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Dimensions and Units English Units of Measurement Units of Weight English units of measurement Today: A system of weights and measures used in a few nations, the only major industrial one Chapter 6 continued: being the United States. It actually consists of Dimensions and Units two related systems—the U.S. Customary System of units, used in the United States and dependencies, and the British Imperial System. Units of Weight The pound (lb) is the basic unit of weight (which is proportional to mass) (how?). Within the English units of measurement there are three different systems of weights. In the avoirdupois system, the most widely used of Question: the three, the pound is divided into 16 ounces (oz) and the ounce into 16 drams. The ton, used Is there a problem? to measure large masses, is equal to 2,000 lb (short ton) or 2,240 lb (long ton). In Great Britain the stone, equal to 14 lb, is also used. “weight is proportional to mass.” Answer: You need to be aware of Force = Mass* Acceleration the law governing that proportionality: Newton’s Law Acceleration is NOT a constant, mass is. Force = Mass* Acceleration Even on earth, g = 9.81 m/s2 is NOT constant, but varies with latitude and Question: elevation. Is there a problem? “weight is proportional to mass.” “weight is proportional to mass.” 1 Another problem arises from the A mass is NEVER a “weight”. common intermingling of the terms “mass” and “weight”, as in: Force = Mass* Acceleration “How much does a pound of mass weigh?” Or: “If you don’t know whether it’s pound- “Weight” = Force mass or pound-weight, simply say pounds.” “Weight” = Force Forces in SI Units …because on earth all masses are 2 exposed to gravity. So 1N = 1 kg* 1m/s Weight =m * g When SI units are used, the factor gc The notion of ‘weight’ is not very is 1. There is no need for gc in the useful in engineering, because SI system. many situations are not static. Forces in US Conversions Customary Units Example: convert meters to miles. Conversion factor: 1 mile = 1609 m Force is a Basic unit 3200m =? miles Mass becomes a derived unit: 1 lb-mass (lbm) Answer: 3200m = 3200m* 1mi/1609m 1lbf = 1lbm*g or 1 lbm = 1lbf/g = 3200/1609 mi = 1.989 miles Where g is approx. 32.2 ft/s2 2 Conversions Conversions In class exercise: convert feet to miles. In class exercise (speed): convert miles/hour Conversion factor: 1 mile = 5280 ft to feet/second. 9800 ft =? miles Conversion factor: 1 mile = 5280 ft 45 mph = ??? ft/s In class exercise (speed): convert miles/hour In class exercise (speed): convert miles/hour to feet/second. to feet/second. Conversion factor: 1 mile = 5280 ft Conversion factor: 1 mile = 5280 ft 45 mph = ??? ft/s 45 mph = ??? ft/s Step by step procedure: Step 2: We can multiply by 45mi 5280 ft 45mi Thus: 45mph = * 45mph = 5280 ft 1hr 1mi 1 = 1hr 1mi In class exercise (speed): convert miles/hour In class exercise (speed): convert miles/hour to feet/second. to feet/second. Conversion factor: 1 mile = 5280 ft Conversion factor: 1 mile = 5280 ft 45 mph = ??? ft/s 45 mph = ??? ft/s Step 3: Step 4: We repeat the process for the time: 45mi 5280 ft 1hr 45mi 5280 ft We get: 45mph = * 45mph = * * 1hr 1mi 1hr 1mi 3600s 1hr Multiply by 1 = We can now simplify the fractions. 3600s The units mi and hr cancel. 3 In class exercise (speed): convert miles/hour to feet/second. 45 mph = ??? ft/s Conversions Pressure = Force per unit area Step 5: 5280 ft ft In class: We get: 45mph = 45* = 66 3600s s Convert 35,000 Pa to psi, using basic units such as feet and pounds-force. Plausibility check: The resulting units must be distance/time. The resulting numbers must be consistent with the original question. In class: In class: Convert 35,000 Pa to psi, using basic Convert 35,000 Pa to psi, using basic units such as feet and pounds-force. units such as feet and pounds-force. Definitions: Solution: 1N 1lbf 35000N 0.2248lbf 1m2 1Pa = 1psi = 35000Pa = 2 * * 2 2 m2 in2 m 1N 39.37 in In class: Convert 35,000 Pa to psi, using basic Some measurements involve units such as feet and pounds-force. multiple units and conversions Solution: Example: Flow speed measurement with a Pitot tube 0.2248lbf Thus: 35000Pa = 35000* = 5.076psi 39.372 in2 4 Pitot tube Pitot tube According to Bernoulli’s law, Total pressure = Static pressure + Dynamic pressure (from flow speed.) The Bernoulli equation states: Pitot tube We can detect the Dynamic pressure by measuring the difference between static and dynamic pressures (e.g. height hm of the water column at right) Pitot tube Pitot tube characteristic Air velocity (ft/minute) Flow speed v is found from the dynamic pressure P reading: 2 dyn P = 1 ρ( ) dyn 2 v 2 P or v = dyn ρ Dyn. Pressure (inches of H2O) 5 SI 2 P SI 2 P v = dyn v = dyn ρ ρ Inserting: Pdyn = 500 Pa 3 In class: Pitot tube measurement analysis ρAir = 1.2 kg/m Given: Pdyn = 500 Pa gives: 3 5 3 ρAir = 1.2 kg/m at 1.01*10 Pa 2 * 500 N m v = 2 Determine the air velocity in m/s 1 . 2 m kg US Cust. SI 2 * 500 N m 3 SI 2 P * g v = dyn c 2Pdyn * gc 2 v = v = 1 . 2 m kg ρ ρ We substitute : 1N with 1 kgm/s2 2 * 500 N m 3 1 kgm In class: Pitot tube measurement analysis v = 2 2 2 1 . 2 m kg 1 N * s Given: Pdyn = 19 lbf/ft 3 Simplification gives: ρAir = 0.0735 lbm/ft at 14.7 psia m 2 Determine the air velocity in ft/s v = 833 = 28 .9 m s 2 s US Customary example 2*19lbf 32.2 lbm * ft ft3 2Pdyn * gc v = v = ft2 1 lbf * s2 0.0735lbm ρ 2 Inserting: Pdyn = 19 lbf/ft Simplification gives: 3 ρAir = 0.0735 lbm/ft ft2 ft 2*19lbf 32.2 lbm * ft ft3 v = 16648 = 129 v = s2 s ft2 1 lbf * s2 0.0735lbm 6 Chapter 8 Example: Statistics of Gambling Statistics Who wins? The player or the casino? Example: Statistics of Gambling Example: Statistics of Gambling We can believe the warm and Or we can do the cold math. fuzzy testimonials: An example follows on the next series of “Bob”: “I won $134,567 at xyz slides. Palace” We’ll discuss the math concepts afterwards. Source: www-ec.njit.edu/~grow/ roulette/sld001.htm Q: What kind of distribution will result? 7 8 Mean µ: Average Value: 1 Mean = µ = ∑ xi N N ( x − x ) σ 2 = i ∑ n − 1 σ 2 Standard Dev σ: 67% of events are within µ+/-σ 9 The preceding analysis did not consider the fact that the casino limits the player’s capital. All casino tables carry a sign e.g. “minimum bet $5 maximum bet $500” Without the sign, a player The sign prevents the player could recover from losing from recovery at the point of streaks by doubling each losing maximum profit to the casino. bet, to lose only the predicted The average table game 5%. winnings at Nevada casinos are 15% of wagered amounts. 10 20 Statistical Analysis Example Playing heads or tails. We toss 20 coins each time, and count the numbers of heads. 1000 plays show the following results: We can plot the number of heads counted in each of the 20 classes vs. the frequency of occurrence. This plot is called a histogram. Histogram The histograms of random distributions exhibit the familiar Gaussian “bell curve” shape. Many distributions are “biased” by deterministic factors. Statisticians can detect such biases and point to their causes. Some examples follow. 11.
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