Unit Cancellation Unit Cancellation Units follow the same division and multiplication rules that numbers do ? Calculations can always be checked
using unit cancellation Fermi Question An exercise in unit cancellation 1.How many hairs are there on a dog? 2.What is the mass of all the worlds fingernail clippings from today? 3.How much has the mass of the human populaton of
Earth changed in the last year? 4.If you life earnings were given out to you by the hour (throughout your life), how much would you receive per hour? 5.If all the people in the world were crowded together, how much area would they cover? 6.How much waste is produced in Canada every year?
What is the unit for k? = 1 2
2 Unit conversion is a special case of unit cancellation You are responsible for knowing: k, M, G, T m, , n, p
Measurements Every measurement is wrong! To a certain degree Examples: Your Birthday Height
Measurement Every measurement has a degree of uncertainty The uncertainty is based on: The measuring device
The skill of the measurer Ways to indicate How Accurate a Number Is 1) Tolerance Intervals 2) Percent Error 3) Significant Digits
Tolerance Intervals Tolerance is the greatest range of variation that can be allowed 25 0.5 would be between 24.51 to 25.49 25 0.05 would be between
24.951 to 25.049 25 0.005 would be between 24.9951 to 25.0049 Tolerance Intervals Used on blueprints and in machining to indicate accuracy
Activity: Tolerance a Part Design a square piece with a square hole through its centre. Key Dimensions: How must the length of the piece be toleranced How big must the square hole through the piece be to accommodate the square peg
Criteria: The square piece must sit inside a container that was toleranced 20m lm on each side The square peg is dimensioned 2m 0.1m Percent Error Indicates a percentage that the value may be out by
Sample Problem You buy an 8 foot 2x4. You measure the board to be 94.5 inches long. What is the percentage error for the length of the board? 12 inches = 1 foot
Significant Digits The accuracy is determined by the right most digit 25 would be between 24.51 to 25.49 25.0 would be between 24.951 to 25.049 25.00 would be between 24.9951 to
25.0049 Significant Digits: Uses the number of digits in the number to express the uncertainty of the number 1 x 101 is less accurate than 12 12 is less accurate than 12.0
12.0 is less accurate than 12.00 Rules for Significant Digits All non-zero digits are significant All zeros contained between non-zero digits are significant i.e. 207 3sd
All trailing zeros to the right of a decimal are significant i.e. 3.000 4sd When are zeroes not significant? All leading zeros to the left of
the first non-zero digit are not significant i.e. 0.00058 2 sd Whole Numbers Exact numbers have infinite significant digits
How many cars do I have? 1 If there are no decimal places and zeroes to the left of the decimal, it is impossible to tell how many significant digits. Example 200, 2010 How accurately is it measured?
Examples a.) 125.9 b.) 230 c.) 0.00658 d.) 500.3 Examples:
a.) 125.9 - 4 s.d. b.) 230 - 3 s.d. c.) 0.00658 - 3 s.d. d.) 500.3 - 4 s.d. Defining Cylindricity - Extra http://www.youtube.com/watch? v=ZuECC8RMZ40
Interchangeable Parts Muskets were originally designed by expert gun smiths who made one gun at a time If it broke and needed servicing, the gun would have to be sent back to an expert
gunsmith Early US Attempts In 1801, Eli Whitney built ten guns from the same parts, disassembled all of them in front of congress and then reassembled them from a pile of parts The concept was great but each part was
still handmade by skilled craftsmen Changes that Allowed Interchangeability New machines (lathes, mills, etc) Development of jigs to control the path of the tool and fixtures to
hold the piece in place Blocks and gauges to determine the accuracy of the finished parts How to Improve Accuracy Improve the measuring instrument
The smaller the unit on the measuring device, the more precise the measurement Know how to measure Always look straight down on the measuring device
How to Improve Accuracy Repeat Measure several times to get a good average value Measure twice, cut once Measure under controlled conditions
Objects can shrink or expand based on measuring conditions Why wouldnt you make everything very, very accurate? Cost Waste of time Not needed
The more accurate a measurement must be, the more it costs to do so Bibliography Britton. "Geometric Dimensioning and Tolerancing." Lecture.
http://synthetica.eng.uci.edu/~mccarthy/ mechanicaldesign101/GDandT.pdf. 1 Feb. 2013 Operations with Significant Digits Adding and Subtracting
Use the lowest number of decimal places from the numbers given in the problem 2.35 + 7.669 = 10.019 According to the rule, we can only keep 2 decimal places = 10.02 12.875 8.3 =
20.876 5 = Multiplying and Dividing Use the lowest number of significant digits form the numbers given in the problem 3.4 x
2SD 2.35 = 7.99 3SD = 2 SD = 8.0 780 / 23 =
200.600 x 0.0012 = Scientific Notation Used to express numbers with the proper number of significant digits 1.32 x 1010 (3 s.d) Only one digit to the left of the decimal place
Only leave the number of sig digs you want in the numerical portion Finding the exponent Every time you move the decimal to the left, the exponent goes up 1 Every time you move the decimal to the right, the exponent goes down 1
3,567,000,000 (with 3 SD) 0.000,000,000,056,0 (with 3 SD) In this course, The number of significant digits in your answer should be the same as the lowest number of significant digits in the
question Ex: If each students has 0.50 cents and there are 100 students, how much money do they have? Math Rules to Know Divide by a term is the same as multiplying by the reciprocal
Simplify: /( Math Rules to Know Solve in two ways: FT + FM = 2 2FT + 5FM = 5
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