### Physics Scientific Measurement Guided Inquiry v2

v2 Physics Scientific Measurement Guided Inquiry v2 Use your prior knowledge, the textbook, and internet research to complete this inquiry. 1. ALL measurements MUST include _____________________ and _____________________. SI Units – scientists all over the world have agreed on a single measurement system called Le Systeme International d’Unites or SI Units Base Quantities & Units – the 7 quantities that can be measured directly and their SI unit. Complete the table below. Base Quantity Quantity Symbol SI Unit Name SI Unit Abbreviation length l
mass m
time t
temperature T
amount of substance n
electric current I
luminous intensity lv Derived Quantities & Units – all other quantities are combinations of multiplying or dividing base quantities and their units. Complete the table below for some of the important derived units in chemistry. Derived Quantity Quantity Symbol SI Unit name SI Unit Abbreviation Area A Volume V Force F Momentum ρ Acceleration a Velocity v Torque τ Pressure P Summarize 2. Why are units important? 3. Why do scientists use the SI units? 4. Compare and contrast base quantities and derived quantities. v2 Scientific Notation (aka Exponential Notation) – A convenient way to write very large or very small numbers. Also, a convenient way to show the number of significant figures in a measurement that has non-­‐significant trailing zeros. Numbers are written in the form M x 10n where, M is a number greater than or equal to 1 but less than 10 n is a whole number Examples • Write 6.5 x 104 km in “normal” notation. The exponent is positive so move the decimal 4 places to the right adding zeros as necessary, 6 5 0 0 0. km. •
Write 2.49 x 10-­‐3 cm in “normal” notation. The exponent is negative so move the decimal 3 places to the left adding zeros as necessary, 0 . 0 0 2 4 9 m •
Write 210400 g in scientific notation. Move the decimal so the number is greater than or equal to 1 but less than ten, 2 1 0 4 0 0 g. Since the decimal moves 3 places to the left the scientific notation is 2.49 x 103 cm. •
Write 0.0000502 in scientific notation. Move the decimal so the number is greater than or equal to 1 but less than ten, 0 . 0 0 0 0 5 0 2 g. Since the decimal moves 5 places to the right the scientific notation is 5.02 x 10-­‐5 cm. Convert the following quantities into scientific notation: 5. 0.00012 mm = 6. 89630000 s = 7. 35.908 kg = 8. 0.23007 g = Convert the following quantities from scientific notation to “normal” notation: v2 9. 3.044 x 10-­‐2 m = 10. 9.9559 x 102 m = 11. 5.20006 x 104 cm = 12. 2.10 x 10-­‐9 kg = Use your calculator to perform the following calculations. Keep the numbers in scientific notation and write your answer with the correct units. 13. (2.76 x 102 m) x (1.09 x 101 m) = 14. (9.00005 x 102 m) x (5.993 x 10-­‐3 m) = 15. (5.55 x 103 m) / (1.50 x 102 s) = 16. (2.33 x 102 m) / (5.9 x 10-­‐2 s) = 17. (9.655 x 103 s) + (5.50 x 102 s) = 18. (3.26 x 101 s) + (1.77 x 10-­‐1 s) = 19. (3.5 x 10-­‐3 s) -­‐ (5.50 x 10-­‐1 s) = Summarize 20. Why do we use scientific notation? 21. How do you convert “normal” numbers into scientific notation? 22. How do you convert numbers written in scientific notation into “normal” numbers? Measurements & Significant Figures (aka Sig Figs) Sig figs tell us the accuracy of a measurement value. When we use sig figs correctly all of the digits are known with certainty except for the last digit (on the right). This last digit contains some uncertainty. Scaled instruments like rulers, balances and graduated cylinders have divisions or markers that indicate the value of the measurement. With these measuring tools, the divisions give us values that we know with certainty. When we estimate a value between the smallest divisions there is some uncertainty. v2 When reading scaled instruments, you should record the value to the smallest division plus one additional digit that is estimated. For example, let’s measure a nail with two rulers. The smallest division for ruler A is tenths of a centimeter (aka millimeters). The smallest division for ruler B is centimeters. With ruler A we know with certainty that the nail is longer than 4.3 cm but slightly less than 4.4 cm. We can estimate that the tip of the nail is 80% to 90% of the way between 4.3 cm and 4.4 cm. So we can estimate the hundredths value as an ‘8’ or ‘9’. Using significant figures we would write the value as either 4.38 cm or 4.39 cm. We know the ‘4’ and the ‘3’ with certainty, but the hundredths value has some uncertainty; we know it’s not 0 through 7, but it could be ‘8’ or it could be ‘9’. With ruler B we know with certainty the nail is longer than 4 cm but less than 5 cm. We can estimate the tenths of a centimeter value as ‘3’ or ‘4’. Using significant figures we would write the value as either 4.3 cm or 4.4 cm. Notice that we have no way to estimate the hundredths value using this ruler. If we wrote down a hundredths value it would be a guess with complete uncertainty, so we don’t include the hundredths value in this case. Remember, the last digit has some uncertainty, but not complete uncertainty. Measuring tools with digital readouts give us the correct number of sig figs. For example, the digital balances in our lab measure mass to the nearest 0.1 gram. This means that the tenths place has some uncertainty. Read the description and write the measured values. 23. You measure a force using a spring scale that has 0.1 Newton as its smallest division. Give an example of a measurement that has the correct number of sig figs and the correct units. Underline the certain digits. 24. You measure a force using a spring scale that has 1 newton as its smallest division. Give an example of a measurement that has the correct number of sig figs and the correct units. 25. You measure a line with a meter stick that has 1 cm as its smallest division. Give an example of a measurement that has the correct number of sig figs and the correct units. Underline the certain digits. 26. You measure the temperature with a thermometer that has 1 °C as its smallest division. Give an example of a measurement that has the correct number of sig figs and the correct units. Underline the certain digits. v2 27. You mass a stapler on the digital scale and the display reads 172.4 g. Which digit(s) are known with certainty and which digit(s) have some uncertainty? Summarizer 28. Why do we use significant figures to record scientific measurements? Which Digits are Significant? In the previous section we learned how to write measured quantities using significant figures. Suppose you are looking at a quantity that someone else has measured. How do you know how many significant figures the number has? There are rules for counting the number of significant figures in a measured value. •
All non-­‐zero digits are significant. Example -­‐ 1225.3 has 5 sig figs •
Zero digits sandwiched between non-­‐zero digits are significant. Examples -­‐ 1002 has 4 sig figs and 1.001 has 4 sig figs •
Zero digits sandwiched between a non-­‐zero digit on the left and a decimal point on the right are significant. Example -­‐ 1200. has 4 sig figs •
Zero digits to the right of non-­‐zero digit and a decimal point are significant. Example – 1200.00 has 6 sig figs •
Zero digits to the right of non-­‐zero digits with no decimal point are NOT significant. Examples -­‐ 1210 has 3 sig figs and 1200 has 2 sig figs •
Zero digits with only non-­‐zero digits on the right are NOT significant. Examples -­‐ 0.0113 has 3 sig figs and 0.00113 has 3 sig figs 29. How many sig figs does each of the following measurements have? 1225 4 sig figs 100.45 5 sig figs 0.001 1 sig fig 1201 4 sig figs 10.45 4 sig figs 0.0010 2 sig figs 10000 1 sig fig 10.4 3 sig figs 0.0019 2 sig figs 1000. 4 sig figs 10.04 4 sig figs 0.113 3 sig figs Sig Figs & Scientific Notation As you’ve already learned, scientific notation is written in the format m x 10n. When a number is written in scientific notation only the m value determines the number of sig figs. For example, 1.00 x 104 has 3 sig figs and 1.0 x 10-­‐3 has 2 sig figs. Why doesn’t the n value help in determining the number of sig figs? The numbers from the previous example written in normal format are 10 000 and 0.0010, respectively. The n number only adds placeholder zeros, which are not significant! 30. How many sig figs does each of the following measurements have? v2 1.0000 x 104 5 sig figs 1 x 104 1 sig fig 1.01 x 10-­‐1 3 sig figs 1.000 x 104 4 sig fig
1.0010 5 sig figs 1.01 x 10-­‐2 3 sig figs 1.0 x 104 2 sig figs 1.0 x 10-­‐2 2 sig figs 1.01 x 10-­‐3 3 sig figs
Scientific notation is helpful when some, but not all of the trailing zeros are significant. Let’s look at the measurement one thousand, if the thousands digit is the only significant digit we can write the number as “1000”. If the digits are significant to the ones digit, then we can write the number as “1000.” However, how do we write the number if the ones digit is not significant? The only way to write one thousand with 3 sig figs is to use scientific notation, 1.00 x 103. Similarly, if only the thousands and hundreds digits are significant, then one thousand would be written as 1.0 x 103. 31. Use scientific notation to write the following numbers with the correct sig figs: 4
12 000 with 3 sig figs 1.20 x 10 4
12 000 with 4 sig figs 1.200 x 10 4
40 510 000 with 5 sig figs 4.0510 x 10 40 510 000 with 7 sig figs 4.051000 x 10
4
Sig Figs & Math Operations 4.38 cm 4.38 cm When we add, subtract, multiply and divide by numbers that contain + 4.4 cm + 4.4 cm uncertainty we need to make sure that our answer represents the amount 8
.78 c
m 8.8 cm of uncertainty we have in the answer. For example, let’s add the two lengths together from the previous nail example. The answer 8.78 cm in the first calculation overstates the accuracy of our answer. Remember the length measured with ruler B could be anywhere from 4.30 cm to 4.40 cm. We do not have a good idea of what the value is in the hundredths place. So we don’t include the hundredths place. The second calculation is a true representation of the accuracy of our answer. The same 9.34 cm 9.34 cm thing holds true if we were subtracting two numbers. The first calculation + 4.4 cm + 4.4 cm overstates the accuracy of our answer. The second calculation is a true 4
.94 c
𝑔
= 47.50 5 𝑝𝑒𝑙𝑙𝑒𝑡𝑠
𝑝𝑒𝑙𝑙𝑒𝑡
If the 5 were a measured number, then our answer could only have 1 sig fig. However, since counted quantities are exact numbers our answer is limited by the number of sig figs for the mass value, which is 4 sig figs. Similarly, if we wanted to convert 2.45 km to meters our calculation would be … 1000 𝑚
= 2450 𝑚 1 𝑘𝑚
The equality 1 km = 1000 m is a definition not a measurement, so both these numbers are exact. Our final answer is limited by the number of sig figs in the measured distance of 2.45 km, which is 3 sig figs. If you aren’t sure whether a number is exact or not, ask yourself, “did we get this number by measuring something, counting something or is it a definition?” If it’s a measurement then it has to be considered when determining the number of sig figs in the answer. If it’s a counted quantity or a definition it is NOT part of the consideration. 2.45 𝑘𝑚×
Sig Figs & Rounding Most of the answers to your calculations will have fewer sig figs than the answer on your calculator. You’ll need to determine if the smallest significant digit needs to be rounded up or left as is. Look at the digit to the right of the smallest significant digit and “if it’s 4 or below leave it go, 5 and above give it a shove.” Examples – Round each number to 2 sig figs. 3.06 m rounds to 3.1 m 3.0499 m rounds to 3.0 m 99.6 km rounds to 1.0 x 102 km 2549 g rounds to 2500 g 33. Round the numbers below to the specified number of sig figs. 77.076 to 4 sig figs 77.08 3
1002 to 2 sig figs 1.0 x 10 0.0044476 to 2 sig figs 0.0044 0.0044476 to 3 sig figs 0.00445 Precision, Accuracy & Percent Error v2 Accuracy and Precision They aren’t the same thing! Study the chart on the right and come up with your definitions for precision and accuracy. Systematic errors are usually a result of the experimental design. There may be one or more steps in the procedure that introduce error or the equipment used to carry out the experiment may have some limitations that are creating the error. Reproducibility errors are often a result of human error, such as slight differences in the way each trial is performed, or mistakes in measuring results. 30. What are your definitions of accuracy and precision? Percent Error A measure of how accurate experimental values are. A low percent error means the experimental value is close to the accepted value. A large percent error means the experimental value is much different than the accepted value. Here is the formula for calculating percent error: 𝑉𝑎𝑙𝑢𝑒!""#\$%#& − 𝑉𝑎𝑙𝑢𝑒!"#!\$%&!'()*
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟 =
×100 𝑉𝑎𝑙𝑢𝑒!""#\$%#&
If you have a hard time remembering if it’s the accepted value minus the experimental value or vice versa just remember “a comes before e, which means accepted minus experimental.” 31. Calculate the Percent errors for the pairs below (show your work): Experimental Value Accepted Value Percent error 3.45 g 3.49 g 105 m 101 m Summarizer 32. What might be a source of error if your results are imprecise? 33. What might be a source of error if your results are inaccurate? Equalities, Conversion Factors & Dimensional Analysis v2 Often we need to convert a measured quantity from one unit to another unit. For example, we measured the mass of an object as 454.3 grams, but we need the measurement in kilograms. We can use the factor label method to do the conversion. Factor Conversion Method Steps • Write an equality equation for the two units, i.e. 1 kilogram = 1000 grams !""" !
! !"
•
Create two conversion factors from this equality, i.e. •
Pick the conversion factor that has the original unit in the denominator (on the bottom) and multiply it by the ! !"
original measurement, i.e. 454.3 𝑔×
!""" !
! !"
𝑎𝑛𝑑 !""" !
•
Cancel out units that appear in both the numerator (top) and denominator (bottom), i.e. 454.3 𝑔×
•
Perform the calculation and include the units that aren’t cancelled, i.e. . 454.3 𝑔×
! !"
!""" !
! !"
!""" !
= 0.4543 𝑘𝑔 The Metric System Equalities Prefix tera Unit Abbreviation T Exponential Factor 1012 Meaning 1 000 000 000 000 Example 1 Tm = 1 x 1012m giga G 109 1 000 000 000 1 Gm = 1 x 109m mega M 106 1 000 000 1 Mm = 1 x 106m kilo k 103 hecto h deka 10 1 dam = 10 m 0
1 1 meter (m), gram (g) -­‐1
1/10 10 dm = 1 m -­‐2
1/100 100 cm = 1 m -­‐3
1/1000 1 000 mm = 1 m -­‐6
10 c milli 1 hm = 100 m 10 d centi 100 1
10 deci 1 km = 1 000 m 10 da 1 000 2
10 m 10 micro μ (mu) 10 1/1 000 000 1 x 106 μm = 1 m nano n 10-­‐9 1/1 000 000 000 1 x 109 nm = 1 m pico P 10-­‐12 1/1 000 000 000 000 1 x 1012 pm = 1 m *** MUST MEMORIZE nano to kilo English & English-­‐Metric Equalities Mass & Weight 1 lb = 453.6 g 1 oz = 28.35 g 1 kg = 2.2 lb Length 1 in = 2.54 cm 1 km = 0.62 miles 1 ft = 12 in 1 mile = 1.6 km 1 yd = 3 ft Volume 1 qt = 0.946 L 1 L = 1.06 qt 1 gallon = 4 quarts = 8 pints = 128 oz Time 60 s = 1 min 1 cm3 = 1 mL 60 min = 1 hr 24 hr = 1 day 365 days = 1 year Use the factor-­‐label method to calculate the unit conversions. Show your work including the correct sig figs. 43. 500. mL = _____________ L 49. 0.22 km = ___________ m 44. 400. mg = _____________ kg 50. 1.25 km = ____________ yds 45. 3500. s = _____________ hr 51. 2400 mm3 = _____________ cm3 46. 1.5 yr = _______________ s 52. 14.33 m3 = _____________ dm3 47. 15 m3 = ________________ mm3 53. 9722 µm2 = _____________ hm2 48. 4.2 ft3 = ________________ cm3 Summarizer 54. How do you set up conversion factors once you have an equality equation between two units? 55. How do you determine which conversion factor to use in your unit conversion calculation? v2 v2 Dimensional Analysis – Factor-­‐Label Method Often we need to convert from one quantity to another quantity to solve a problem. For example, we need to figure out the volume of gas we can put in the car if we have \$20.00 and gas costs \$3.929 per gallon. We use the factor-­‐label method for these types of calculations. Steps for Dimensional Analysis with Factor-­‐Label Method • Look at the original units, the units for the answer, and find an equality equation between the two. Note that there may not be one equation that relates the two; rather you may need to use several equality statements to go from your starting units to your final units. In our example, we are starting with dollars and need an answer in gallons. We have an equality 1 gallon = #3.929 \$".!"!
! !"#
•
Create two conversion factors from this equality, i.e. •
Pick the conversion factor that has the original unit in the denominator (on the bottom) and multiply it by the original measurement, i.e. \$20.00×
! !"#
\$".!"!
! !"#
𝑎𝑛𝑑 \$".!"!
•
Cancel out units that appear in both the numerator (top) and denominator (bottom), i.e. \$20.00×
•
Perform the calculation and include the units that aren’t cancelled, i.e. \$20.00×
! !"#
\$!.!"!
! !"#
\$!.!"!
= 5.090 𝑔𝑎𝑙. Note that the 1 gallon and \$3.929 are exact numbers so our answer is limited to the number of sig figs in \$20.00, which are 4. Use dimensional analysis and the factor-­‐label method to solve these problems. Show your work and include the correct units and sig figs in your answer. 56. The ladle at an iron foundry can hold 4500 kg of molten iron; 446 metric tons of iron is needed to make rails. How many ladleful’s of iron will it take to make 446 metric tons of iron? (1 metric ton = 1000 kg) 57. A baker uses 1.75 tsp of vanilla extract in each cake. How much vanilla extract in liters should the baker order to make 600 cakes? (1 tsp = 5 mL) v2 58. A person drinks six glasses of water each day, and each glass contains 300 mL. How many liters of water will that person consume in a year? What is the mass of this volume of water in kilograms? (Assume one year has 365 days and the density of water is kg/L.) 59. Mr. Good drives 50 miles round trip to school each day. There are 5 school days per week, his car averages 22.1 miles per gallon and gasoline costs \$3.929 per gallon. How much money does Mr. Good spend on gasoline each week? Summarizer 60. What are the three key steps to solving problems with dimensional analysis and the factor-­‐label method? 61. How do you determine how many equality statements you need to convert from one quantity to another quantity? Dimensional Analysis with Units We can use dimensional analysis and to determine the mathematical relationship between variables. v2 Example 1: How are time, distance and velocity related mathematically? Step 1 – What are the units for each of the variables? We know that the unit for velocity is meters per second, the unit for time is seconds, and the unit for distance is meters. Step 2 – Create an equality statement with the units. 𝑚𝑒𝑡𝑒𝑟𝑠 ÷ 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 = 𝑚𝑒𝑡𝑒𝑟𝑠 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 Step 3 – Substitute the variables for the units and change the equal sign to a proportional sign (𝛼). 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ÷ 𝑡𝑖𝑚𝑒 𝛼 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 We have to use the proportional sign in place of the equal sign because dimensional analysis with units does not tell us if there are any numerical constants in the equation. In this example there aren’t, but we know that from previous knowledge, not from dimensional analysis. Example 2: How are acceleration, velocity and distance mathematically related to each other? Step 1 – We know the unit for acceleration is meters per second squared, the unit for velocity is meters per second, and the unit for distance is meters. Step 2 -­‐ This is a trial and error step. You need to try different combinations until you get the correct equality statement. 𝑚
𝑚 !
× 𝑚
=
𝑠!
𝑠
Step 3 – 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 × 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝛼(𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦)! In this case, there are some constants involved and the actual equality is 2𝑎𝑑 = 𝑣 ! − 𝑣!! . Dimensional analysis could not tell us that the product of acceleration and distance is doubled, nor could it tell us that that is equal to the change in velocity squared (𝑣! is the symbol for initial velocity). Use dimensional analysis with units to determine the mathematical relationship between the variables in each problem below. 62. What is the relationship between time, acceleration and distance? 63. What is the relationship between mass, acceleration and force? The SI unit for force is Newtons, which is the same as kg•m/s2. 64. What is the relationship between mass, velocity and momentum? The unit for momentum is kg•m/s. 65. What is the relationship between force, the change in momentum (∆ρ), and time? v2 Solving Equations by using Non-­‐Numerical Values Most of us are very comfortable solving equations by substituting numerical values for the variables and get a numerical answer. This is an important skill in science and engineering. However, it is equally important if not more important to be able to solve equations using non-­‐numerical values. This allows us to conceptually understand the impact changing one variable has on another variable. Example 1 – Car A travels at twice the speed of Car B for one-­‐third the amount of time. How far did Car A travel compared to Car B? Step 1 – Find the correct equation or set of equations. In this case, Distance = velocity x time. Step 2 – Write the distance equation for both cars. Using subscripts is very helpful. 𝐷! = 𝑣! × 𝑡! 𝑎𝑛𝑑 𝐷! = 𝑣! × 𝑡! Step 3 – Write down the relationships between the non-­‐numerical values that are given in the word problem. !
𝑣! = 2𝑣! 𝑎𝑛𝑑 𝑡! = !𝑡! Step 4 – Substitute the “B” values for the “A” values in the DA equation and solve the equation. !
!
𝐷! = 2𝑣! × !𝑡! = !𝑣! 𝑡! Step 5 – Use the “B” equation to substitute DB for vBtB. !
𝐷! = ! 𝐷! ∴ Car A travelled two-­‐thirds as far as Car B. Use non-­‐numerical values to solve the following problems. 66. Object A has twice the mass of Object B, but only one-­‐fourth the volume. How dense is Object A compared to Object B? 67. The distance a car travels under constant acceleration is calculated using the equation d = ½ at2. A Mini Cooper roadster accelerates twice as fast as a Ford F-­‐150 pick-­‐up truck. How far does the roadster travel compared to the pick-­‐up truck if the roadster accelerates for twice as long as the pick-­‐up truck? 68. The formula for calculating the surface area of a sphere is 𝑎𝑟𝑒𝑎 = 4𝜋𝑟 ! . Planet A has a radius that is two-­‐
thirds the radius of Planet B. What is the surface area of Planet A compared to Planet B? 69. Newton’s 2nd Law states that force = mass x acceleration or F = ma. In #67 the force propelling the pick-­‐up truck is 1 ½ times the force propelling the roadster. What is the mass of the pick-­‐up truck compared to the mass of the roadster?