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V0.6
LAB EXERCISE COTR – L-M2
MEASUREMENT AND ERROR ANALYSIS USING PARALLAX
Lab format: This lab is performed with lab kit and student supplied materials.
Relationship to theory: This lab corresponds to the analysis of error in data.
OBJECTIVES
 The student will be able to use parallax to determine the distances to various objects.
 The Student will be able to identify and quantize introduced sources of error due to the way
in which measurements are taken.
o The student will be able to estimate the error inherent in the distances derived
from the parallax angles where the distant object is not at an infinite distance.
EQUIPMENT LIST
Student Supplied
 Table, floor, or other sufficiently large flat level surface
 Flat piece of wood or metre stick
 Duct tape
 Carpenter’s square (for the simplified methods)
 Metric tape measure with at least mm accuracy.
 A parallax angle measuring device (If you intend to use the device we suggest here, see Appendix 1
for additional equipment and materials.)
Instructor Supplied
 There is a data analysis spreadsheet that your instructor may or may not want you to use. Check
with your instructor.
INTRODUCTION
The distances to the nearest stars are known more accurately because we can measure them directly
without depending on other less precise data. One way of doing this is to measure the angular
separation between a nearer object and a more distant object from 2 different locations. This assumes
the more distant object is ‘infinitely’ far away. The fact that the more distant object is not truly infinitely
far away introduces some error.
In this lab exercise you will measure the angular separation between a nearer object and a more distant
object and determine how far away the more distant object must be before the error introduced by the
fact that it isn’t infinitely far away is acceptable.
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WARNINGS
 If you construct the parallax measuring device described here it will require the use of some
tools which may be power tools. When using these tools the usual safety advice for such tools
should be adhered to.
 There are no special warnings with regard to this lab exercise proper.
THEORY
Parallax is any apparent shift in the direction to an object due to the motion of the observer. By
measuring this angular shift and knowing the length of the base line over which the shift occurred, the
distance to objects as far away as a few hundred light years can be determined with reasonable
accuracy. Beyond a few hundred light years the angular shift is so small as to be immeasurable so other
methods must be employed. The actual distance determination is a process known as triangulation. By
measuring angles the direction vector makes with the ends of the base line and measuring the length of
the base line we can determine distances from the ends of the baseline to the distant object using the
law of sines. The law of sines states that given a triangle, XYZ , where X, Y, and Z are the interior
vertex angles and x, y, and z are the lengths of the sides opposite each angle respectively then:
sin X sin Y sin Z


x
y
z
(EQ L-M2.01)
See figure 1 below:
Figure 1: Basic triangulation
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Since
X  Y  Z  180  Z  180  X  Y
(EQ L-M2.02)
with some algebraic manipulation of EQ L-M2.01 and EQ L-M2.02 we get:
x
z sinX
sin 180  X  Y 
(EQ L-M2.03)
y
z sinY
sin 180  X  Y 
(EQ L-M2.04)
and
x and y are the distances to the object from the 2 observation points. In the case of a star any difference
in these two quantities will be negligible compared to the vast differences being measured, but for
closer objects the differences between these lengths could become significant. Since we will not initially
be lining our object up on the perpendicular bisector of the baseline (z) and our objects will be a lot
closer than a star, x and y could be significantly different.
Sometimes measuring the angles X and Y is not convenient. In this case we can pick an object that is
“much” further away assume it is an infinite distance away, and use the change in its angular separation
from the nearer object as we move a known distance left or right from our original position. Assuming
the far object is infinitely far away we now have this picture:
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Figure 2: Parallax angles measured relative to an infinitely far away object.
Now object P2 is infinitely far away and object P1 is the object we want to know the distances x and y to.
Since P2 is infinitely far away, rays j and k are parallel. We measure X , Y , and z.
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Angles Ax and Ay are supplementary angles so:
Ax  Ay  180  Ay  180  Ax .
(EQ L-M2.05)
X  Ax  X  X  Ax  X
(EQ L-M2.06)
Y  Ay  Y  Y  Ay  Y  Y 180  Ax  Y
(EQ L-M2.07)
Z  180  X  Y  180   Ax  X   180  Ax  Y 
(EQ L-M2.08)
Z  X   Y .
(EQ L-M2.09)
Now
and also
Substituting these into equations EQ L-M2.03 and EQ L-M2.04 we now get:
x
z sin  Ax  X 
sin  X  Y 
(EQ L-M2.10)
and
y
z sin 180  Ax  Y 
sin  X  Y 
(EQ L-M2.11)
In reality no star is infinitely far away, although some might as well be. In this lab exercise you are going
to determine how close the distant background object can be in terms of baseline length before the
error introduced by choosing a background object that is too close is unacceptable. To that end we will
analyse the actual situation. In figure 3, below, ΔXYZ has been ‘squashed’ down to make the angles we
are measuring easier to see.
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Figure 3: Parallax angles measured relative to a far object that is a large, but finite distance away.
Note that we are still measuring X , Y , and z in an attempt to determine distances x and y.
However, we can no longer assume that rays j and k are parallel to each other. In figure 3, above, we
have drawn the ray j as if it is equivalent to ray j in figure 2. This leaves us with only one angle, M ,
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that will represent the error in our parallax measurements that we have introduced. As we move point
P2 further and further away ray k will become closer and closer to coinciding with ray m and M will
approach zero. As we move P2 closer in the opposite will be the case.
Now Angles Ax and Ay are no longer perfectly supplementary angles. However:
Ax  Ay  M  180  Ay  180  Ax  M
(EQ L-M2.12)
So the way have drawn our diagram:
X  Ax  X  X  Ax  X
(EQ L-M2.13)
Y  Ay  Y  Y  Ay  Y  Y 180  Ax  M  Y
(EQ L-M2.14)
Z  180  X  Y  180   Ax  X   180  Ax  M  Y 
(EQ L-M2.15)
Z  X  Y  M .
(EQ L-M2.16)
and:
Now substituting these into equations EQ L-M2.03 and EQ L-M2.04 we now get:
x
z sin  Ax  X 
sin  X  Y  M 
(EQ L-M2.17)
and
y
z sin 180  Ax  Y 
sin  X  Y  M 
(EQ L-M2.18)
These are the general equations for finding the distance. We assume no particular orientation for z, so
we still have to measure Ax as well as X , Y , and z. You can proceed with this lab as it stands here,
but there are 2 possible ways to simplify these equations and reducing the measurements you need to
make in addition to X , Y , and z.
Simplification 1:
Make sure you orient z such that Ax = 90°. Now equations EQ L-M2.17 and EQ L-M2.18 become:
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x
z sin  90  X 
(EQ L-M2.19)
sin  X  Y  M 
and
y
z sin  90  Y 
(EQ L-M2.20)
sin  X  Y  M 


We can simplify this further using the trig identity: sin        sin  cos      cos  sin     .
If we set   90 , the right-hand side becomes sin 90 cos     cos90 sin     1 cos     0
and cos     cos  . Now equations EQ L-M2.19 and EQ L-M2.20 read:
x
z cos  X 
sin  X  Y  M 
(EQ L-M2.21)
y
z cos  Y 
sin  X  Y  M 
(EQ L-M2.22)
and
Using this simplification you only need to measure X , Y , and z. M is a small, but unknown
quantity and not necessarily insignificant. However, you need some way to make sure angle Ax = 90°.
Simplification 2:
A more fundamental simplification could be made if you can line the near object up with the distant
object so that the distant object appears to be directly behind the near object. Then to get your parallax
measurement you move a distance (perpendicular to your line of sight) to one side or the other.
Because of the way we have developed equations EQ L-M2.17 and EQ L-M2.18, we’ll assume we move
to the right, but you could also make this work if you moved to the left. Now you have the situation
below in figure 4:
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Figure 4: Parallax angles measured relative to a far object that is
a large, but finite distance away using a right triangle.
Now
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X  0 and Ax  X  90
(EQ L-M2.23)
Ax  Ay  M  180  Ay  90  M
(EQ L-M2.24)
Y  Ay  Y  Y  Ay  Y  Y  90  M  Y
(EQ L-M2.25)
Z  90  Y  90   90  M  Y   M  Y
(EQ L-M2.27)
and:
Now substituting these into equations EQ L-M2.19 and EQ L-M2.20 we now get:
x
z sin  90  0

sin  Y  M 
z
sin  Y  M 
(EQ L-M2.28)
and
y
z sin  90  Y 
sin  Y  M 

z cos Y
sin  Y  M 
(EQ L-M2.29)
If you analyse the right triangles directly, these are the equations you would expect. Notice that x is
now the hypotenuse of our right triangle, so we aren’t really interested in the results of equation EQ LM2.28. On the other hand, y is now the distance to P1 directly so EQ L-M2.29 is the equation we are
interested in.
(It’s interesting to note that if M  0 as it would in the case where P2 is an infinite distance away
equation EQ L-M2.29 reduces to:
y
z
 z cot Y .
tan Y
(EQ L-M2.30)
Again this is what you would expect.)
Using this simplification you now only need to measure Y , and z. M is again a small, but unknown
quantity and you need some way to make sure angle Ax = X = 90°.
PROCEDURE
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Use parallax to determine the distance to an object at a known distance. Compare the distance
calculated using parallax to the measured distance. Determine the experimental error inherent in the
use of parallax in terms of how far away the distant reference object is.
1. Construct the parallax measuring device described in Appendix 1 or acquire another instrument
to allow you to measure the angular separation of 2 objects. If you are using a different
instrument, be sure to record it in your lab report.
2. For simplicity it is recommended that you use simplification 2 for the first time through this lab
exercise. This will result in a much simpler job to make your calculations and determine your
distances. If you do this lab more than once or if your instructor asks you to, you can then use
the general equations or simplification 1. The rest of this procedure assumes you are using
simplification 2. If you are using simplification 1 or the general equations then you will need to
adapt the steps given here and make the additional angle measurements.
3. If you are working indoors you will want a fairly short baseline, but if you are using more distant
objects outdoors you will want a longer baseline. It might be good to run through this lab first
indoors where you have more control over your environment. The principles are the same and
it will be easier for you to check the actual distances with a modest tape measure so you can
compare them with the distances that you will measure using parallax. From here on, these
instructions assume you are working indoors, so you will have to adapt them if you are working
outdoors. Use a long room or hallway. The longer it is the farther away you can place your
distant object. For a working surface use a floor, long table, or a number of small tables (or
chairs) all the same height.
4. Lay a straight piece of wood on your working surface to represent your baseline. A metre stick
or a nice piece of 1x2 would be ideal. This is your baseline board. Press the carpenter’s square
against your base-line board such that the left end points in the direction where you intend to
locate your near and far objects. Tape your baseline board to your flat working surface so it
won’t move accidentally. Duct tape would work well for this, but so would several pieces of
masking tape if you are concerned about not damaging the finish on your table. See figure 5.
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Figure 5: Baseline, baseline board, and carpenter’s square placement
5. Decide on your base-line length. If you have limited space then a baseline of 20 or 30 cm would
be adequate. (Figure 5 shows a baseline of 30 cm.) If you will be working with larger distances
then correspondingly longer baselines would be appropriate. Measure from the left end of the
carpenter’s square along the baseline board and mark you base-line board where the ends of
your base-line will be. Record this measurement as z.
6. Now measure along the edge of the carpenter square perpendicular to your baseline to the
position where you will place your nearer object. Again if you have limited space a smaller
measurement will be better than a longer one. Place your nearer object out at a distance of 2 to
5 times your baseline length depending on the size of your work space. A piece of masking tape
with a mark on it at that spot will allow you to save that position. Record this distance as yactual.
See Figure 6 below.
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Figure 6: Object Placement
7. Now measure out along the same line from the left end of your baseline to about 6 or 7 times
the baseline length and place another piece of masking tape here. Record this length as j1.
Repeat this for positions j2, j3, j4, and j5 that are each about 2 more baseline lengths beyond the
previous j position all along the same perpendicular line from the left end of your baseline. If
you have room you may want to include additional j lengths. Record these lengths accordingly.
See Figure 6 above.
8. Next, set your near object at the point yactual and your far object at j1. When you site from the
left end of your baseline board along the carpenter’s square, you should see your near object
obscuring your far object. This will make sure they are all on the same perpendicular line from
your baseline. Your objects can be made of any long narrow object, such as narrow tallish
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bottles, ‘fat’ marking pens, etc. (In figure 6, you can see we used marking pens stood on end.)
They should be as narrow as possible while still being able to stand up vertically. Move to the
right-hand end of your baseline and using your parallax measuring device, measure the
apparent angle that you see between your near object and your far object. Record this as Y1 .
(See Appendix 1 (or the instructions for your particular parallax angle measuring tool) for
instructions on how to measure this angle.) Move the far object to each position from j2 to j5
(and further if your workspace allows) making sure that they are still on the same line from the
left end of your base line. Record each measurement for Y2 to Y5 etc.
9. If your instructor directs you to, repeat this lab exercise using the simplification 1 equations
and/or the general equations.
Figure 7: The lab exercise entirely set up.
Note – The lines representing Y do not appear to connect with yactual and the right end
of baseline because
Y
is being measured several cm above the level of the table.
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ANALYSIS AND QUESTIONS
1. Using equation EQ L-M2.29, set M  0 and calculate the y value for each Y you recorded.
Compare each value you get from your parallax measurements with yactual. If you took your
measurements carefully, you should find that the value you calculate for y should be getting
closer and closer to yactual as your distant object position becomes further and further away.
2. Analyze your data and predict how far from the left end of your baseline the distant object
would have to be in order to have 20% or less error in your calculated value for y. How far
would it have to be to get 10% or less error? How far would it have to be to get 1% or less
error? Explain.
3. In each case, what values of M would be required to correct each of your parallax
measurements so you would get yactual?
4. Can you calculate what your Y measurements should have been in an ideal situation? If so,
tell what your random error is for each Y measurement you took.
5. Discuss the difference between your systematic error and your random error and how each
affected the results of this lab exercise.
6. If you re-did this lab exercise using the simplification 1 or general equations, then are your
results similar to what you found using the simplification 2 equations? Explain.
Note – There may be a spreadsheet available to help you with some of this analysis. Ask your instructor.
REFERENCES
Include a list of all references mentioned or utilized to create the lab. You may wish to add some
optional sources of information for further study.
This original lab exercise was authored by Ron Evans, MSc in response to a request by Jim
Baily, PhD of College of the Rockies under the
Remote Science Labs for Second Year Physics Project funded by BCcampus
2012 – 2014
All other images were produced by Ron Evans and are covered by the CC license of this document.
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Appendix 1
A Parallax Measuring Device
Figure A1: Parallax Measurement Device
There are several ways you can measure parallax angles. One very good way would be with a transit
and if you are working outside a surveyor’s transit would be excellent. However transits are expensive
and for a once-off lab exercise like this, are probably not economically viable. You could construct a low
quality transit using cardboard, nails, and a protractor. This is a bit too crude for this lab exercise. You
are welcome to try any of these devices if you have access to them, but since most people will not have
access to these high quality devices we are suggesting a parallax measuring device like the one in Figure
A1 above. You already have a digital caliper in your lab kit that was used in other lab exercises, so some
low cost material from your local hardware store should give you a serviceable parallax measuring
device. The materials required are:
 From your lab kit:
o digital calipers
 From your local hardware store:
o 3’ length of square ¾ x ¾ wood (could be hard wood)
o 3: 2” Right angle brackets
o 2: cup hooks
o 2: 2” long small diameter bolts with nut and 2 flat washers each.
 From any source:
o 1: rubber band
This parallax device uses the principle that given the length of the 2 legs of a right triangle you can use
the inverse tangent to find the acute angles of the triangle. See figure A6 below for details.
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Figure A2: General placement of hardware
1) Figure A2 (above), shows the general placement of the hardware on the slightly more than 50
cm long wood. Note that the back sight Bracket is mounted flush to the left side of the piece of
wood as is the back bracket. The alignment of these 2 brackets is important as they define one
line of sight that you will need when using this instrument. Figure A3 (below) shows the detail
of the front of the Parallax measurement device.
Figure A3: Front detail
Figure A4: Caliper
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2) The back bracket is aligned to the left edge of the wood. The front bracket had to be offset so
that it would not put pressure on the red and blue control buttons on the caliper. (Refer to
figure A4 (above) to see a picture of the caliper that we used. If your caliper is designed
differently, then you will want to vary this design to accommodate your own caliper.) The
spacing of the front and back bracket is such that it supplies a snug fit on the body of the caliper.
If you find that your caliper fits too loosely you will want to add some tape to the back bracket
to give the caliper a snug fit without damaging the caliper plastic parts. The caliper is mounted
upside down on the board (see figure A1) so a hole was drilled to accommodate the set screw.
You’ll notice that the outside measure closest to the board will remain stationary when the
caliper is opened. It is important that the inside edge of this fork of the caliper line up close to
the left edge of the board and the left edge of the back bracket. We discovered that this would
result in the inside measure butting up against the left side of the wood thus preventing the
outside measure forks of the caliper from closing completely. To accommodate the inside
measure forks a notch was cut in the left side of the wood. This way the caliper could close
completely. It is important that the caliper be able to close completely so you can set your zero
correctly.
3) First cut a piece of the ¾ by ¾ wood about 50 cm long. Slightly longer would be better. Then
hold the caliper at a location where the front bracket will fit on the board in front of it. Make
sure it is positioned such that the inside edge of the outside measure is close to in line with the
left edge of the wood. Mark this position. Drill out the screw holes with a drill bit that is slightly
smaller than the outside diameter of the screws so that the wood will not split. (If you look
carefully you’ll see that we did actually split the wood when we inserted the outside screw to
hold the front bracket in place … oops!) Check that the position of the caliper against the front
bracket is still appropriate and that the front bracket does not interfere with any of the caliper
control buttons. If either of these conditions is not met then you will have to reposition the
front bracket.
4) Now holding the caliper in place, position the back bracket such that it will be snug against the
caliper and flush with the left edge of the wood. Mark the screw hole positions carefully. Again
drill the screw holes so you won’t split the wood when tightening the screws and then fasten
the back bracket to the wood. Make sure the caliper fits snugly between the brackets and that
the inside of the right caliper fork is close to in line with the left edge of the back bracket. If the
fit is not snug then you will have to apply some tape to make it snug, but not so tight that the
plastic parts of the caliper will be damaged.
5) Make sure that the outside measure of the calipers can close completely when the caliper is
held in position. We found that it wouldn’t close completely and had to notch the left edge of
the wood where the inside measure butted up against the wood. You may need to do the same.
6) In this configuration with the caliper in place the instrument is unstable and will tend to tilt with
the weight of the tail of the caliper. So we cut a ~15 cm piece of the wood and bolted it to the
right side of the wood. It is important that the instrument be able to lay flat on a table so make
sure the bottom of the 2 pieces of wood are flush with each other. We used bolts, but if you
have patience and some clamps you could also use wood glue for the fastening. This additional
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piece of wood will probably interfere with the set screw of the caliper so you’ll need to carefully
mark where the set screw will touch the wood and drill a hole to accommodate it. See figure A3
above. If this hole is sufficiently small and positioned correctly it will provide additional restraint
to keep the caliper from sliding to the left or right during your parallax measurements.
7) Now fasten the back sight bracket to the back end of the ~50 cm long piece of wood. Make sure
it is flush with the left side of the wood. See figure A2
8) Insert the cup hooks as shown in front of and behind the caliper position. Now you should be
able to hold the caliper in place with a rubber band as you can see in figure A1. The caliper
should be close to perpendicular to the top surface of the wood, If it is obviously not, then use
some tape on the brackets or bend the front and back brackets slightly to insure it is close to
perpendicular. Be careful to maintain a snug fit for the caliper.
9) Now that your parallax measurement device is complete, you need to know the length of the
long leg of the right triangle you will be working with. Make sure the caliper is positioned
correctly and open the outside measures of the caliper a couple centimetres. Carefully measure
the distance from the top left corner of the back sight bracket to the inside edge of the inside
fork of the outside measure on the caliper. This length will pass just over the top of the back
bracket. See the yellow line in figure A5. This distance will be the long leg of your right triangle.
Record it in mm. Our instrument had a long leg of 446 mm which we wrote right on the
instrument.
Figure A5: Measure this length
Using your parallax measuring tool:
10) To use your parallax measurement device, close the outside measures of the caliper completely
and turn the caliper on. Make sure it is measuring in mm and if necessary set it to zero. Line the
left side of the device at the position of the back sight bracket at end of the baseline. See figure
A6 below.
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Figure A6: Measuring the parallax angle
11) Sight from the top left corner to the inside edge of the inside fork of the outside measure. Line
these 2 points up with the far object. Now hold the parallax measuring device in place while you
open and adjust the caliper so that the inside edge of the outside fork aligns with the near
object. Now read the caliper. This is the length of the short leg of your right triangle.
12) You can now calculate your parallax angle. Remember that in a right triangle:
tan  
opposite
adjacent
so in this case
tan  parallax angle  
 short leg 
short leg
 parallax angle  tan 1 
 .
long leg
 long leg 
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PHYSICS COURSE NAME
LAB x
In this lab exercise we have called the parallax angle Y .
Appendix 2
Note to Instructors
There is a spreadsheet, “Lab Exercise L-M2 – Calculation Spreadsheet.xlsx” that you may or may not wish to give to
your students. Whether you use this spreadsheet or not depends on how much original work you want your
students to do for this lab exercise. If you do want to give this spreadsheet to your students you may want to
delete the sections under Theoretical Error because it answers some of the questions asked in the analysis section.
M Calculation and the Random Error Calculation sections for the same
reason. This is left up to you. You will probably want to keep the entire spreadsheet intact for yourself
to help with the marking of these lab reports.
You may also want to delete the
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21