Materials Included In Kit

Additional Materials Required

Safety Precautions

The materials in this laboratory are considered nonhazardous. Students should wear safety glasses when sighting through the straws to avoid contact with their eyes. Caution students to avoid looking directly at the sun. Remind students to wash their hands thoroughly with soap and water before leaving the laboratory.

Lab Hints

• Enough materials are provided in this kit for 30 students working in pairs or for 15 groups of students. Both parts of this laboratory activity can reasonably be completed in one 50-minute class period. The prelaboratory assignment may be completed before coming to lab, and the data compilation and calculations may be completed the day after the lab.
• Be sure students know how to use the tangent key on their scientific calculators. If calculators are not available, most computers have a calculator program with a scientific view. Simply type in the numeral for the angle and click on the tangent key. Tangent tables for whole-number angles from 0 to 90 degrees may be found online. See http://math.com/tables/trig/tables.htm for one example (accessed November, 2009).
• For practice indoors, tape a picture of a tree or other object near the ceiling of the classroom. If the school gym is available, students may also determine the height of the basketball rim or the top of the backboard.

Teacher Tips

• The student-made clinometers are useful for practicing measurement, finding the height of tall objects, and may also be used to determine the maximum altitude of a launched model rocket.
• Use this activity to emphasize that direct observations and measurements of many objects, such as very small, very large or faraway objects, are not always possible in science and that indirect measurements are frequently necessary. Challenge students to give examples.

Answers to Prelab Questions

1. When using a clinometer, why is it important to know the height of the observer’s eye?

   The adjacent side of the imaginary right triangle formed when using a clinometer is at eye level, parallel to the ground. Therefore the height of the observer’s eye must be added to the length of the opposite side to determine the total height of the object being measured.

2. The height of a football goal post crossbar is 10 feet. Use the information in the Background and Procedure sections to describe how you would determine the height of the upright posts.

   Use a clinometer to sight the top of one post and note the angle indicated by the weighted string. Measure the distance from the observer to the base of the post to find the length of the adjacent side of the imaginary right triangle. Next, calculate the length of the opposite side by multiplying the length of the adjacent side by the tangent of the angle. Finally, measure the height of the observer’s eye from the ground and add that distance to the length of the opposite side to determine the total height of the upright post.

3. A bird nest is near the top of a large tree. You stand 10 meters away from the tree and look at the nest through the sight of a clinometer. The angle indicated by the clinometer is 60 degrees and the distance from the ground to your eye is 1.5 meters. Calculate the height of the bird nest above the ground.

   10 m (tan 60°) + 1.5 m = 10 m (1.7321) + 1.5 m = 17.32 m + 1.5 m = 18.82 m

Sample Data

Answers to Questions

1. Draw a right triangle and label the following: right angle, elevation angle (θ), hypotenuse, adjacent side and opposite side.
   {12033_Answers_Figure_4}
2. Using a scientific calculator with a tangent key, find the tangent to angle θ for each object. Record the tangent of each angle in the data table.

   See Sample Data.

3. Use Equation 1 from the Background section to determine the length of the opposite side for each object. Record the length of each opposite side in the data table.

   See Sample Data.

4. Add the height of the observer’s eye to the length of the opposite side (Equation 2) to determine the total height of each object. Record the total height for each object in the data table.

   See Sample Data.

5. Two lab partners take turns using the same clinometer to sight the top of the school flagpole. They each obtain different angles from the instrument. Assuming the partners are using the clinometer correctly, explain how this is possible.

   The eyes of the two partners may not be the same height or the partners may not be standing the same distance away from the flagpole. Either of these variables or both together would change the angle of sight to the top of the flagpole.

References

Bilash, B.; Maiullo, D. A Demo a Day—A Year of Physics Demonstrations; Flinn Scientific: Batavia, IL, 2009; p 2.

Introduction

While on a hike you see a very tall pine tree ahead. Just how tall is the tree? Construct a handy instrument known as a clinometer and use it to indirectly measure the height of tall objects such as trees and buildings.

Concepts

• Indirect measurement
• Triangulation

Background

The height of an object, such as a tree or building, may be determined using a technique called triangulation. This method involves creating an imaginary right triangle (one angle of the triangle is 90°) in which the observer stands at a distance “x” from the point that lies directly underneath the object. If the object is a tree or building, x represents the distance between the observer and the base of the tree or building. The line of sight between the observer and the height of the object forms the elevation angle, θ (Greek small letter theta), with respect to line x. The line of sight represents the side of the triangle—opposite the right angle—known as the hypotenuse. With x representing the base of the triangle and the line of sight representing the hypotenuse, then the height of the object from the observer’s eye, h, is the third side of the triangle (see Figure 1).

To determine the height of an object, first the distance from the observer to the object is measured with a meter stick or tape measure. Next, angle θ is measured with a clinometer. The line of sight is determined by looking through a tube or site on the clinometer. In order to calculate the height of the object, the tangent of θ (tan θ) must be known. The tangent of an angle is a ratio determined by dividing the side opposite the angle (in this case the height of the object) by the side adjacent to the angle (distance x). Since the length of the opposite side of the triangle is unknown, a scientific calculator with a tangent key or a tangent table may be used to find tan θ.

The height is then calculated using Equation 1. With the clinometer held at eye level, the value of h is actually the height of the object from the height of the observer’s eye, not from ground level (see Figure 1). Therefore, one more step is needed to determine the total height, H, of the object. The distance from the ground to the observer’s eye must be added to h (Equation 2).

Experiment Overview

The purpose of this laboratory activity is to construct a clinometer. The clinometer will be used to determine the elevation angle of various tall objects and a method of triangulation will be used to determine the height of the objects.

Materials

Prelab Questions

1. When using a clinometer, why is it important to know the height of the observer’s eye?
2. The height of a football goal post crossbar is 10 feet. Use the information in the Background and Procedure sections to describe how you would determine the height of the upright posts.
3. A bird nest is near the top of a large tree. You stand 10 meters away from the tree and look at the nest through the sight of a clinometer. The angle indicated by the clinometer is 60 degrees and the distance from the ground to your eye is 1.5 meters. Calculate the height in meters of the bird nest above the ground.

Safety Precautions

The materials in this laboratory are considered nonhazardous. Use caution when sighting through the straw to avoid contact with the eye and do not look directly at the sun. Wear safety glasses. Wash hands thoroughly with soap and water before leaving the laboratory. Please follow all laboratory safety guidelines.

Procedure

Part I. Make a Clinometer

1. Obtain a 30-cm piece of string and a washer. Tie one end of the string to the washer, knotting securely.
2. Tie the free end of the string through the hole in the protractor or around the base at the 3-inch mark. Make sure the weighted string will hang on the front side of the protractor.
3. Position the knot of the string at the hole and tape the looped part of the string to the protractor. Note: When the protractor is held with
   {12033_Procedure_Figure_2}
4. Lay the straw across the back side of the protractor so the straw crosses the 90° mark and the 3" mark (see Figure 3).
   {12033_Procedure_Figure_3}
5. Position the straw so that approximately one centimeter extends beyond the straight edge of the protractor.
6. Use cellophane tape to secure the straw to the protractor.

Part II. Determine the Height of an Object

7. Using a meter stick or tape measure, measure the height of the observer’s eye above the ground in meters. Record the height in the data table on the Make Your Own Clinometer Worksheet.
8. Measure the horizontal distance from the observer to the base of the object (ground level). Record the distance in meters for the adjacent side of the triangle in the data table.
9. The observer holds the clinometer so the weighted string hangs freely as in Figure 2. The straw should be parallel to the ground. Note: Make sure the observer is standing at the distance measured in step 8.
10. Holding the protractor, the observer looks through the long end of the straw (the sight) and tilts the clinometer so the top of the object is visible through the sight (see Figure 1). Caution: Hold the end of the straw away from the eye while sighting the object.
11. Once the top of the object has been sighted, the partner notes the degree mark through which the string passes. Note: This angle will be between 0° and 90° and is referred to as angle theta (θ). Record the angle in the data table.
12. Complete the calculations on the worksheet to determine the height of the object.
13. Repeat steps 7–12 for a second object, trading places with your partner.

Student Worksheet PDF

12033_Student1.pdf