Download Physics 2120 Lab Manual Fall 2011

Department of Physics
and Astronomy
Physics 2120
Lab Manual
Fall 2011
Contents
Lab Reports and Marks
1 Error Analysis
5
8
2 Rotational Motion
15
3 Conservation of Angular Momentum
20
4 Kater’s Pendulum
26
5 Mechanical Equivalent of Heat
34
6 Electrical Equivalent of Heat
39
7 Ruchardt’s Method
43
8 Stefan-Boltzmann Law
47
9 Stirling Engine
51
10 Atomic Spectra
55
A Reading Vernier Scales
62
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.
5
6
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?
ABSENCE FROM LAB SESSIONS
7
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.
LAB
1
Error Analysis
Background
When taking experimental data, it is imperative to understand that no instrument is completely accurate. No matter how precise an instrument may be, there will always be a
degree of uncertainty associated with the results. When presenting these results, the degree
of precision must also be presented. Consequently, the data obtained from experimental
work, and therefore all results based on that data, are not simply values; they are a range
of values. For example, the results of a study on the gravitational constant g at sea level
may result in a value of 9.76 m/s2 , with an uncertainty of 0.07 m/s2 . This result should
be interpreted as a statement that the author(s) have ascertained the value to be between
9.69 m/s2 and 9.83 m/s2 , assuming that the data is subject to normal random error.
There are two general methods available to arrive at these uncertainties. The first is based
on the properties of the instrumentation used to take the measurements. The most common
example of this are length measurements taken using a ruler. Consider the situation depicted
in figure 1.1. It is fair to say that anyone would conclude that the length of the nail is
closest to 5.5 cm; this is clear from the markings on the ruler. However, beyond this level
of accuracy, it would be more difficult to find a consensus among a number of observers.
Some may say that the length is 5.47 cm, others 5.52 cm. In general, when dealing with
a common measuring device, such as a scale, ruler, etc., the uncertainty in the reading is
assumed to be ½ of the smallest division. In this example, the length could be read as
5.50 ± 0.05 cm. This also satisfies the common experimental aphorism “the error is in the
last digit”.
This method proves adequate when dealing with most pre-lab measurements. These measurements are typically static, and involve measuring the lengths, weights, etc., of experimental components. Within the process of taking data, however, this method of determining
uncertainty becomes less suitable. Experimental data is often affected by many more factors
than simply the resolution of the instrument. These factors will vary from experiment to
8
BACKGROUND
9
Figure 1.1: Measurements taken with a ruler.
experiment, but can include such things as air flow, temperature, pressure, background radiation, electric fields, magnetic fields, etc.. All of these factors can result in data taken from
what it is believed to be an identical initial condition to yield different measurements. Unfortunately, since these errors vary from one trial to the next, they are generally impossible
to quantify and deal with from a theoretical perspective.
How exactly can these errors be predicted? The simple answer is that they can’t; in these
circumstances, we must rely on the field of statistics to provide an estimate of the uncertainty. If an experiment is subject to random error, then it should obey Gaussian statistics,
a theory that you should have discussed in previous lab courses. A measurement should be
taken over a number of trials with controlled initial conditions, which will result in a set of
data for each value of the controlled variable. The ‘best’ value among this set is assumed to
be the mean x̄, and this value is typically used to represent the result from the set of trials.
Another statistical measure of note is the standard deviation σx , which is representative of
the ‘spread’ in the data. For precise data, this value should be small, representing a set
of data that does not deviate greatly from the mean. A measurement with less precision
would have a larger standard deviation. If further review of Gaussian statistics is required,
refer to the Statistics lab within the Physics 1000 manual.
In general, the error in a particular variable x is represented as δx. In cases where a
measurement is subject to random error, as described above, it suffices to let δx = σx . If
a greater degree of confidence is required, letting δx = 2σx is more appropriate. In either
case, the approach of taking many measurements and allowing the error in the reported
value to be a function of the standard deviation of the set of results is generally preferred
to using the resolution of the measuring instrument. The result for a single measurement
can then be presented as x̄ ± σx .
A slightly different quantity is used when expressing the error in the mean. Recall that
the mean is the ‘best guess’ for the actual value corresponding to a set of measurements
of that value. It is arrived at by combining all measurements of the set together, each one
having individual errors δx. If we assume that the mean is a better measurement than each
single measurement, shouldn’t the error in this value be less? This can be proved using the
methods in the following section, but for the time being we shall simply assume this is the
case. The equation for the standard deviation (or error) in the mean itself is
σx
σm = √
N
(1.1)
10
LAB 1. ERROR ANALYSIS
Note that the standard deviation of the mean decreases with an increasing number of
measurements; this is very important. In all cases, the result of a measurement subject to
random error should be presented as x̄ ± σm .
Propagation of Error
A significant portion of the field of error analysis focusses on the propagation of uncertainty
throughout the analysis of a set of data, which leads to a more substantial uncertainty in
the final results. Consider a simple case of adding together two measurements of length,
each taken with the same instrument having a precision of 1 mm. One measurement yields
a result of 3.1 cm, the other 5.7 cm. When added, the final result is 8.8 cm; however,
since our initial measurements had a degree of uncertainty, the actual sum could have a
value anywhere between 8.6 cm and 9.0 cm. Note now that the uncertainty of 1 mm has
doubled to 2 mm. Similar examples can be shown for measurements that are multiplied,
divided, or indeed subject to any mathematical operation. Considering that the final results
of an experiment are often calculated from data in this manner, it should be clear why the
application of a standardized practise of error analysis is necessary within a laboratory
environment.
The uncertainty in a particular variable that is derived from experimental data can be found
using methods drawn from calculus. The general equation for the error δq in a variable q
that is a function f of N variables x1 , x2 , . . . , xN is
s
δq(x1 , x2 , . . . , xN ) =
∂f
δx1
∂x1
2
+
∂f
δx2
∂x2
2
+ ... +
∂f
δxN
∂xN
2
(1.2)
This equation may appear somewhat daunting, and should typically only be used if the
function f involves more complex elements like trigonometric or exponential functions. If
the function f involves only addition, subtraction, multiplication, or division, we can derive
simpler equations from this general equation.
Exercise 1. Assume that q = f (x) = ax. Let the error in x be δx, while the error in a is
zero (a is a constant). Find δq and δq
q .
Exercise 2. Assume that q = f (θ) = a sin θ. Let the error in θ be δθ. Find δq and
Exercise 3. Assume that q = f (x, y, z) =
δz, respectively. Find δq and δq
q .
x+y
z .
δq
q .
Let the errors in x, y, and z be δx, δy, and
Error Bars
When displaying results in a graph, the uncertainities in the y-values are typically represented using error bars. These bars appear as vertical (and occasionally horizontal) bars
ERROR BARS
11
that are centered on each individual point on the graph. Each bar terminates in a horizontal
bar, leading to a distinct ‘I’ shape. The length of these bars is a direct representation of the
uncertainty in that particular point. Given that each point can have a different value for
the uncertainty, the length of the bars can vary from point to point. Consider the following
data set:
Height(cm)
10
20
30
40
50
Velocity (m/s)
3.4, 3.2, 3.5, 3.8
8.9, 9.3, 9.5, 10.1
18.6, 19.8, 20.5, 19.3
32, 33.4, 33.8, 35.2
50.2, 51.3, 51.6, 53.6
v̄
3.48
9.45
19.6
33.6
51.7
δv = σm
0.13
0.25
0.4
0.7
0.7
Table 1.1
It is clear to see from this data set that the error in each individual point of the data varies,
thus leading to a variance in the size of the error bars around each point. When graphed,
the data appears as in figure 1.2.
Error bars should be included on all of the graphs contained in your lab reports. It is a fairly
simple procedure to get these error bars to appear in common spreadsheet programs such as
Excel or OpenOffice. Within both of these programs, simply right click on a data series and
select the appropriate option; ‘Format Series’ for Excel, ‘Add Error Bars’ for OpenOffice.
There is a tab named ‘Y Error Bars’ within the ‘Format Series’ dialog in Excel that contains
the necessary options for adding error bars. The interface should be self-explanatory for
the most part. Note that when the uncertainty is estimated through the precision of the
instrument, choosing a ‘Fixed’ error amount is appropriate. If the uncertainty is found
using the standard deviation of a number of trials, then a column of values containing the
errors in each point should be made, and referenced using the ‘Custom’ error amount.
Figure 1.2: Plot of the data from Table 1.1, with error bars.
12
LAB 1. ERROR ANALYSIS
Note that as mentioned briefly above, it is possible for error bars to appear horizontally,
implying uncertainty in the controlled (x) variable. Within this lab course, we will assume
little to no uncertainty in this variable, thus eliminating the need for these horizontal bars.
This is not an unjustified assumption, and is in fact fairly common in experimental work, as
ideally there should be a great deal of control over the controlled variable. In circumstances
where this is not possible, horizontal error bars should certainly be included.
Errors in Linear Regression
Linear regression is a common tool for analysing data. It allows the calculation of a variety of
physical measurements or coefficients given the correct axes, in a way that minimises error.
However, it is more difficult to find the error in the slope and intercept using conventional
means. Recall the equations for the slope and intercept of a linear least squares fit. Given
P
P
∆ = N x2i − ( xi )2 ,
Slope = (N
Intercept = (
X
X
x2i
xi yi −
X
X
yi −
xi
X
X
xi
yi )/∆
X
xi yi )/∆
(1.3)
(1.4)
It is a difficult task to apply equation 1.2 to these expressions, and successfully doing so does
not necessarily lead to the correct result. In order to find the errors in these quantities, we
must find a quantity known as the standard error σy . This error represents the variance of
the points predicted by our calculated slope and intercept from the actual points measured.
It can be found using the equation
σy2 =
1 X
(yi − (mxi + b))2
N −2
(1.5)
Note that the quantity mxi + b can be represented as ypred , the predicted y values given our
calculated slope and intercept. The standard error can be viewed as a measure of how well
the linear regression line fits the data. If there are a large number of points with values that
deviate significantly from the predicted points, then the sum in equation 1.5 becomes large,
as does the error. Using this quantity, we can find the error in the slope m and intercept b
using the following equations:
δm2 =
δb2 =
N σy2
∆P
σy2 x2i
∆
(1.6)
(1.7)
Note that popular spreadsheet programs have built-in functions that can evaluate most of
these quantities easily.
Exercise √
4. Using the data from
√ table 1.1, construct a new table with columns for 2Height
and v̄. Create a plot of v̄ vs Height (h). Create additional columns for xi , xi yi ,
ypred , and (yi − ypred )2 . Calculate all the values in the x2i and xi yi columns, and find
WEIGHTED AVERAGES
13
the sums necessary to evaluate the slope and intercept. Once these are found, use
these values to calculate the ypred column using ypred = mxi + b. Once these values
are determined, find the (yi − ypred )2 column and calculate its sum. Use all of this
information to calculate the standard error and the quantities δm and δb.
Weighted Averages
Through the course of taking data, it may become apparent that certain points contain a
large degree of error due to the presence of one or more particularly erroneous trials. These
points may lead to concern about how they will affect the outcome of the experiment as
a whole. A question then arises of how to properly deal with this data. Simply removing
the trials that are considered or indeed known to be erroneous might appear to be a simple
solution, but also an obviously dubious one. Ideally, we would like to have a system that
allows us to keep all the data yet not give the questionable points the same weight as other,
more reliable points. The calculation of weighted averages and weighted regression lines
involves such a system.
The uncertainty in a particular point is the criteria that will be used to determine the
reliability of that point. The presence of a higher degree of random error will typically (but
not certainly) create measurements that are further away from the ‘true’ value. This is turn
will lead to a greater value for the standard deviation, if this trial is indeed an anomaly.
This also agrees with the assertion that precise measurements possess a lower degree of
uncertainty. Note that if a point appears erroneous yet has a low degree of uncertainty, this
would indicate the presence of systematic error, as opposed to random error. In these cases,
there is little to do but re-take the data or continue the analysis with the point included.
In order to allow the reliable points to have a more substantial affect on the results, we will
introduce weighting factors for each point. These weighting factors wi can be found using
the following equation:
wi =
1
(δyi )2
(1.8)
It should be easy to see that this equation will lead to higher weighting factors for precise
points, since it is inversely proportional to δy. It should also be apparent that the weighting
factors should be easy to find using a spreadsheet program.
Once these weighting factors have been determined, we can use them to evaluate certain
quantities in a manner that emphasizes those values with greater weight. One such quantity
is the weighted mean, which can be found using
14
LAB 1. ERROR ANALYSIS
P
wi yi
ȳ = P
(1.9)
wi
Note that if all the weighting factors are equal, this reduces to the equation for the mean
(try it). This weighted mean will not be used frequently within the course of the lab, but
serves as a simple introduction to the application of weighting factors.
A more useful application is found in the determination of the slope by linear regression.
Cases may arise during the course of analysing data where one or two points appear clearly
out of place with the assumption that the data is linear. If these points are indeed less
precise, as determined by the error associated with these points, then we would like them
to have a diminished impact on our determination of the slope and intercept of the graph.
This can be achieved by introducing the weighting factors into the equations for the slope
and intercept as follows:
Slope = (
X
Intercept = (
X
wi
X
wi x2i
wi xi yi −
X
wi yi −
X
X
wi x i
X
wi yi )/∆
(1.10)
wi x i
X
wi xi yi )/∆,
(1.11)
where ∆ =
wi wi x2i − ( wi xi )2 . Note the differences between these equations and
equations 1.3 and 1.4; each sum contains an additional term wi , and each appearance of N
P
is replaced with
wi . These changes ensure that the terms with larger weighting factors
are given due consideration, while being normalised to ensure the correct scale of values in
the final result.
P
P
P
Exercise 5. Find the slope and intercept of the data from table 1.1 using weighted linear
regression, and compare to the slope and intercept found in Exercise 4. Plot the data
again and include both linear regression lines, clearly labelled. Which one appears
like the better fit?
LAB
2
Rotational Motion
Objective
To determine the relationship between the rotational dynamics variables, and verify conservation of energy for a rotational system.
Apparatus
• Rotary Motion Sensor
• Motion Sensor
• Brass disk to attach to Rotary Motion Sensor
• String or thread
• Hanging mass (piece of packing foam)
• Science Workshop 500 interface (if using SW Rotary Sensor)
Background
It is conventional to describe the motion of a rotating body using different variables than
those used to describe translational motion. While the translational variables are still
valid, they become more difficult to analyse when one considers that different parts of the
rotating object are moving at different translational rates. Consider the example shown in
Figure 2.1. If the wheel is rotating at constant velocity, both the inner and outer paths
will take the same amount of time to traverse. It is clear from the diagram that these two
paths represent drastically different distances, so even though the wheel’s rotation can be
described as constant, the translational velocity changes depending on what part of the
object we are describing.
15
16
LAB 2. ROTATIONAL MOTION
Figure 2.1: Two paths traversed by two points on the same wheel, through a complete
rotation.
We can remove this difficulty by using different variables. Even though the two paths
from Figure 2.1 traverse different distances, the angle covered in each case is the same, 2π
radians. The same can be said of any part on the wheel. Thus instead of measuring the
distance travelled, we can measure the angle rotated. In a similar manner, we can define
a variable for the amount of rotation per unit time, known as the angular velocity ω. The
rate of change of the angular velocity is known as the angular acceleration, α. These two
variables are derived from the angle rotated in exactly the same way that the velocity and
acceleration are derived from distance travelled, that is
ω =
α =
=
dθ
dt
dω
dt
d2 θ
dt2
(2.1)
(2.2)
(2.3)
These variables obey precisely the same kinematic equations as their translational counterparts. In order to find the rotational analogue, all that must be done is to substitute
the rotational variable for its translational counterpart. Thus the equation v = v0 t + 21 at2
becomes ω = ω0 t + 21 αt2 . This process extends to more than just the kinematics equations,
and similar analogues exist for laws such as Newton’s 2nd Law, conservation of energy,
conservation of momentum, etc.
In order to analyse a rotational system, you will be using a rotary motion sensor that can
measure angular position, velocity, and acceleration. This system will be connected to that
of a falling mass (a linear system), and the linear motion of this mass tracked using a motion
sensor.
By analysing a system containing both translational and rotational parts, we will attempt
to determine the relationship between the variables corresponding to these motions. We
PROCEDURE
17
Figure 2.2: Alignment of the attachable pulley with the rotary motion sensor.
will also examine conservation of energy from a rotational perspective to test out the theory
that every translational equation has a rotational analogue.
Procedure
1. Measure and record the mass and radius of the brass disk. Attach the 3-step pulley
onto the rotary motion sensor shaft as shown in Fig. 2.3. Place the disk on top of the
pulley, and screw it into place.
2. Measure a length of string so that when the string is attached yet not wound around
the upper pulley, the end of the string is roughly 30 cm from the ground.
3. Attach a mass to one end of the string, and attach the other end to the hole in the
pulley attached to the shaft of the rotary motion sensor. The mass should have a flat
surface facing downwards, if possible. The mass must be quite small, 15 g at most. A
piece of packing foam is quite good for this purpose.
4. Place the motion sensor underneath the hanging mass, pointing upwards.
5. Start DataStudio, and add the appropriate sensor to the display using the ‘Experiment
Setup’ button (if this wasn’t done automatically when the sensors were plugged in).
Figure 2.3: Side view of the rotary motion sensor with the brass disk attached. The 3-step
pulley should slide on and off the shaft of the sensor if it is not aligned properly. Ensure
that the disk is screwed firmly in place.
18
LAB 2. ROTATIONAL MOTION
6. Display a graph of the motion sensor input, and of the rotational apparatus input.
Set both sensors to record data at 10 Hz.
7. While holding the top disk in place to ensure that it does not move, hang the mass
over the edge of the bench, with the string securely in the groove on the front pulley.
8. Test the alignment of the motion sensor by clicking ‘Start’ in DataStudio and moving
the mass up and down. Ensure that the sensor is indeed detecting the mass and only
the mass. Stop taking data when you are satisfied that this is the case.
9. The 3-step pulley attached to the rotary sensor shaft has 3 pulleys of different radii.
Measure the diameter of the two smallest using a caliper, and record your results,
with uncertainty. Wind the string around the smallest pulley first, this will be the
first one we shall test.
10. Click ‘Start’ in DataStudio, and let go of the mass. The mass will fall, then start
to rise abruptly as it reaches the end of the string. Once the mass has reached its
minimum height, click ‘Stop’.
11. Export your position and angular position data as text files.
12. Repeat the experiment using the slightly larger step (the middle step) on the 3-step
pulley. Note that when the string is attached to the largest pulley, the mass will fall
too quickly to take data.
Analysis
Open your data files using Excel, and combine them into a single data file. In a separate
worksheet, copy the position data from one of your trials that corresponds to the fall of the
mass. Find the angular position data that corresponds to the fall and copy this in to an
adjacent column. It may be helpful to look at the angular velocity data from DataStudio
to see the approximate time frame of the fall.
Add a column to your table for the distance fallen, which is simply each height value
subtracted from the initial height (d = h0 − h). Also add columns for the angle in radians,
and the translational and rotational velocity. The velocities should be found using the
conventional formula v = ∆h/∆t, ω = ∆θ/∆t. Add three extra columns for the energy
data, which will be calculated purely from your translational data. The potential energy U
can be found using mgh, where h is the height of the mass above the sensor. The kinetic
energy K is found using ½mv 2 . Your table should look something like this:
h (m)
θ (°)
d (m)
θ (rad)
v (m/s)
ω (rad/s)
U (J)
K (J)
U + K (J)
DISCUSSION
19
Create graphs of d vs. θ for each pulley. Find the slopes of each of these lines and present
each slope clearly, with uncertainty. Also create graphs of v vs. ω for one of the pulleys
(your choice), and find the slope.
Discussion
What should the slope of your position vs. angle and velocity vs. ω graphs be equal to? Do
your results agree with predicted theory? Are there any sections of your graph that look
non-linear? What experimental details would lead to this kind of result?
Are the energy values in the last column of your table constant? Why not? What source of
energy is not being considered in this column? Calculate this energy for a number of points
on your table (or simply change the appropriate column in your table), and see if your new
results agree with conservation of energy. Was energy conserved in this experiment?
LAB
3
Conservation of Angular Momentum
Objective
The objective of this experiment is to verify the conservation of angular momentum during
a rotational collision.
Apparatus
• Rotational Dynamics Apparatus
• Computer with DataStudio
• Science Workshop interface/adapter
Background
The angular momentum (L) for a particle is defined as
L = mvr
(3.1)
where m represents the mass of the particle, v its velocity, and r the distance of the particle
from the axis of rotation. If an object rotating about some fixed point is thought of as
a conglomerate of many particles, then the previous definition of L may be extended to
describe the angular momentum of any rotating body by a simple summation.
L=
X
mi vi ri
(3.2)
i
The total angular momentum of the body will simply be the sum of the angular momenta
of the particles in the body. Since all the particles comprising the body move with the same
20
BACKGROUND
21
angular velocity ω, the angular momentum of the body may be expressed as
L=ω
X
mi ri 2
(3.3)
i
In equation 3.3 the term mi ri2 represents the masses and positions of all the particles in
the body. The reader should recognize this as the moment of inertia of the body. Therefore
the angular momentum of the body may be expressed as
P
L = Iω
(3.4)
To this point, we have considered L as a scalar quantity. It is important to remember that
~ is in fact a vector quantity, similar to the translational momentum
the angular momentum L
~ involves the cross product operator.
p~. The vector definition for L
~ = ~r × m~v n̂
L
(3.5)
Recall the definition of the cross product
~×B
~ = |A| |B| sin θ n̂
A
(3.6)
For introductory physics θ is usually 90°, making sin θ = 1. Therefore
~×B
~ = |A| |B| n̂
A
(3.7)
Applying the same restriction in θ to equation 3.5 yields
L = |~r|m|~v | n̂
(3.8)
(Compare equations 3.8 and 3.1.)
~ must denote
Consider a rotating disk. To fully describe its angular momentum the vector L
both the magnitude direction of the angular momentum. The magnitude is denoted by
~ the direction, of which there are only two possibilities, is indicated by the
the length of L,
~ and the right hand rule (RHR).
direction of L
~ will always be perpendicular to the plane of rotation of the disk. If the thumb of the
L
~ pointing in the same direction, then the fingers of the right
right hand is placed along L,
hand will curl in the direction of rotation of the disk (clockwise or counterclockwise).
Conservation of Angular Momentum
The principles of conservation (energy, and momentum) are some of the most fundamental
laws of nature. Conservation is easily pictured using the vector description of angular
~ is a function of two quantities, I (moment of inertia) and ω
momentum. Recall that L
(angular velocity). Consider a figure skater executing a stationary spin on the ice with
~ = I~
arms extended. The skater will have an angular momentum of L
ω . As the skater’s
22
LAB 3. CONSERVATION OF ANGULAR MOMENTUM
~ increased also? No, the
arms retract the skater’s angular velocity will increase. Has L
~
skater’s moment of inertia has decreased and therefore L is unchanged (Note: Energy is
not conserved; do you know why?).
Consider two separate disks rotating with different amounts of angular momentum. If the
disks are coupled together (ideally) then the principle of conservation of momentum dictates
~ after the collision must be the same as L
~ before the collision, i.e.
that L
L1i + L2i = L1f + L2f
I1 ω1i + I2 ω2i = (I1 + I2 ) ωf
NOTE: Directional information must also be considered.
If both disks were rotating in the same direction the vector picture would be:
If they were rotating in opposite directions the vectors would resemble:
This experiment involves the verification of the conservation of angular momentum in both
of these scenarios.
Setup
1. Remove each disk from the apparatus and ensure that all surfaces are clean.
2. Ensure that the digital display is connected to AC power using a 9 V adapter.
PROCEDURE
23
3. Plug the 2 audio jacks from the apparatus into a Digital Adapter. The adapter itself
should be plugged in to a Pasport USB interface, and this interface plugged into a
computer running Datastudio. Note which port (#1 or #2) the yellow plug is inserted
into; this is the jack corresponding to the top disk.
4. Upon plugging the interface in, the software should prompt you for the type of sensor
you have just plugged in. Choose "Photogate and Picket Fence" twice (once for each
disk).
5. In the setup window, remove the position and acceleration data from each sensor
(simply uncheck the box).
6. While still in the setup window, select the "Constants" tab for one of the sensors. The
value shown is what the program assumes is the distance between bars on the picket
fence. Using the fact that there are 200 bars around the exterior of the disk, calculate
the distance between bars. Enter this value into the box provided for each sensor. Be
sure to record your calculation and corresponding result in your report.
Procedure
1. Record the radii and masses of any disks you use in this experiment. Uncertainties
should also be approximated and recorded. Replace the disks onto the apparatus once
done. Turn on the air supply so that the pressure shown on the gauge is roughly 9
PSI.
2. Place the pin in the centre of the top disk. This will create a cushion of air between
the two disks that allows them to spin independently.
3. Spin the top disk in one direction. (Note: the effects of friction increase with rotational
speed – keep the speeds below 600 Hz.) Record the direction of travel.
4. Spin the bottom disk in the same direction.
5. Start Datastudio. After a couple of seconds, remove the pin to collapse the air cushion
between the disks. Stop take data shortly after the collision. From looking at the
graph of the data, make a note of approximately when the collision occurs. This will
help find the appropriate data when sifting through your data files.
6. Export your data as a text file. You will need to export the top and bottom disk data
separately. Be sure to label your file with the direction of rotation of the two disks
(same or opposite), and the type of top disk (steel or aluminum).
7. Repeat the experiment rotating the disks in opposite directions. If using the steel top
disk, try to spin the disks with substantially different angular velocities.
8. Repeat the experiment with the aluminum top disk.
24
LAB 3. CONSERVATION OF ANGULAR MOMENTUM
As an alternate means of saving your data, you may choose to display the data in a table
in Datastudio. Simply double click on the "Table" button on the left side of the screen and
choose the appropriate sensor/run. Data can be selected, copied, and pasted from these
tables directly into a single Excel file. This removes the need for multiple text data files. In
order to obtain data from the other disk, it may be necessary to close the table and reopen
it, selecting the other sensor. Consult the Analysis section to see what portion of the data
to select and copy.
Analysis
For each trial, find the time at which the collision occurs. In the case of the disks spinning
in opposite directions, the exact point may not be very pronounced. Find the times before
and after the collision when the velocities appear stable and use these as your data points.
Copy 10 values of the velocity of each disk from before and after the collision. Paste these
into a spreadsheet file, clearly labelled. You may use either data file to obtain the velocity
after the collision, since the disks are rotating at the same rate. You may disregard the
time column, since the times will most likely not line up exactly.
For each trial, you should now have 3 columns of data representing the linear velocities of
each disk before and after the collision. Find the mean and standard deviation of the mean
for each column and report these as the velocities before and after the collision. This will
give you 3 velocities for each trial, with errors: vtop , vbottom , and vf inal .
Convert all your velocities into angular velocities using the radius of the disk. Record these
velocities (with the appropriate sign!) in a simple table for each trial:
ω
L
K
Top
Bottom
Final
Calculate the moment of inertia of the disks using I = 1/2mr2 . Record each value with
uncertainty, clearly labelled. Use these values to find the L and K columns of your tables,
using L = Iω and K = 1/2Iω 2 . Be sure to include uncertainty in all results. Note that the
motion of inertia of the 2 disks when they are rotating as one is simply equal to the sum of
the individual I values.
Using the top and bottom L values, calculate the final angular momentum using conservation of momentum. Also find the final kinetic energy using conservation of energy.
DISCUSSION
25
Discussion
Discuss whether or not your results show conservation of angular momentum within error,
for each case. Which cases appear to be better examples of conservation of momentum?
Is there any reason why some trials should be better than others? What was the primary
source of error in this experiment? Repeat this same discussion for conservation of energy.
LAB
4
Kater’s Pendulum
Objective
To investigate the theory of the pendulum and determine g to within 0.1%.
Apparatus
• Kater’s Pendulum w/2 attachable weights
• Wall mount
• Photogate
• PASCO Digital Adapter
• Computer w/DataStudio
• Screwdriver (Flathead)
Background
An accurate value of g, the acceleration due to gravity, is needed to calculate an object’s
motion in the earth’s gravitational field or to measure the mass of the earth using the
universal gravitational constant, G, such as in Cavendish’s original experiment. Local
variations in g are important for the study of geological formations, and thus for locating
mineral deposits. Theoretically, one can determine g from the measurement of the period
of a simple pendulum.
In practical terms, the use of a single pendulum for such a measurement is difficult. The
theory is based on a point mass connected to a massless support pivoting about a single
26
BACKGROUND
27
Figure 4.1: Free body diagram for a pendulum of arbitrary shape. The gravitational force
acts at the centre of mass and has a component in the negative θ direction equal to M gd sin θ.
The dashed line indicates θ = 0.
point in space, all 3 of which are extremely difficult to realise experimentally. An alternate
method was devised by Henry Kater in the early nineteenth century, following a suggestion
by Bessel. A pendulum was constructed with 2 knife edges on either side of the centre
of mass, which served as pivot points upon which the pendulum could be swung (see Fig.
4.3). The centre of mass itself was adjustable via weights which could be fixed at different
positions on the pendulum. When the period of the pendulum is identical when swung
from either pivot point, the equation for g is identical to that of a simple pendulum with
length equal to the distance between the pivot points. Since this length is measurable to a
high degree of accuracy, this permits a very precise determination of the local value of g.
This method was used in the 1930’s to determine the value of g in Washington as 980.080
±0.003 cm/s2 , which is accurate to within 1/1000th of a percent.
In order to arrive at the appropriate equation describing the relation between the value for
g and the length and period of the pendulum, we shall begin with first principles. Since
this a rotational system, the simplest place to begin is with the rotational form of Newton’s
2nd Law, τ = Iα. The torque here is due to gravity, which acts as a restoring force (see
Fig. 4.1). Substituting in the appropriate component of gravity gives
M gd sin θ = −Iα,
(4.1)
where M is the mass of the pendulum, d is the distance to the centre of mass, and the negative sign is due to the torque always acting towards θ = 0o . Since the angular acceleration
α is simply the 2nd time derivative of θ, we can express this equation in terms of θ as
∂2θ
+ M gd sin θ = 0.
∂t2
If the angle is small, then we can use the small angle approximation sin θ ≈ θ:
I
I
∂2θ
+ M gdθ = 0.
∂t2
(4.2)
(4.3)
28
LAB 4. KATER’S PENDULUM
Figure 4.2: The distance between the sharp edge of each pivot and the CoM are labelled as
d1 and d2 . We will assume d1 < d2 . Note that d1 + d2 = L, where L is simply the distance
between the knife edges. This measurement is completely independent of the location of
the centre of mass, and reasonably easy to measure accurately.
This is a second order differential equation in θ, with a simple solution θ = sin ωt, where
s
ω=
M gd
,
I
(4.4)
where ω is the angular frequency of the pendulum. The period T is related to this frequency
by the equation T = 2π/ω, thus the period of the pendulum is
s
T = 2π
I
.
M gd
(4.5)
For Kater’s pendulum, there are 2 knife edges that the pendulum can pivot about. If the
centre of mass of the pendulum is not exactly halfway between the two pivots, then it is
possible that the pendulum can have two different periods when swung from either end. We
shall label these two different periods T1 and T2 . The moment of inertia about each pivot
can be found using the parallel axis theorem,
I = Ic + M r 2 ,
(4.6)
where Ic is the moment of inertia about axis going through the centre of mass (CoM), and
r is the perpendicular distance between this axis and a parallel axis that the pendulum is
pivoting about (see University Physics, section 9.5). If we let the distance between each
pivot and the CoM be d1 and d2 , as in Fig. 4.2, then we can combine equations 4.5 and 4.6
to express T1 and T2 as
s
T1 = 2π
Ic + M d21
M gd1
s
T2 = 2π
Ic + M d22
M gd2
(4.7)
By solving each equation for Ic , equating the results, and expressing the final result in terms
of the differences and sums of the distances and periods, we can arrive at
8π 2
g = T 2 +T 2 T 2 −T 2 1
2
1
2
d1 +d2 + d1 −d2
(4.8)
SETUP
29
This is the correct expression for determining g using this apparatus. If we are able to
adjust the center of mass so that T1 = T2 , then this equation simplifies to
g=
4π 2 L
,
T2
(4.9)
where L = d1 + d2 is the distance between the knife edges. This is exactly the same
equation as a simple pendulum of length L, and is completely independent of the location
of the centre of mass!
Before we get carried away with this elegant result, we should of course note that experimentally it is simply not possible to have T1 = T2 . There will always be a small discrepancy
between the two, and we should account for this discrepancy. This leaves us with equation
4.8, which unfortunately is much more complex, while also containing the measurements d1
and d2 . This is a problem, since d1 and d2 are extremely difficult to measure accurately, as
doing so requires a precise measurement of the center of mass.
We can attempt to deal with this problem by expressing equation 4.8 as a series expansion.
If we let ∆T = T1 − T2 , and T = (T1 + T2 )/2 (the average period), then this equation
becomes
8π 2
.
g = 2 (4.10)
2T
2T ∆T
+
L
2d1 −L
By factoring out a 2T 2 /L term from the denominator and using a series expansion for the
remaining term, we arrive at
4π 2 L
∆T
L
1+
,
2
T
T L − 2d1
g=
(4.11)
to first order. This is very similar to equation 4.9, but includes a corrective term that is
proportional to ∆T . If ∆T < 0.0001s, then this term will be extremely small, allowing us
to determine g to a very high level of accuracy if L is measured precisely. Note that d1 is
still present in this equation, but only in the small corrective term, implying that the center
of mass need not be determined to a great deal of accuracy.
For the experiment, we will use 2 adjustable brass bobs to alter the center of mass of the
pendulum. The larger bob will be fixed at one end of the pendulum, while the smaller bob
will be adjusted so that pendulum will have a similar period when swung from either knife
edge. The timing of the period will be done via photogate and computer, as explained in
the setup section below.
Setup
In order to accurately measure the period of the pendulum, we will use a photogate. This
device emits a constant beam between an emitter and a receiver, conveniently placed into
a ‘gate’ shape. When an object is placed between the two, the beam is broken and a signal
30
LAB 4. KATER’S PENDULUM
Figure 4.3: A simple diagram of the setup for this experiment. The two bobs are adjustable
using a screwdriver, allowing for a variable centre of mass. The knife edges create a very
precise pivot point. When the period of the pendulum is approximately the same when
swung from both knife edges, a value for g can be found.
is sent, usually accompanied by a time stamp. The difference between two time stamps
can be used as a measure of time between 2 consecutive events. Since a pendulum travels
periodically along the same path, a single photogate is all that is required to time its motion.
1. First measure and record the weight of the bobs and the pendulum. Include an
estimate of the error.
2. Suspend the pendulum on the wall support at the back of the lab. The pendulum
should swing parallel to the wall, adjust the knife edge as necessary to ensure that
this is the case. This adjustment should be done any time the pendulum is placed
on the support. Also note that whenever you suspend the pendulum, the knife edge
need not be placed in the exact same spot each time. A small deviation to the left or
right is of little consequence. Damp out any motion when you are satisfied with the
placement of the pendulum.
3. Place a photogate at the bottom of the pendulum so that the photogate beam is broken
by the tip of the pendulum at the bottom of its swing. The height of the photogate
may need to be adjusted, this should be straightforward. You do not need to be
extremely precise with this placement, anywhere near the bottom of the pendulum
will suffice.
4. Connect the photogate to a PASCO Digital Adapter. Connect this adapter to a USB
PROCEDURE
31
link, and connect the link to the USB port on a computer running Datastudio. Upon
connection, a window should pop up in Datastudio asking you how you want to use the
sensor. Choose the ‘Photogate and Pendulum’ option. This option will only display
a time after the beam has been broken twice, which corresponds to the period of the
pendulum.
5. Remove the pendulum from the support. Slide the large bob on to one end of the
pendulum, and tighten it in place using a screwdriver. Leave at least 5 cm between
the end of the beam and the edge of the bob to ensure that when this bob is on the
bottom of the pendulum, it does not interfere with the photogate beam.
You should now be ready to take data.
Procedure
We wish to adjust the masses on the pendulum so that when swung from either knife edge,
the period is identical (within a fraction of a percent). Instead of simple trial and error, we
will use a very simple root-finding procedure to establish a proper starting point, and then
adjust from there.
1. First record the distance between the knife edges, L. This should be printed on the
pendulum.
2. Measure and record the distance between the flat edge of the knife edge and the edge
of the large bob. Using calipers would probably be easiest, but note that accuracy is
not overly important.
3. Slide the small bob on to the opposite end of the pendulum and tighten it using the
screws. Adjust the screws so that the bob can be loosened and tightened simply
using the central pressure screw. Position the bob roughly 5 cm from the end of the
pendulum. Measure the distance between the flat edge of the knife edge and the edge
of the small bob, and record this distance.
4. Place the pendulum on the knife edge (carefully!). Be sure to take note of whether the
large bob is at the top or bottom of the pendulum, and label your times appropriately
(e.g. Tu for up, Td for down).
5. Displace the pendulum by a small angle and release. Click start on DataStudio. Once
you have 10 periods sampled, stop taking data. You should attempt to ensure that
the magnitude of the oscillations are similar for each trial, as the period has a small
dependence on amplitude.
6. Copy the results into a spreadsheet (highlight them within the table and hit Ctrl-C
to copy). When pasted, two columns will appear in the spreadsheet. The first column
containing times may be deleted, it is unnecessary.
32
LAB 4. KATER’S PENDULUM
7. Find the average and standard deviation of the mean of your periods. Label the trial
using the distance of the small bob from the knife edge. It may be worthwhile to keep
a separate table going, with columns for the small bob distance, and the two times
for that distance, clearly labelled.
8. Carefully remove the pendulum from the support, invert it, and place the other knife
edge on the support. Repeat the previous 3 steps.
9. Place the small bob close to the knife edge, roughly 1 cm away. Measure the distance
using a caliper, and record it. Repeat the previous 5 steps.
10. Using these two initial times, find the period as a function of distance, assuming a
linear fit between the two. You will need two such equations, one for the ‘up’ times,
and one for the ‘down’. Recall that you only need two numbers to characterise a
straight line, the slope and the intercept; both should be easily found. Alternatively,
you may plot the initial two points and add a trendline between them to find the
equation of each line.
11. Find the point of intersection of the two lines. This will be the position where the
periods are equal. Place the small bob at this distance and again find the periods,
as above. Note that it is possible to get a negative position; in this case, the small
bob must be placed between the two knife edges. If this is the case, you may want to
repeat the above steps to find a more appropriate starting point.
12. You will likely still need to adjust the position of the small bob until the periods are
almost equal. Continue changing the position of the small bob until the difference
between the periods is less than 0.0001 s. Note that the position measurements for the
bob are now of less consequence, they are merely to ensure that you do not measure
the period for a particular position twice. Accuracy in the position is therefore of
little importance.
13. Once you have the correct period, we need to approximate the center of mass of the
pendulum. Remove the pendulum from the supports and place it on a table, without
moving the bobs. Let the sharp end of the knife edge closest to the large bob be the
x = 0 point. Measure the distance to the center of each bob and record them (see
Fig. 4.4).
Analysis
Calculate T , the average of T1 and T2 , and ∆T = T1 − T2 . Approximate the error in
each of these results using error analysis. Note that the error in T is likely much greater
than the result given by simple error analysis using δT1 and δT2 , but it will suffice for this
experiment. Using your value for T and L, find an approximation to g using equation 4.9.
Include an estimate of the error.
In order to improve the result, we need to find a value for the corrective term in equation
4.11. This equation requires us to know d1 , the distance to the centre of mass. Using your
DISCUSSION
33
Figure 4.4: Coordinates needed to calculate the center of mass of the pendulum. Here xL
is the coordinate of the large bob, and xS is the coordinate of the small bob, measured to
the center of the bob. Note that these are in fact coordinates, not distances, so the sign
does matter.
measurements of the coordinates of the bobs, find the centre of mass. Assume that the
CoM of the pendulum is directly at the middle of the bar. Once you have found d1 , find a
value for the corrective term and use this term to find an improved value for g.
In order to evaluate how good your result is, we will find a theoretical value for g in
Lethbridge. Note that the local value for g is based primarily (but not exclusively) upon
two factors, the altitude and the latitude. The altitude correction is simply due to the
increased distance from the centre of mass of the Earth. Since the gravitational force
decreases with distance, being ‘farther away’ from Earth will reduce the force. The latitude
correction is due to Earth’s rotation, which acts as a fictitious centrifugal force. This is more
pronounced at the equator, yet is not a factor at the poles. Consult University Physics,
section 12.7 for more information on this rotational effect. According to the handbook for
the American Institute of Physics, the local value of g is given by
g = 9.780490(1 + 0.0052884 sin2 φ − 0.0000059 sin2 2φ) − 0.000003084h,
(4.12)
where φ is the latitude, h is the altitude in metres, and g is in m/s2 . Find the latitude and
altitude of Lethbridge using an online source and use them to find a theoretical value for g.
Discussion
How do the theoretical and experimental values for g compare? Do they agree, within error?
If not, what experimental details might lead to the discrepancy? One thing to consider is
how accurate the theoretical calculation is; what might cause this value to be incorrect?
If a resistive term were present, from either air resistance or friction between the knife edge
and support, would your experimental value for the period be too high or too low? What
effect would this have on your value of g? Based on your comparison between theory and
experiment, do you believe that a resistive term explains any discrepancy?
Why is it important that the pendulum be asymmetric? Equation 4.11 might provide some
insight.
LAB
5
Mechanical Equivalent of Heat
Objective
The objective of this experiment is to measure the ratio of mechanical work done to heat
produced, known as the mechanical equivalent of heat.
Apparatus
• Mechanical Equivalent of Heat Apparatus
• Weights totalling at least 3 kg
• Calipers
• Balance
• Ohmmeter and Wiring
Background
The principle of conservation of energy dictates that heat arising from a given amount
of work must be equivalent to the amount of work performed. Therefore a quantifiable
relationship between work measured in Joules and thermal energy measured in Calories
must exist. The relationship is called the mechanical equivalent of heat.
In the late 18th century Benjamin Thompson (Count Rumford) observed while boring
cannons that as the boring bits dulled the amount of heat increased even though less
material was removed. This contradicted the theory of the day which suggested that the
heat was the result of a fluid called "caloric" removed from the material during the boring
process. Rumford established the mechanical equivalent (J) of heat to be 1 cal = 5.700
34
BACKGROUND
35
Figure 5.1: Setup for the MEH experiment
Joules.
After a series of precise experiments (1840-1870), James Prescott Joule concluded the relationship to be 1 cal = 4.240 Joules. By definition the relation is 1 cal = 4.186 Joules.
In this experiment work is performed by turning a hand crank. In another part of the
apparatus this work is turned into heat by friction. By calculating the work done and
the change in temperature of the apparatus, the relationship between the two may be
established.
A measurable amount of work is performed by turning the hand crank, which turns the
aluminum cylinder. A nylon rope is wrapped around the cylinder several times to support
the hanging mass. The friction between the rope and the cylinder is just enough to maintain
the mass at a constant height. This ensures that the torque on the cylinder is constant and
measurable.
As the cylinder turns, the friction between the cylinder and the rope converts the work done
into heat which raises the temperature of the cylinder. The temperature of the cylinder
can be measured by a thermistor embedded in the cylinder. By monitoring the change
in temperature (change in resistance of the thermistor) of the cylinder the thermal energy
transferred to the cylinder can be measured. Knowing the work done and the change in
thermal energy the ratio between work and energy may be calculated.
The work done on the cylinder by turning the crank equals τ , the torque acting on the
cylinder, times θ, the total angle through which the torque acts. It is difficult to measure the
torque directly, however, since the motion of the cylinder is more or less constant throughout
the experiment we know that the torque supplied just balances the torque provided by the
friction of the rope. The torque can be calculated as:
τ = M gR
(5.1)
36
LAB 5. MECHANICAL EQUIVALENT OF HEAT
where M is the hanging mass, g the acceleration of gravity and R the radius of the cylinder.
The angle through which the torque acts is simply 2π radians multiplied by the number of
revolutions, N . Therefore, the work done in each trial can be calculated as:
W = 2πN M gR
(5.2)
The heat absorbed by the thermistor can be found by using the formula:
Q = mc∆T,
(5.3)
where m is the mass, c is the specific heat capacity, and ∆T is the change in temperature.
All of these quantities refer to the object being heated, in this case the thermistor. The
specific heat capacity of the thermistor is 0.220 cal/g°C.
Once these quantities are found, the mechanical equivalent of heat (MEH) can be found
easily by using the definition of this quantity:
MEH = Work/Heat
(5.4)
Procedure
1. Record the mass M and diameter d of the cylinder, with uncertainty.
2. Weigh the hanging mass.
3. Set up the apparatus as per Fig. 5.1.
4. A starting resistance of between 110 kW and 125 kW or less is recommended. If the
resistance is too low, cool the cylinder in an ice bucket. If too high, warm it with your
hands.
5. Pick the number of revolutions (N ) at which to stop, preferable between 20 and 50
(more is better).
6. Record the resistance of the thermistor as displayed on the multimeter (Ri ). Also
record the uncertainty.
7. While applying light tension to the free end of the string, crank slowly to lift the mass
approx. 10 cm. off the ground, and continue holding the end of the string to maintain
this height.
8. Set the counter on the apparatus to zero by turning the knob clockwise.
9. Begin to turn the crank while releasing the tension in the string. If the string is
simply resting on your finger, this should be sufficient tension to prevent the mass
from falling while you turn the crank. Crank at a steady consistent pace until you
have reached the desired number of revolutions. At this point stop cranking and watch
the resistance. Record the minimum resistance achieved (Rf ).
ANALYSIS
37
10. Record the number of turns (N ) of the crank. If you went through a partial revolution
on the final turn, estimate the fraction of a revolution and the uncertainty.
11. Repeat the experiment 4 more times from step 4, using a different number of revolutions N for each trial.
Analysis
Present your results in a suitable table. Include columns for the number of cranks, the initial
and final resistances and temperatures, the work done, the heat produced, and finally the
MEH.
Calculate values for the initial and final temperature by using the data sheet for the thermistor. The data sheet lists values for the resistances at which the temperature is an integer.
The resistance values measured will not correspond exactly with those on the sheet, so an
approximation must be made. If, for example, you measure a resistance value of 123,000
W, then according to the data sheet the temperature lies between 20°C (126,740 W) and
21°C (120,810 W). To obtain a more exact value, we will assume that the change is linear
between each set of points on the data sheet. The temperature can then be found by using
the formula:
R1 − R
T = T1 +
,
(5.5)
R1 − R2
where T1 is the lower temperature of the range that the resistance lies within (20°C), R1 is
the resistance corresponding to T1 (126,740 W), R is the resistance measured (123,000 W),
and R2 is the resistance corresponding to the higher temperature of the range in question
(120,810 W). Substituting these numbers from the above example in to this formula gives a
result for T of 20.63°C. Calculate all the temperatures from each trial using this formula.
Use equations 5.2 and 5.3 to find the work done and heat absorbed for each trial, with error.
Use these values to find the MEH using equation 5.4.
After finding the MEH for each trial, find the average and standard deviation of the mean
of your results and present this as the final result.
Discussion
Compare your value for the MEH with the accepted value of 4.186 J/cal and comment.
Discuss any sources of error that you feel may have affected your results.
Which individual trials were the best? Is there anything particularly special about the
initial and final temperatures of these trials? How could the initial and final temperatures
by chosen so as to minimise the effects of heat transfer from the environment?
38
LAB 5. MECHANICAL EQUIVALENT OF HEAT
Is it experimentally possible that the heat absorbed by the cylinder could be greater than
the work done on it? Is it theoretically possible?
LAB
6
Electrical Equivalent of Heat
Objective
To determine the relationship between the heat emitted by an electrical system and the
work done by that system.
Apparatus
• Calorimeter
• DC Power Supply
• 2 Multimeters
• Temperature Probe
Background
When heat is added to a solid or a liquid, that energy goes into increasing the materials
internal energy, and thus increasing its temperature. The relationship between the heat
and the resulting change in temperature is given by
Q = mc∆T,
(6.1)
where Q = heat, m = mass, c = specific heat, and ∆T= change in temperature.
In this experiment, heat is added to the calorimeter by the use of a heating resistor. Work
is produced by a power supply providing an electrical current.
There are two things changing temperature in this experiment: The water and the aluminum
39
40
LAB 6. ELECTRICAL EQUIVALENT OF HEAT
Figure 6.1: Setup
calorimetry cup. The water and cup have different masses and specific heats, but we assume
that they both have the same change in temperature since they are in thermal contact.
The generated electrical energy is converted into heat in the resistor, increasing the temperature of the water and the cup. Historically, this has been referred to as the Electrical
Equivalent of Heat (hereafter referred to as EEH).
Electrical power is determined by the voltage produced by the generator, and the resulting
current that is produced.
P = IV,
(6.2)
where I = Current through the resistor (Amps), V = Voltage across the resistor (Volts),
and P = Power (Watts = Joules/sec).
Power is the rate at which energy is generated or used: Power= Energy/time. Thus Energy
can be calculated by E = P t, or in a case where the power is not constant, energy is the
area under a power vs. time graph.
Procedure
1. Weigh the inner aluminum cup and record the mass, with uncertainty.
2. Pour cold water (use a little ice) into the cup until it is at least half full and reweigh
the cup.
3. Find the mass of water by evaluating the difference between the 2 masses; make sure
there is at least 100 g of water in the cup.
4. Record the initial temperature of water by clicking Start in DataStudio and noting
the constant value. Ensure that the water is in thermal equilibrium! (Stir a little to
make sure) Record the uncertainty. Recall that if the temperature value is fluctuating
slightly, it is not appropriate to let the uncertainty be the precision of the instrument!
ANALYSIS
41
5. Turn on the power supply and set it to output between 4 and 5.5 Volts. Record the
voltage, with uncertainty.
6. Create a circuit that consists of the ammeter, voltmeter, heating element, and power
supply. Refer to figure 6.1 for more details on the setup.
7. Simultaneously start DataStudio and switch on the power supply.
8. Record the current shown on the ammeter. Watch the voltage and current on the
meters; they will change over time. Record their values at regular intervals (every
minute, for example) and assume an average value for your calculations. For the
uncertainty, use the standard deviation of the mean or the precision of the instrument,
whichever is greater.
9. Let the current flow for roughly 5 minutes, stirly occassionally. Do not stir too vigorously, as this will actually add energy to the water (conversion of mechanical energy
to heat).
10. Turn off the power, and record the time in seconds. Do not stop taking data! Be
aware that the time you require for your analysis is the amount of time that current is
flowing through the resistor, which may or may not be identical to the time that the
software was taking data. With this in mind, provide an estimate of the uncertainty
in your time value.
11. Stop taking data after the temperature has clearly peaked. Record the highest temperature achieved, with uncertainty. Note that thermal equilibrium within a fluid
can be difficult to achieve, so different areas of the water may have slightly different
temperature values. Be liberal with your estimation of the uncertainty.
Analysis
Using your values for the current, voltage, and time, calculate the electrical work done using
the equation
W = IV t
(6.3)
Also find the error in the work.
Find the heat transferred into the water using the equation
Q = mc∆T
(6.4)
The mass used in this equation must include the mass of the cup. However, the cup has a
different heat capacity than the water, so you must convert the aluminum mass into water
mass using the ratio of their specific heats. The specific heat capacity for aluminum is 0.215
cal/(g °C), while that of water is 1 cal/(g °C).
Find the electrical equivalent of heat, W/Q, with error.
42
LAB 6. ELECTRICAL EQUIVALENT OF HEAT
Discussion
Does your value for the electrical equivalent of heat agree with the established value, 4.186
J/cal, within error? If not, to what do you attribute the difference?
Explain what forms of error could affect the work done and the heat absorbed by the water.
Which of these errors do you believe is dominant? Could you test your theory? From your
conclusion, would you expect your result to be higher or lower than the expected value?
LAB
7
Ruchardt’s Method
Objective
To determine a value for
Cp
CV
(γ) using Ruchardt’s Method.
Apparatus
• Gas Law Apparatus
• Pressure Sensor and interface, either
– Low Pressure Sensor and Science Workshop interface OR
– Dual Pressure Sensor and PASport USB link
• Tubing
• Computer with DataStudio software
Background
In Ruchhardt’s Method, a cylinder of gas is compressed adiabatically by plucking the piston.
The piston will then oscillate about the equilibrium position. Gamma, the ratio of specific
heat, can be determined by measuring the period of oscillation.
If the piston is displaced downwards a distance x, there will be a restoring force due to the
resulting pressure change which forces the piston back toward the equilibrium position.
Just like a mass on a spring, the piston will oscillate. The piston acts as the mass and the
air acts as the spring. The period of oscillation of a mass on a spring (or for the piston and
43
44
LAB 7. RUCHARDT’S METHOD
air) is
r
T = 2π
m
k
(7.1)
To determine the spring constant, k, for air, calculate the force when the piston is displaced
a distance x. When the piston is displaced downward a distance x, the volume decreases by
a very small amount compared to the total volume: dV = xA where A is the cross-sectional
area of the piston.
The resulting force on the piston is given by F = (dP )A, where dP is the small change in
pressure. To find a relationship between dP and dV , we assume that if the oscillations are
small and rapid, no heat is gained or lost by the gas. Thus the process is adiabatic and
P V γ = constant
where
γ=
Cp
= Ratio of Molar Specific Heats
CV
(7.2)
(7.3)
For a diatomic gas, CV = 5/2R and Cp = 7/2R, so γ = 7/5 = 1.4. Taking a derivative of
Equation 7.2 gives
V γ dP + γP V γ−1 dV = 0
(7.4)
Solving for dP ,
dP =
−γP
−γP A
dV =
dx
V
V
(7.5)
−γP A2
dx
V
(7.6)
γP A2
.
V
(7.7)
Plugging into F = (dP )A gives
F =
Comparing this to F = −kx shows that
k=
Substituting into the period equation for k gives
T2 =
4π 2 mV
γP A2
(7.8)
The total volume is Ah + V0 , where h is the height measured on the labeled scale and V0 is
the unknown volume of gas below zero on the label and contained within the tubing. Thus,
if the square of the period is plotted versus the piston height, the resulting graph will be a
2m
straight line with a slope of 4π
γP A . Therefore the ratio of specific heats is given by
γ=
4π 2 m
,
(slope)P A
(7.9)
where m = mass of piston, A = cross-sectional area of piston, P = atmospheric pressure,
and the slope is from the graph of T 2 vs. h.
PROCEDURE
45
Procedure
1. If using the Science Workshop interface, ensure that it is plugged in and switched on.
2. Start DataStudio.
3. Click on the Setup button.
4. Select "Choose Interface".
Select Science Workshop 500, click ok.
5. Click on the yellow circle surrounding port that the sensor is plugged in to.
6. Select Low Pressure Sensor.
7. Set sample rate to 1000 Hz or higher (use arrows).
8. Double click the "Graph" button.
The program is now ready to take data.
1. Set the height of the piston to 90 mm (ensure that open valve is open), using the
screw on the Gas Law Apparatus to hold the piston in place.
2. Close the valve and release the screw. The piston should remain in place (it might
drop a small amount).
3. Click the Start button in DataStudio.
4. With the tip of your finger, depress the platform by roughly 0.5 to 1.0 cm and release.
5. The piston will snap back and oscillate very rapidly.
6. Click Stop on DataStudio.
7. Ensure that the "Align matching X scales" button is unselected.
8. Zoom in on the oscillatory portion of the graph using "Zoom Select".
9. Using the "Smart Tool", find and record the period of the oscillations. Record multiple
values, one for each clearly identifiable period of oscillation.
10. Repeat this procedure for the same height.
11. Lower the height of the piston by 10 mm and repeat, continuing until you reach a
height of 10 mm.
12. Record the pressure in the lab.
46
LAB 7. RUCHARDT’S METHOD
Analysis
Construct a table of your heights and corresponding periods (include all of them). Find the
average and standard deviation of the periods for each height, and present these in separate
columns.
Make a plot of T 2 vs. h. You will need to construct a column for T 2 based on your period
data. Include error bars for your T 2 data. Use the linear regression function in Excel to
evaluate the slope and intercept of the best fit line, as well as the error in each of these
values. Also include the best fit line on your plot.
Using your value for the slope, find γ using equation 7.9. Also determine the uncertainty
in your result.
Predict the volume of air contained within the bottom of the apparatus and the tubing.
To do this, you will need to measure the inner diameter of the tubing used (use calipers),
as well as its length. Record your values for these measurements. Also record your best
guess for the height beneath the zero mark in the cylinder. Using these values, calculate
the volume of air.
We will now calculate this same volume using your intercept from the T 2 vs. h graph and
equation 7.8. Let V = V0 + Ah and write the resulting equation. Note that the first term
involving V0 should equal the intercept of your graph. Set the first term equal to your
intercept, and solve this equation for V0 . Careful with units!
Discussion
Does your value for γ agree with the theoretical value of 1.4, within error? If not, to what
do you attribute the difference?
Compare the 2 values you found for V0 . How well do they agree? What could possibly be
the reason for any difference?
Include the answer to one of the following questions in your discussion:
1. If the piston were filled with a different (monatomic) gas, how would the values for
the period change for each height value?
2. Describe the experimental details of using a different gas. Address how would you
introduce gas into the tube, evacuate the previous gas, change height, etc.
3. How could the experiment be altered so that the period could be measured using a
stopwatch (a minimum period of 0.5 seconds)?
LAB
8
Stefan-Boltzmann Law
Objective
To determine the intensity of light as a function of distance, and to show the relation
between the intensity and temperature of an emitting body.
Apparatus
• Stefan-Boltzmann Lamp
• Radiation Sensor (Thermopile)
• Metre stick
• 3 Multimeters
Background
Electromagnetic radiation absorbed and emitted by any substance is dependent on the
temperature of the substance. Josef Stefan showed experimentally in 1879 that for a perfect
emitter (a "‘black body"’) the rate at which energy is emitted is related to the object’s
temperature by
P = σAT 4
(8.1)
where A is the surface area of the radiator, T is the temperature in Kelvin, and σ is the
Stefan-Boltzmann constant, which has a value of 5.6703 × 10−8 W/(m2 K 4 ). is a constant
that assumes a value of 1 for a perfect blackbody, and 0 for a perfect whitebody (perfect
reflecter). This relation was subsequently derived theoretically by Ludwig Boltzmann, who
had been a student assistant of Stefan’s at the University of Vienna, and later succeeded to
Stefan’s professorial post there.
47
48
LAB 8. STEFAN-BOLTZMANN LAW
Figure 8.1: Setup for Part I
If we examine a particular object of constant area, then this Law states that the total power
radiated from that object will be proportional to the 4th power of the temperature. Recall
that the power of a light source is a measure of the amount of energy it radiates per unit
time, and is purely a function of the source itself. The rate of energy received by a remote
object or location is referred to as the intensity, which is simply power per unit area (of the
object, not the source). Since the energy emitted will become less dense as it radiates from
the source, this implies that the intensity at a particular point is a function of the distance
to the source.
In order to find the relation between the distance and the intensity, and the relation between
power and temperature, we will use the apparatus shown in figure 8.1. The radiation sensor
used is actually a thermopile, a collection of thermocouples that respond electrically when
heated. When exposed to radiation, the thermopile will have a potential difference across its
output jacks that is proportional to the intensity of light incident on its surface, which can be
read using a voltmeter. The lamp used has a specific profile of resistance to temperature,
so by calculating the resistance using Ohm’s Law, the temperature of the lamp can be
obtained. You will be provided with a table and graph with the necessary relations to
calculate the temperature.
Procedure
Part I
1. Secure a metre stick to the table using tape.
2. Set the lamp at one end of the metre stick, assuring that the zero mark of the metre
stick aligns with the centre of the filament.
ANALYSIS
49
3. Adjust the height of the Radiation Sensor so it aligns with the lamp.
4. Connect the sensor to the voltmeter and the lamp to the power supply.
5. Place the sensor at the 10 cm mark of the ruler, pointing directly towards the lamp.
Ensure that the lamp is OFF.
6. Record the reading of the voltage from the sensor. This measurement is the background radiation present at this point.
7. Repeat this background measurement for each 10 cm mark of the metre stick. Find
the average of your results and use this result as the average background radiation.
8. Turn on the power to the lamp. Ensure that the voltage to the lamp does not exceed
13 V.
9. Using the same method that was used to find the background radiation, take readings
of the voltage of the sensor at the following distances:
From 5 cm to 20 cm in steps of 2.5 cm. (7 readings)
From 20 cm to 50 cm in steps of 5 cm. (6 readings)
From 50 cm to 100 cm in steps of 10 cm. (6 readings)
10. Ensure that the readings are taken quickly, and that the radiation shield is between
the sensor and the lamp while adjusting the sensor to a new distance. This will prevent
the sensor from being heated between readings.
Part II
1. Place the radiation sensor a fixed distance away from the lamp, from 5 to 10 cm.
2. Adjust the voltage to the filament to 1 V. Read the current to the filament and the
voltage output from the sensor and record them.
3. Repeat this procedure in 1 V steps, up to 10 V.
4. Turn off the lamp, let both the lamp and sensor cool for roughly a minute, and repeat
the procedure.
Analysis
Part I
Prepare a suitable table for your results. In addition to the columns for position and voltage,
you will need a column for the adjusted voltage, which is simply the measured voltage minus
the background voltage.
50
LAB 8. STEFAN-BOLTZMANN LAW
Plot the adjusted voltage vs. distance. Using a power regression, find the relation between
the two. Based on this relation, create a new plot that will appear linear by applying the
appropriate power you just found to one of your columns of data. Perform linear regression
on this plot to find the slope and the uncertainty in the slope. Note that the value of the
slope should be related to the power of the source. Find the electrical power dissipated by
using the equation P = IV . Compare the two results.
Part II
Construct a table of your results as shown. Note that the intensity is the voltage from the
sensor, which is proportional to the actual intensity, as discussed in the background section.
V (V)
I(A)
Intensity (mV)
R (W)
T (K)
T 4 (K4 )
Plot T 4 against intensity. Perform linear regression on this plot, and find the percentage
error in the slope. This will be used as an indicator of how well the linear regression fits
the data.
Prepare a second plot using only temperature values of greater than 1500 K, and again
perform linear regression. How does this plot compare to the first?
Discussion
From your data from Part I, can you conclude that the lamp is radiating light in a spherical
manner? Is the lamp truly a point source of radiation? If not, how might this affect your
results? Do you see such an effect in the data you have taken? Does the electrical power
dissipated equal the power of the source from your data?
For Part II, what can you conclude about the relation between temperature and the intensity? Is it possible to determine the values for the constants in the Stefan-Boltzmann Law?
Explain your reasoning. What sources of thermal radiation, other than the lamp filament,
might have influenced your measurements? What effect would you expect these sources to
have on your results?
LAB
9
Stirling Engine
Objective
To view the operation of an actual heat engine, and calculate the efficiency of the engine.
Apparatus
The apparatus listed will be set up for you.
• Heat Engine/Gas Law Apparatus
• Aluminum Cannister with rubber stopper
• Tubing
• 2 Pyrex 1L containers
• Large rod stand
• Small Mass Hanger (5 g)
• Weights, up to 230g
• Rotary Motion Sensor
• 2 Temperature Sensors
• Low Pressure Sensor
• Science Workshop/PASport interface
51
52
LAB 9. STIRLING ENGINE
Background
The theoretical maximum efficiency of a heat engine depends only on the temperature of the
hot reservoir, TH , and the temperature of the cold reservoir, TL . The maximum efficiency
is given by
TL
=1−
.
(9.1)
TH
The actual efficiency is defined as
=
W
,
QH
(9.2)
where W is the work done by the heat engine on its environment and QH is the heat
extracted from the hot reservoir.
At the beginning of the cycle, the air is held at a constant temperature while a weight
is placed on top of the piston. Work is done on the gas and heat is exhausted to the
cold reservoir. The internal energy of the gas (∆U = nCV ∆T ) does not change since the
temperature does not change. According to the First Law of Thermodynamics, ∆U = Q−W
where Q is the heat added to the gas and W is the work done by the gas.
In the second part of the cycle, heat is added to the gas, causing the gas to expand, pushing
the piston up, doing work by lifting the weight. This process takes place at constant pressure
because the piston is free to move. For an isobaric process, the heat added to the gas is
Q = nCp ∆T , where n is the number of moles of gas in the container, Cp is the molar heat
capacity for constant pressure, and ∆T is the change in temperature. The work done by the
gas is found using the First Law of Thermodynamics, ∆U = Q − W , where Q is the heat
added to the gas and ∆U is the internal energy of the gas, given by ∆U = nCV ∆T , where
CV is the molar heat capacity for constant volume. Since air consists mostly of diatomic
molecules, Cp = 27 R and CV = 25 R.
In the third part of the cycle, the weight is lifted off the piston while the gas is held at the
hotter temperature. Heat is added to the gas and the gas expands, doing work. During this
isothermal process, the work done is given by
W = nRT ln
Vf
Vi
(9.3)
where Vi is the initial volume at the beginning of the isothermal process and Vf is the final
volume at the end of the isothermal process. Since the change in internal energy is zero for
an isothermal process, the First Law of Thermodynamics shows that the heat added to the
gas is equal to the work done by the gas:
∆U = Q − W = 0
(9.4)
In the final part of the cycle, heat is exhausted from the gas to the cold reservoir, returning
the piston to its original position. This process is isobaric and the same equations apply as
in the second part of the cycle.
PROCEDURE
53
Procedure
1. Ensure that one port on the apparatus is open. Adjust the height of the piston to
between 20 and 40 mm, and record the height. Estimate the height below the zero
mark on the scale, and record an estimate of the error involved in these measurements.
2. Measure the length of all the tubing involved and record them individually. Include
the length of the tubing on the apparatus itself. Be sure to include a reasonable
estimate of the error.
3. The volume of the cannister with the stopper inserted firmly is 180.5 cm3 ± 1 cm3 .
You may make your own measurement if you prefer.
4. Place the aluminum cannister in the cold water bath.
5. Reconnect any tubing you disconnected in the process of your measurements.
6. Choose an amount of mass to be lifted by the engine, between 100 g and 200 g. It is
advisable to use a single mass, as this will make transferring the mass much easier.
7. Before initiating your data collection, ensure that you are familiar with the steps
outlined below, as each must be carried out in quick succession.
8. Start DataStudio, and perform the following steps:
(a) Place the mass on the platform.
(b) Transfer the cannister from the cold reservoir to the hot.
(c) Remove the mass from the platform.
(d) Transfer the cannister from the hot reservoir to the cold.
9. Print the resulting P vs. position graph.
10. Export the tables of position, pressure, and temperature vs. time.
Analysis
Label the points A,B,C,D on your plot. State which processes are isobaric, and which are
isothermal. Also state the ideal gas laws (Boyle’s, Charles’, etc.) which should hold in each
stage of the engine and write these laws down.
Find the initial volume of air in the engine, VA . This should include the volume of the
tubing, the cannister, and the original volume in the cylinder. Include an estimate of the
error in your result.
Open your exported tables in Excel and combine them into one table of data with a shared
time variable. Construct a new table from this data as follows:
54
LAB 9. STIRLING ENGINE
Point
A
B
C
D
h(cm)
Pgauge (kPa)
Pabs (kPa)
V (cm3 )
T (K)
The volume column can be calculated using V = VA + πr2 h, where r is the radius of the
piston, and h is the height of the piston at that point. We need to find the efficiency of
the engine, so we need to calculate the amount of work done by the engine, and the heat
generated within it. Complete the following calculations to determine these quantities:
(note: use absolute pressure, not gauge pressure)
1. There is heat added to the cylinder in the process from B→ C. This heat can be found
using the equation
7 PC VC
QB→C =
(TC − TB )
(9.5)
2 TC
2. Find the heat involved in the process from C→ D using the equation
VD
= PD VD ln
VC
QC→D
.
(9.6)
3. Find the total heat QH = QB→C + QC→D .
4. Find the work done by the cycle by finding the area under the curve.
Using the quantities found above, find the theoretical and actual efficiencies of the engine
using equations 9.1 and 9.2.
Discussion
How do the theoretical and actual efficiencies of the engine compare? How do you account
for the disparity? Explain what this implies for heat engines in general.
Find the actual work done against gravity, W = mgh, where h is the height that the weight
was lifted from its lowest point. Does this agree with the work done by the engine found
above? If not, explain why the additional work is required.
Using the ideal gas laws you wrote down for the analysis, find VC and VD . What conclusions
can you draw from comparing these values to the ones obtained previously?
LAB
10
Atomic Spectra
Objective
To investigate the quantum mechanical phenomenon of atomic spectra in a variety of gases,
and determine a value for the Rydberg constant.
Apparatus
• Grating spectrometer
• Diffraction grating
• Eyepiece/lens/magnifying glass for viewing Vernier scale
• Discharge tube holder w/power supply
• Various discharge tubes (hydrogen and helium)
Background
Our current model of atomic structure assumes that the electrons surrounding an atom are
confined to specific energy levels, or orbitals. This is a form of quantisation, as the electrons
have certain fixed allowable energies, while intermediate energy levels are forbidden. At
the lowest energy state of the atom (absolute zero), these electrons are in a particular
configuration, with all the lowest energy levels being filled. If the electrons are excited
somehow, the resulting change in energy will cause them to ‘jump’ and occupy a higher
energy orbital, if the energy gained is sufficient to overcome the forbidden gap. There are
a variety of physical factors that will excite an electron, including temperature, interaction
with photons, and the presence of an electric field. Once one or more electrons associated
55
56
LAB 10. ATOMIC SPECTRA
Figure 10.1: A simplified view of double slit diffraction. When the path length difference,
given by d sin θ, between 2 almost parallel rays of light is nλ, the waves will constructively
interfere, and a maxima will appear. In between these maxima the waves exhibit varying degrees of destructive interference, reducing the intensity. The effect is much more
pronounced if there are more slits, such as in a diffraction grating.
with a given atom become excited, the lower energy levels are no longer filled. This enables
these excited electrons to ‘fall’ back to lower energy levels, and in the process lose energy.
This energy is manifested as a photon of a particular wavelength, as given by the equation
E = hc/λ.
(10.1)
This relation between the wavelength of electromagnetic radiation emitted from electron
orbitals and the configuration of the orbitals themselves is extremely useful. If the wavelength of radiation emitted from a substance can be somehow measured experimentally,
then immediately information about the energy levels is gained. Since the configuration of
orbitals is unique to a particular substance, the material emitting the radiation can be determined simply by examining its spectrum. This type of analysis, known as spectroscopy,
has been used to determine the chemical composition of materials ranging from the microscopic to the astronomical. The composition of stars light years away can be found simply
by examining the light we receive from them!
In the lab, all that is required to perform this type of analysis is a way of measuring
wavelength. In addition, the gas must be excited in some way. The latter can be easily
accomplished by applying a large electric field to the sample. The former can be done
using a common optics technique that you have likely already examined; diffraction. When
EM radiation is incident upon two narrow slits separated by distance d, the interference of
the wave patterns after encountering the slits will result in alternating regions of low and
high intensity (minima and maxima, respectively). The maxima will occur when the waves
constructively interfere, satisfying the equation
nλ = d sin θ,
(10.2)
SETUP
57
where n is the order of the spectral line, λ is the wavelength, d is the slit spacing, and θ is
the deflection of the light measured normal to the slits (see Fig. 10.1). If the number of slits
is increased while maintaining the same slit separation d, the equation will remain the same,
but the clarity of the interference pattern will improve. A diffraction grating, then, with
several thousand lines per cm will produce a very distinct pattern with narrow well-defined
maxima. These maxima will manifest themselves as a bright band of colour if the incident
radiation is indeed in the visible spectrum, enabling us to measure the deflection of the
wave without resorting to more advanced detectors. By measuring the angle of deflection
of a particular band using an optical spectrometer, the wavelength can be determined.
Repeating this procedure for each line will give the entire spectrum for the gas, which can
be used to identify it, since the spectrum is unique to a particular gas.
We will be using this technique to study a number of gases, including hydrogen. Hydrogen
was one of the first elements to be studied using this analysis, primarily because several
lines appear in the visible spectrum. In the 1850s, Angstrom had measured the wavelengths
of the 4 visible lines down to a hundredth of a nanometer (quite a feat!), yet why they had
these particular wavelengths was as yet unknown. It was Balmer, a Swiss math teacher,
who found that the wavelengths obeyed the equation
1
1
1
= R0
−
,
λ
n2 m2
(10.3)
where R0 is known as the Rydberg constant and m, n are integers (note: this was not
the actual equation put forth by Balmer, but it is very closely related to it). Using this
equation with n = 2 and m = 3, 4, 5, 6 perfectly recreates the 4 lines found by Angstrom.
This set of lines with n = 2 is known as the Balmer series. The equation was also used
to predict additional hydrogen lines outside the visible spectrum by setting n and m to
different integers, and these predictions were verified experimentally. The problem with the
equation is that noone had any idea why it worked; the physics was not understood. It was
Bohr who realised that this equation made perfect sense in terms of electron orbitals. In
his interpretation, m and n were simply the initial and final orbitals of an excited electron,
which explains their integer representation. This was one of the major discoveries that
prompted the quantum revolution.
In this lab, we will attempt to determine a value for the Rydberg constant R0 by examining
the hydrogen spectrum. We will find the wavelengths of the visible lines by using the
optical spectroscopy technique outlined above, and by fitting the results to equation 10.3,
an estimate of this constant can be found.
Setup
Perform the following steps before making any measurements with your apparatus. Consult
Fig. 10.2 to see how the various parts of the spectrometer are labelled.
1. While looking through the telescope, slide the eyepiece in and out until the cross-hairs
58
LAB 10. ATOMIC SPECTRA
Figure 10.2: A diagram of one of the spectrometers used for this experiment. Not all will
be the same, but the telescope and collimator tubes should remain very similar for each
apparatus.
come into sharp focus. Loosen the graticule lock ring, and rotate the graticule until
one of the cross-hairs is vertical. Retighten the lock ring and then refocus if necessary.
2. Focus the telescope at infinity. This is best accomplished by focusing on a distant
object (e.g.; out the window).
3. Check that the collimator slit is partially open (use the slit width adjust screw).
4. Align the telescope directly opposite the collimator.
5. Ensure that the discharge power supply is plugged in. Plug the discharge tube holder
into the power supply. Insert a helium tube into the holder, and turn the power on.
6. Place the discharge tube directly in front of the collimator of the spectrometer. With
the viewing tube at an angle of 0o (see Fig. 10.3, look through the eyepiece at the
tube. Adjust the focus knob on the collimator (NOT the telescope) so that the image
of the slit is in focus.
7. Adjust the slit width using the attached screw so that the image is bright and clear.
Make a note of which side of the slit image is fixed, and which moves when the screw
is adjusted. All measurements should be done relative to the fixed side of the slit.
Also take note of what happens to the slit image when the opening is wide or narrow;
this knowledge may be useful later.
PROCEDURE
59
Figure 10.3: Aligning the telescope and collimator to provide a clear image of the slit.
8. Clamp the diffraction grating in place on the spectrometer table using the clips so that
the writing on the grating is at the top. This will ensure that the slips are oriented
vertically, which results in the vertically oriented spectral lines that we require. Align
the grating by eye so that it is perpendicular to the collimator.
Procedure
After following the setup steps above, your apparatus should now be ready to take data.
Perform the following steps for each gas that you will study. You should start with helium
and hydrogen, and continue to one of the ‘mystery’ gases. Note that for the mystery gas,
the spectrum can be quite complex, and it may be difficult to explicitly measure all the
spectral lines. Measure at least 4 and then simply describe the rest of the spectrum.
1. Place the appropriate gas tube in the discharge tube holder and turn on the power.
Place the tube directly in front of the spectrometer slit.
2. Unlock the rotation mechanism for the telescope. While looking through the eyepiece,
rotate the telescope until you can clearly see one of the spectral lines. Align the
crosshairs with the edge of the line using the fine adjust knob, and lock the telescope
in place. Refer to the setup steps if you are unsure of which edge to measure to.
3. Read and record the angle θ from the Vernier scale on the spectrometer. You will
likely need a magnifying glass or eyepiece to properly read the scale. Note that the
scale gives readings below one degree in minutes of arc, as opposed to a decimal scale.
One minute of arc equals 1/60 of a degree (60" = 1o ).
4. Unlock the telescope and measure the same spectral line from the other side of 0o (see
Fig. 10.4). Again record your result.
5. Repeat the previous 2 steps for each spectral line you can see. Refer to the table in the
analysis section to find out how many lines you should see for helium. For hydrogen,
you should be able to see 3 distinct lines, and possibly a 4th. Note that the pattern
does repeat, with each repetition of a spectral line possessing a higher value of n (the
‘order’ of the line). Be sure to record the order of the line you see. Recall that those
lines closest to θ = 0o have order n = 1, with n increasing as the angle increases. Try
and find as many hydrogen lines as you can, preferably all the n = 1 and n = 2 lines.
60
LAB 10. ATOMIC SPECTRA
Figure 10.4: Every spectral line will be repeated on either side of 0o . For every measurement,
it is important to measure the same line on each side, as this eliminates errors associated
with improper alignment.
Analysis
The diffraction grating you have chosen will most likely have a value for the number of
lines per cm or mm printed on it (typically 6000 lines/cm). Due to imperfections in the
manufacturing process, this number is likely up to 2% off. We would obviously prefer not
to have this level of uncertainty in our final result, so our first task should be to accurately
determine the correct number of lines per cm in your grating. To accomplish this, we will
use the data from the helium spectrum.
Take the average of your 2 results for each line and express the results as a decimal. Using
the known wavelegths of the helium spectrum and equation 10.2, calculate the inter-slit
distance d and convert this number to lines/cm. Your analysis should appear in a table like
the following:
n
1
Colour
Violet*
Violet*
Violet
Blue
Cyan
Orange
Orange*
Red
Red
Reading 1
15o 33”
Angle
Reading 2
15o 29”
Average
15.517o
Wavelength
(nm)
438.793
443.755
447.148
471.314
492.193
501.567
504.774
587.562
667.815
Lines/cm
5999.4
Note that the lines marked with a * are very weak and may not be visible; adjust your
table accordingly if you can not see them. The angles given in the table are purely for
demonstrating the correct method of displaying your angle, they are not meant to be accurate. Using the results in the table, find the average and standard deviation of the mean
of your results for the slit spacing of your grating. You can use this result in the following
DISCUSSION
61
portions of analysis to evaluate the wavelength of the light from the other sources you have
examined.
Construct a similar table to the one above for the hydrogen spectrum. Replace the final
column with energy, in eV. In this case, you will evaluate the wavelength from your angle
result using equation 10.2 as opposed to copying it from a table. Once you have the
wavelength, the energy can be found by using equation 10.1.
To determine the Rydberg constant, assign the visible lines you found an integer value for
m. Recall that for the Balmer series, m will have values of 3, 4, 5, and 6; it is up to you
to establish which line should be assigned which m value. Using equation 10.3 with n = 2,
create a plot whose slope will be R0 . Find this slope, with uncertainty.
For your mystery element, compare your spectrum to an online source (include which source
you used in your report). Compare the wavelengths given by your source to those you found,
and specifically mention which spectral lines agree.
Discussion
Compare your Rydberg constant result to the accepted value. Comment on the agreement
with the known result R0 = 1.09737316 × 107 m−1 , and attempt to explain the primary
sources of error.
Discuss your determination of the mystery element. Explain your reasoning behind choosing
the element you did, using the supporting evidence from your analysis. What problems, if
any, did you experience when measuring the spectra?
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.
62
63
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.