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Measuring Angular Diameter and Angular Diameter of an Ojbect

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Author: Janice VanCleave

Physics experiments involve the measurement of a variety of quantities. Generally we measure an object directly by placing it on or in a measuring instrument or by placing an instrument against the object. But some things are either too large, too small, or too far away to measure directly. Sizes of these objects are determined by a method called indirect measurement, which relies on mathematical calculations.

In this project, you will determine the relationship between apparent diameter (how large an object appears to be from a specific distance) and angular diameter (the apparent diameter of an object measured in radians or degrees). You will learn how to use the angular diameter of an object to indirectly measure its actual diameter. You will also learn how to construct an instrument called an astrolabe and use it to measure the angular diameter of an object. This angle will be used to calculate the actual linear diameter of an object.

Getting Started

Purpose: To demonstrate the relationship between apparent diameter and angular diameter.

Materials

• 36-inch (l-m)-long strip of adding machine tape
• yardstick (meterstick)

Procedure

1. Use the masking tape to secure the adding machine tape to a wall at about eye level, as shown in Figure 30.1.
2. Stand 10 feet (3 m) in front of the adding machine tape on the wall.
3. Close one eye and hold your thumb in front of the open eye.
4. Move your thumb toward and away from the wall in front of your eye until your thumb just blocks your view of the adding machine tape.

Results

At a certain distance from your eye, the width of your thumb appears to be the same size as the length of the paper strip.

Why?

It's clear when you hold the paper strip in your hand that the width of your thumb is not the same as the length of the paper strip. Your thumb and the paper only appear to be the same size when they are viewed at different distances from your eye. When your thumb and the paper are at a certain distance from your eye, they have the same apparent diameter (how large an object's diameter appears to be from a specific distance), so they look like they are the same size. This is because at the point where your thumb blocks the view of the paper tape, your thumb and the paper tape have the same angular diameter (the apparent diameter of an object expressed in degrees or radians). When two objects placed one in front of the other have the same apparent diameter, they will have the same angular diameter. In Figure 30.1, the angular diameter for the thumb and the paper strip is angle A (A°).

Try New Approaches

How does the apparent diameter of an object relate to how close or far away from the observing point it is? Repeat steps 2 through 4 of the experiment: First, while keeping your thumb in position, take four steps forward. Notice how large or how small the paper strip appears as compared to your thumb. Return to your original position. Then repeat the experiment, taking four steps backward.

1. How does distance from the observer affect the angular diameter of an object? Design a way to model the affect of distance on the angular diameter of an object. One way is to draw a line across the center of a 12-inch (30-cm) square of white poster board. With a paper holepunch, make a hole in the poster board at one end of the line. Cut three 24-inch (60-cm) pieces of string, each of a different color. Hold the strings together and fold them in half. Thread about 1 inch (2.5 cm) of the folded end of the strings through the hole in the poster board, and tape the ends to the back side (side without the line) of the poster board. Cut three l-inch-by-inch (2.5-by-l5-cm) strips from a sheet of colored poster board and draw a line across the center of each strip. Center the colored strips across the line on the poster board at 3 inches (7.5 cm), 6 inches (15 cm), and 9 inches (22.5 cm) from the end of the poster board with the hole. Stretch two of the same-color strings, called A1 and A2, across the poster board so they touch the top and bottom corners of paper strip A, as shown in Figure 30.2. Tape the strings at the edge of the poster board and cut off any excess string extending past the edge of the poster board. Repeat this procedure for the remaining strings B1, B2, C1, and C2 and paper strips Band C. Label the angles formed by the strings A°, B°, C°. This model can be displayed to represent how distance from a viewer affects the angular diameter of objects of equal sizes.

2. How can you use the angular diameter of an object to determine its actual diameter? One way is to use the principle of similar right triangles (right triangles with the same angles but different-length sides), which states that the tangents of similar angles of two similar right triangles are equal. The tangent of one of the acute angles in a right triangle is the ratio of the length of the two perpendicular lines making up the right angle, with the side opposite the acute angle divided by the length of the angle's adjacent side. For example, in Figure 30.3, if the hypotenuse (the side of a right triangle that is opposite the right angle) and adjacent side of angle A (A) for triangle CAB are extended, triangle EAD will be formed. Since the angles of the two triangles are the same triangles, CAB and EAD are similar right triangles; thus the tangent of one triangle is equal to the tangent of the other.

You can demonstrate this method by calculating the height of an object, such as a wall in a room. Do this by holding a 12-inch (30-cm) ruler at arm's length from your face. Record the length of the ruler, 12 inches (30 cm), as distance BC (see Figure 30.4). With the ruler extended, close one eye and look at the ruler with your open eye. Walk toward or away from the wall until the ruler's apparent height is the same as that of the wall. This means that they both appear to be the same height. Ask a helper to measure the distance from the ruler to your eye. Record this distance as AB (see Figure 30.4). Next, ask your helper to measure the distance from the wall to your eye. CAUTION: For distances AB and AD, measure near but not touching your eye. Record this distance as AD (see Figure 30.4). The ratio of the tangent of angle A for the two similar triangles is:

or in metric

Most walls in a home are 8 feet (96 inches or 240 cm) high—so either the measurement is off by 1.2 inches (3 cm) or the wall is actually 97.2 inches (243 cm) high. You could measure the wall's height to determine its true measurement and evaluate the accuracy of your measurement. Also, the example shows measurements for only one trial. Repeat the measurements four or more times at the same distance and average the results.

3. Angular diameter can be measured using an astrolabe, which is an instrument that can be used to measure angular distances (the apparent linear measurements between two points expressed in degrees or radians). Use the astrolabe to measure angular distances, and use it to determine the actual size of an object, such as the height of a tree.

Make an astrolabe by tying one end of a 12-inch (30-cm) piece of string through the center hole in the base of a protractor. Attach the free end of the string to a washer. Tape a drinking straw along the straight edge of the protractor. Without covering the lines, place pieces of masking tape on the protractor and write 0 to 90 on the pieces of tape (see Figure 30.5). Stand outdoors at a measured distance from a tall tree. Record this as distance AB. Ask a helper to measure from your eye to the ground, and record this as distance AD. Close one eye, and use the other eye to look through the viewing end of the straw. Sight the top of the tree through the straw. Ask a helper to read the angle where the string crosses the protractor.

Record this as angle A (A). Calculate the height of the section of the tree labeled BC in Figure 30.5 using this formula:

tangent A = BCIAB

Thus,

BC = tangent A × AB

Since AD = BE, the total height of the tree is equal to BC + AD. (See Appendix 1 for the value of tangent A.)

Get the Facts

During a solar eclipse (when the Moon moves between Earth and the Sun, thus blocking the Sun's light), the angular diameter of the Moon and the Sun are equal. During an annular eclipse, the Moon moves between Earth and the Sun, covering all of the Sun except a small ring around its edges. Find out more about the real and apparent diameters of the Sun and the Moon. What causes the changes in the apparent sizes of the Moon? For information, see Janice VanCleave's A+ Projects in Astronomy (New York: Wiley, 2002), pp. 9–15.