Lecture 2 Engineering Calculations I and HW _1-1

8/8/2019 Lecture 2 Engineering Calculations I and HW _1-1

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Lecture # 2

Engineering Calculations


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Engineering Calculations

Units and Dimensions

Force and Weight

Numerical Calculations and Estimation Data Representation and Analysis


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Systems of Units

SI system: mass (kg), length (m), time (s).

CGS system: mass (g), length (cm), time (s).

American engineering system:mass (lbm), length (ft), time (s).


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Conversion of Units:


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Dimensional Homogeneity

When the units are the same, values can be added, subtracted, multiplied or

divided. If the units are different, values may be multiplied or divided only

(never added or subtracted), for example:

Same units:

4 m 1 m = 3 m correct

2m x 1m = 2 m2 correct

2g / 1g = 2 (dimensionless) correct

Different units:

5.0 km / 2 h = 2.5 km/h correct

10.0 kg/s / 0.2 kg/m3 = 50.0 m3/s correct

3 cm + 1mm = ? Incorrect


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Class question:

What are the units of the constant 9 in the valid

equation below if B is in kg?

D (ft) = 3t(s) + 4 + 9B2


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Class question:

What are the units of the constant 9 in the valid

equation below if B is in kg?

D (ft) = 3t(s) + 4 + 9B2

Answer : ft / kg2


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Force and Weight


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Force and Weight

Remember:

F = M (a/gc) W = M (g/g

c)

F and W have units of force e.g. Newton,

dyne, Ibf, etc


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Significant figures and Scientific notation


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Determination of the number of significant figures

Counting from the left to the right, starting with the first nonzero digit:

- to the last non zero digit if there is no decimal point.

- to the last digit if there is a decimal point

Examples: 12,200 has 3 s.f. while 0.003040 has 4 s.f.

Numbers may be expressed in decimal or scientific notation:

123,000,000 is 1.23x108 or 0.123x109 ( note the number

multiplied by the power of ten is typically between 0.1 and 10).

If the number is counted (i.e. not measured or calculated) its

value implicitly contains an infinite number of significantfigures, e.g. 5 cows means 5.00000. cows, and 60 min in one

hour means 60.0000000..min in one hour,etc) This is called a

pure integer.


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Rules for mathematical operations

(2) In addition and subtraction, the accuracy with which you quote ananswer does notdepend directly on the number ofS.F. in the inputnumbers as above. Instead, the result must not have more decimalplaces than the component with the fewest number of decimalplaces, regardless of the number of significant figures. Here aresome examples.

2.34 2.34 2.34 2 2.341234

+0.18 +2.8 -2.32 +8.43 -2.0

-------- ------- -------- -------- --------------2.52 5.1 0.02 10 0.3

Note that the answer can have fewer, more, or the same number ofS.F.as the various input numbers in the problem!


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Rules for mathematical operations

(3) In operations involving both addition (or subtraction) and

multiplication (or division), both rules apply and the final answer

must not be more accurate than the least accurate number in the

whole operation.

(4) In conversion problems, the actual conversion factor is not to be

considered in the rules for rounding off since the reason for

expressing numbers to a certain number of significant figures is

based on accuracy of an observed measurement.


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Rules for rounding off numbers

(1) If the digit to be dropped is greater than 5, the last

retained digit is increased by one. For example,

12.6 is rounded to 13

(2) If the digit to be dropped is less than 5, the last remaining

digit is left as it is. For example,


(3) If the digit to be dropped is 5, the last remaining digit is

increased by one if it is odd, but left as it is if even. For example,


12.5 is rounded 12 (not 13 !)

This rule means that if the digit to be dropped is 5, the result isalways rounded to the even digit.


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General guidelines for using calculators

In a long calculation involving mixed operations, carry as many digits

as possible through the entire set of calculations and then round the

final result appropriately. For example,

(5.00 /1.235) + 3.000 + (6.35 / 4.0)

= 4.04858 ... + 3.000 + 1.5875

= 8.630829... The accuracy of this result can not be more than theaccuracy of the least accurate number in the calculation, which is 4.0

(one decimal), hence we round the final answer to one decimal.

Answer = 8.6


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Class Exercises

37.76 + 3.907 + 226.4

319.15 32.614

125 0.23 + 4.109

2.02 x 2.5


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Class Exercises

37.76 + 3.907 + 226.4 = 268.1= 268.067 > no more than one decimal

319.15 32.614 = 286.54= 286.536 > no more than 2 decimals

125 0.23 + 4.109 = 129

= 128.879 > no decimals

2.02 x 2.5 = 5.0= 5.05 > no more than 2 s.f.., round to 5.0 ,

last digit 0 is even


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Class Exercises

600.0 / 5.2302

(5.5)3

0.556 x (40 32.5)

45 x 3.00


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Class Exercises

600.0 / 5.2302 = 114.7= 114.71836 > no more than 4 s.f.

(5.5)3 = 170= 166.375 > no more than 2 s.f.

0.556 x (40 32.5) = 4= 4.17 > no decimal

45 x 3.00 = 140= 135 > no more than 2 s.f.


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Validating

Results


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Home Work # 1


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Home Work # 1

Wednesday, August 25, 2010

Problems:

2.1 page 31

2.2 page 31

2.3 page 31

2.6 page 312.7 page 32

2.8 page 32

2.9 page 32

2.10 page 32

2.16 page 32


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Solutions


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Documents

Lecture 2 Engineering Calculations I and HW _1-1