Expt 1: Measuring from a Distance
Physics E-1a
Fall 2006
Due Tuesday, Oct 10, by 6 pm in your lab TF’s mailbox
Introduction
There are many objects in the universe that simply aren't easy to measure. You can not go
out with a ruler and easily determine the radius of the Earth (although, it can be done!).
Similarly, the size of the Moon is beyond the reach of your meter stick, and even the
heights of the trees in the Yard are hard to sample with even the longest of tape measures.
Since direct measurement isn't always possible, people have found indirect techniques for
getting at various measurements. In this experiment, we will explore how an object's
angular size can be used to determine its physical size, or its distance.
Objectives
The measurements for this experiment will not be performed in the lab. You will conduct
them on your own (or with a partner), on your own time. We call this a “home experiment” and hope to accomplish the following:
• If you have been away from math for a while, this experiment will help you review
the geometry and trigonometry.
• It will start you off on the right track concerning error analysis — learning to identify
and deal with uncertainties in your measurements.
• You'll also get some practice in plotting data and extracting information from a graph.
• And finally, you will discover that it’s possible to measure seemingly difficult things
with simple tools (this is the fun part). It’s actually pretty amazing what one can learn
about our physical world using only the simplest of apparatus, good observations, and
mathematics.
Method
The tool you will make is a quadrant. The quadrant uses the basic principles of surveying:
that light travels in straight lines, and that some lengths and angles of a triangle (or other
shape) can be measured so that unknown lengths, often not directly measurable, can be
calculated. The simplest cases are (1) those in which one angle of a triangle is very
“small,” or (2) one angle is 90˚. By “small” we mean that the angle (measured in
radians) differs from its sine or tangent by so little that, for all practical purposes, these
three quantities can be considered equal — this small angle approximation will be used in
the moon measurement (and many times in this course throughout the year). Secondly, if
one of the angles is 90˚, we simply apply the rules governing right triangles.
Order of Tasks:
1
First make your quadrant (page 10) and learn how to use it by measuring a window. Next
measure the height of the Science Center with this new tool. Then, using only your arm, a
piece of paper, and a ruler, measure the size of the moon. Please don’t leave the moon
measurement until the last minute as the weather and/or phases of the moon may not
cooperate with you. Finally, answer all the bold numbered questions on a separate paper
with a few concise sentences. You will hand in these answers along with a copy of your
graph from the procedure section (there is a check-list for this report on page 8).
Quadrant
Surveying began with instruments like the one you’ll make. The basic quadrant consists
of some kind of angle measuring device (like a protractor) that you can sight along and
determine the angle between a reference point (or direction) and the object you’re
sighting. It has been improved by adding telescopes with cross-hairs, graduated circles
with magnifiers and verniers, magnetic compasses, levels and plumb-bobs, and recently
lasers and corner reflectors. Ours shall remain simple; instructions on how to make it are
on the last page of this write-up.
To use the quadrant, sight along the edge of the
cardboard — the angular height of the object being
sighted is read off where the plumb bob string
intersects the protractor scale. A sample quadrant
will be shown in lecture. [Tip: Instead of putting
your eye up close to the cardboard edge, try moving
your head back a bit so that the entire edge will be
in focus as you try to sight along it. If you find
sighting to be tricky, tape a drinking straw along the
edge and sight through it.]
Preliminary
Analysis
Observations
and
As with any new tool, you should try it out first and measure the size of things that you
already know (or can easily verify) to make sure “it works.” This means learning how to
use your new tool well; learning the techniques and tricks, and coming to understand its
limitations. Test your quadrant by trying to determine the height of a window from afar.
Measure the distance between you and the window; call it D. To determine the angular
height of the window, start by sighting the bottom of the window with the quadrant level
(you’ll probably have to adjust your viewing height by crouching down). Then, without
changing your position, sight the top of the window and record the angle; call it θ.
Practical Hint: When your line-of-sight is on the point of interest, pinch the string against
the quadrant (to hold it in place) and then look to see what angle it is lined up with. Think
about what you can do to improve your technique to minimize the uncertainty in your
angle measurements. If you're doing this with a partner, take turns practicing making
measurements and critique each other's style so that you can better understand how to
avoid measurement errors. Repeat your angle measurements to see how reproducible they
are. Having measured θ, the angle subtended by the window, you can calculate the height
of the window using H = D tanθ. Now measure the height of the window with a tape
measure and compare this measurement with the calculated value of H.
2
1 → What is the percentage error in your calculated value of H?
2 → Is this error in H consistent with the uncertainty you found in your angle
measurements? If not, why not?
The Height of the Science Center
Now for something a little more challenging... measuring the height of the Science
Center. First make a few order of magnitude estimates of its height. Though you may be
more comfortable thinking in feet, most scientific measurement is done with the metric
system, so try to make your estimates in meters as well. Remember, 1 m ≈ 3 feet.
Here are some possibile ways of estimating the height of the Science center — you need
not do them all and you may have better ways of estimating:
• “Guessing gives the height of the Science Center as more than _____ft (or m) and less
than ____ft (or m).”
• “It is about ____ft (or m) as judged by the number of stories.”
• “It is about ____ft (or m) from a knowledge of the height of a stair and the number of
stairs I need to climb to the top.”
3 → What is your estimate of the height (in meters) of the Science Center and how
did you arrive at this number?
3
Geometry
The measurement scheme is similar to your preliminary observations, except that in this
case you don't make a direct measurement of the distance from the origin (the base of
Science Center) to your observation points. To measure the length of the baseline, use
your normal stride as the unit of length. [You can figure out now, or later, what the
conversion is between your stride length and the number of meters by measuring off, say
ten paces.] As you can see from the geometry in the figures, it would be easy to
determine D if we knew L. But we don’t know L, and there’s no easy way to measure it,
because you don’t know where the origin is (the point directly below the dome).
Now, if you measure several angles at various distances, you could turn this into a
geometry problem and deduce D by using appropriate triangles and trigonometric
identities. However, a better solution to this little conundrum is to use all the angle and
distance measurements and plot them on a graph. The reason this is “better” is because,
by incorporating many measurements in your analysis, you will be minimizing the
uncertainty in the final result. By drawing a suitable graph, you can make use of all your
data to obtain a single “best” value for the height of the Science Center.
The parameters needed for the analysis are shown in the following diagram.
4
D
φ
4
φ
φ
3
2
φ
1
origin
starting point
x3
L
According to the diagram above, if your point of measurement is a distance (L+x) from
the origin, then
D
L+x
tan φ = L+x
and thus
cot φ = D .
Here, x is the distance from the starting point and φ is the angular height of the Science
Center dome at that value of x. 1 L is the unknown distance between the starting point and
the origin. This expression can be rewritten in the form of a straight line, (y = ax + b):
1
L
cot φ = D x + D .
When we plot the cotangent of φ vs the distance from the starting point, we obtain a
1
L
straight line whose slope is
and whose y-intercept is
. The height D can be
D
D
determined from the graph of such a line. Note that the quantities x/D and L/D are both
pure numbers, i.e. they have no units. If x, D, and L are all given in meters, then meters
over meters cancel. This means, of course, that cotangent φ (like all trigonometric
functions) is a pure number. However, the slope, 1/D, does have units. If x is in meters,
D and L should also be in meters, and the slope has units of reciprocal meters (1/m=m-1).
Further note that it is not necessary to know L, although it can also be determined from
your data.
4 → How can you determine L from the graph?
1
A reminder: the cotangent is 1/tan, not tan-1. Don't make that mistake on your calculator!
5
Procedure
Walk far enough outside the Science Center so that you can see the telescope dome on
the roof. Let this spot be your starting point (see figure), a distance L away from the
origin (the foot of the perpendicular from the telescope dome to your baseline). Measure
(and record) the angle between the sight-line to the top of the telescope dome and the
vertical. Repeat the angle measurement two or three times and find the average angle
from this point of observation. Now move to another observation point farther away
from the building (say 10 or 15 paces - record how many) and make a new set of angle
measurements. You need to move far enough to produce a significant change in the
angle. Repeat this procedure until you have measured angles from at least 4 different
points.
Graph your data in the following way: Take, as horizontal axis (abscissa), the distance x
from your starting point to each point of observation along the base-line. You can use, as
your unit of x, one stride; you do not need to convert distances into meters until later.
Take, as a vertical axis (ordinate), the cotangent (1/tan) of your measured angles.
Choose scales such that your graph occupies most of the width and height of a page of
graph paper. (But use sensible scales based on multiples of 5 or 10. Avoid, for example,
using 3 or 4 squares to represent a distance of 10 units.) For each point of observation,
plot cot φ against x. Draw a small circle around each point.
5 → What are the sources of uncertainty and explain by how
much they are likely to affect your measurement? To show the cot
effect of uncertainties in the angle measurements, plot, for each
observation point, not only the cotangent of the average angle φ but
also the max and min values cotangent of φ from your data. Do not
circle these additional points, but connect them with a short vertical
line — this is your so-called error bar. There will also be an
uncertainty in converting from number of paces to meters. Add
horizontal error bars to each data point to indicate that uncertainty.
Each point will then look like a little cross as shown in the figure to
the right.
φ
slope = 1/D
Fit, by eye, the best straight line through the points. The magnitude
of the slope of the graph is the reciprocal of the height of the Science Center, measured in
whatever units you used for x.
6 → What do you determine the height of the Science Center to be?
7 → From the slopes of the steepest and least-steep lines that you can draw through
your data points (while staying within your error bars), estimate the percentage
error in your determination of D.
6
x
Measuring the Size of the Moon
If you know the distance to the moon2, it’s relatively easy to determine its size
(diameter). This calculation makes use of the “small angle approximation” (see
Introduction: Method) diagrammed below. Remember that the angle must be expressed in
radians (1 radian = 57.3o ).
Moon
width of paper
D
ß the angle
subtended
R
distance to eye
L
distance to the Moon
D ≈ Rß, ß measured in radians
The angle subtended by the moon is pretty small. Incidentally, if you saw the movie
Apollo 13, you may remember Tom Hanks (Lovell) lying in a lawn chair after a party one
evening comparing the size of the moon with his outstretched thumb. Try it yourself.
Your outstretched thumb subtends an angle a little less than 2˚ (measure it and apply the
small angle approximation to check this). The moon’s diameter is between 1/4 and 1/3
the thickness of your thumb. Now you can make a reasonable guess as to the angle
subtended by the moon (order of magnitude approximation). Going back to the movie
Apollo 13, Lovell compared the size of the earth with his thumb when he came around
the moon.
8 → How did the size of the earth compare with the size of his thumb? If you didn’t
see the movie, how should the sizes compare?
To actually measure the angle subtended by the moon (ß), cut yourself a rectangular strip
of paper. Make the narrow dimension (D in the drawing) small — say, 1/2 cm or 1/4”.
The longer side can be any length. Hold the piece of paper up at arm’s length and line up
the narrow side (D) across the moon’s diameter. Move the paper closer to your eye until
the width of the paper just matches the width of the moon. Measure the distance between
your eye and the piece of paper — this distance is R. Now you can determine the angle ß,
which is the angle subtended by the moon.
2Giancoli
gives a value of 384x103 km (inside front cover). Actually, the distance to the moon varies by about 14%
from its apogee to perigee. Since you (presumably) don't have the appropriate astronomical data at hand that tell in
which part of the orbit the moon is at this time, you might as well round off this number to 400,000 km, or 4x108 m.
For your information, the moon's diameter is 3.48x106 m.
7
However, before measuring the moon, test your technique by measuring the thickness of
a tree. You can confirm the accuracy of the method with a direct measurement of the
tree’s thickness.
9 → What is the thickness of the tree using this method?
10 → How does this value compare to the direct measurement?
Knowing the distance to the moon L, you can now calculate its diameter.
11 → What is the size of the moon?
12 → How does your number compare to the known size? Again, quote uncertainties.
Be careful here and don’t be too optimistic. Remember that no matter how good your
measurements are, the final result is no better than the least accurate number in your
calculation.
Check List for the Report
•
Answer the 12 bold face questions on a separate paper. Be sure to answer all
parts of each question.
•
Include tables of all measurements as well as calculations.
•
Include the graph used to determine the Science Center height. Prepare the graph
following the directions on pp. 11-13 of the Lab Companion.
8
Lunar Myths and Measurements
You may have noticed that the moon looks much bigger when it rises over the horizon —
it’s particularly dramatic with a full moon. If you have a chance, measure it and
determine how much bigger it appears to be. You’ll be surprised.
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Quadrant Pattern
To construct the quadrant, cut out the pattern below and paste it onto a piece of card stock or
cardboard. Attach the top side of the quadrant (labeled top), to a pencil or small rod. Tie a key to
one end of a 1.5 ft piece of string, and attach the other end of the string to the pencil on the right
side where the circle with the cross is located on the pattern. Tape the string to the pencil so that
it can't slide. You are done! Your finished quadrant should look like the quadrant in the figure on
page 2.
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