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Why units matter!

Flight 173 ran out of fuel in flight…. You may ask, “So, how does a jet run out of fuel at 26000 ft ???"

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Why units matter!

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  1. Flight 173 ran out of fuel in flight…. You may ask, “So, how does a jet run out of fuel at 26000 ft ???" 1.  A maintenance worker found that the fuel gauge did not work on ground inspection.  He incorrectly assured the pilot that the plane was certified to fly without a functioning fuel gauge if the crew checked the fuel tank levels. 2.  Crew members measured the 2 fuel tank levels at 62 cm and 64 cm.  This corresponded to 3758 L and 3924 L for a total of 7682 L according to the plane's manual. Why units matter! 3.  The ground crew knew that the flight required 22,300 kg of fuel.  The problem they faced was with 7,682 L of fuel on the plane, how many more liters were needed to total 22,300 kg of fuel? 4.  One crew member informed the other that the "conversion factor" (being the fuel density) was 1.77.  THE CRUCIAL FAULT BEING THAT NO ONE EVER INQUIRED ABOUT THE UNITS OF THE CONVERSION FACTOR.  So it was calculated that the plane needed an additional 4,917 L of fuel for the flight. Alas, that was too little.

  2. Lecture 2 Goals: (Highlights of Chaps. 1 & 2.1-2.4) • Conduct order of magnitude calculations, • Determine units, scales, significant digits (in discussion or on your own) • Distinguish between Position & Displacement • Define Velocity (Average and Instantaneous), Speed • Define Acceleration • Understand algebraically, through vectors, and graphically the relationships between position, velocity and acceleration • Perform Dimensional Analysis

  3. Lecture 2 Reading Assignment: • For next class: Finish reading Ch. 2, read Chapter 3 (Vectors)

  4. Length Distance Length (m) Radius of Visible Universe 1 x 1026 To Andromeda Galaxy 2 x 1022 To nearest star 4 x 1016 Earth to Sun 1.5 x 1011 Radius of Earth 6.4 x 106 Sears Tower 4.5 x 102 Football Field 1 x 102 Tall person 2 x 100 Thickness of paper 1 x 10-4 Wavelength of blue light 4 x 10-7 Diameter of hydrogen atom 1 x 10-10 Diameter of proton1 x 10-15 See http://micro.magnet.fsu.edu/primer/java/scienceopticsu

  5. Time IntervalTime (s) Age of Universe 5 x 1017 Age of Grand Canyon 3 x 1014 Avg age of college student 6.3 x 108 One year3.2 x 107 One hour3.6 x 103 Light travel from Earth to Moon 1.3 x 100 One cycle of guitar A string 2 x 10-3 One cycle of FM radio wave 6 x 10-8 One cycle of visible light 1 x 10-15 Time for light to cross a proton 1 x 10-24

  6. Mass Stuff Mass (kg) Visible universe ~ 1052 Milky Way galaxy 7 x 1041 Sun 2 x 1030 Earth 6 x 1024 Boeing 747 4 x 105 Car 1 x 103 Student 7 x 101 Dust particle 1 x 10-9 Bacterium 1 x 10-15 Proton 2 x 10-27 Electron 9 x 10-31 Neutrino <1 x 10-36

  7. Order of Magnitude Calculations / EstimatesQuestion: If you were to eat one french fry per second, estimate how many years would it take you to eat a linear chain of trans-fat free french fries, placed end to end, that reach from the Earth to the moon? • Need to know something from your experience: • Average length of french fry: 3 inches or 8 cm, 0.08 m • Earth to moon distance: 250,000 miles • In meters: 1.6 x 2.5 X 105 km = 4 X 108 m • 1 yr x 365 d/yr x 24 hr/d x 60 min/hr x 60 s/min = 3 x 107 sec

  8. Dimensional Analysis (reality check) • This is a very important tool to check your work • Provides a reality check (if dimensional analysis fails then there is no sense in putting in numbers) • Example When working a problem you get an expression for distance d = v t 2( velocity · time2 ) Quantity on left side d  L  length (also T  time and v  m/s  L / T) Quantity on right side = L / T x T2 = L x T • Left units and right units don’t match, so answer is nonsense

  9. Exercise 1Dimensional Analysis • The force (F) to keep an object moving in a circle can be described in terms of: • velocity (v, dimension L / T) of the object • mass (m, dimension M) • radius of the circle (R, dimension L) Which of the following formulas for Fcould be correct ? Note: Force has dimensions of ML/T2 orkg-m / s2 F = mvR (a) (c) (b)

  10. F = mvR Exercise 1Dimensional AnalysisWhich of the following formulas for Fcould be correct ? •  •  •  Note: Force has dimensions of ML/T2 Velocity (n, dimension L / T) Mass (m, dimension M) Radius of the circle (R, dimension L)

  11. Tracking changes in position: VECTORS • Position • Displacement (change in position) • Velocity (change in position with time) • Acceleration

  12. 10 meters Joe +x -x 10 meters O Motion in One-Dimension (Kinematics) Position / Displacement • Position is usually measured and referenced to an origin: • At time= 0 seconds Pat is 10 meters to the right of the lamp • origin = lamp • positive direction = to the right of the lamp • position vector :

  13. 10 meters 15 meters xi Pat Position / Displacement • One second later Pat is 15 meters to the right of the lamp • Displacement is just change in position • x = xf - xi Δx xf O xf = xi + x x = xf - xi = 5 metersto the right ! t = tf - ti= 1 second

  14. Average Speed & VelocityChanges in positionvsChanges in time • Average velocity = net distance covered per total time, includes BOTH magnitude and direction • Speed, s, is usually just the magnitude of velocity. • The “how fast” without the direction. • Average speed references the total distance travelled Active Figure 1 http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=282

  15. Representative examples of speed Speed (m/s) Speed of light 3x108 Electrons in a TV tube 107 Comets 106 Planet orbital speeds 105 Satellite orbital speeds 104 Mach 3 103 Car 101 Walking 1 Centipede 10-2 Motor proteins 10-6 Molecular diffusion in liquids 10-7

  16. x t Instantaneous velocityChanges in positionvsChanges in time • Instantaneous velocity, velocity at a given instant • If a position vs. time curve then just the slope at a point http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=230

  17. Exercise 2Average Velocity x (meters) 6 4 2 0 t (seconds) 1 2 3 4 -2 What is the magnitude of the average velocity over the first 4 seconds ? (A) -1 m/s (B) 4 m/s (C) 1 m/s (D) not enough information to decide.

  18. Average Velocity Exercise 3What is the average velocity in the last second (t = 3 to 4) ? • 2 m/s • 4 m/s • 1 m/s • 0 m/s x (meters) 6 4 2 -2 t (seconds) 1 2 3 4

  19. Exercise 4Instantaneous Velocity x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 What is the instantaneous velocity at the fourth second? (B) 0 m/s (A) 4 m/s (C) 1 m/s (D) not enough information to decide.

  20. x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 turning point Average Speed Exercise 5What is the average speed over the first 4 seconds ?0 m to -2 m to 0 m to 4 m  8 meters total • 2 m/s • 4 m/s • 1 m/s • 0 m/s

  21. x t vx t Key point: position  velocity • (1) If the position x(t)is known as a function of time, then we can find instantaneous or average velocity v • (2) “Area” under the v(t) curve yields the change in position • A special case: If the velocity is a constant, then x(Δt)=v Δt + x0

  22. “2D” Position, Displacement y time (sec) 1 2 3 4 5 6 position -2,2 -1,2 0,2 1,2 2,2 3,2 (x,y meters) x position vectors origin Continued on Monday….

  23. Assignment Recap • Reading for Monday’s class on 9/13 • Finish Chapter 2 & all of 3 (vectors)

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