Name: Measurements and Calculations Introduction Much of what we do in this laboratory exercise may seem rather distantly related to astronomy. After all, you won’t be looking at a single star all night. So why do it? Because you’ll be making many measurements and doing many calculations over the course of this semester, and we need to make sure that you’re comfortable with how to make measurements, how to use your calculators, how to do unit conversions, and how to report your answers. What you’ll need • • • • • Meter stick Scientific calculator (the same one that you will be using all semester) This lab exercise. A ball from the front table. Note paper. Procedure • • • • READ the lab. Make sure that you understand what it says. Answer the questions in the spaces provided. Hand in these sheets. This is an informal lab, and so you do not need to follow the procedures for a formal lab write-up. A) Using scientific notation Scientific notation is a form of shorthand. For example, it would rapidly cause writer’s cramp if I had to constantly write the mass of the Sun as: 2,000,000,000,000,000,000,000,000,000,000 kg That’s a 2 followed by 30 zeroes. It’s much easier to write it as: 2x1030 kg What the shorthand (scientific notation) form says is that I’m multiplying 2 by 10, thirty times. Let’s look at this using a simpler example, say the number: 2000 I can re-write that as: 2x10x10x10, because 1000 is 10 multiplied by itself three times. Well, that’s actually more painful than writing 2000, and so we replace the three multiplications by ten as 2x103 To convert 2000 into scientific notation is pretty straightforward. I start by actually putting in the decimal point where it belongs, as 2000. I then count the spaces that I would shift the decimal to put it just right of the leftmost number, or: 2000. The number of spaces becomes the exponent of ten in my scientific notation (three, in this case). Most numbers, however, will look more like this: 15348.226 To convert that to scientific notation, follow the same rule: count the number of spaces between the decimal point and the leftmost number, or: 15348.226 It took four shifts to get the decimal to the right of the “1,” and so the scientific notation version would be: 1.5348226x104 For numbers less than one, the exponent of the ten is negative. For example, the number one one-hundredth: 0.01 = 1 1 = = 1×10−2 100 10 ×10 The negative exponent is just a shorthand way of indicating that we’re actually dividing by ten twice. To convert a number less than one into scientific notation, I again count the € number of spaces that I have to move the decimal in order to put it to the right of the leftmost non-zero number. For example: 0.000254 Because I had to shift the decimal four places to put it to the right of the “2,” the scientific notation form of the number above is: 2.54x10-4 Exercise: Convert the following into scientific notation. Show your work in the space provided. 1. 15.448 2. 0.0314156 3. 1666684.2 4. 0.00000449 B) Computing with scientific notation Multiplying numbers written in scientific notation by hand takes a bit more thought than normal multiplication. When multiplying two powers of ten, you add the exponents. For example: 103 x 104 = 103+4 = 107 Or just one thousand times ten thousand equals ten million. When the number has both parts of the scientific notation, you multiply the non-exponent parts as you would normally. For example: 1.5 ×10 4 ∗1.6 ×10 5 = 1.5 ∗1.6 ∗10 4 ∗10 5 = 2.4 ∗10 4 +5 = 2.4 ×10 9 To help myself out, I re-arranged the numbers to put the powers of ten next to each other. € In many cases, multiplying the non-exponents together gives me another power of ten. For example: 3.5 ×10 3 ∗ 4 ×10 2 = 3.5 ∗ 4 ∗10 3 ∗10 2 = 14 ∗10 3+2 = 14 ×10 5 = 1.4 ×101 ×10 5 = 1.4 ×10 6 Exercise: Do the following computations by hand (you can check your answer using a calculator). Write your answers in scientific notation. Show your work in the space provided. € 5. 1.2x106 * 2.8x107 6. 3.4x104 * 2.2x10-5 7. 4.5x103 * 6.0x108 8. 8.8x104 * 5.5x10-5 C) Make sure that you’re familiar with your calculator Your scientific calculator has its own shorthand way of handling scientific notation, which you should use. For example, in doing the following: 1.33 ×10 3 ∗1.4 ×10 4 many people will press the literal sequence of keys on their calculator: € “1.33” “x” “10” “^” “3” “x” “1.4” “x” “10” “^” “4” Sometimes this works, and sometimes your calculator will give you odd results. Avoid the potential of odd results by using the “E” or “EE” key on your calculator. The “E” or “EE” stands for “enter exponent,” and tells the calculator that you’re entering a number in scientific notation. So, to enter the number 1.33x103, you would press the following keys: “1.33” “EE” “3” What your calculator displays depends on the model. On some (expensive models), you will see the actual scientific notation form of the number. On others, it may look like: 1.33E3 That’s your calculator’s shorthand version of the shorthand. Instead of writing “x10” it just writes “E” (for “exponent”) in order to save display space, and you have to realize that the 3 after the E is the exponent of 10 in the scientific notation. Do the following calculations on your calculator, and make sure that you agree with the answer provided in parentheses. See your instructor if you do not see how to make your number agree. In the space provided, indicate whether or not your answer agrees, and write in what key your calculator uses to enter powers of ten. 9. 4.3312x1021 * 6.6602x10-23 (2.885x10-1) 10. 2x1030/1.67x10-27 (1.198x1057) D) Accuracy and precision Accuracy and precision often get confused with each other, so let’s start with two rough definitions: Accuracy: This is describes of how well your measurements compare with the actual value. The closer your measurements are to reality, the more accurate they are. Precision: This describes how well you are able to repeat your measurements. The closer your repeated measurements are to each other, the more precise they are. It is possible to be accurate, precise, neither, or both. To picture it, think of a shooting target, where you want to hit the bullseye. Let’s say that you take twenty shots. The following is neither accurate nor precise: The shots are all over the place, and the scatter isn’t even centered on the bullseye (though one hit, by dumb luck). The following would represent high precision, but low accuracy: The shots are tightly grouped, but nowhere near the bullseye. The following, however, would represent both high precision, and high accuracy: We can usually get a sense of the precision of our measurements. Let’s use the meter stick for that: Look at a meter stick and answer the following questions in the space provided. 11. What is the smallest unit of measurement on the meter stick? 12. What is the smallest length that you could measure accurately using the meter stick? 13. What is the precision of measurements that you make using the meter stick (huge hint: it’s half of the smallest division of the meter stick)? Accuracy is tougher to get a handle on. Lots of things can mess up your accuracy, without you even knowing it. For example, does your meter stick go all the way to zero, or is the end a bit banged up? When you weigh yourself on some scales at the gym, are you sure that they are properly calibrated (they might not measure pounds properly, or their zero point might be off)? Human factors can enter in as well. For example, somebody might not be very careful in using the meter stick, or they might view it at an angle and so get a bad reading. 14. Consult with your group members, and/or members of other groups. Imagine that you are timing the swing of a pendulum with a stopwatch (you’ll do this in a lab later this semester). Think of at least three factors that might affect the accuracy of your measurements. Write them here. E) Significant figures Significant figures are just that--numbers that have significance, as determined by your accuracy and precision. Let’s say that I measure the width of three identical tables that are side-by-side. I use a meter stick with a precision of 0.5 mm, and I measure the width as 3423.4 mm. To get the width of one table, I divide my answer by three. If I blindly do this on my calculator, it gives me the following answer: 1141.133333 mm If I record that number, then I’m claiming that I know the width of one table to an accuracy of 10-6 mm. This is more than 100 times smaller than the size of the average bacterial cell! If I write down that kind of precision, I’ve automatically lost a point on my lab report (or if I do it at work, my boss laughs at me, and my next Annual Performance Evaluation suffers). I cannot claim that my measurement for one table is any better than one-third of my original precision (because there are three tables, the possible measurement error is “spread out” across the three). So, the possible error of my measured size of one table is 0.5/3 mm, which rounds off to 0.2 mm. Because my precision is at the level of tenths of a millimeter, I cannot quote my measurement out further than that, and I have to round off my answer to: 1141.1 mm This indicates the precision of my measurement. The number of significant figures here is 5, the number of digits that I can report as real, measured numbers. Note, though, that if I thought that my meter stick measurements were off by as much as 1 mm, then I would have to round my answer off to 1141 mm. Scientific notation helps with reporting significant figures. For example, how many significant figures are in the following number? 2000 The answer is that we don’t know. The zeroes might be real measured numbers, or just placeholders for the two. If, on the other hand, I had written it as 2000.0 then the added zero after the decimal tells everybody that there are actually five significant figures here (the zero after the decimal is one of them). On the other hand, let’s say that there are only two significant figures there. Then it would be better for me to use scientific notation to include only the significant figures: 2.0x103 The 10 and its exponent don’t count as significant figures; all they do is tell everybody how large the number is. It’s the numbers ahead of the 10 that are significant figures (two figures in this case). For numbers less than one, leading zeros again are only placeholders, and not significant figures. For example, the following has only three significant figures: 0.000936 Round off the following numbers to two significant figures. Write your answers in scientific notation. 15. 1.372x105 16. 3042.5 17. 0.049887 18. 1.989x1030 F) Uncertainty ALL measurements have uncertainty. The uncertainty is an estimate of accuracy. Go back to the table measurement example above. If I’m good at reading it, and the meter stick is in good condition, then the smallest possible uncertainty in my ability to make a single measurement is 0.5 mm (the same as my precision). That was the uncertainty for the width of three tables. For one table, I divide that by three, which rounds off to an uncertainty of 0.2 mm. I would therefore quote my answer as: 1141.1 +/- 0.2 mm The last number, after the +/- sign, is the amount by which my measurement is uncertain. This might not be an honest representation of my true accuracy, though. What if the meter stick is 20 years old, and very battered? What if I have poor eyesight? What if I tend to look at an angle? I would have to think of these and probably increase my quoted uncertainty to make a more honest assessment of my measurement. If, for example, I didn’t have confidence that I could actually make a measurement to better than 1 mm, then I would need to quote my answer as: 1141 +/- 1 mm Note that I never quote a number to a higher level of precision than my accuracy. So, in the above example, if my uncertainty is 1 mm, I don’t quote my measurement to a smaller decimal place than that. Do the following (answer questions in the space provided): 19. Take a ball from the front table. Using your meter stick, measure its diameter. Repeat for a total of three measurements, and record each in the space below. Also, compute the average of your measurements, and record it in the space below. What is the uncertainty of each one of your measurements? We take more than one measurement to decrease our uncertainties. With three measurements, the uncertainty of your average is the uncertainty of each measurement, divided by three. Record that number below (round off to one significant figure). What is your final answer (written as I did for the table width above)? F) Unit conversions Another source of problems for many of us is unit conversions. The best way to avoid problems here is to treat units as if they were numbers, and can be cancelled in numerators and denominators. Let’s say that I want to convert a measurement in inches to cm. There are 2.54 cm in 1 in. If I measured something to be 11 in. in length, then the conversion is: 11 in × 2.54 cm = 28 cm 1 in I treat the inch units as if they were numbers, and cancel the one in the numerator with the one in the denominator, leaving cm. You’ll also notice that I write 28 cm, and not € 27.94 cm, which my calculator told me. That’s because I can’t quote the converted number to more significant figures than the original number. You can run large strings of these. Let’s say that I want to convert a measurement of 2.0 miles to centimeters. Well, there are 5280 feet in a mile, 12 inches in a foot, and 2.54 cm in an inch, so the answer is: 2.0 miles × € 5280 ft 12 in 2.54 cm × × = 3.2 ×10 5 cm 1 mile 1 ft 1 in Do the following conversions. Show your work and write your answers in the space provided. 20. Convert a measurement of 5.2 pounds to kg (there are 2.2 lb in one kg). 21. Convert a time of 2.1 years to fortnights (there are 365.24 days per year, and 14 days in one fortnight). 22. Convert a weight of 19 stone into ounces (1 stone = 14 lb, and 16 ounces = 1 lb.).