1. Chapter 1 Physics Scientific Measurement

2. Accuracy, Precision, and Error However, the measurement is no more reliable than the instrument used to make the measurement and the care with which it is used and read. The thermometer used by a doctor may be better than a 50 cent one from the gas station!

3. Accuracy, Precision, and Error Accuracy is how close a measurement comes to the actual value. If you are perfectly healthy, an accurate measurement from a thermometer would say your body temperature is 98.6°F.

4. Accuracy, Precision, and Error Precision is how close a series of measurements are to one another. If you measure your temperature, a precise series of measurements would be 98.6, 98.6, 98.6, 98.6.

5. Accuracy, Precision, and Error

6. To determine accuracy, it’s necessary to know the actual value. For example, we know that water boils at 100°C. If you stick a thermometer in boiling water and it reads 97°C, you know that isn’t an accurate measurement.

7. Significant Figures Take the measurement of the water in your beaker. Tell it to me as close as you can estimate.

8. Significant Figures The known digits in a measurement plus the estimated digit are known as significant figures. It is highly important to report measurements with the correct number of significant figures. Your calculations will be very different based on your carefulness with significant figures.

9. Significant Figures There are rules for determining the number of significant figures in a number. You will need to know these very well!!!!

10. Significant Figures Rule 1: All nonzeros are significant. The number 231 as 3 significant figures. The number 18 has 2 significant figures. How many significant figures do the following numbers have? 25587991983753

11. Significant Figures Rule 2: All zeros between nonzeroes are significant. The number 302 has 3 significant figures. How many sig figs do the following numbers contain? 101100110001

12. Significant Figures Rule 3: Zeros to the left of nonzeros are not significant. These are called placeholders. The number 0.0045 has 2 significant figures. How many sig figs do the following numbers have? 0.0010.0023410.00000000000000000000002

13. Significant Figures Rule 4: Zeros to the right of a nonzero are significant only if they come after a decimal point. 100 has only 1 significant figure. 1.00 has 3 sig figs (the two zeros are after a decimal point) How many sig figs do the following numbers have? 43.001.0109.000

14. Significant Figures We can perform operations with significant figures; each operation has specific rules. Addition and subtraction: your answer must have the same number decimal places as the number with the least number of decimal places.

15. Significant Figures That sounds confusing, but it’s not: 100.11) You can see that the top number stops +101.145 at the tenth’s measurement. Draw a 201.245 line after the shortest number. 201.22) Add the numbers like normal. 3) Your answer cannot be any longer than that line, so you must round. 4) Since the following numbers are less than 5, it stays at 2. Thus our answer is 201.2.

16. Significant Figures Multiplication and division: Your answer cannot have more significant figures than your least significant number. 9 x 451)Since 9 only has one significant figure, your answer can only have one significant figure. 2) Multiply like normal; 9 x 45 = 405 3) Round to the nearest number with one sig fig. 405 rounds down to 400

17. Significant Figures One more example with division: 2.4526 / 8.4 = 0.291976 *We can only have 2 sig figs (8.4) * Round to the nearest 2 sig figs 0.291976 would round to 0.29

18. Round each of the following to the number of sig figs indicated. A. 67.029 to 3 sig figs B. 0.15 to 1 sig fig C. 52.8005 to 5 sig figs How many sig figs are there in each of the following measurements? A. 0.4004mL B. 6000g C. 1.00030 D. 400.mm Calculate the sum of 6.078g and 0.3329g. What is the product of 0.8012m and 3.44m? Divide 94.20g by 3.16722mL.

19. Scientific Notation In science class and life, you’ll see both really, really big and really, really small numbers (mostly small, though). Rather than writing 0.000000012, scientists have developed a shorthand way of writing this.

20. Scientific Notation Scientific notation uses two numbers: a coefficient and a 10 raised to a certain power. For example, let’s go back to 0.000000012. In scientific notation, we would write this as 1.2 x 10 -8 1.2 is the coefficient; let’s see where that 8 came from.

21. Scientific Notation 0.000000012 To write a number in scientific notation, count the number of digits between the decimal place and the first nonzero number. If we do that with the above number, you’ll find that there are 8 places.

22. Scientific Notation To get the coefficient, write the nonzero numbers with a decimal after the first number: 1.2 Then write the x10 after it: 1.2 x 10 All that’s left is the power we are raising that 10 to; this comes from the number of digits you just counted between the original decimal place and the first nonzero number, which was 8. 1.2 x 10 -8 You put the negative sign when you are dealing with a number less than one; this tells you that it’s a decimal rather than a number greater than one.

23. Scientific Notation Example: Convert 0.0000178 to scientific notation.

24. Scientific Notation With big numbers, the procedure is slightly different. 1897548 This number could also be written as 1897548.0 Once again, count the number of digits between the decimal place and the first number. There are 6 numbers between the decimal and the 1.

25. Scientific Notation 1897548.0 Now, we need to determine the coefficient. Usually, we just want two significant figures in the coefficient, so we need to do some rounding. For two sig figs, we just want to look at the first 3 numbers: 189. We would round this to 1.9 Now we can put it all together: 1.9 x 10 6

26. Scientific Notation Example: Convert 987543221 to scientific notation.

27. Scientific Notation You add, subtract, multiply, and divide numbers in scientific notation quite easily in your calculator. Example: 1.4 x 10 -4 X 2.3 x 10 -8 To do this in your calculator, type the following (1.4 --- 2 nd --- EE -4) x (2.3 --- 2 nd ---EE -8) MAKE SURE YOU ALWAYS ENCLOSE EACH TERM IN PARENTHESES LIKE I HAVE ABOVE!!!! Failure to do so will almost always result in a wrong answer.

28. Scientific Notation Divide the following with your calculator: 9.4 x 10 -7 / 2.7 x 10 -4

29. Practice 1. Convert 245000 to scientific notation. 2. Convert 0.0000234 to scientific notation. 3. Convert 2.3x10 4 to standard notation. 4. Convert 8.2 x 10 -4 to standard notation. 5. Divide 4.8x10 6 by 8.0x10 6.

30. 1. How many sig figs are in the number 400? 2. How many sig figs are in the number 2002? 3. How many sig figs are in the number 40.00? 4. Convert 0.0000822 to scientific notation. 5. Convert 1.25 x 10^4 to standard notation.

31. SI Units It’s not enough to simply take a measurement and attempt to communicate information with just a number. For example, let’s say you measure the distance between a tree and a water source. You can’t simply tell a thirsty traveler to “walk 5 in that direction.” 5 what? Steps, feet, yards, miles, kilometers?

32. SI Units In Physics (and most other sciences) we use the international system of units; otherwise known as the metric system. This system uses multiples of 10 to distinguish measurements.

33. SI Units To measure length, we use the meter. To measure mass we use grams. To measure volume we use liters.

34. SI Units However, these base units sometime aren’t enough. For example, one paperclip is roughly equal to a gram. If you wanted to measure paperclips, grams would be fine; however, if you wanted to measure the weight of me, it would be silly to use grams.

35. SI Units To simplify things, we can use prefixes that allow us to shorten numbers. For example, I weigh 190 pounds. (1lb. = 454g) In grams, I would weigh 86182.550 g. This is a large number. Instead, we can express this in a different way.

36. SI Units The metric system uses prefixes to express really big or really small numbers more simply: Kilo (k) = 1000 / 10 3 Hecto (h) = 100 / 10 2 Deka (da) = 10 / 10 1 Base (m, l, or g) = 1 Deci (d) = 0.1 / 10 -1 Centi (c) = 0.01 / 10 -2 Milli (m) = 0.001 / 10 -3

37. SI Unit An easy way to remember the order is using the phrase: King Henry Died By Drinking Chocolate Milk Kiss Her/Him Daily Because Divorce Cost Money These prefixes tell you how big your number is. For example 1kg = 1000 grams Remember, kilo = 1000 (previous slide) All you do is change the prefix to the number it represents and there’s your number.

38. SI Units How many meters are in 1km? How many liters are in 3mL? How many grams are in 1cg?

39. Converting Numbers To convert between different units, we use a process called dimensional analysis. Units are the measurement that identify your number: meters, pounds, dogs, apples, etc. For example, if I just tell you 4, you have no idea what I’m talking about. I could mean 4 dogs, 4 cats, 4 apples, 4 inches, etc. Always pay attention to your units, they are very, very important!

40. Converting Numbers In dimensional analysis, you cancel out your units to achieve the ones you want. For example, let’s say I want to express the number of eggs in dozens. How many dozens of eggs do I have if I have 36 eggs? Well, we know that 1 dozen = 12 eggs. This is known as our conversion factor.

41. Converting Numbers So we have 36 eggs, and we know that 12 eggs = 1 dozen. Using this number and our conversion factor, we can use dimensional analysis to figure out how many dozen we have. All you do is multiply your number by the conversion factor. 36 eggs x 1 dozen = 3 dozen 12 eggs

42. Conversion As you learned in algebra, you always cancel out units top to bottom, never across. Therefore, if you’re trying to get from eggs to dozens, you want to make sure you eliminate eggs, therefore you place your egg units diagonal rather than across from each other.

43. Converting Numbers 1 kilogram = 2.2 pounds. How many kilograms are in 22 pounds? How many pounds are in 11 kg?

44. Converting Numbers We can use the same process to convert between SI prefixes. K H D B D C M Convert 1 kg to hg. 1 kg = 10 hg 1 kg x 10hg = 100kg 1kg

45. Converting Numbers However, there’s a much easier way that you’re free to use.

46. Converting Numbers Converting between SI prefixes is really nothing more than adding or subtracting zeros. K H D B D C M As you go from left to right, the value increases by 10x; as you go right to left, it’s reduced by 10x.

47. Converting Numbers K H D B D C M So, if you are converting from left to right, just add a zero; if you’re moving from right to left, move your decimal place to the left. Example: Convert 1km to m. We have to make 3 moves to get from K to our base units which includes meters, therefore 1km = 1000 meters.

48. Converting Numbers Convert 45cm to km. K H D B D C M

49. Density Density is the measure of an object’s mass compared to it’s volume: Density = mass / volume Units for mass are grams (g) Units for volume may be cm 3 or L. Thus, density will have the units of g/L or g/ cm 3

50. Density What is the density of a 10cm 3 piece of lead with a mass of 114g? Mass = 114 g Volume = 10 cm 3 Density = mass / volume Density = 114g / 10cm 3 = 11.4 g/cm 3

51. Density The density of water is 1g/cm 3. What will happen to an object that is less dense than water when placed in a beaker? What will happen to an object that is more dense?