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CHAPTER 2 UNITS OF MEASUREMENT ACCURACY AND PRECISION (DART BOARD EXAMPLE) Accuracy How close you are to the mark you are trying to hit (bulls-eye) Precision How close the measurements are to each other Three students were each asked to prepare 5 samples of sodium chloride crystals that weigh ~10.0 g each. After preparing the samples, each sample was weighed and the following measurements were recorded. Student 1 Student 2 Student 3 10.0 10.4 g 8.1 g 10.1 g 12.3 g 8.0 g 9.9 g 8.4 g 7.9 g 9.8 g 6.2 g 7.8 g 9.7 g 14.1 g 7.7 g Which set of measurements are the most accurate? The least accurate? Which set of measurements are the most precise? The least precise? PERCENTAGE ERROR Can be calculated for one experimental/measured data point or a set of data Compares the experimental value/set of values to the correct or accepted value (given or look up) % Error = Value experimental – Value accepted x 100% Value accepted When would percentage error be negative? Positive? Sample problem on p.43 HOMEWORK Complete Measurements and Calculations: Sample Problem C: Percentage Error worksheet MEASUREMENT ACTIVITY Using a ruler, measure a small object in centimeters (pen, pencil, book thickness, etc.) Record your measurement. How certain are you that _____________is ______________ long? Is it exactly _____________ long or is it a little more or a little less? RULE FOR MEASUREMENT Report measurements to one decimal place past the smallest graduation on the instrument. If you say someone is 6.0000 feet tall, you are saying that the instrument you are using to measure height can accurately measure to the thousandths place! UNITS OF MEASURE SI (Le Système International d’Unités)/ Metric System World’s most widely used system of measurement 7 base units (length, mass, time, temperature, amount of substance, electric current, luminous intensity- see p.34) U.S. Customary System (developed from English system) We need to be able to convert units within each system and between the two systems. DIMENSIONAL ANALYSIS (UNIT CONVERSIONS) Conversion factors Ratios derived from the equality between two different units that can be used to convert from one unit to the other See Metric Prefixes and Conversion Factors handout 1 ft = 12 in. Write this as 1 ft or 12 in 12 in 1 ft Convert 78.0 inches to feet. Convert 78.0 inches to centimeters. 6.5 feet or ~198 cm HOW DO I START? One given: (examples to follow) First step is always to start with the given/1. Multiple givens: Try to find a given with no denominator if possible and start with that given/1 HOW TO GO TO 2ND, 3RD, ETC. STEPS Each step should cancel out a previous unwanted unit and bring in other units that are progressing toward the desired units of the answer. 5 gallons of water weighs _________ kg? 5 gal 4qt 1L 1000ml 1g 1 kg ------ x ------ x ---------- x ---------- x ------ x ------- = 1 1 gal 1.06qt 1L 1ml 1000g CALCULATOR USE The math of the previous problem is most easily handled as a “chain operation”. All numbers in the numerator are multipliers and all numbers in the denominators are divisors. Example from previous slide: 5 x 4 / 1.06 x 1000 / 1000 = 18.9 CLASS WORK Complete Measurements and Calculations: Sample Problem B-Conversion Factors worksheet MASS AND WEIGHT- WHAT IS THE DIFFERENCE? Mass Measure of the quantity/amount of matter Does not depend on gravity Weight Measure of the gravitational pull on matter In space there is no gravity so you are weightless, but you still have the same mass as you do on earth. On other planets with different gravitational pull, you would weight more or less. DENSITY Density = Mass Volume D = m/ V Densities of common materials given on p.36 in Figure 2.8. DENSITY PROBLEM A sample of aluminum metal has a mass of 8.4g. The volume of the sample is 3.1 cm3. Calculate the density of aluminum. What are we given? What are we trying to find? D = m/ V 8.4g = 2.7 g/cm3 3.1 cm3 VARIATIONS ON DENSITY PROBLEMS Given the density and mass, find the volume. V = m/ D Given the density and volume, find the mass. m=D*V PROBLEM The volume of a copper wire is 1000 cm3. The density of copper is 8.92g/cm3. What is the mass of the copper wire? m=D*V 1000 cm3 * 8.92g/cm3 = 8920 cm3 CLASS WORK Complete Measurements and Calculations: Problem A- Density worksheet Now we need to know how to write our answers using scientific notation and the correct number of significant figures. SIGNIFICANT FIGURES Rules (from p.45 in the textbook) Zeros appearing between nonzero digits are significant. Zeros appearing in front of all nonzero digits are not significant. Zeros at the end of a number and to the right of a decimal point are significant. Zeros at the end of a number but to the left of a decimal point may or may not be significant. Zeros as placeholders are not significant. Use a decimal after the zeros if the zeros are considered significant. HOW MANY SIGNIFICANT FIGURES ARE IN EACH OF THESE MEASUREMENTS? 35.2 m 10.04 g 0.00020 kg 2000 lb 2000. lb 1.25000 m3 SIGNIFICANT FIGURES (P.44-49) Rules for Rounding- do not follow rules given in Table 6 (too complicated!) Rules for Addition/ Subtraction with Decimals Answer must have same number of digits to the right of the decimal place as there are in the measurement having the fewest digits to the right of the decimal point Rules for Multiplication/ Division Answer can have no more significant figures than are in the measurement with the fewest number of significant figures EXAMPLES 5.44m – 2.6103m = 2.83m (2 decimal places) 2.4 g/ml x 15.82 ml = 38g (2 sig figs) Complete Measurements and Calculations: Significant Figures Worksheets D and E Four Step Process to Solving Problems ADDITIONAL EXAMPLE Polycarbonate plastic has a density of 1.2 g/cm3. A photo frame is constructed from two 3.0 mm sheets of polycarbonate. Each sheet measures 28 cm by 22 cm. What is the mass of the photo frame? 1.2 g/cm3 x 3.0 m x 1 cm x 28 cm x 22 cm x 2 10 mm = 440 g SCIENTIFIC NOTATION & SIGNIFICANT FIGURES 1000 = 1.00x103 This form is taught in math. This form is not recognized by computers, however. Therefore, we will use the following: Always put the decimal after the 1st number and using the 4th number, round back to the 3rd number for 3 significant figures. 1400 = 1.40e3 .00001256 = 1.26e-5