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Chapter 2: Measurement and Problem Solving The SI System • All units in the SI system are related to the standard unit by a power of 10 • The power of 10 is indicated by a prefix • The prefixes are always the same, regardless of the standard unit Common Prefixes in the SI System Prefix Symbol Decimal Equivalent Power of 10 1,000,000 Base x 106 1,000 Base x 103 mega- M kilo- k deci- d 0.1 Base x 10-1 centi- c 0.01 Base x 10-2 milli- m 0.001 Base x 10-3 micro- m or mc 0.000 001 Base x 10-6 nano- n 0.000 000 001 Base x 10-9 Length • = Measure of the two-dimensional distance an object covers • SI unit = meter About 3½ inches longer than a yard • 1 meter = one ten-millionth the distance from the North Pole to the Equator = distance between marks on standard metal rod in a Paris vault = distance covered by a certain number of wavelengths of a special color of light • Commonly use centimeters (cm) 1 m = 100 cm 1 cm = 0.01 m = 10 mm 1 inch = 2.54 cm (exactly) Mass • = Measure of the amount of matter present in an object • SI unit = kilogram (kg) about 2 lbs. 3 oz. • Commonly measure mass in grams (g) or milligrams (mg) 1 kg = 2.2046 pounds, 1 lbs. = 453.59 g 1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103 mg 1 g = 0.001 kg = 10-3 kg, 1 mg = 0.001 g = 10-3 g Volume • = Measure of the amount of three-dimensional space occupied • SI unit = cubic meter (m3) a Derived Unit • Commonly measure solid volume in cubic centimeters (cm3) 1 m3 = 106 cm3 1 cm3 = 10-6 m3 = 0.000001 m3 • Commonly measure liquid or gas volume in milliliters (mL) 1 L is slightly larger than 1 quart 1 L = 1 dL3 = 1000 mL = 103 mL 1 mL = 0.001 L = 10-3 L 1 mL = 1 cm3 Common Units and Their Equivalents Length 1 kilometer (km) = 0.6214 mile (mi) 1 meter (m) = 39.37 inches (in.) 1 meter (m) = 1.094 yards (yd) 1 foot (ft) = 30.48 centimeters (cm) 1 inch (in.) = 2.54 centimeters (cm) exactly 1 kilogram (kg) = 2.205 pounds (lb) 1 pound (lb) = 453.59 grams (g) 1 ounce (oz) = 28.35 (g) Mass Common Units and Their Equivalents Volume 1 liter (L) = 1000 milliliters (mL) 1 liter (L) = 1000 cubic centimeters (cm3) 1 liter (L) = 1.057 quarts (qt) 1 U.S. gallon (gal) = 3.785 liters (L) Problem Solving and Dimensional Analysis • Arrange conversion factors so starting unit cancels Arrange conversion factor so starting unit is on the bottom of the conversion factor • May string conversion factors So we do not need to know every relationship, as long as we can find something else the beginning and ending units are related to unit 1 x unit 2 unit 1 = unit 2 Dimensional Analysis Exercise 1: Convert 7.8 km to miles 1. Write down the Given quantity and its unit 2. Write down the quantity you want to Find and unit 3. Write down the appropriate Conversion Factors 4. Write a Solution Map Given: 7.8 km Find: ? miles Conversion Factors: Solution Map: Solution: 1 km = 0.6214 mi km mi 0.6214 mi 1 km 5. Follow the Solution Map to Solve the problem 6. Sig. Figs. and Round Round: 7. Check Check: Units & Magnitude are correct 7.8 km 0.6214 mi 4.84692 mi 1 km 4.84692 mi = 4.8 mi Dimensional Analysis Exercise 2: Convert 20.0 lbs to kg 1. Write down the Given quantity and its unit 2. Write down the quantity you want to Find and unit 3. Write down the appropriate Conversion Factors 4. Write a Solution Map Given: 20.0 lbs Find: ? kg Conversion Factors: Solution Map: Solution: 1 kg = 2.205 pounds lbs kg 2.205 lb 1 kg 5. Follow the Solution Map to Solve the problem 6. Sig. Figs. and Round Round: 7. Check Check: Units & Magnitude are correct 20 lbs 1 kg 9.07029 kg 2.205 lbs 9.07029 kg = 9.1 kg Dimensional Analysis Exercise 3: How many cups of cream is 0.75 L? 1. Write down the Given quantity and its unit 2. Write down the quantity you want to Find and unit 3. Write down the appropriate Conversion Factors 4. Write a Solution Map Given: 0.75 L Find: ? cu Conversio n Factors: Solution Map: Solution: 1 L = 1.057 qt 1 qt = 4 cu L qt 1.057 qt 1L cu 4 cu 1 qt 5. Follow the Solution Map to Solve the problem 6. Sig. Figs. and Round Round: 3.171 cu = 3.2 cu 7. Check Check: Units & Magnitude are correct 0.75 L 1.057 qt 4 cu 3.171 cu 1L 1 qt Converting Quantities Involving Units Raised to a Power Convert Cubic Inches into Cubic Centimeters 1) Find Relationship Equivalence: 1 in = 2.54 cm 2) Write Solution Map in3 cm3 3) Change Equivalence into Conversion Factors with Starting Units on the Bottom 3 2.543 cm3 16.4 cm3 2.54 cm 3 3 1 in 1 in 3 1 in Convert 2,659 cm2 into square meters 1. Write down the Given quantity and its unit 2. Write down the quantity you want to Find and unit 3. Write down the appropriate Conversion Factors 4. Write a Solution Map 5. Follow the Solution Map to Solve the problem Given: 2,659 cm2 ? m2 Find: Conversion Factors: Solution Map: Solution: 1 cm = 0.01 m cm2 m2 0.01 m 1 cm 2,659 cm 2 2 1 10 4 m 2 1 cm 2 0.2659 m 2 6. Sig. Figs. and Round Round: 0.2659 m2 7. Check Check: Units & Magnitude are correct Volume vs Mass of Brass y = 8.38x 160 140 120 Mass, g 100 80 60 40 20 0 0.0 2.0 4.0 6.0 8.0 10.0 Volume, cm3 12.0 14.0 16.0 18.0 Density Ratio of mass:volume Solids = g/cm3 1 cm3 = 1 mL Liquids = g/mL Gases = g/L Mass Density Volume Volume of a solid can be determined by water displacement – Archimedes Principle Density : solids > liquids >>> gases except ice is less dense than liquid water! Density • For equal volumes, denser object has larger mass • For equal masses, denser object has smaller volume • Heating objects causes objects to expand does not effect their mass!! How would heating an object effect its density? • In a heterogeneous mixture, the denser object sinks Why do hot air balloons rise? Density Solution Maps: Volume Density Mass m, V D Density Volume Mass m, D V V, D m Mass Density Volume Density as a Conversion Factor • The gasoline in an automobile gas tank has a mass of 60.0 kg and a density of 0.752 g/cm3. What is the volume? • Given: 60.0 kg • Find: Volume in L • Conversion Factors: 0.752 grams/cm3 1000 grams = 1 kg Density as a Conversion Factor Exercise: • A 55.9 kg person displaces 57.2 L of water when submerged in a water tank. What is the density of the person in g/cm3?