Lab 1 Measuring and Scales

AST 195 Measuring and Scales Objectives: After completing this lab, you should be able to: 1. Use the metric system to a. Make length measurements b. Convert between measurements within the metric system 2. Design and construct scale models (drawings) 3. Understand the use of multiple model representations 4. Understand and use units specific to Astronomy and the various structures the units are associated with Introduction: Metric Measuring The metric system is fundamental to work in science. In this class, you will be using the metric system exclusively. Once you are used to it, the metric system is actually easier to use than the everyday English system (technically called the US Customary System). In reality, you are familiar with the metric system; you use it every time you spend a penny. The “cent” is from the fact that there are 100 cents in one dollar. In the same way, there are 100 centi meters in one meter. Centi is the prefix that means 1/100. Each metric prefix simply represents a power of 10. Each prefix can be used with any base unit (meters, seconds, grams, etc.) So, just as one can say “1 millimeter,” one could say “1 millisecond.” In both cases, there would be 1/1000 of the meter or the second. The most common metric prefixes you will be using are centi and milli. They both represent parts of the unit you are using. Generally, you will be measuring lengths, so you need to be familiar with centimeters (cm) and millimeters (mm). There are two different ways to think about the meaning of these two prefixes. They are summarized in table 1 Prefix Substitution Meter stick centi 1 cm = 1/100 meter 100 cm = 1 meter milli 1 mm = 1/1000 meter 1000 mm = 1 meter With prefixes representing numbers greater than 1, the substitution process works better. For example, you could say 1 km = 1000 m. The k stands for kilo which represents 1000, so just substitute the 1000 for the k in the expression. Page 1 of 12 Table 1

Reading and Using a Metric Ruler Generally, you will be using a ruler (just a short meter stick) to make your measurements. Metric rulers have large divisions marked in centimeters (cm). Remember, centimeters are hundredths (1/100 = .01) of a meter. Each centimeter is divided into ten smaller divisions that are millimeters (mm). Remember, millimeters are thousandths of a meter (1/1000 = . 001). At times you will notice that your measurements land in between two tick marks. In this case split the difference and choose the value halfway between them. The example below would be written 17.5 mm. It is more than 17 mm but not quite 18 mm, but we don’t have any finer tick marks to judge whether it is 17.6 mm or 17.8 mm, so we record it as 17.5 mm ± 0.5 mm. Scales In Astronomy, you will be looking at object sizes and distances that are so great, they cannot fit into the room (or even on the earth). To help understand objects that cannot be dealt with directly, scientists develop models to help visualize the objects. In many cases, a scale model is the most useful. The simplest scale model is a drawing. An example of an everyday scale drawing is a map. Different scales give you different sizes of maps representing the same area. One of the problems in astronomy is finding the scale that is most appropriate when dealing with various objects. For example, a scale that would allow you to put all the planets in the solar system on a single sheet of paper, would make all the planets so small that you couldn’t even see them. Assume on a map, you look at the legend (scale) and (in metric!) you find 1 cm = 100 km. The scale factor then is: 100 km/cm. The scale factor is simply a ratio that compares one size to another. In this case, you are comparing the size on the scale to the actual size of the Page 2 of 12 Millimeter divisions Figure 1 Centimeter cm

object. Using a scale factor keeps all the sizes in the proper relationship with each other. This is what allows the scale drawing to be a useful model. There are two ways to use a scale factor. One is to take a measurement on the scale drawing to find the actual distance. The other is starting with the actual size of the object and finding the distance on the scale to represent it. Scale to actual example: If you measure the distance between two cities as 2.5 cm, how far apart are the cities? In this example, you will assume you know the scale factor: 1 cm is equal to 100 km. So, to find the distance, use actual distance = (distance on paper) x scale factor In this example, You have converted the scale distance to the actual distance. Notice that the Scale Factor has units of km on top and cm on bottom. You will always use the scale factor with the units of your final answer (in this case, the actual distance in km ) on top. In the example below, the Scale Factor gets flipped so cm are on top because we are finding the distance on paper in cm . Actual to scale: When you are converting the actual distance to a scale distance, you will need to find how large to draw something. Assume you are working on the above map (whose scale was 1 cm = 100 km) and you know the distance between two cities is 1500 km. You can find the scale distance on your map between the cities. This time, you need to use distance on paper = actual distance x scale factor For this example, You would draw the line 15 cm long in order to represent the actual 1500 km distance. Note that you have used the same scale factor in both cases, but the numerator and denominator switched depending whether you are converting from the scale distance to the actual distance, or the actual distance to the scale distance. Page 3 of 12 actual cm km cm km distance    ( . ) 2 5 100 1 250 scale km cm km cm distance    1500 1 100 15

Determining the Scale The final problem generally encountered with making a scale drawing of something is deciding what scale to use. To determine the scale to use, you need to think about the largest object or distance to be represented and the size of the model you can use. For example, let’s assume you want to draw a scale model of a room on a sheet of paper. Assume the room is 9 m by 14 m. Step 1: Measure the piece of paper. The longest side is just under 28 cm. Step 2: The longest size you need to fit is 14 m, so your scale factor is 28 cm = 14 m. This can also be expressed as 2 cm = 1 m. Or, 1 cm on your scale will represent ½ meter in reality and this can be expressed as 1 cm = 0.5 m. Step 3: Determine the size you will draw the model: 9 m × 1 cm 0.5 m = 18 cm or 9 m × 2 cm / m = 18 cm 14 m × 1 cm 0.5 m = 28 cm or 14 m × 2 cm / m = 28 cm The two different ways of writing the scale factors and even the actual processes are equivalent. Making a scale model As your first example in making a scale drawing, you will look at a selected portion of the solar system. The size of a planet is generally given in terms of its diameter or its radius. Page 4 of 12 Figure 2 A diameter is the distance across a circle or sphere through the center. A radius is the distance from the center to the edge. The radius will equal one- half the diameter.