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Namas ChandraIntroduction to Mechanical engineering Chapter 8-1
EML 3004C
CHAPTER 8
Dimensions, Units, and Error
Namas ChandraIntroduction to Mechanical engineering Chapter 8-2
EML 3004C
Objectives
Understand the difference between a unit and dimensionsConvert quantities between the different systems of measurementUnderstand the types of error in a measurement Express large numbers in scientific notation
Sections
8.1 Introduction8.2 Dimensions and Units8.3 Conversion between Different Units8.4 Errors and Accuracy8.5 Significant Digits8.6 Summary
Namas ChandraIntroduction to Mechanical engineering Chapter 8-3
EML 3004C
Basic Units and Dimensions
Unit is the basic quantity in a given system of measurement yard, meter, inch
Some units can be a combination of different basic units Example: Unit of force depends on the choice of the units
of mass and acceleration Dimension describes the quantity in terms of Mass, Length, Time
and Temperature without specifying any unit.Example:
-22
-2
La LT
T
F m a MLT
Namas ChandraIntroduction to Mechanical engineering Chapter 8-4
EML 3004C
Basic Units and Dimensions…2
One of the fundamental tenet for any physically valid equation is that dimensions on either side of the equation should be identical.Now F=ma=kx. Now to get the units of stiffness k,
-2
-2MLTk = = MT
L
In fact dimensional analysis is used to check any proposed equation is valid or not.
Groups of quantities can be combined in a manner that the group becomes dimensionless and they oftentimes represent very important fundamental quantities.
Namas ChandraIntroduction to Mechanical engineering Chapter 8-5
EML 3004C
8.2 Systems and Measurement
There are three major systems of measurement They are
BGS (British Gravitational System)EES (English Engineering System)SI System (The current worldwide standard)
BGS Here force (not mass), feet (for length) and second (time) are the
fundamental units.
Since, ,mass is derived unit. 2 -1FM FT L
a
Namas ChandraIntroduction to Mechanical engineering Chapter 8-6
EML 3004C
Systems of Measurement…2
The unit of mass is a slug
Since weight is a force acting on mass of 1 slug
A body with a mass of 1 slug has a weight of 32.174 lb.
2lb sec1 slug = 1
ft
2
ftW = 1 slug 32 = 32.174 lb
sec
Namas ChandraIntroduction to Mechanical engineering Chapter 8-7
EML 3004C
Systems and Measurement…3
EES Here the unit of mass is defined such that weight is 1 lbf Mass is 1 lbm
For this system use
SI SI is the metric system. Fundamental units are grams (weight),
cm ( length), and seconds (time).
2
ft1 lbf = 1lbm 32.174
sec
c 2c
ma ftF = with g = 32.174
g sec
Namas ChandraIntroduction to Mechanical engineering Chapter 8-8
EML 3004C
Conversion between Units
We are faced with converting a quantity from one system to another. Use the logic that units should cancel on both sidesExample:
6
3
3
mile m m30 30 0.447 13.411
hour sec sec
m 10 m 1sec13.411
sec 1m 10 millisec
m13.411 10
millisec
mile mile 5280 ft 0.3048 m 1 hour1 1
hour hour 1 mile 1 ft 3600 sec
m0.447
sec
Conversion of FPS SI
Force 1 lb = 4.4482 NMass 1 slug = 14.5938 kgLength 1 ft = 0.3048 m
Namas ChandraIntroduction to Mechanical engineering Chapter 8-9
EML 3004C
Comparison of Units in BGS, EES, and SI
Quantity BGS EE SIMass Slug lbm kilogram
Length ft ft meter (m)
Time sec (s) sec (s) sec (s)Area
Velocity Ft/s Ft/s M/sAceleration
Density Force Lbf Lbf Newton(N)
Pressure Pascal (Pa)
Energy Ft x Lbf Ft x Lbf Joule
Volume
Power Ft x Lbf/s Ft x Lbf/s Watt (W)
2ft2m
2ft/s 2m/s3Slug/ft 3lbm/ft 3Kg/m
2Lbf/ft
3m3ft
2ft
2ft/s
2Lbf/ft
3ft
Namas ChandraIntroduction to Mechanical engineering Chapter 8-10
EML 3004C
Order of 10 in metric system
Mult. Factor Prefix SI Symbol tera T giga G mega M kilo k
hecto h deka da deci d centi c milli m micro nano n pico p
femto f atto a
12
9
6
3
2
1
1
2
3
6
9
12
15
10
10
10
10
10
10
10
10
10
10
10
10
10
μ
1810
Namas ChandraIntroduction to Mechanical engineering Chapter 8-11
EML 3004C
Errors and Accuracy
Every quantity is measure in some form or another Accuracy is defined as the difference between true value and
measured value All measurements contain error Error has two parts
Bias error-inherent in the measurement systems. E.g. graduation in a tape
Precision error- Error that arises due to lack of repeatability. Arises due to operator or environmental conditions