I II III Unit Conversions MEASUREMENT Today's Objectives 1) Importance of unit conversions 2) Parts of a measurement 3) Units in equations 4) Documenting unit conversions 2 Are Units important? 3 "The 'root cause' of the loss of the spacecraft was the failed translation of English units into metric units in a segment of ground-based, navigation-related mission software, as NASA has previously announced," said Arthur Stephenson, chairman of the Mars Climate Orbiter Mission Failure Investigation Board. "The failure review board has identified other significant factors that allowed this error to be born, and then let it linger and propagate to the point where it resulted in a major error in our understanding of the spacecraft's path as it approached Mars."4 International System of Units (SI) Fundamental Dimensions: Derived Dimensions: Length = m Force = N (newton) = kg*m/s 2 Mass = kg Energy = J (joule) = N*m Time = s Power = W (watt) = J/s SI prefixes listed in table need to have prefixes with a memorized for exams!!! 8 U. S. Customary System (USCS) Fundamental Dimensions: Derived Dimensions: Length = ft Mass = slug = lb f *s 2 /ft = lb m Force = lb f Energy = ft*lb f Time = s Power = ft*lb f /s 9 A. SI Prefix Conversions 1.Find the difference between the exponents of the two prefixes. 2.Move the decimal that many places. To the left or right? A. SI Prefix Conversions mega-M10 6 deci-d10 -1 centi-c10 -2 milli-m10 -3 PrefixSymbolFactor micro- nano-n10 -9 pico-p kilo-k10 3 move left move right BASE UNIT A. SI Prefix Conversions 1) 20 cm = ______________ m 2) L = ______________ mL 3) 45 m = ______________ nm 4) 805 dm = ______________ km ,000 32 = A. SI Prefix Conversions NUMBER UNIT NUMBER UNIT 532 m = _______ km 0.532 B. Dimensional Analysis The Factor-Label Method Units, or labels are canceled, or factored out Converting units Factor label method Regardless of conversion, keeping track of units makes things come out right Must use conversion factors - The relationship between two units Canceling out units is a way of checking that your calculation is set up right! B. Dimensional Analysis Steps: 1. Identify starting & ending units. 2. Line up conversion factors so units cancel. 3. Multiply all top numbers & divide by each bottom number. 4. Check units & answer. Common conversion factors Factor EnglishFactor 1 gallon = 4 quarts4 qt/gal or 1gal/4 qt 1 mile = 5280 feet5280 ft/mile or 1 mile/5280 ft 1 ton = 2000 pounds 2000 lb/ton or 1 ton/2000 lb Common English to Metric 1 liter = quarts qt/L or 1 L/1.057 qt or L/qt 1 kilogram = 2.2 pounds 2.2 lb/kg or 1 kg/2.2 lb or kg/lb 1 meter = yards yd/m or 1m/1.094 yd or 0.917m/yd 1 inch = 2.54 cm 2.54 cm/inch or 1 in/2.54 cm B. Dimensional Analysis Lining up conversion factors: 1 in = 2.54 cm 2.54 cm 1 in = 2.54 cm 1 in 1 in = 1 1 = Line Mole Method Process to convert from one unit to another Example: Convert 3.00 m to inch: ? = 3.00 m 100 cm 1 in 1 m 2.54 cm ANSWER = 118 in 5 Line Mole Method Process to convert from one unit to another Example: Convert 3.00 m/s to m/hr: ? = 3.00 m 60 s 60 min s min hr ANSWER = 10,800 m/hr 6 Example Metric conversion How many milligrams are in a kilogram? B. Dimensional Analysis How many milliliters are in 1.00 quart of milk? 1.00 qt 1 L qt = 946 mL qtmL 1000 mL 1 L B. Dimensional Analysis You have 1.5 pounds of gold. Find its volume in cm 3 if the density of gold is 19.3 g/cm 3. lbcm lb 1 kg 2.2 lb = 35 cm g 1 kg 1 cm g B. Dimensional Analysis How many liters of water would fill a container that measures 75.0 in 3 ? 75.0 in 3 (2.54 cm) 3 (1 in) 3 = 1.23 L in 3 L 1 L 1000 cm 3 B. Dimensional Analysis 5) Your European hairdresser wants to cut your hair 8.0 cm shorter. How many inches will he be cutting off? 8.0 cm1 in 2.54 cm = 3.1 in cmin B. Dimensional Analysis 6) Taft football needs 550 cm for a 1st down. How many yards is this? 550 cm 1 in 2.54 cm = 6.0 yd cmyd 1 ft 12 in 1 yd 3 ft B. Dimensional Analysis 7) A piece of wire is 1.3 m long. How many 1.5 cm pieces can be cut from this wire? 1.3 m 100 cm 1 m = 86 pieces mpieces 1 piece 1.5 cm As an Individual (on a piece of paper)... Use the Factor Label Method to convert the following: 1) 34.6 m/s to mph 2) 15 ft 3 to Liters 3) Can you use FLM when converting from Celsius to Kelvin? Why or why not? Conversion Factors: Handout 14 Units in Equations Addition/Subtraction: All terms must have the same units Multiplication/Division: Units may cancel, like variables Homogeneity: Units on both sides of the "=" sign must be the same Transcendental Functions: (sin x, ln x, e x ) the argument (x) does not have a dimension 7 Valid Units? 10 m 20 m 30 m 2 ss 12 mile 28 mile 40 mile hour 8 H:\classes\eng1101\fall05\lectures\spring05\en1.13b.sketch.amy.sxi Converting Area and Volume Caution: Make sure the units cancel Area: 150 ft 2 to yd ft 2 1 yd 150 ft 2 (10)2 yd2 OR 3 ft (3) 2 ft 2 Volume: 12 ft 3 to Liters 12 ft 3 (12) 3 in 3 (2.54) 3 cm 3 (1) 3 m L (1) 3 ft 3 (1) 3 in 3 (100) 3 cm 3 1 m 3 16 Chemical Herbicide Spill Line Mole Method - Example Problem: The permeability of sand is 1.0x10 -4 cm/s. If a chemical herbicide is dumped on a sandy soil, how long (in hours) will it take for the contaminant to reach the well 150 feet away. Diagram: Herbicide 150 feet Well Permeability of Sand = 1.0x10 -4 cm/s t = Time (hours) 1.0x10 -4 cm/s = __?__ ft/hr 10 Chemical Herbicide Spill Factor Label Method - Example Theory: Permeability = Distance/Time Assumptions: Sand has constant permeability in area Herbicide moves per permeability of sand Solution: cm s 11 Chemical Herbicide Spill Line Mole Method - Example Theory: Permeability = Distance/Time Assumptions: Sand has constant permeability in area Herbicide moves per permeability of sand Solution: 1.0x10 -4 cm 1 in1 ft60 s60 min s2.54 cm12 in 1 min1 hr = ft/hr 12 Chemical Herbicide Spill Line Mole Method - Example Solution: Permeability = ft/hr Time = Distance / Permeability t = 150 ftOR t = 150 fthr ft/hr ft t = hours = hours How many years is that? t = hr1 day1 yr= 1.4 yr 24 hr365 day 13 As an individual, solve... Water Tower Problem Problem Statement: Your home town is growing so rapidly that another water tower is necessary to meet the needs of the community. Civil and environmental engineers predict that the water tower will need to hold 1.00 x kilograms of water. The engineers also estimate the density of the water to be 999 kilograms per cubic meter. If this tower is 50.0 meters high and spherical, what volume (gal) of water will the tower hold and what will the diameter (ft) of the tower have to be? 14 Diagram: mass of water = 1.00 x 10 6 kg density of water = 999 kg/m 3 tower height = 50.0 m ? volume of water (L) ? diameter (ft) Theory: 4 Volume of a sphere r3r3 3 diameter 2 r2 r V4 Assumptions: tower is spherical 15 Solution: 1.00 x10 6 kg 1 m L volume of water == 1.00 x 10 6 L 999 kg 1 m 3 4 Volume of a sphere r3r3 diameter2 r2 r V x 10 6 L ft 3 volume of water == 3.53 x 10 4 ft 3 1 L 3.53 x 10 4 ft 3 diameter2 r2 r ft 4 16 Golf Ball Design Minimum allowed diameter of a golf ball is 42.67mm Maximum Mass = 45.93g The surface usually has a pattern of dimples designed to improve the ball's aerodynamics. The method of construction and materials greatly affect the ball's playing characteristics such as distance, trajectory, spin and feel. Have a two-, three-, or four-layer design constructed from various synthetic materials Harder materials, such as Surlyn, usually result in the ball's traveling longer distances, Softer covers, such as Balata, tend to generate higher spin, more "feel" and greater stopping potential. Golf balls are separated into three groups depending on their construction: two-, three-, or four-piece covers. 17 Golf Ball Design Minimum allowed diameter of a golf ball is 42.67mm Assuming a golf ball has a spherical shape What is the golf ball diameter in inches? What is the volume of a golf ball in cubic centimeters and cubic inches? Maximum Mass = 45.93g What is the mass of a golf ball in pounds? What is the density of a golf ball in g/cm 3 and lb/in 3 ? 18 Golf Shafts Golf shafts are what connects the grip to the golf head The profile of the golf shaft is circular in shape and is usually thicker at the grip end than at the club head end. Any strong and light material may be used to make the golf shaft. Almost all shafts today are made of either graphite or tempered steel The shaft is a tapered tube made of metal (usually steel), or graphite fiber. The shaft is roughly 1/2 inch in diameter (12 mm) near the grip and between 35 to 45 inches ( cm) in length. 19 Golf Shafts Almost all shafts today are made of either graphite or tempered steel Graphite: g/cm 3 Steel: 7, kg/m (0.284 lb/in) How much would the shaft of a golf club weigh in pounds if it were constructed from graphite or steel? Assume: Shaft Diameter = 1/2 inch and solid Shaft Length = 40 inches Why would you choose a graphite club over a steel club or vice versa? What is tempered steel? 20 I II III Roentgen Pronounced rentgen with a hard g Limitations only applies to photons only applies in air only applies to energies less than 3 MeV l Named after Wilhelm C. Roentgen (thus the abbr... is capital R) I II III rad 1 rad = 1 Roentgen I II III rem The unit of dose equivalent for any type of ionizing radiation absorbed by body tissue in terms of estimated biological effect - Unit of dose equivalent Dose in health record is in units of rem 1 rem = 1 Roentgen l Roentgen Equivalent Man Quality Factor (Q) The specific value that accounts for the ability of different types of ionizing radiation to cause varying degrees of biological damage X-rays, gamma rays, & beta particles 1 Neutrons & High energy protons 10 Alpha Particles 20 Curie (Ci) Radiation hazard does not solely depend on the activity. It also depends on the type of decay (alpha, beta, photon, etc.) l Named in honor of Pierre Curie SI Radiation Protection Units Becquerel (Bq) for Curie 1 Ci = 3.7 x Bq Gray (Gy) for rad 1 Gy = 100 rad Sievert (Sv) for rem 1 Sv = 100 rem Unit Analysis BASE UNIT CONVERSION TABLE Unit Unit Conversion 1 Bq 2.7 x Ci 1 Ci3.7 x Bq 1 Bq1 dis/sec 1 dis/sec2.7 x Ci 1 Ci3.7 x dis/sec Unit Analysis (Cont.) BASE UNIT CONVERSION TABLE Unit Unit Conversion 1 rem 0.01 Sv 1 Sv 100 rem 1 rad 0.01 Gy 1 Gy 100 rad 1 R 2.58 x C/kg 1 meter 3.28 ft (39.37in) Summary Radiation Protection unit definitions (including SI units) Unit Prefixes Unit conversions Old Terms Roentgen- Based on the quantity of electrical charges produced in air by X or Gamma photons 1R=2 billion pr RAD- Radiation Absorbed Dose is the work energy resulting from the absorption of one ROENTGEN or 6.24 E5 Mev More Old Terms REM- Roentgen Equivalent Mammal is equal to the absorbed does in RADS multiplied by a quality factor Quality Factors Beta = 1 Gamma & X ray photons = 1 Alpha = 10 Neutrons = 20 New Terms sort of International Units have replaced the RAD and REM GRAY (Gy) = 100 RAD SIEVERT (Sv) = 100 REM Same Quality Factors apply to the Sv Units of Radioactivity Curie (Ci) = 2.22 E12 dpm or 3.7E10 dps Becquerel (Bq) = 1 dps Maximum Dose/year = 5 REM or 50 mSv Maximum Dose/year for Declared Pregnant Woman & Minors= 0.5 REM or 5 mSv