontal
and vertical directions. At the exact moment it reaches its maximum height (point B) and
16
BACKGROUND
17
begins to fall earthward, an identical ball is dropped from the same height (point C).
If the horizontal motion is actually independent of the vertical motion, the balls will strike
the ground at the same instant. One expects this to be the case since the only force acting on
the balls is gravity (neglecting air resistance) and as a result, Newton’s second law predicts
that the horizontal motion of ball 1 will remain at its initial constant velocity while both
balls will accelerate downward with acceleration g.
A strobe photograph of the falling balls would therefore reveal the following.
The equations describing projectile motion are easily derived. Beginning with a background
in one dimensional motion (see Free Fall experiment) and applying the requirements for
projectile motion, a complete description of the trajectory, maximum height, and maximum
range of a projectile can be found as follows.
Given that the motion of a projectile is two separate and distinct one dimensional motions,
and after launch the only force acting on the projectile is the force of gravity, start by using
the equations for one dimensional motion separately for the vertical and horizontal motions
of the projectile.
Vertical Motion (y axis)
Horizontal Motion (x axis)
Acceleration = constant = −g
y = y0 + v0y t − 1/2gt2
vy = v0y − gt
Acceleration = 0
x = x0 + v0x t
vx = v0x
The path of the projectile (trajectory) is a set of points (x,y) which can be calculated by
combining the two one dimensional equations for x and y. For simplicity use the origin (0,0)
of the coordinate system as the launch point of the projectile, therefore x0 = y0 = 0. At
launch the projectile’s path will have an angle θ0 with the x axis, therefore v0x = v0 cos θ0
and v0y = v0 sin θ0 . To describe the trajectory of the projectile simply use the expression
for the x coordinate to define t (equation 3.2) and then substitute this definition for t into
18
LAB 3. PROJECTILE MOTION
the expression for the y coordinate (equation 3.3).
x = v0x t
x
t=
v0x
(3.1)
(3.2)
1
y = v0y t − gt2
2
(3.3)
By substituting equation 3.2 into 3.3 as described above, the latter equation becomes
y = v0y
x
v0x
1
x
− g
2
v0x
2
,
(3.4)
This equation is made more useful by replacing v0x and v0y by their equivalents in terms of
v0 ,
y = v0 sin θ0
x
v0 cos θ0
1
− g
2
x2
(v0 cos θ0 )2
!
x sin θ0
gx2
− 2
cos θ0
2v0 cos2 θ0
gx2
= x tan θ0 − 2
2v0 cos2 θ0
=
(3.5)
The maximum height and range equations of a projectile can be derived in a similar manner.
At its maximum height the vertical velocity of a projectile will become zero. The time at
which this occurs can be defined from vy = v0y − gt by setting vy equal to zero.
0 = v0y − gtymax
v0y
v0 sin θ0
∴ tymax =
=
g
g
(3.6)
Substituting this value of t into the expression for y (eq. 3.5) results in an expression for
the maximum height of the projectile.
1
y = v0y t − gt2
2
v0 sin θ0
1
v0 sin θ0 2
ymax = v0 sin θ0
− g
g
2
g
2
2
2
2
v sin θ0 v0 sin θ0
= 0
−
g
2g
2
2
v sin θ0
= 0
2g
(3.7)
PROCEDURE
19
The maximum range of the projectile occurs (assuming launch and landing height are the
same) when the trajectory passes through the x-axis. This occurs at a value of t which is
two times the value of t for the maximum height calculation, txmax = 2tymax .
2v0 sin θ0
g
2
v 2 sin θ0 cos θ0
= 0
g
2
v sin 2θ0
= 0
g
xmax = v0x txmax = (vo cos θ0 )
(3.8)
Note that 2 sin θ cos θ = sin 2θ.
Procedure
All necessary equipment is assembled in the dark room. You will be instructed in its
operation and will obtain a digital photograph of a projectile.
1. Record the flash rate (flashes/minute) from the dial of the strobe light.
2. Have one group member take the picture while the other drops the ball through the
tube. With the strobe light flashing, depress the shoot button on the camera. You
will hear a beep shortly thereafter, indicating that the multi-second exposure is now
being taken. Immediately upon hearing this beep, the ball should be dropped into
the tube. The camera will beep again when the exposure has finished.
3. Check your picture to ensure that the path of the ball was in fact recorded. Hit the
replay button and use the zoom buttons on the camera to do this. Your picture should
appear similar to Fig. 3.2. If it does not, try again.
4. Once satisfied, note the file name of your picture.
5. Once each group has taken a picture, your lab instructor will distribute your picture
to you in some fashion (either directly downloading it to your computer, or emailing
it to you).
Figure 3.2: A sample strobe picture depicting the parabolic flight of a ball in projectile
motion.
20
LAB 3. PROJECTILE MOTION
6. Open the picture using the Paint program. Find the first image of the ball as it
leaves the tube and place the mouse pointer over it. The coordinates of the pointer
will appear just underneath the bottom right corner of your picture, as an x,y pair.
Record these numbers.
7. Repeat the previous step for each successive image of the ball. You should get at least
12 points. Make sure to record the coordinates in sequential order (as the ball travels
from left to right)!
8. Place the mouse pointer at the top of the metre stick shown in your picture. Record
just the y coordinate, the x coordinate is inconsequential.
9. Repeat the previous step at a point 10 cm below the top of the metre stick. The
stick should have alternating bands of colour for every 10 cm section to facilitate this
measurement. Repeat this procedure until you have 6 total coordinates.
Analysis
Enter your data into a spreadsheet as 2 columns. Create 2 new columns that have the
first point scaled to (0,0). To do this simply subtract the first x coordinate from all the
x coordinates, and subtract the first y coordinate from all the y coordinates. This is very
easy to do using a spreadsheet equation; don’t resort to using a calculator! Your instructor
can give you assistance should you require it.
Construct the following table using your scaled set of coordinates as the first 2 columns.
You do not need to include your unscaled set of data in your report.
x (pixels)
0
y (pixels)
0
x (cm)
0
y (cm)
0
vx (cm/s)
v1x
vy (cm/s)
v1y
ax (cm/s2 )
a1x
ay (cm/s2 )
a1y
In order to convert your coordinates from pixels to cm, you will need to find a scale factor.
To do this, use your results from the measurement of the metre stick. You should have
6 different y-coordinates corresponding to the ends of 5 separate 10 cm sections of ruler.
The difference between these coordinates is the number of pixels per 10 cm. Simply find
the difference between successive coordinates (which should give you 5 similar numbers),
average all your results together, and use this result as the scale factor. Include all your
calculations. Once you have the scale factor, you should be able to easily convert your pixel
values into cm. Writing out the correct unit conversion might help.
The velocity and acceleration of the projectile are calculated by dividing the change in two
successive positions of the projectile by the corresponding elapsed time. The time between
flashes is gained from the strobe rate that you recorded. Note that this strobe rate is given
DISCUSSION
21
in flashes/minute, which is an example of a frequency. What you require is seconds/flash; a
simple unit conversion and a little ingenuity will give the correct ∆t. Also note that similar
to the free fall lab, you will have 1 less velocity value than position, and 1 less acceleration
than velocity. This will create blank spaces in the bottom right of your table.
From your data and calculations construct the following graphs:
Graph 1
Horizontal position vs. elapsed time.
Vertical position vs. elapsed time.
Graph 2
Horizontal velocity vs. elapsed time.
Vertical velocity vs. elapsed time.
Graph 3
Horizontal acceleration vs. elapsed time.
Vertical acceleration vs. elapsed time.
Perform linear regression on both series of your velocity graph. What do the slopes tell you
in each case?
From your data determine v0 and θ0 . As an initial guess for these values, use your first values
of vx , vy from your table as v0x and v0y and use simple trigonometry. Using the equation for
trajectory, equation 3.5, add a new column to your table that finds the theoretical y-value
based on your x-values (in cm). Construct a plot of the trajectory of your projectile (y
vs. x), with 2 series, one for the actual y values, and one for the theoretical y values. Use
equations 3.7 and 3.8 to find the theoretical maximum height and range of the projectile.
Extract the experimental values from your trajectory graph as best you can. Note that the
maximum range is the horizontal distance covered when the ball crosses y = 0.
Discussion
From your results, what can you conclude about the independence of motion in the vertical
and horizontal direction? Consult your free fall lab results. Do your results in the vertical
direction appear similar (or identical) to the free fall lab? What exactly does this prove, if
anything?
You should have found a value for g in your analysis; how does it compare to the actual
value? Is this result better or worse than previous values you have found in the lab? If
your value for g is not within error of the actual value, what do you think is responsible for
this? Think about how your scale factor and value for ∆t are involved in this value; what
if they’re incorrect? What effect would this have on your results?
Comment on your trajectory graph, noting the agreement or disagreement between your
theoretical plot and the experimental plot. Adjust your values for v0 and θ0 until you arrive
at better agreement between the 2 plots. Did you have to increase or decrease the values
22
LAB 3. PROJECTILE MOTION
to arrive at better agreement? Does this make sense? Think about the initial guesses for
v0 and θ0 ; why would these guesses be not quite correct? Hint: Consider the actual time
at which the velocity values from your table occur.
LAB
4
Centripetal Acceleration
Objective
The objective of this experiment is to determine the mass of an unknown object by calculating the centripetal acceleration it imparts to a puck orbiting on an air table.
Background
In order for an object to maintain a stable circular orbit about some fixed point it must
be constantly accelerated toward that point. This acceleration is known as centripetal
acceleration and is characterized by the expression,
a=
v2
r
(4.1)
where v is the speed of the object and r is the radius of its orbit.
In this experiment the force necessary to supply the required centripetal acceleration is
provided by a weight hanging under the air table on a cord that passes through a hole in
the table and is attached to the puck (see Figure 4.1).
By determining a value for the centripetal acceleration of the system you can calculate the
mass of the weight.
Neglecting friction, we observe that the only force acting on the puck is the tension T that
is directed radially inward. Hence, by Newton’s second law (the vector sum of the forces
acting on a body is equated to the mass of that body multiplied by its acceleration) we
have
T = Ma
23
(4.2)
24
LAB 4. CENTRIPETAL ACCELERATION
Figure 4.1: A puck attached via string to a hanging mass through a hole in the air table.
The puck is travelling in a circular orbit around the central hole, with the tension in the
string providing the necessary centripetal acceleration.
where M is the mass of the puck. Since the puck is undergoing circular motion we can
substitute Eq. 4.1 for the acceleration, i.e.,
T =
M v2
r
(4.3)
If we now apply Newton’s second law to the unknown hanging mass m, we obtain the
equation
T = mg
(4.4)
Combining Eqs. 4.4 and 4.3 by eliminating T , we obtain an equation that permits us to
determine m, the unknown mass.
M v2
m=
(4.5)
rg
In order to determine the radius of the orbit r and the speed of the puck v as it orbits the
central hole in the air table, we will use a camera and strobe light to take record images of
the puck at regular time intervals.
Figure 4.2: The information to be gained from the strobe photograph. Each dot represents
the position of the centre of the puck at each flash of the strobe light. The C variable shown
represents the straight-line distance between successive images of the puck.
PROCEDURE
25
Figure 4.3: The relation between an isosceles triangle of sides r − C − r, and the arc length
s of a circular segment of radius r and angle θ.
The average speed v in Eq. 4.5 can be found by dividing the arc length s of the circular
orbit between two images by the corresponding elapsed time. In other words,
s
v=
(4.6)
∆t
The arc length s is found using the equation s = rθ, where θ is the angle between 2 successive
points. Note that this s is not the same as the straight line distance C between successive
points as shown in figure 4.2. The angle θ can be found using the values of C and r as
follows. Consider the triangle r − C − r, shown in Fig. 4.3. From the law of cosines, we can
write the relation between the variables r, C, and θ shown in this figure as
C 2 = 2r2 − 2r2 cos θ
(4.7)
If r and C are known, then θ can be found using this equation. This will subsequently give
us values for s using the equation s = rθ, and therefore v using equation 4.6.
Procedure
In order to use Eq. 4.5 to determine the unknown mass m it is necessary to determine the
speed v, the mass M , and the orbital radius r of the puck. To determine M use one of the
top loading balances. To find r and v, obtain a photograph of one revolution of the puck
(illuminated by a strobe light) as follows.
1. Record the frequency of the strobe light. Note that the strobe light gives the strobe
rate in flashes/minute; you may have to convert this to other units.
2. Measure the diameter of the puck, preferably with calipers.
3. Turn on the air supply and ensure that the puck glides smoothly over the surface of
the air table. Practice throwing the puck into its orbit with constant velocity, making
sure that the puck does not touch the sides of the air table. Note that you will not
have to throw the puck very hard in order to do this.
26
LAB 4. CENTRIPETAL ACCELERATION
4. Take a picture within the first 3 revolutions of the puck, leaving the shutter open
for slightly less than one revolution. The shutter speed should be set to roughly 3
seconds.
5. When a satisfactory photograph has been obtained, transfer the photograph on to one
of the computers and open the file with Paint.
6. Determine the scale factor by using the radius of the puck. Make 5 measurements
of the radius in pixels using the mouse pointer, and find the average and standard
deviation. Comparing this to your measured value in cm for the radius of the puck
will give the scale factor (the conversion ratio between pixels and cm).
7. Centre the mouse pointer on the central hole of the air table through which the string
passes and record its coordinates.
8. Centre the mouse pointer on the centre of each image of the puck, and record the x
and y positions as shown in Paint. It is important to record the proper sequential
order of the images; if you need assistance determining which is the first and last
image, ask your lab instructor.
Analysis
Scale your position data for the puck so that the centre of the air table becomes the origin.
To do this, simply subtract the coordinates of the center of the table from each point.
This involves subtracting the x coordinate of the centre from each x coordinate measured,
and performing a similar procedure for the y coordinates. This is easiest to do using a
spreadsheet table.
In order to find the speed of the puck as it orbits, we need to find values for the radius of
the puck’s orbit and the angle θ traversed by the puck with each successive image. In order
to do this, construct the following table.
Position
x
y
x1
y1
x2
y2
x3
..
.
y3
..
.
∆x
∆Position
∆y
∆x1 = x2 − x1
∆y1 = y2 − y1
q
∆y2 = y3 − y2
..
.
..
.
q
∆x2 = x3 − x2
..
.
..
.
θ
Radius
C
∆x21 + ∆y12
∆x22
+
..
.
..
.
∆y22
q
2
2
θ1
x22
y22
θ2
..
.
..
.
x + y1
q 1
+
..
.
..
.
To find values for θ, use equation 4.7. You will have to rearrange this equation to solve for
θ yourself. Make sure that your angles are given in radians, not degrees. The arc length
formula is valid only if the angle is in radians.
Your values for C, r, and θ should be similar for each entry in your table. Take the average
and standard deviation of each of these columns and present each value as mean ± standard
DISCUSSION
27
deviation of the mean. Convert the C and r values from pixels to cm by using the scale
factor you found.
Using these values for M , C, r, and θ, and equations 4.5 and 4.6, find the mass of the hanging
mass m. Be sure to include an estimate of the uncertainty in your result by performing the
necessary error analysis.
Discussion
How large is the uncertainty in your result? Does this represent a good experimental
design? What do you think could be done to improve it? In particular, what was the least
precise measurement (highest uncertainty) that you took? Could this be measured more
accurately?
Look at your values for C, r, and θ; do they remain the same throughout the picture you
took? Do you notice any trends in the data? If there is, explain why these values would
follow this trend, and what this trend implies physically.
LAB
5
Principles of Mechanics
Objective
The objective of this experiment is to illustrate several principles of mechanics, primarily
concerned with conservation of energy, using a simple collision as an example.
Apparatus
• Ramp
• Clamp
• Ball Bearing, or Mouse Ball
• Newsprint
• Meter Stick(s)
• 2 small pieces of Carbon Paper
Background
This simple experiment provides an opportunity to apply a variety of mechanical principles.
We will roll a ball down an incline and examine its path as it flies, strikes the floor, and
bounces. With a few simple distance measurements we should be able to examine energy,
projectile motion, kinematics, and collisions.
28
PROCEDURE
29
Figure 5.1: A simple experiment to test kinematics and energy, with all the appropriate
measurements.
Procedure
Figure 5.1 depicts the experimental set-up and measurements to be made.
1. Place the ramp on a lab bench with the end of the ramp parallel to the floor (shim
the stand if necessary), and clamp it in place.
2. Measure the heights h and H and record them. Refer to Fig. 5.1 to see exactly what
these heights represent.
3. Measure the mass (m) of the ball using the scales at the back of the lab.
4. Roll the ball bearing down the ramp, allow it to hit the floor and bounce. Place 1 or
2 sheets of newsprint on the floor (tape them down) such that the bearing will strike
the paper on impact and first bounce (points C and D in Figure 5.1). In some cases,
only 1 sheet is required, in most others 2; you’ll simply have to experiment in order
to determine if 1 sheet will be adequate.
5. Place a small piece of carbon paper face down at each point of impact. These do not
need to be taped down.
6. Place the ball on the ramp and release it. The ball should strike the back of each
piece of carbon paper and therefore leave a mark on your newsprint. Verify that this
does indeed occur, if not, then repeat the trial.
7. Repeat the experiment 5 more times, making sure that the ball is released from the
same point on the ramp each time. Measure the horizontal distance to each point
using a metre stick (c and d in Fig. 5.1), and record the distances.
30
LAB 5. PRINCIPLES OF MECHANICS
8. The height of the bounce (e) can then be found by placing a piece of paper mounted on
a book mid-way between points C and D. Place the leading edge of the book exactly
at the midway point, place a piece of carbon paper on the surface of the book (a piece
of tape may be useful in this case), and do 6 more trials. The ball will leave a mark
on the paper at the maximum height. Measure and record these heights.
Analysis
Using kinematics, calculate the following quantities:
1. The time required for the ball to strike the ground after leaving the ramp. Hint: What
is the vertical velocity of the ball as it leaves the ramp?
2. The speed of the ball as it leaves the ramp (point B). Note that this speed is the
horizontal velocity of the ball, which can be calculated using the time from question
1 and the distance the ball travelled before reaching the ground (c).
3. The vertical velocity of the ball as it strikes the ground (point C).
4. The speed at point C.
5. The speed at point D. By examining the path of the ball from the maximum height
of the bounce to point D, this problem is solved in an identical manner to questions
1-4.
Using these values for the speeds, construct a table that lists the kinetic, potential, and
total mechanical energy at points A, B, C, and D. Your table should resemble the following:
Height
Speed
KE
PE
Mech Energy
A
B
C
D
Calculate the speed of the ball at point C using energy considerations. Does it agree with
the result you found using kinematic methods?
Discussion
Is the mechanical energy in this experiment conserved? If not, where is it lost (between A
and B, B and C, etc.)? Is the collision at point C elastic or inelastic?
DISCUSSION
31
To this point we have not considered the energy of rotation as the ball rolls down the ramp.
The kinetic energy of rotation (Krot ) can be calculated as,
1
Krot = Iω 2 ,
2
where I is the moment of inertia ( I = 25 mr2 for a solid sphere), and ω = vr is the angular
velocity. By substituting these expressions for I and ω into the equation for Krot , find
an equation for Krot that is a function of mass and speed. Compare this equation to the
usual kinetic energy equation. Using the speed as the ball leaves the ramp as v, calculate
this rotational energy. Now add this number to the energies at points B and C. Does
the inclusion of the energy of rotation in your calculation reconcile the differences in the
mechanical energies at A, B, and C?
LAB
6
Conservation of Mechanical Energy
Objective
The objective of this experiment is to test the validity of the principle of conservation of
mechanical energy by making independent measurements of potential and kinetic energy in
a system in which frictional forces are negligible or at least very small.
Background
The principle of conservation of mechanical energy states that in an isolated system where
only conservative forces are present, the algebraic sum of the potential and kinetic energies
is a constant. A simple air track is a reasonable approximation of an isolated system.
In this experiment you are free to assemble a system of your own choosing using a combination of gliders, weights, springs, strings, etc. Once assembled, you will put the system
in motion and track the motion of the system with sonar. By knowing the positions and
velocities of all the objects in your system it is possible to calculate the potential and kinetic
energy of the system and therefore investigate the principle of conservation of mechanical
energy.
Consider the following possible system as an example. The system is composed of one glider
and two springs assembled as per Fig. 6.1. If the glider is pulled to one end of the track
and released it will begin to oscillate back and forth as it is pulled first by one spring and
then by the other.
At the beginning of the experiment just before the glider is released all the energy in the
system is in the form of potential energy stored in the extended spring. At the instant of
release, the stored energy begins to transform into kinetic energy in the glider and potential
energy in the second spring as it begins to stretch. As the glider reaches the opposite end
32
BACKGROUND
33
Figure 6.1: A simple system to track conservation of energy using an air track.
of the air track it slows, finally stopping momentarily, then changes direction and returns
along the air track. At the point the glider stops all the energy in the system is stored in
the second spring. The energy in the system continues to be exchanged between the glider
and the springs during subsequent cycles.
To calculate the energy of this system we must know the following:
• The velocity of the glider
• The spring constants of the springs
• The extended lengths of the springs
The velocity of the glider is obtained from the position data provided by the sonar. Using
the definition of average velocity (vave = ∆Position/∆Time) a one difference calculation of
consecutive positions divided by the sampling rate of the sonar yields the velocity of glider.
The kinetic energy of the glider at any point is given by
1
K = mv 2
2
(6.1)
Hooke’s law is used to determine the spring constants k1 and k2 . This law states that the
restoring force on a spring (which is equal in magnitude to the force applied) is proportional
to the elongation of the spring, F = −kx. The spring constant (k) can be thought of as the
stiffness of the spring. If k is large it takes a large force to stretch or compress the spring
(think of the coil springs which support a car 2000 N/M). If k is small the force required to
move the spring is quite small (spring inside a ball point pen 2 N/M). k can be determined
simply by hanging the spring in question from a solid support and measuring its extension
under various weights. The extended length of the spring (x) is the overall length of the
spring under load minus the natural length of the spring (no load). A plot of force versus
the extended length of the spring, x (overall length minus natural length) should yield a
straight line of slope k.
The final piece of necessary information is the extended lengths of the springs during the
experiment. The sonar system records the distance between the sending unit and the sonar
34
LAB 6. CONSERVATION OF MECHANICAL ENERGY
target on the glider at up to 60 times per second. All that is necessary is to subtract all
extra distances from the data. Consider one spring in our example system:
Figure 6.2: A schematic of the different distances required to find the stretched length x.
The constant distances a and b must be subtracted from the reading from the motion sensor
in order to arrive at the actual stretched length of the spring. The stretch x can then be
found by taking this measured length and subtracting the constant unstretched length.
Knowing the extended length of the springs and their spring constants allows you to calculate the potential energy (U ) as follows:
1
U = kx2 ,
2
(6.2)
where the k’s and the x’s are the spring constants and the extended lengths of the springs.
Procedure
Conservation of Energy on an Air Track
This section requires an air track, of which there are only 2 or 3 available. If your group
is waiting to do this section, proceed to the Hooke’s Law section below. You can take the
measurements for that section and even begin analysing the resulting data without using
an air track.
1. As a group design a system to study. You may use one of the examples shown in
Fig. 6.3, or devise something of your own. As you design your system be sure that is
possible to extract all necessary information for calculation from the sonar data. Also
be sure that your system is free to move the entire length of the air track. Note there
are some things you must include in your system:
• At least one spring.
PROCEDURE
35
Figure 6.3: Examples of different systems you can examine using the air track. Feel free to
use any of these, or create your own.
• At least one glider.
• A hanging mass (less than 500g), or a tilted air track, or both.
2. Level the air track unless you are intentionally raising one end. If you are, then
measure the height to the top of the air track itself at both ends, and record these
heights.
3. Assemble your system on the air track and test it for speed (1.5 m/sec is a reasonable
upper limit), make adjustments as necessary.
4. Set up motion sensors as necessary to track the position of any gliders used. One
sensor is necessary for each glider in your setup. The positions of hanging masses can
be extrapolated from your glider data, assuming they are connected to a glider via
string, not a spring.
5. Make appropriate measurements to find the stretch of the spring x, as shown in Fig.
6.2.
6. Be sure to measure any distances, masses, angles, etc. that are important to your
calculations before dismantling your system.
Hooke’s Law
In the system you design to study conservation of energy, you are required to use at least
one spring. In order to obtain the correct amount of potential energy in this spring, you will
need to know the spring constant k. Complete the following steps to measure this constant.
36
LAB 6. CONSERVATION OF MECHANICAL ENERGY
1. First measure the unstretched length of the string. This should be done on a tabletop,
so that the weight of the spring itself does not cause it to stretch.
2. Obtain a stand, a metal rod, and an S-clamp to connect the two. Assemble these so
that the rod is horizontal. Hang your spring from the rod.
3. Obtain and weigh a mass hanger. Hook the hanger onto the end of your spring and
measure the resulting length of the spring (including the hooked ends of the spring).
Your spring may ‘jiggle’ a little, try and dampen this motion.
4. Add some mass to the hanger and repeat the length measurement. Do this for 5 total
amounts of mass (including your original measurement with just the hanger). Do not
exceed 100g of total mass!
Analysis
Using your Hooke’s Law data, find the spring constant for your spring(s). To do this, create
a graph of F vs. l, where F is the force stretching the spring, and l is the length of the
spring. The slope of this graph should be the spring constant. Refer to the Background
section of the manual if you require more information.
Using the spreadsheet program on your computer construct an appropriate data table. The
table should include a column for every quantity that must be calculated. The table shown
below would be appropriate for the system shown in Fig. 6.1 at the beginning of this outline.
xglider
x1
x2
Uspring1
Uspring2
UT otal
vglider
Kglider
ET otal
Your table may not be the same if you used a slightly different system. Make sure to include
the following:
• The kinetic energy of each moving object. A velocity column for each moving object
is optional.
• The potential energy of any object that changes height. This can be a hanging mass,
or a glider if the track is not level.
• The potential energy of each spring used. This requires knowledge of the stretch of
the spring x, which may require a separate column. Refer to Fig. 6.2 for assistance
in calculating the required x values.
Construct the following graphs:
• Plot all potential energies (including total) on a single graph as functions of time.
DISCUSSION
37
• Plot all kinetic energies (including total) on a single graph as functions of time.
• Plot total potential, total kinetic, and total energy on a single graph as functions of
time.
Discussion
Does your system indeed show conservation of energy? If not, where do you believe the
primary losses of energy are? Describe any elements of your graphs that seem incorrect,
and give possible explanations as to why they appear as they do. Are there any sources of
energy you are not measuring?
For your Hooke’s Law graph, what would the x-intercept of this slope represent, theoretically? Note that this physical quantity was actually measured during the experiment;
compare the 2 values. Do they agree? Remark on any disagreement. Why we would choose
not to use this theoretical value, and instead measure it?
LAB
7
Conservation of Momentum
Objective
The objective of this experiment is to study the vector nature of momentum and the principle of conservation of momentum by studying air track and air table collisions.
Apparatus
• Air Track
• Dual Beam Sonar System
• Air Table
Background
The momentum of an object is a vector quantity defined as the product of the mass and
the velocity of the object:
p~ = m~v
(7.1)
In a system of objects (1,2,3,...,n), the total momentum p~tot of the system is simply the
vector sum of the momenta of the individual objects.
p~tot = m1~v1 + m2~v2 + m3~v3 + . . . + mn~vn .
(7.2)
We saw in the conservation of energy experiment that if we isolate a system of objects (with
conditions), the total mechanical energy of that system does not change; it is conserved.
Likewise if we isolate a system of objects, under certain conditions the momentum of that
system will not change. This is the principle of conservation of momentum.
38
PROCEDURE
39
The necessary condition for conservation of momentum is that no external forces may act
on the system. The designation "external" is very important and is linked to something
called impulse. Imagine a one dimensional collision between two rigid objects. For there to
be conservation of momentum there must be no external forces. In other words, once set
in motion no outside agent may interact with the objects; this means no "pushes" or "pulls"
and no friction. The two objects comprise a system for which the total momentum of the
system is the sum of the individual momenta of each object. According to the principle of
conservation of momentum the momentum before the objects collide will be the same as
the momentum after the collision.
p~(total before) = p~(total after)
(7.3)
If you observe one of the objects during the collision you would conclude that its momentum
changes due to the collision, likewise so does the momentum of the other object. There are
two ways view these momentum changes with respect to the principle of conservation of
momentum. First, take the point of view that the forces involved between the objects are
internal to the system, cancel out,and therefore do not alter the total momentum of the
system. Or, rewrite the expression for conservation of momentum to include the forces
that act on individual objects in the system (this is known as the Impulse - Momentum
Theorem).
p~(total before) + J~ = p~(total after)
(7.4)
In this expression J~ represents a quantity that is referred to as the impulse. Impulse is a
force that acts for a given amount of time, like the forces that are present during a collision.
Procedure
We will complete this experiment in three parts:
Part I Part II Part III -
Single one dimensional collisions.
Multiple one dimensional collisions.
An elastic two dimensional collision.
In each part the objective will be to describe the momentum of the objects in the collision,
define the forces involved, and examine the collision(s) for conservation of momentum.
The calculations for these experiments are quite simple and only involve calculating p~ = m~v
for each object in the collision then summing to find the total momentum. From these
calculations we will plot graphs of momentum vs. time for each case.
For parts I and II we will use the air tracks and sonar systems to collect the data. Once
collected the data is transferred to a spreadsheet for analysis. The spreadsheet is ideal for
constructing data tables and graphs.
In each experiment you will need to know the masses of the objects. Use the top loading
40
LAB 7. CONSERVATION OF MOMENTUM
balances located on the counter by the sink. Since the data for part III is on video you will
need to scale the data back to real life scale. This can be done by placing a metre stick in
the video or by measuring the diameter of the puck(s) used in the collision.
The motion of the gliders on the air track is in one dimension, therefore the vector part of
the momentum can be handled with the use of +/- signs. The motion of the pucks on the
air table is in two dimensions and you will need a grid and an origin to define directions
and distances. This is easiest to do when the frame of the video is square with the air table.
Part I
Using the air track create:
• A single elastic collision between two gliders.
• A single inelastic collision between two gliders.
• A single elastic collision one glider and the end of the air track.
Part II
Using the air track create multiple collisions between two gliders and the ends of the air
track. When plotted your position data should look like the plot below.
ANALYSIS
41
The traces represent the distance of each glider from the sonar head as a function of time.
Since the sonar head is located at one end of the air track, the upper trace was made by
the glider furthest away from the sonar head (far glider) and the lower trace by the glider
nearest the sonar head (near glider). The peaks on the top trace are indicative of collisions
between the far glider and the far end of the air track. The valleys on the lower trace
indicate collisions between the near glider and the near end of the air track. The peaks
of the lower trace and the valleys of the upper trace always occur at the same time and
therefore indicate collisions between the two gliders.
Part III
Using the air table create a two dimensional elastic collision between two pucks. Record
the collision with the video camera and transfer the video clip to the Videopoint software
for analysis.
Analysis
Set up a data table in which you calculate the momentum of each glider and the total
momentum of the system. The table below is an example for Part I with two gliders. All
other tables will have the same structure but differ depending on the number of gliders and
dimensions. Note that MKS units have been used in this table.
Time (s)
Position (m)
Glider 1 Glider 2
Momentum (kg m/s)
Glider 1 Glider 2 Total
Construct a plot of momentum vs. time for each experiment. Include traces for each object
in the system and the total momentum of the system.
For each case in Part I identify any forces that are present and classify them as external,
internal, or impulse.
For Part II, plot the position data for each glider and the total momentum of the system
on one graph. Number each collision, and determine whether momentum was conserved in
each collision.
Analyze Part III using unit vectors. Separate the momenta into x and y components and
sum these components separately.
42
LAB 7. CONSERVATION OF MOMENTUM
Discussion
Determine whether momentum is conserved in all of your collisions. Support your conclusions as to whether or not momentum was conserved. For your Part III analysis, is
momentum conserved in each direction? Compare your conservation of momentum results
to your conservation of energy results from any previous lab(s). Which quantity (momentum
or energy) seems to show conservation more completely?
LAB
8
Impulse and Momentum
Objectives
To examine the force involved in a collision as a function of time, and show that the
maximum force that occurs during a collision can be manipulated by cushioning the impact.
Apparatus
• Air Track with Air Supply
• Glider, with attached spring on one side
• Force Sensor
• Motion Sensor
• Additional materials for damping collision
Background
In the previous lab, we examined simple systems where momentum was conserved. In this
lab, we will examine cases where it is not, or more specifically, what causes a change in
momentum of an object or system of objects. The quantity that we associate with this
change is known as the impulse. The relation between impulse and momentum is quite
simple; in order for an object to go from initial momentum p1 to final momentum p2 , an
impulse J is necessary. Hence the equation
p1 + J = p2 ,
43
(8.1)
44
LAB 8. IMPULSE AND MOMENTUM
Figure 8.1: The magnitude of the force experienced by both objects involved in a collision
as a function of time. The area under this curve is the impulse.
where p1 = m1 v1 and p2 = m2 v2 . Note the similarity to conservation of energy and the
work-energy theorem, Ei + W = Ef . The work W can be thought of as an amount of
external energy added to a system, whereas the impulse J can be thought of as an amount
of external momentum added to the system.
Clearly there is some relation between force and impulse. Assuming the mass of an object
remains constant, if an object experiences a change in momentum, then its velocity must
have changed, implying an acceleration. We already know that such an acceleration demands a force, as dictated by Newton’s 2nd Law. However, the impulse on an object is not
only related to the force exerted on it, but the amount of time over which that force acts.
Think of trying to push a person on a toboggan; if you want them to go faster, thus giving
them more momentum, you simply push for a longer period of time. In a simple case where
the force is constant, the relation between the impulse J, the force F , and the time t is
simply
J = F ∆t (Constant force).
(8.2)
In a collision, the situation is generally more complicated, and the force is not constant.
Typically the force as a function of time is similar to that shown in Fig. 8.1. The general
equation for J, which can be used if the force is constant or not, is then
Z
J=
F dt,
(8.3)
where the integral on the right-hand side can also be evaluated by finding the area under
the F -t curve shown in Fig. 8.1.
The concepts of impulse and momentum are of considerable use when examining collisions.
Conventional applications of analysis involving force and acceleration tend to be difficult
to understand since they do not include an explicit time or velocity dependence. Consider,
for example, hitting a golf ball with a club. You know from experience (or you would at
least assume) that swinging a club faster would result in the ball travelling faster after the
collision. How do we express this in terms of Newton’s Laws? Drawing a free body diagram
for the ball would show that there is a force acting on the ball from the face of the club,
causing it to accelerate in the direction that the club is swung. The question now becomes,
SETUP
45
Figure 8.2: The setup for this experiment. The motion sensor should be tested to ensure
that it does indeed record just the motion of the glider.
how does the velocity enter into the equation? Imagine we initially provide some force to
accelerate the club, but are now swinging it with constant speed. The equation F = ma
contains no mention of how fast the object is travelling currently, so this makes the problem
difficult to understand or solve. We know there is indeed a force acting between the 2
objects, but how in fact can we solve for this force in the end?
This problem can be addressed by examining the initial and final momentum of the ball.
Initially, the ball is at rest, therefore the momentum is zero. After the golf club has made
contact, it imparts some of its momentum to the ball just as in any other collision. This
allows us to include the momentum of the club in the expression for the final speed of the
ball, which would include the club’s velocity. The force created depends upon the time that
the objects are in contact. If this time is known, then the force can be calculated through
equation 8.2. Note that this would in fact be the average force on the ball over this period
of time, as the force-time relationship is similar to that shown in Fig. 8.1.
In this experiment, we will attempt to examine the force vs. time relationship of a simple
collision between an air glider and a force sensor. The sensor can take 1000 measurements
per second, allowing us to examine the collision in detail to see if the impulse does in fact
appear as in Fig. 8.1. We will look at 2 different types of collision, one involving a spring,
and one not, to see what changes this makes to the overall shape of the curve.
Setup
1. Mount a force sensor on one side of the air track by screwing the bolt through both
the metal plate connected to the track and the hole at the back of the sensor.
2. Using a lab jack, raise the side of the air track opposite the sensor so that this end is
roughly 20 to 25 cm higher than the other.
3. Connect an air supply to the track but leave the supply off for the time being.
4. Place a motion sensor near the force sensor, pointing up the track at the glider. Ensure
that the sensor path is not obstructed by the force sensor, and points parallel with
the air track.
46
LAB 8. IMPULSE AND MOMENTUM
Figure 8.3: Measurement of the incline of the track. The heights h1 and h2 are measured
to the middle "ridge" of the track, as this is typically the most convenient point to measure.
The distance L is 200 cm if the ends of the track are used, but any 2 points along the track
can be used, as long as L is known. Note that by examining the triangle formed by the
dotted lines, it is easy to show that sin θ = (h2 − h1 )/L.
5. Connect the sensors to a nearby computer and launch the DataStudio software (the
software should launch automatically).
6. In the DataStudio ‘Setup’ window, ensure that the sample rate for the force sensor is
set to 1000 Hz, and the motion sensor to either 10 or 20 Hz.
7. The graph window should contain a plot with force on the y-axis, while the position
data should be displayed in a table or digit display. If this is not the case, ask your
instructor for assistance.
8. Turn on the air supply. Place a glider on the track and hold it still. Start DataStudio,
and manually move the glider up and down the track to ensure that the sensor is
reading the glider properly. If not, adjust the height and/or angle of the motion sensor,
or ask your instructor for assistance. Turn off the air supply and stop DataStudio
when you are satisfied that the motion sensor is working properly.
Procedure
Part I
In this section we will examine the different ways we can calculate impulse, and compare
the impulse found in 2 different collisions with different F − t relationships.
1. Weigh the glider and record the mass. Measure the height to the middle ridge of the
track at both ends, and record the heights. These heights will be used to calculate
the incline of the air track. See Fig. 8.3 for details.
2. Place the glider on the track with the non-spring just touching the force sensor. Start
DataStudio and record the motion sensor reading as d (the distance between the sensor
and the sail on the glider). See Fig. 8.4(b) for an illustration.
PROCEDURE
(a) Initial Position
47
(b) Collision with Sensor
(c) Final Position
Figure 8.4: A depiction of the 3 distances necessary to carry out the analysis. The glider
begins at initial position xi in (a), then travels down the track and collides with the sensor,
as in (b). The distance d can be found prior to starting your trial, but note that it is different
for both the spring and non-spring cases. After the collision in (b), the glider travels back
up the track until stopping at the maximum position xf , as in (c). This distance must be
found from a table of position values provided by the motion sensor. These are the only 3
distances required to carry out the analysis.
3. Move the glider roughly 20 to 40 cm up the track.
4. Click start in Datastudio, and record the initial position of the glider xi . Turn on the
air supply. The glider should slide down the track and impact with the force sensor.
Allow the glider to rebound to its maximum height before stopping the sensors.
5. Export your force data as a text file.
6. Find the numerical value for the maximum distance reached after the collision by
viewing a table of the position data in DataStudio. Note that the position values
will decrease as the glider approaches the sensor, then increase after the collision with
the force sensor, until the maximum height after the collision is reached. It is this
maximum distance xf you wish to record (see Fig. 8.4(c)).
7. Turn the glider around so that the spring side is facing the sensor and repeat the
procedure, placing the glider the same distance up the track as before (Step 3). Be
sure to repeat step 2 with the non-spring side, also.
Part II
In this section we will examine the maximum force achieved during a collision for different
heights along the air track, and thus different impact velocities. We will use the results to
attempt to find the relationship between maximum force and velocity.
1. Set the glider on the track with the non-spring side pointing towards the force sensor.
The motion sensor will not be used in this part. Turn on the air supply and click
start in DataStudio.
48
LAB 8. IMPULSE AND MOMENTUM
2. Place the leading edge of the glider at a distance of at least 5 cm away from the force
sensor. Record this distance.
3. Release the glider and let it strike the sensor. With Datastudio still running, repeat
the measurement twice more, then stop the sensor.
4. The graph of force vs. time should appear on the screen as at least 3 well-defined
peaks, corresponding to the 3 collisions with the sensor. There may be more if the
glider was allowed to rebound and strike the sensor multiple times from one trial;
disregard these peaks. Using the ‘hand’ tool, locate the maximum force reached
during each impact and record the results.
5. Repeat the procedure 5 additional times at 5 different heights. Smaller heights are
generally preferred, as the maximum force that the sensor can read is 58.8 N. If you
notice your measurements reaching this threshold, try a different height.
Analysis
Part I
Open your data files for the force sensor (you don’t need data files for the position data)
in Excel and graph all the data. Using the graph, look for the time when the first collision
occurs, which should appear as a spike in the force data. Find this portion of data in your
table. Copy and paste the section of data that corresponds to the collision into a new
worksheet. Label the columns appropriately. Convert your times into milliseconds (1 s =
1000 ms), and set the first time in your table to be zero. Graph this data.
The graphs should appear as inverted parabolas. Add a trendline to each graph, and
choose a polynomial of degree 2 from the menu. Display the equation on the graph, and
make sure there are at least 3 significant figures for each coefficient of the polynomial (ask
your instructor for help doing this if you need it). You now have an equation for the force
between the glider and the sensor as a function of time, for both the spring and non-spring
cases. Print both the tables and graphs (1 for spring side, 1 for non-spring), and include
them in your report. Do not print all of your original data!
Using these equations, we can find the impulse. Recall that the value of the impulse
is
R
the area under the F − t curve, which can be found by evaluating the integral J = F dt.
Using your equations for force as a function of time, find the impulse for both the spring
and non-spring cases by performing this integration for each of your F (t) equations. The
units for J will be the units of the x-axis multiplied by the units of the y-axis, e.g. N · ms,
if you used ms for the units of time. Convert this to kg · m/s by dividing by 1000 (yes,
dividing).
We can also find the impulse by using the equation J = ∆p, which requires finding the
momentum of the glider immediately before and after the collision. In order to find the
DISCUSSION
49
momentum at these times, the velocity of the glider must also be known. To find these
velocities, we must use your measurements for xi , xf , and d (see Fig. 8.4). In addition, we
need to calculate the acceleration down the track.
The component of the weight down the incline is mg sin θ, therefore the acceleration is given
by a = g sin θ, where θ is the angle of the track. Since sin θ = (h2 − h1 )/L (see Fig. 8.3),
the acceleration can be found using a = g(h2 − h1 )/L. Use your measurements for h1 and
h2 to find a using this equation, and be sure to show your work.
Using your values for xi , d, and a, the velocity just before impact vi can be found using
q
vi = − 2a(xi − d)
(8.4)
The velocity just after the impact can be found similarly using your measurement for xf :
vf =
q
2a(xf − d)
(8.5)
Note the difference in sign between the 2 velocities, since the glider changed direction during
the collision.
Using the relation between impulse and momentum J = ∆p = m(vf − vi ), and your results
for vi and vf , find the impulse J. Remember to include the correct signs for the velocities!
You now should have
2 impulse values for each case (spring/non-spring), one gained from
R
the integral J = F dt, and one gained from the equation J = ∆p, giving a total of 4
impulse results. Summarise all 4 impulse results clearly in your report (a simple table
would suffice).
Part II
Present your results in a table as follows:
Initial Position
Distance Travelled
Velocity
Max. Force
Find the average of the 3 maximum force values found for each height to present in your
force column. Calculate the distance travelled by simply subtracting the distance along the
air track the glider rests when touching the sensor. Find the velocity using basic kinematics
(the same equation used for Part I). Once you have your table completed, plot the maximum
force as a function of velocity.
Discussion
From Part I, compare your impulse values for the 2 cases (spring, non-spring) examined.
Do they give similar results? Should they? Also compare your results from the integration
50
LAB 8. IMPULSE AND MOMENTUM
calculation to the impulse found using the impulse-momentum equation. Do they also
agree?
From your results for Part II, what can you say about the relationship between max force
and velocity? Can you explain these results in any way, using theory? Given that maximum
force is the cause of breakage of an item during a collision, what can you say about what
speeds are most dangerous?
LAB
9
Motion of Centre of Mass
Objective
The objective of this experiment is to show that in the absence of an external force, the
centre of mass of a 2 object system will follow a straight line, regardless of the complexity
of the motion of the 2 objects.
Apparatus
• Small Air table, with pucks of various masses
• Newsprint
• Metre stick
Background
Given your experience with the impulse and momentum experiments you should now be
confident in the statement, "in the absence of an external force the motion of an object does
not change." This principle raises an interesting question with respect to the motion of the
centre of mass of an object, or for that matter, the centre of mass of a group of objects.
Centre of mass is an extremely useful concept. In effect, you can take all the mass of an
object of any shape and consider it as concentrated at the location of the centre of mass.
Imagine holding a metre stick. There is one point (the 50 centimetre mark) at which you
could balance the metre stick on your finger. It is as though all the forces of gravity acting on
the various parts of the metre stick can be gathered at the 50 centimetre mark in opposition
the reaction force of your finger.
51
52
LAB 9. MOTION OF CENTRE OF MASS
The metre stick is after all a collection of atoms held together by chemical bonds. The forces
between the atoms are not external forces but rather internal forces (Newton’s third law)
and therefore exert no net force on the metre stick. You can regard the centre of mass of
the metre stick not as the centre of mass of a single object but rather the centre of mass of
the collection of atoms that comprise the metre stick. The individual motions of the atoms
are the result of internal forces. If you apply these ideas to two pucks on a level air table
the result suggests that once you put the pucks in motion, i.e. no more external forces, the
centre of mass of the two pucks should travel in a straight line. This is true even if the two
pucks collide.
To completely understand this situation you must be careful about labeling forces as external
or internal. Imagine an elastic collision between two objects. If you focus on one of the
objects and its centre of mass then the contact force of the collision is an external force
to that object and the motion of that object will change. The same is true of the other
object in the collision. Remember, the collision forces are equal and opposite to each other
(Newton’s third law). If you consider the two objects to be parts of a system, then the
forces of the collision are internal to that system. The system has a centre of mass and
no external force has been exerted on that centre of mass as a result of the collision. The
centre of mass of the system should therefore travel in a straight line.
We will not discuss the details of rotation about the centre of mass only to say that it follows
the same reasoning as the preceding discussion. In summary, the motion of the centre of
mass is a powerful analytical tool because every possible motion of a system can be reduced
to the translational motion of the centre of mass and a rotational motion about the centre
of mass.
The motion of the centre of mass is the ’simplest’ motion of an object. To fully appreciate
the possibilities imagine the motion of a tennis racket thrown into the air with a spinning
motion, as in Fig. 9.1.
If you traced the motion of the tip of the handle of the racket you would find a combination
of translation and rotation. However, if you followed the location of the centre of mass of
the racket it would be a smooth parabola (no rotation) as shown in the picture. After the
racket left your hand the only force (neglecting air resistance) acting on the racket was the
force of gravity. Gravity acts on each part of the racket.
If you divide the racket into many small pieces (particles) and draw vectors on each piece
representing the force of gravity you will find that there is one point on the racket where
the forces of gravity are balanced. This is the location of the centre of mass of the racket.
(A more formal way of defining the balance point is to say that the sum of the gravitational
torques is zero.)
Another definition of the centre of mass is the ‘weighted average position of the mass of a
system’ (system can mean a single object or a collection of objects that are not necessarily
physically connected such as a group of particles).
BACKGROUND
53
Figure 9.1: The motion of a thrown tennis racket that is spinning. Note that while the
motion may seem complex, the centre of mass of the racket is travelling in the same manner
as a thrown ball, as shown in the picture.
Consider two equal masses as a system. If they were joined togther by a rigid rod the centre
of mass would be halfway between them. You could easily verify this by balancing the rod
on your finger. We can calculate the position of the centre of mass with respect to some
reference point by finding the ‘average’ position of the mass in the system. Let’s pick the
position of M1 as our reference and determine the distance (Xcom ) from that point to the
centre of mass (see Fig. 9.2)
To find the average position of the mass of the system we must calculate the product of
distance and mass for each mass in the system with respect to our reference point, sum the
results, and divide by the total mass in the system. For this example we would get:
Xcom =
M1 x1 + M2 x2
M (0) + M L
L
=
=
M1 + M2
M +M
2
(9.1)
Exactly the answer you would expect. Note here that M1 = M2 = M since the masses of
the pucks are equal.
Figure 9.2: The centre of mass of a simple 2 object system lies along the length L, closer
to the heavier mass.
54
LAB 9. MOTION OF CENTRE OF MASS
(a) Trial 1
(b) Trial 2
(c) Trial 3
(d) Trial 4
Figure 9.3: 4 different trials for the motion of the centre of mass. Note that in (a) and (b),
the 2 pucks are identical mass, while in (c) and (d) the 2 pucks have different masses. This
mass discrepancy is depicted by 2 different sized pucks in the figure, but their actual size is
in fact the same.
Procedure
We will study how the centre of mass of a two puck system moves during a collision using
a small Air table. The air table is equipped with a spark generator. As the pucks move
across the table they leave a trail of burn marks on a sheet of newsprint. Your instructor
will show you how to operate the air table. You will create 4 different collisions, as depicted
in Fig. 9.3.
In each case, start the pucks in the specified positions and place your foot over the pedal
that controls the spark generator. It is important that you not push down the pedal while
touching the metal parts of the pucks (you’ll get a shock). Start the pucks in motion and
push down the pedal immediately after letting go. Release the pedal before the pucks leave
the paper surface; this is especially important for Case 2. Attempt a few practice runs for
each case without using the pedal to make sure you get a satisfactory collision.
Analysis
Label the points on your newsprint for each case. The first point of the path of each puck
should be labeled ‘1’, with each successive point being labelled in numerical order. Label at
DISCUSSION
55
least 15 points, preferably more. This will leave you with 15+ ‘pairs’ of points. Join each
pair with a straight line. For Case 2, it is easier to label the points in reverse chronological
order, starting with the last pair. This can only be done if you released the pedal before
the pucks left the surface of the paper.
Now you need to find the centre of mass of the system and mark it on each of these lines.
For Cases 1 and 2, the two pucks have equal mass, so the centre of mass will be halfway
between them. For Cases 3 and 4, the two pucks have unequal mass. Use equation 9.1
to find the centre of mass of the system in terms of the length L between them. Show
your result and the corresponding work in your report. Note that there are two different
equations that you can get, depending on whether you choose to measure from the smaller
or larger puck. For example, if one puck is twice as large, then Xcom = 0.667L, measured
from the smaller puck, or Xcom = 0.333L, measured from the larger puck (9.1 can verify
this result). Obviously these 2 equations refer to the exact same point.
Once you have found the expression for the centre of mass, measure the distance between
each pair of points and mark the location of the centre of mass. Once you have done this
for all pairs of points, plot the path of the centre of mass by joining these new points with
a single best-fit line. Your sheet for each case should look like Fig. 9.4. If your best-fit line
does not fit the points well, check your calculations again.
Figure 9.4: The motion of the centre of mass can be found by joining the points, finding
and marking the centre of mass of each line, then finally joining all these marks.
Discussion
Look at the motion of the centre of mass in each case; does it in fact move in a perfectly
straight line? If not, try and reason what could have gone wrong. If the centre of mass
does appear to move in a straight line in each case, can you conclude that there are indeed
no external forces on this system? What would be an example of an external force in this
case?
LAB
10
Angular Acceleration of a Disk
Objective
The objective of this experiment is to determine the relation between torque and angular
acceleration, and to verify that the moment of inertia of a complex body can be found by
summing the moments of inertia of the individual parts.
Background
The analogy of Newton’s second law F~ = m~a as applied to rotational motion states that the
angular acceleration of a body (α) is proportional to the torque τ and inversely proportional
to the moment of inertia or rotational inertia I of the body.
τ = Iα,
(10.1)
where [τ ] = N m, [I] = kg m2 and [α] = rad/s2 .
Just as the mass of a body is a measure of the tendency of the body to resist linear acceleration, so the moment of inertia is a measure of a body’s resistance to angular acceleration.
If a body is free to rotate about an axis, then a torque is required to change the rate of
rotation. The resulting angular acceleration is proportional to the torque as per Equation
10.1 and exists only during the time the torque acts on the body.
The moment of inertia of a body is dependent on the mass of the body and the distribution
of the mass about the axis of rotation. To determine I, the body is divided into a number of
infinitesimal elements mi , each at some distance ri from the axis of rotation. The moment
of inertia is given by the sum of all the products mi ri2 calculated for each element.
I=
N
X
i=1
56
mi ri2
(10.2)
BACKGROUND
57
Figure 10.1: The forces and distances present in the simple disk-mass apparatus.
If the body is a simple geometric shape, e.g., a disk, sphere, or cylinder, I can be easily
calculated; results for such calculations are given in tables. For a solid disk the expression
is
1
I = mr2 ,
(10.3)
2
where m is the total mass of the disk and r its radius. For a ring of inner radius r1 and
outer radius r2 (imagine a disk with a central hole cut out), the moment of inertia becomes
1
I = m(r12 + r22 ).
2
(10.4)
In this experiment the linear acceleration a of a mass m suspended on a cord is measured.
This cord is wound around the hub of the disk (see Fig. 10.1) and as a result, the magnitude
of the linear acceleration of the rim of the hub is also a.
By calculating the linear acceleration on the rim we can determine the angular acceleration
α of the hub (which is also the angular acceleration of the disk) through the equation
a = rα,
(10.5)
where r is the radius of the hub.
The torque ~τ (a vector) resulting from a force F~ applied about an axis is given by the
expression:
τ = rF sin θ.
(10.6)
58
LAB 10. ANGULAR ACCELERATION OF A DISK
In this experiment the torque is supplied by the tension in the cord from which the mass is
suspended (see Fig. 10.1). The torque can be calculated as
τ = rT sin 90o = rT,
(10.7)
and is considered to be a vector pointing into the paper.
Combining Eqs. 10.1 and 10.7, we obtain
T =
Iα
r
(10.8)
Applying Newton’s second law to the linear motion of the falling weight we can write
T = mg − ma
(10.9)
Eliminating T in Eqs. 10.8 and 10.9 yields
mgr − mr2 α = Iα
(10.10)
This equation is linear, so a plot of mgr − mr2 α vs α should yield a straight line with slope
I, assuming the frictional torque is constant. In order to find α, we will use stopwatches to
measure the time required for the mass to fall a certain distance from rest. Using kinematics,
this time and distance are related to the acceleration of the falling mass by the equation
1
d = at2 .
2
(10.11)
Since a can be related to α using Eq. 10.5, we can substitute for a in Eq. 10.11 and find
that
2d
α = 2.
(10.12)
rt
Procedure
There are 2 different types of apparatus used for this experiment, the wall-mounted apparatus at the front of the lab, and a number of similar desktop apparati which must be placed
near the forced air outlets. The procedure for each apparatus is slightly different, consult
the appropriate section below in order to obtain your data. Note that you are only required
to take data for ONE of the two types of apparatus, not both. Each apparatus requires 3
to 5 people, so you should team up with another group to take your data.
Wall-Mounted Apparatus
1. Using a metre stick, measure the distance between the ground and the bottom of the
mass hanger as it rests on the platform provided.
PROCEDURE
59
2. Record the masses of the hanger, the disk, and the hub. These are printed on the
apparatus.
3. Using calipers, measure the inner radius of the hub, the outer radius of the hub (where
the string in actually wrapped around, NOT the lip), and the radius of the disk itself.
Note that it is generally much easier to measure diameter with a caliper, so this is
probably the most suitable technique. The disk itself is too large for this method,
however; you’ll most likely have to simply do your best to measure the radius of the
disk with the calipers.
4. Place an amount of mass between 20g and 200g on the mass hanger as it rests on the
platform.
5. Arm 3 group members with stopwatches, and let the other be in charge of dropping
the mass. Use the lever or the string attached to the lever to remove the platform
from beneath the hanger, and time the hangers descent using the stopwatches.
6. Reset the apparatus and repeat the previous step for the same mass. This should give
you 6 times per mass.
7. Repeat the previous 3 steps for 5 additional different masses between 20g and 200g
(6 masses total).
Desktop Apparatus
1. First ensure that your apparatus is connected to the forced air supply. Your lab
instructor can assist you if this is not the case. These first few steps should be done
with the air supply off.
2. Remove the hub by first removing the central screw. Measure the radius of the hub
using calipers, and the mass using one of the scales. Note that there is a lip on the
hub around the outer radius; be sure to measure the radius to the inner surface, not
the lip.
3. Remove the top disk and measure its radius and mass in a similar manner. Reassemble
the apparatus after you are done, including the central screw. Note that you do not
need to make any measurement for the lower disk, as this disk will spin independent
of the top disk, and thus is not part of the experiment.
4. Tie a piece of thread around the central screw and lead the thread through the groove
on the hub. Wrap the thread around the hub a few times, then lead the thread over
the pulley attached to the apparatus and to the ground. Cut the thread so there it
will easily reach the ground, with at least a foot to spare.
5. Obtain and weigh a mass hanger, then attach the hanger to the end of the thread.
6. Check to see if the other group has completed their measurements and is on this same
step, if not, then simply wait until they are.
60
LAB 10. ANGULAR ACCELERATION OF A DISK
7. Turn on the air supply so that the pressure valve reads between 9 and 12 PSI, then
wind the thread around the hub by rotating the disk with your hand until the top of
the mass hanger is just below the pulley, and hold the disk in place.
8. Using a metre stick, measure the distance between the ground and the bottom of the
mass hanger as it hangs below the pulley. This height can technically be any distance
you choose, but you must place the hanger at this same height with each trial, so
make sure that this height is repeatable.
9. Place an amount of mass between 10g and 100g on the mass hanger while you hold
the disk still.
10. Arm 3 group members with stopwatches, and let the other be in charge of dropping the
mass. Simply let the disk spin by removing your hand, and time the hangers descent
using the stopwatches. As soon as the hanger hits the ground, stop the disk from
spinning with your hand. If you don’t, the hanger will lurch upwards after reaching
the end of the thread, which may cause your thread to break.
11. Reset the apparatus by spinning the disk to raise the hanger and repeat the previous
step for the same mass. This should give you 6 times per mass.
12. Repeat the previous 3 steps for 5 additional different masses between 10g and 100g
(6 masses total).
Analysis
Construct a table similar to the following, using equations 10.10 and 10.12 to calculate
values for τ and α. Note that the equations for both of these variables contain r, which in
all cases refers to the outer radius of the hub (where the string is actually in contact with
the hub). Also be sure to include the mass of the hanger in your mass column!
Mass
(kg)
t1
t2
t3
Time(s)
t4 t5 t6
t̄
σt
α
(rad/s2 )
τ
(N·m)
Plot a graph of τ vs. α. Determine the slope of this graph, and label this value as the
experimental value of the moment of inertia (Iexp ).
The theoretical moment of inertia Ith can be calculated using your measurements of the
mass and radius of the disk/hub system and Equations 10.3 and 10.4. The apparatus
consists of a ring attached to a disk, both of which share the same axis of rotation. When
objects are attached in such a way, their combined moment of inertia is simply the sum of
the individual moments of inertia of each object. By combining equations 10.3 and 10.4 in
such a manner, it can be shown that
Ith =
Md Rd2 + Mh (Rh2 + rh2 )
,
2
(10.13)
DISCUSSION
61
where
Md
Rd
Mh
Rh
rh
=
=
=
=
=
mass of disk
radius of disk
mass of hub
outer radius of hub
inner radius of hub.
Note that if using the desktop apparatus, the value for rh is 0.
Discussion
Compare Iexp with Ith , and record the percentage difference between the 2. Do these two
values agree within reason? Given that the equation for Ith given above is in fact found
by summing the moment of inertia of the ring and disk, do your results appear to confirm
that the moment of inertia of a complex body can be found by summing the moments of
the individual parts?
On your graph of τ vs. α, does the line pass through the origin of your plot? In other
words, is your y-intercept equal to zero? Should it be? Note that when we were discussing
the rotational form of Newton’s 2nd Law, we said that τ = Iα, but really this should be
Στ = Iα, just like the translational version. We only considered one torque in our analysis,
that created by the tension in the string; is this correct? If another torque were present,
how would it affect this graph?
LAB
11
Nuclear Half-life
Objective
To determine the half-life of
137 Ba.
WARNING: No food or refreshments are allowed in the lab during this experiment.
Apparatus
• Ba-137 Liquid Sample
• Geiger Tube
Background
Most elements present in nature exist in more than one nuclear configuration. The slightly
different configurations of an element are known as isotopes. Consider neon which has
several different natural forms:
20 Ne
10
21 Ne
10
22 Ne
10
The atomic number (10) identifies each as being the element neon, while the mass number
denotes differing numbers of neutrons in the respective nuclei (20, 21, 22). Other isotopes
19
24
of neon exist, i.e.: 18
10 Ne 10 Ne 10 Ne. These have half-lives ranging from 1.5 sec to 3.38 min.
Unstable isotopes rearrange themselves into stable configurations primarily through nuclear
emission. In this experiment we will study some of the emission characteristics of various
62
BACKGROUND
63
isotopes, shielding properties of various materials, and examine different detection methods.
Types of Nuclear Radiation
We are interested in three types of radioactive nuclear emission:
Alpha (α) These are positive particles which are identical to helium nuclei except for their
nuclear origin. They travel about 0.1 the speed of light and possess great ionizing
power because of their relatively slow speed and great charge. All the alpha particles
emitted from a given species of nucleus are mono-energetic (identical in energy). Alpha
particles may be stopped by a thin sheet of paper.
Beta (β) Unlike alpha particles, they are negatively charged and are identical to the electron except for their nuclear origin. Commonly encountered beta particles travel up to
0.9 the speed of light. Unlike alpha particles, beta particles are emitted in a wide range
of energies, hence the variation in their speed. Beta particles are more penetrating
than alpha particles but may be stopped by several thin sheets of metal.
Gamma (γ) Gamma emission is similar to x-rays and light rays but have a shorter wave
length. These rays are emitted because of energy transitions within the nucleus itself,
while x-rays are emitted because of energy transitions in orbital electrons. Since
gamma rays have no charge, they do not react with matter in the same manner as
alpha and beta particles. They travel at the speed of light and have great penetrating
power.
Nuclear Emission
While it is impossible to predict which nucleus in a collection of radioactive nuclei will decay
next, it is possible to establish the rate at which the collection of nuclei decays. It can be
shown that a collection of N radioactive nuclei will decay at a rate proportional to the
number of nuclei yet to undergo emission in the sample. The rate of decay is characteristic
of the isotope in question and can be expressed either as the half-life (T1/2 ) or the mean life
(τ ) of the isotope.
Mathematically the rate of decay (R) is expressed as,
R = λN
(11.1)
where N is the number of nuclei yet to undergo decay and λ is the decay constant characteristic of a particular radionuclide. The rate of decay as a function of time follows an
exponential pattern expressed as
R = R0 e−λt ,
(11.2)
where R0 is the rate of decay at t = 0.
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LAB 11. NUCLEAR HALF-LIFE
Figure 11.1: Radioactive activity (R) of a decaying species as a function of time.
By taking the natural log of both sides of equation 11.2 a linear relationship results which
can be used to determine λ.
ln R = −λt + ln R0
(11.3)
λ can therefore be found by calculating the slope of a plot of ln R vs. t.
The rate at which a sample decays is called the activity and carries the SI unit of becquerel
(Bq). One becquerel is one decay per second. Another unit of nuclear decay rate is the
curie (Ci). One curie is 3.7 × 1010 Bq. When it is not clear whether a detector is recording
all the disintegrations from a sample it is common practice to use units of counts per unit
time.
Nuclear Half-Life
The half-life (T1/2 ) of a radionuclide is a measure of how long a particular isotope will
continue to decay. The value of the half-life of an isotope is the time required for the initial
rate of decay (which can be any value) to drop to one half that rate.
The half-life of a radionuclide is calculated by relating the decay constant to the time for
one half of the nuclei in a sample to disintegrate.
1
R0 = R0 e−λT1/2
2
(11.4)
Taking the natural logs of each side of this equation yields an expression for the half-life of
a radionuclide.
ln 2
T1/2 =
(11.5)
λ
PROCEDURE
65
Figure 11.2: The decay process of Cs-137. 2 emissions are possible, both γ and β. The
Ba-137 compound which is leached from the sample decays only by γ emission, and with a
very short half-life.
The barium isotope used in this experiment is generated from the beta decay of Cs-137 with
a half-life of 30 years. At this point the Ba-137 nucleus is unstable and quickly undergoes
a decay by gamma emission (662 KeV) to become stable. The half-life determined in this
experiment is the half-life of the gamma decay to form stable Ba-137.
Procedure
You will be given a liquid sample of Ba-137 on a planchette. This isotope decays quickly so
you must be ready to work as soon as the sample is received.
1. Place the sample under the Geiger-Muller tube.
2. Set the DataStudio program to display readings every 30 seconds by clicking on
‘Setup’, then selecting the sensor in the setup window. There should be a window
shown below the sensor where you can set the amount of time between each reading,
simply type in 30.
3. Hit ‘Start’. A table should be displayed that gives the number of counts every 30
seconds. Let the program run to 600 seconds then click ‘Stop’. Copy this data into a
spreadsheet.
66
LAB 11. NUCLEAR HALF-LIFE
Analysis
Construct a data table containing the number of counts within each 30 second interval
and the time. The first time entry in this case should be 30 sec, with the counts then
representing the number of counts registered after 30 seconds. Create an additional column
for the natural log of the counts.
Construct 2 plots from your data, the number of counts vs. time, and the natural log of
the counts vs. time. Perform linear regression on your log plot. The slope of this line is the
decay constant (λ) for this isotope. Using equation 11.5, find the half-life of Ba-137.
Discussion
Look up the half-life of Ba-137. Do your results agree? As discussed in the background
section, 2 types of decay are possible from Cs-137, both γ and β, each with their own
half-life. The Geiger-Muller tube will detect both kinds, but is much better at detecting β
radiation. If some Cs-137 happened to be introduced into your sample, what effect, if any,
would this have on your results?
The exponential decay relationship is ubiquitous in the sciences, but it can also be used to
describe many different kinds of real life processes that occur outside science. In general, this
relation describes any process where the rate of the process is proportional to the current
amount of the reactant, and once a particular reactant has participated in the process, it
can no longer participate. Can you think of another process outside of physics that would
exhibit this behaviour?
APPENDIX
A
Reading Vernier Scales
A vernier scale consists of a stationary scale (main scale) and a sliding scale (vernier scale).
The divisions on the vernier scale are smaller than those on the stationary scale. N divisions
of the vernier scale equal N − 1 divisions of the main scale, (see Fig. A.1, where N = 10)
Figure A.1: Divisions on a Vernier scale.
In Fig. A.1, it may be seen that both the zero and the 10 mark on the vernier coincide with
some mark on the main scale. The first mark beyond the zero on the vernier is 1/10 of a
main scale division short of coinciding with the first line to the right of the main scale zero.
This difference between the lengths of the smallest divisions on the two scales represents the
least amount of movement that can be made and read accurately. This amount is called the
least count of the instrument. If the vernier scale is moved a distance 1/10 the distance of
the difference between the divisions of the main scale, the next mark of the vernier coincides
with a mark on the the main scale. Only one mark on the vernier will align with a mark
on the main scale for a given measurement. See Fig. A.2.
67
68
APPENDIX A. READING VERNIER SCALES
Figure A.2: Reading a Vernier scale. Note how only a single pair of lines on the main scale
and sliding scale line up perfectly.
As you can see accurate measurements of 1/10ths of a main scale division are possible.
Consider the scale in Fig. A.3 to have units of millimeters. In Fig. A.3a we see the zero of
the vernier lies between the 6 and 7 mm marks and the second mark on the vernier coincides
with a mark on the main scale. The reading is therefore 6.2 mm. By similar reasoning the
reading in Fig. A.3b is 3.7 mm.
(a)
(b)
Figure A.3: Reading a Vernier scale. The left reading is 6.2mm, while the right is 3.7mm.
Nearly all verniers have N divisions of the vernier scale equal to N − 1 divisions of the main
scale and the method of determining the reading is similar to that described above. The
least count is always 1/N of the length of the smallest main scale divisions. In the example
above, the least count is 0.01 cm.
APPENDIX
B
Error Analysis
Error analysis is the evaluation of precision in measurement. As such, the word "error"
takes on the connotation of "uncertainty" rather than "mistake". Since no measurement can
be completely free of error it becomes the experimenter’s task to minimize all uncertainties
as far as reasonably possible. In our experiments this will be a three-fold process:
• Minimization of uncertainties in experimental apparatus.
• Optimization of experimental methods and procedures
• Calculation of uncertainties in final results.
Every experiment should begin with a thorough assessment of the apparatus including:
condition, resolution, precision, calibration, etc. The apparatus should then be assembled,
and procedures developed that will provide the most accurate results possible. Finally,
the precision of the outcome should be calculated. Following this procedure allows the
experimenter to assess the worth of the experiment even before it is actually done.
The experimenter must have a reasonable knowledge of the types of uncertainties that exist
and how they will propagate in a given experiment. The remainder of this section will deal
with those two subjects.
Types of Errors
Systematic Errors
A systematic error is one which repeats itself to the same degree in each trial of an experiment. As such it cannot be discovered simply by repeating a measurement several times.
Due to their reproducibility, systematic errors are very difficult to trace.
Systematic errors usually involve some aspect of the experimenter’s apparatus or methodol69
70
APPENDIX B. ERROR ANALYSIS
ogy (improperly calibrated equipment, poor experimental design, and/or procedure are the
most likely places to find systematic errors). Consider an experiment to measure voltage
in an electrical circuit. If the voltmeter used in the experiment reads 1.5 volts when no
voltage is applied (calibration error), each reading of the voltage will be 1.5 volts too large.
This is typical of the most common types of systematic errors.
Random Errors
Simply stated, random errors are errors which can be found by repeating a measurement
several times. Random errors can be treated statistically (systematic errors cannot) and
therefore repeated measurement allows one to minimize the effect of random error in experiments. Random error arises from our inability to hold all the variables constant in addition
to the random character inherent in nature.
Reading Errors
Reading error is associated with any directly measured quantity and reflects the limits of
the devices used in the measurement process. For example, a ruler graduated in millimetres
has much less reading error than a ruler graduated in centimetres. The resolution of the
scale plus the interpolation of the measurement between the finest graduations of the scale
constitute the major contributors to reading error.
In the case of digital readings the error is considered to be ½ of the final digit. For example,
if a display reads 1.09, and there is no other information, the reading would be considered
to be 1.09 ± .005.
Summary
All the factors discussed, systematic, random, and reading errors combine to yield the
overall error in a measurement. Experimenters strive to minimize error through careful
design and thoughtful execution of experiments.
Data gathered from experimental work must carry with it some indication of uncertainty.
This estimate of uncertainty follows the data through any calculation or manipulation and
is always quoted with the final results. The next section deals with the propagation of error
during calculations. It is more involved than the previous discussion and thus demands
more attention.
PROPAGATION OF ERRORS
71
Propagation of Errors
Definitions
Absolute Error: The total uncertainty in a quantity, x; usually expressed as δx (i.e.
x ± δx).
Relative Error: The uncertainty in a quantity, x in fractional form. Usually expressed as
δx/x, where δx/x is dimensionless. The relative error can be expressed as percentage
error by multiplying δx/x by 100%.
Example:
An iron block is found to weigh 3.44 ± .03 kg.
Absolute error δm = .03 kg
Relative error δm/m = .03/3.44 = .01 = 1%
Possible Error: The maximum possible uncertainty in the outcome of a mathematic operation involving values and their uncertainties.
Example: Consider the addition of two values A and B.
If A is known to a precision of ±δA, and B to a precision of ±δB, the sum of the two
values could be as large as (A + δA) + (B + δB), or as small as (A − δA) + (B − δB).
If C = A + B then the possible error in C, δC would be δC = δA + δB.
Possible error assumes the most extreme values in the uncertainty combine in the
worst possible combination. For example, if A = 10 ± 1 and B = 20 ± 2 then for A + B
the largest possible value would be 11 + 22 = 33 while the smallest value would be
9 + 18 = 27, in other words 30 ± 3. Using the rule δC = δA + δB we arrive at the
same value for error in C namely, δC = 1 + 2 = 3.
Probable Error: A more conservative estimate than possible error. Probable error assumes that quantities used in calculations do not combine in the worst possible fashion (applies to mutually exclusive events) but rather with a statistical probability.
Estimates of probable error will be lower than those of possible error.
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APPENDIX B. ERROR ANALYSIS
Rules for the Propagation of Errors in Calculations
A + B = C, A − B = C, where A ± δA, B ± δB
Addition / Subtraction:
Possible Error: δC = δA + δB
Probable Error: δC =
Multiplication / Division:
p
(δA)2 + (δB)2
A × B = C, A/B = C, where A ± δA, B ± δB
Possible Error: δC/C = δA/A + δB/B
Probable Error: δC/C =
Exponents:
p
(δa/A)2 + (δb/B)2
C = An , where A ± δa
Error: δc/C = n δa/A
General Expression for Error Propagation
The relationship for error propagation can be expressed and manipulated as the total differential for a dependent variable which is a function of one or more independent variables.
Suppose we had a function f (x, y, z), its total differential would be expressed as
df = ∂f /∂x dx + ∂f /∂y dy + ∂f /∂z dz,
if we exchange the differentials df, dx, dy, dz for the uncertainties δf, δx, δy, δz we have
an expression which allows us to calculate error propagation for more than the simple
arithmetic operations shown previously.
s
δf (x, y, z) =
2
∂f
δx
∂x
+
∂f
δy
∂y
2
+
∂f
δz
∂z
2