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Units of Measure

Units of Measure.

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Units of Measure

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  1. Units of Measure Samuel Clemens obtained his pseudonym, Mark Twain, from a Mississippi River phrase. A weighted rope would be lowered to measure the depth of the river. The rope was marked in fathoms, and the marks would be called out. Mark Twain meant you had passed the second mark, or were two fathoms deep. How deep was the river at “Mark Twain?”

  2. Fundamental Units • Length • Mass • Time • Electric Current • Temperature • Amount of substance • Luminous intensity Supplemental Units • Plane Angle • Solid angle

  3. Derived Units • Velocity • Acceleration • Time • Frequency • Pressure • Force • Moment (torque) • Energy • Mass Density

  4. Units: Length What is the length of this line? Units we use are arbitrary. Key is that we agree on a unit so that we all know what we are talking about.

  5. US Customary System (USC) • Based on things that made sense to people • previously known as English (or British) • 1 inch = 3 dry, round, barleycorns end-to-end • foot = length of King Edward I’s foot • mile = 1000 double paces of Roman soldier • 12 in/ft; 4 in/hand; 3 ft/yd; 5280 ft/mile

  6. Systeme Internationale (SI) • Commonly called metric system, although different • attempted to be less arbitrary • 1 meter • original: one ten-millionth of the distance from the equator to either pole • current: based on wavelength of light Conversion between systems: Exact: 25.4 mm / 1 inch Approximate: 1 mile / 1.6 kilometers

  7. Prefixes Only prefixes with powers of three are officially part of SI system. We will use centimeter, as it is the same order of magnitude as an inch. What is 101 cards? How about a 10-6 scope? Or a 10-2 pede?

  8. Example: Length conversion How many yards are there in 10 km? How many fathoms in a furlong?

  9. Units: Area Area often described as a length-squared, but applies to any shaped area. What is the area of the teal letters in the engage logo? engage • SI • square meter (m2) • hectare (ha) = 104 m2 • USC • square inch, square foot • acre: land a team of oxen could plow in a day • acre is 40 x 4 rods, or 43560 ft2

  10. Example: Area conversion What is the area (in in2) of a 1m x 2m area? How many 1 inch x 1 inch squares fit into a 1 m x 2 m area (get area in terms of in2)?

  11. Units: Volume Volume often described as a length-cubed, but applies to any shaped volume. We are really talking about the number of unit cubes that fit into something. What is the volume of the potato? • SI • Liter; 1 L = (10 cm)3 = 1000 cm3 • USC • Gallon; 1 gallon = 231 in3

  12. Example: Volume conversion What is the volume of a 2m x 2m x 1m box in ft3? What is the volume of a 0.2m x 0.2m x 0.1m box in ft3?

  13. Units: Angles • Most common unit: degree • Why are there 360 degrees in a circle? • convenience • divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18 • Degrees are sub-divided into 60 minutes • minute subdivided into 60 seconds. • Notation 43°1837 = 43 degrees, 18 minutes, 37 seconds

  14. r s θ Units: Angles – Engineering • The unit of radians often used to measure angles • draw angle at the center of a circle • angle defines an arc on the circle’s circumference • ratio of arc length to circle radius is angle in radians • dimensionless • How many radians in 30°?

  15. Measurements Measure width of block using three different instruments. Child’s ruler Metal ruler Caliper How do we communicate the difference between the different measurements? SIGNIFICANT DIGITS Which measurement is better?

  16. Accuracy and Precision Accuracy – measure of the nearness of a value to the correct value Precision – repeatability of the measurement

  17. Estimation • What is the volume of Estabrook 111 in cubic meters? • 30 second – use it to get the right order of magnitude • 5 minute – use quick measurements • 30 minute – use more accurate measurements, may need a more detailed problem definition • Use units that are readily available • Don’t worry about details – this is an estimate • Include an estimate of the accuracy of your estimate • Height Estimation • Use ratios • Arm-length / ruler • Shadows

  18. Significant Digits Significant digits: communicate level of uncertainty. If we say the length of a line is 12.7”, what does that mean? Length between 12.65” and 12.75” How is it different if we say the length of a line is 12.70”? Length between 12.695” and 12.705” The numbers you put down communicate the level of precision, or how sure you are of the number.

  19. Significant Digits: Summary • Most engineering data are assumed to have 3 significant digits • Unless told otherwise, give answers to 3 significant digits • Carry 4 to 5 significant digits in calculations • Be realistic about the number of significant digits you report • How precisely do you know the reported value? • Full credit on exams requires appropriate use of significant figures

  20. Reasonableness • Think about what your answer means • Take five seconds to think about every answer • Use estimation and simplification of the problem to get an order-of-magnitude estimate

  21. EF 101 Final Example • A passenger jet is taking off from Hartsfield International airport in Atlanta. Assuming it starts from rest, and accelerates at a constant rate of 2.6 m/s2, how long does it take to cover the 3.8 km runway? • A) 1.5 s B) 38 s C) 54 s • D) 99 s E) 140 s F) 1460 s Which two answers are unreasonable?

  22. Define the problem Identify the critical data of the problem. Do not be misled by data that is extraneous, erroneous, or insignificant. Diagram A diagram or schematic of the system being analyzed is often very helpful, and may be required. Governing equations Determine what type of problem is being solved. Recognize when certain equations apply and when they do not apply. The governing equations should be written out in symbolic form before substituting in numerical quantities. Calculations Carry out your calculations only after you have completed the first three steps. Check to make sure units are consistent. Solution check Make sure you solved the problem that was posed. If possible, use an independent method or equation to check your result. Check to see that your solution is physically reasonable. Make sure both the magnitude and sign of the answer makes sense. Problem Solving

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