Transcript Dimensions, Units, and Error

EML 3004C
CHAPTER 8
Dimensions, Units, and Error
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-1
EML 3004C
Objectives
Understand the difference between a unit and dimensions
Convert quantities between the different systems of measurement
Understand the types of error in a measurement
Express large numbers in scientific notation
Sections
8.1
8.2
8.3
8.4
8.5
8.6
Introduction
Dimensions and Units
Conversion between Different Units
Errors and Accuracy
Significant Digits
Summary
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-2
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:
 L 
 a    2   L T -2
T 
 F    m  a   M L T -2
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-3
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,
 M LT -2 
-2
k = 
 =  M T 
 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 Chandra
Introduction to Mechanical engineering
Chapter 8-4
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.
F
2 -1
M


FT
L


 Since,
a 
 
Namas Chandra
Introduction to Mechanical engineering
,mass is derived unit.
Chapter 8-5
EML 3004C
Systems of Measurement…2
 The unit of mass is a slug
1 slug = 1
lb  sec
2
ft
 Since weight is a force acting on mass of 1 slug
W = 1 slug  32
ft
sec
2
= 32.174 lb
A body with a mass of 1 slug has a weight of 32.174 lb.
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-6
EML 3004C
Systems and Measurement…3
EES
 Here the unit of mass is defined such that weight is 1 lbf
 Mass is 1 lbm
1 lbf = 1lbm  32.174
 For this system use
F=
SI
ma
gc
ft
sec
w ith g c = 32.174
2
ft
sec
2
 SI is the metric system. Fundamental units are grams (weight),
cm ( length), and seconds (time).
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-7
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 sides
Example:
1
m ile
1
hour
m ile  5280 ft

hour  1 m ile
 0.447
  0.3048 m   1 hour 
 
 

1
ft
 
  3600 sec 
m
sec
30
m ile
m
 30  0.447
hour
 13.411
sec
sec
 10  m
 13.411

sec  1m
6
m
 13.411  10
3
m
m
 
1sec





3
10
m
illisec



Conversion of
FPS
Force
1 lb
Mass
1 slug
Length
1 ft
=
=
=
SI
4.4482 N
14.5938 kg
0.3048 m
m illisec
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-8
EML 3004C
Comparison of Units in BGS, EES, and SI
Quantity
Mass
Length
Time
Area
Velocity
Aceleration
Density
Force
Pressure
BGS
Slug
ft
sec (s)
EE
lbm
ft
sec (s)
ft
2
ft
Ft/s
2
Ft/s
ft/s
2
ft/s
Slug/ft
3
Lbf
Lbf/ft
Energy
Ft x Lbf
Ft x Lbf
Volume
ft
Power
Ft x Lbf/s
3
2
lbm /ft
Lbf
2
Lbf/ft
Namas Chandra
Introduction to Mechanical engineering
SI
kilogram
meter (m)
sec (s)
2
m
M/s
ft
2
3
2
3
Ft x Lbf/s
m /s3
K g/m
Newton(N)
Pascal (Pa)
Joule
m
3
Watt (W)
Chapter 8-9
EML 3004C
Order of 10 in metric system
Mult. Factor
10
12
10
9
10
6
10
3
10
2
10
1
10
10
10
10
10
10
1
2
3
6
9
12
15
10
 18
10
Namas Chandra
Introduction to Mechanical engineering
Prefix
tera
giga
mega
kilo
hecto
deka
deci
centi
milli
micro
nano
pico
femto
atto
SI Symbol
T
G
M
k
h
da
d
c
m
μ
n
p
f
a
Chapter 8-10
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
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-11
EML 3004C
CHAPTER 8..concludes
Dimensions, Units, and Error
Namas Chandra
Introduction to Mechanical engineering
Chapter 8-12