Classroom Activity: Pinhole ProtractorGene Byrd and Renato Dupke
For objects just around us, out to about 50 feet, we perceive distances with our binocular vision. The brain compares the two views of our eyes and estimates distances from the differences between the two. You can see these two views by holding your finger in front of your face and closing each eye in turn. Because each eye has a slightly different view, your finger appears to jump back and forth.
But for more distant objects, our binocular vision becomes inaccurate, because the two views are nearly identical. We cannot directly perceive the distances or sizes of far-away objects. Instead, our eyes only detect the angular size of objects. In other words, we cannot estimate the physical size of an object (in feet or meters), but only the angle it covers (in degrees). If an object is familiar to you -- that is, you know its actual physical size -- your brain can use this knowledge and the object's angular size to estimate its distance. But if you don't know its physical size, you cannot estimate the distance.
Suppose you see a UFO in the sky. Since you don't know much about UFOs, you can't estimate its distance. It could be a small object close up or a large object far away. This is one reason that astronomers tend to distrust UFO reports. People have reported seeing a saucer hovering over a nearby building when in fact it was a blimp many miles away. But if you are able to see something you recognize -- say, your friend waving to you from the blimp gondola -- you can estimate the distance more easily. Based on your friend's apparent angular size and your knowledge of his actual size, you can estimate his distance. Astronomers use a similar process to determine distances of stars and galaxies.
In principle, you could measure angular size with the same device normally used in the classroom to measure angles: a protractor. Imagine if you put a protractor next to your face, with its center at one of your eyes. To measure the angle between the two sides of an object, you would turn your head until the zero-degree mark pointed in the direction of one side of the object. Then, you would read off the angle of the direction to the other side. With an actual protractor, however, this technique takes two people -- the numbers and divisions are too difficult for one person to see.
A pinhole protractor solves this problem in a convenient, inexpensive way. Traditionally, astronomy teachers have used a cross staff to measure angles and explore parallax. But the pinhole protractor is cheaper, easier, and more compact. All it takes is a sheet of card stock and the photocopy-ready pattern on p. 7. Each student can make an instrument. When we have tested the device in our classrooms, we have found that students grasp the concepts of angular size and parallax more easily than when they used the cross staff.
In Part II, which will appear in a future issue of The Universe in the Classroom, students will see how astronomers use parallax -- an extension of binocular vision -- to determine the distances of stars in the Sun's neighborhood. Parts I and II can be done separately.
Looking at your pinhole protractor, you can see that the protractor angle scale is on the curved strip, which is at right angles to a circular arc with its center at the pinhole. Under normal daytime lighting outdoors or indoors, this pinhole gives a reasonably clear view not only of the distant object, but also of the up-close angular scale.
Diagram of constructed protractor
If you wear glasses, try taking them off when using the pinhole protractor. The pinhole actually takes over the function of the lens of the eye. Note that there is an ordinary protractor printed on the flat bottom of the instrument. By means of the outwardly extended lines on the flat sheet, you can see that 0 degrees, 5 degrees, and so on correspond to the same values on the curved strip.
To measure angles, follow these steps:
The pinhole protractor is good for measuring angles to an accuracy of half a degree. Students can use the protractor to measure the angular sizes of various objects. After some exploration, students should be able to tell that angular size, physical size, and distance are related.
distance = 57.3 x physical size ÷ angular size (in degrees).
The angular size is expressed here in degrees. The physical size and distance must be in the same units, such as inches or centimeters. This formula is not good for large angular sizes, but for angles less than about 20 degrees, it will give good results. The factor 57.3 is a conversion factor that changes the angular size in units called radians to a value in degrees. Students familiar with trigonometry can even derive this formula. We will test the above formula by measuring the distance to an ordinary 12-inch (30-centimeter) ruler from various positions with both a yardstick and the protractor.
distance = 57.3 x 12 ÷ 6 = 114.6 inches = 9.5 feet
Now that students have tested the procedure, they can use it to measure the distance to other objects of known size -- even at distances they cannot measure directly with a yardstick, such as buildings, cars, even the Moon.
GENE BYRD is an astronomy professor and RENATO DUPKE is a graduate student at the University of Alabama in Tuscaloosa. Their email addresses are firstname.lastname@example.org and email@example.com. They would like to thank the University of Alabama astronomy teaching assistants for trying out the protractor and identifying bugs in its use.
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