1. I II III Unit Conversions MEASUREMENT

2. Are Units important? 3 On 9/23/99, $125,000,000 Mars Climate Orbiter entered Mar’s atmosphere 100 km (62 miles) lower than planned and was destroyed by heat. 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 1 lb = 1 N 1 lb = 4.45 N

3. 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 8

4. U. S. Customary System (USCS) Fundamental Dimensions: Derived Dimensions: Length = ft Mass = slug = lb f *s 2 /ft = 32.174 lb m Force = lb f Energy = ft*lb f Time = s Power = ft*lb f /s 9

5. Dimensional Analysis  The “Factor-Label” Method  Units, or “labels” are canceled, or “factored” out

6. 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!

7. 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.

8. 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 = 1.057 quarts 1.057 qt/L or 1 L/1.057 qt or 0.946 L/qt  1 kilogram = 2.2 pounds 2.2 lb/kg or 1 kg/2.2 lb  or 0.454 kg/lb  1 meter = 1.094 yards 1.094 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

9. Example Metric conversion How many milligrams are in a kilogram?

10. Dimensional Analysis  How many milliliters are in 1.00 quart of milk? 1.00 qt 1 L 1.057 qt = 946 mL qtmL 1000 mL 1 L 

11. 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 3 1.5 lb 1 kg 2.2 lb = 35 cm 3 1000 g 1 kg 1 cm 3 19.3 g

12. 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

13. Dimensional Analysis 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

14. Dimensional Analysis 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

15. Dimensional Analysis 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

16. 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

17. Converting Area and Volume Caution: Make sure the units cancel Area: 150 ft 2 to yd 2 150 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 3 1000 L (1) 3 ft 3 (1) 3 in 3 (100) 3 cm 3 1 m 3 16

18. 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

19. 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: 10 -4 cm s 11

20. 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 = 0.011811 ft/hr 12

21. Chemical Herbicide Spill Line Mole Method - Example Solution: Permeability = 0.011811 ft/hr Time = Distance / Permeability t = 150 ftOR t = 150 fthr 0.011811 ft/hr0.011811 ft t = 12700 hours = 13000 hours How many years is that? t = 12700 hr1 day1 yr= 1.4 yr 24 hr365 day 13

22. 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 10.0 6 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

23. 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 2 3 3 V www.algonquin.org/pw.htm 4 Assumptions:  tower is spherical 15

24. Solution: 1.00 x10 6 kg 1 m 3 1000 L volume of water == 1.00 x 10 6 L 999 kg 1 m 3 4 Volume of a sphere r3r3 diameter2 r2 r 2 3 3 V 3 4 1.00 x 10 6 L0.035315 ft 3 volume of water == 3.53 x 10 4 ft 3 1 L 3.53 x 10 4 ft 3 diameter2 r2 r 2 3 3 40.7 ft 4 16