Transcript Dimensional Analysis PPT - Glasgow Independent Schools

MEASUREMENT
Unit Conversions
I
II
III
Are Units important?
3
Are Units important?
"The 'root cause' of the loss of the spacecraft was the
failed translation of English units into metric units in a
segment of ground-based, navigation-related mission
software, as NASA has previously announced," said
Arthur Stephenson, chairman of the Mars Climate Orbiter
Mission Failure Investigation Board. "The failure review
board has identified other significant factors that allowed
this error to be born, and then let it linger and propagate
to the point where it resulted in a major error in our
understanding of the spacecraft's path as it approached
Mars."
http://mars.jpl.nasa.gov/msp98/orbiter/
4
International System of Units
(SI)
Fundamental
Dimensions:
Derived Dimensions:
Length = m
Force = N (newton) = kg*m/s2
Mass = kg
Energy = J (joule) = N*m
Time = s
Power = W (watt) = J/s
SI prefixes listed in table need to have prefixes with a
memorized for exams!!!
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U. S. Customary System
(USCS)
Fundamental
Dimensions:
Length = ft
Force = lbf
Time = s
Derived Dimensions:
Mass = slug = lbf*s2/ft
= 32.174 lbm
Energy = ft*lbf
Power = ft*lbf/s
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A. SI Prefix Conversions
1. Find the difference between the
exponents of the two prefixes.
2. Move the decimal that many places.
To the left
or right?
A. SI Prefix Conversions
move right
move left
Prefix
mega-
Symbol
M
Factor
106
kilo-
k
103
BASE UNIT
---
100
deci-
d
10-1
centi-
c
10-2
milli-
m
10-3
micro-

10-6
nano-
n
10-9
pico-
p
10-12
A. SI Prefix Conversions
1) 20 cm =
0.2
______________
m
32
2) 0.032 L = ______________
mL
3) 45 m =
45,000
______________
nm
0.0805
4) 805 dm = ______________
km
A. SI Prefix Conversions
532 m
NUMBER
UNIT
0.532 km
= _______
=
NUMBER
UNIT
B. Dimensional Analysis
 The “Factor-Label” Method
 Units, or “labels” are canceled, or
“factored” out
cm 
3
g
cm
3
 g
Converting units
 Factor label method
• Regardless of conversion, keeping
track of units makes things come out
right
• Must use conversion factors
 - The relationship between two units
• Canceling out units is a way of
checking that your calculation is set up
right!
B. Dimensional Analysis
 Steps:
1. Identify starting & ending units.
2. Line up conversion factors so units
cancel.
3. Multiply all top numbers & divide by
each bottom number.
4. Check units & answer.
Common conversion factors
 English
 1 gallon = 4 quarts
 1 mile = 5280 feet
 1 ton = 2000 pounds
 Common English to Metric

1 liter = 1.057 quarts





1 kilogram = 2.2 pounds
1 meter = 1.094 yards
1 inch = 2.54 cm
Factor
4 qt/gal or 1gal/4 qt
5280 ft/mile or 1 mile/5280 ft
2000 lb/ton or 1 ton/2000 lb
1.057 qt/L or 1 L/1.057 qt
or 0.946 L/qt
2.2 lb/kg or 1 kg/2.2 lb
or 0.454 kg/lb
1.094 yd/m or 1m/1.094 yd
or 0.917m/yd
2.54 cm/inch or 1 in/2.54 cm
B. Dimensional Analysis
 Lining up conversion factors:
1 in = 2.54 cm
=1
2.54 cm 2.54 cm
1 in = 2.54 cm
1=
1 in
1 in
Line Mole Method

Process to convert from one unit to
another
 Example: Convert 3.00 m to inch:
? = 3.00 m 100 cm 1 in
1m
2.54 cm
ANSWER = 118 in
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Line Mole Method

Process to convert from one unit
to another
 Example: Convert 3.00 m/s to m/hr:
? = 3.00 m
s
60 s
min
60 min
hr
ANSWER = 10,800 m/hr
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Example Metric conversion
How many milligrams are in a kilogram?
1kg  1000 g
1 g  1000 mg
1kg 
1000 g
1kg

1000 mg
g
 1, 000 , 000 mg
B. Dimensional Analysis
 How many milliliters are in 1.00 quart of
milk?
qt
mL
1.00 qt

1L
1000 mL
1.057 qt
1L
= 946 mL
B. Dimensional Analysis
 You have 1.5 pounds of gold. Find its
volume in cm3 if the density of gold is
19.3 g/cm3.
cm3
lb
1.5 lb 1 kg 1000 g 1 cm3
2.2 lb
1 kg
19.3 g
= 35 cm3
B. Dimensional Analysis
 How many liters of water would fill a
container that measures 75.0 in3?
in3
L
75.0 in3 (2.54 cm)3
(1 in)3
1L
1000 cm3
= 1.23 L
B. Dimensional Analysis
5) Your European hairdresser wants to cut
your hair 8.0 cm shorter. How many
inches will he be cutting off?
cm
in
8.0 cm 1 in
2.54 cm
= 3.1 in
B. Dimensional Analysis
6) Taft football needs 550 cm for a 1st
down. How many yards is this?
cm
550 cm
yd
1 in
1 ft 1 yd
2.54 cm 12 in 3 ft
= 6.0 yd
B. Dimensional Analysis
7) A piece of wire is 1.3 m long. How many
1.5 cm pieces can be cut from this wire?
m
pieces
1.3 m 100 cm
1m
1 piece
1.5 cm
= 86 pieces
As an Individual
(on a piece of paper)...
Use the Factor Label Method to convert
the following:
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1)
34.6 m/s to mph
2)
15 ft3 to Liters
3)
Can you use FLM when converting from Celsius to
Kelvin? Why or why not?
Units in Equations
Addition/Subtraction: All terms must
have the same units
Multiplication/Division: Units may
cancel, like variables
Homogeneity: Units on both sides of the
"=" sign must be the same
Transcendental Functions: (sin x , ln x ,
ex ) the argument (x) does not have a
dimension
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Valid Units?
10 m
s
12 mile
hour
8
20 m
28 mile
hour
30 m 2
s
40 mile
hour
H:\classes\eng1101\fall05\lectures\spring05\en1.13b.sketch.amy.sxi
Converting Area and Volume
Caution: Make sure the units cancel
Area: 150 ft2 to yd2
150 ft2 1 yd 1 yd
3 ft 3 ft
150 ft2 (10)2 yd2
OR
(3)2 ft2
Volume: 12 ft3 to Liters
12 ft3
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(12)3 in3 (2.54)3 cm3 (1)3 m3
1000 L
(1)3 ft3 (1)3 in3
(100)3 cm3 1 m3
As an individual, solve...
Water Tower Problem
Problem Statement:
• Your home town is growing so rapidly that another
water tower is necessary to meet the needs of the
community. Civil and environmental engineers
predict that the water tower will need to hold 1.00 x
10.06 kilograms of water. The engineers also
estimate the density of the water to be 999
kilograms per cubic meter.
• If this tower is 50.0 meters high and spherical,
what volume (gal) of water will the tower hold
and what will the diameter (ft) of the tower have
to be?
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Diagram:
 mass of water = 1.00 x 106 kg
 density of water = 999 kg/m3
 tower height = 50.0 m
 ? volume of water (L)
 ? diameter (ft)
Theory:
Volume of a sphere
diameter
Assumptions:
3
2r 2 3V
4
 tower is spherical
15
4
3
r3
www.algonquin.org/pw.htm
Solution:
6
volume of water = 1.00 x10 kg
Volume of a sphere
4
3
r3
1 m3
999 kg
1000 L
1 m3
= 1.00 x 106 L
diameter 2 r 23 3 V
4
6 L 0.035315 ft3
1.00
x
10
volume of water =
= 3.53 x 104 ft3
1L
3.53 x 104 ft3
2
3
3
diameter 2 r
40.7 ft
4
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http://en.wikipedia.org/wiki/Golf
Golf Ball Design
Minimum allowed diameter of a golf ball is 42.67mm
Maximum Mass = 45.93g
The surface usually has a pattern of 300-400 dimples
designed to improve the ball's aerodynamics.
The method of construction and materials greatly affect
the ball's playing characteristics such as distance,
trajectory, spin and feel.
Have a two-, three-, or four-layer design constructed
from various synthetic materials
Harder materials, such as Surlyn, usually result
in the ball's traveling longer distances,
Softer covers, such as Balata, tend to generate
higher spin, more "feel" and greater stopping
potential.
Golf balls are separated into three groups
depending on their construction: two-, three-,
or four-piece covers.
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http://en.wikipedia.org/wiki/Golf
Golf Ball Design
Minimum allowed diameter of a golf ball is
42.67mm
Assuming a golf ball has a spherical shape
What is the golf ball diameter in inches?
What is the volume of a golf ball in cubic centimeters
and cubic inches?
Maximum Mass = 45.93g
What is the mass of a golf ball in pounds?
What is the density of a golf ball in g/cm3 and
lb/in3?
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Golf Shafts
Golf shafts are what connects the grip to the golf
head
The profile of the golf shaft is circular in shape and is
usually thicker at the grip end than at the club head
end.
Any strong and light material may be used to make the
golf shaft.
Almost all shafts today are made of either graphite or
tempered steel
The shaft is a tapered tube made of metal (usually
steel), or graphite fiber. The shaft is roughly 1/2
inch in diameter (12 mm) near the grip and
between 35 to 45 inches (89-115 cm) in length.
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Golf Shafts
Almost all shafts today are made of either
graphite or tempered steel
Graphite: 2.09-2.23 g/cm3
Steel: 7,861.093 kg/m³ (0.284 lb/in³)
How much would the shaft of a golf club weigh in
pounds if it were constructed from graphite or
steel?
Assume:
Shaft Diameter = 1/2 inch and solid
Shaft Length = 40 inches
Why would you choose a graphite club over a
steel club or vice versa?
What is tempered steel?
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