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Dimensional Analysis
Day 1
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
Travel to Korea
US Dollar 1181.87 Won
A physical quantity must include:A physical quantity must include:
NumberNumber + Unit+ Unit
Calculation Corner: Unit ConversionCalculation Corner: Unit Conversion
1 dollar = 1181.87 won1 dollar = 1181.87 won
$1 $1
1181.87 won1181.87 won
1181.87 won1181.87 won
$1$1
Calculation Corner: Unit ConversionCalculation Corner: Unit Conversion
$1$1
1181.87 won1181.87 won
1181.87 won1181.87 won
$1$1
“Conversion factors”“Conversion factors”
3 dollar3 dollar1181.87 won1181.87 won
1 dollar1 dollar= 3545.51
won= 3545.51
won(( ))(( ))
A 10k run is 10 Kilometers. If 1 meter = 1.094 yards, use dimensional analysis to find the length of the race
in miles. ( hint: 1mi = 1760yds)
applicable conversion factors:
equality:
10 km
x
1 m = 1.094 yds
______1.904 yds 1 m
We use the idea of unit cancellation
to decide upon which one of the two
conversion factors we choose.
______ 1 km1000m
( )___ 1 km1000 m 6.2 mi) 1 m
1.094yds ( ( )______ 1 mi1760yds
______1 mi1760yds
A car travels a distance of 100feet in about 2.8 seconds. What is the velocity of the car in miles per hour?
applicable conversion factors:
100 feet
2.8sec
______60 sec 1 min
Again, the units must cancel.
______1 hr60 min
( )______ 24mph60 sec1 min
______5280 feet 1 mi
______1 hr60 min( )______ ( )____________
5280 feet 1 mi
How many feet is 39.37 inches?
applicable conversion factors:
equality:
or
X ft
39.37 in
=
1 ft = 12 in
______1 ft 12 in
Again, the units must cancel.
( )____ 3.28 ft1 ft12 in
______1 ft
12 in
Scott found a rare video game on an online auction site priced at 35 Australian dollars. If the exchange rate is $1 US = $1.24 in Australian, find the cost of the video game in US dollars. Round to the nearest cent.
$35 A
1 $28.22
US( )____ $1.24 A
$1 US
How many seconds is 4.38 days?
=
1 h60 min24 h
1 d 1 min60 s____( ) ( )____( )_____ 4.38 d
x
378,432 s
Try: Dimensional Analysis• How many milliliters are in 1.00 quart of milk?
1.00 qt 1 L
1.057 qt= 946 mL
qt mL
1000 mL
1 L
Try: Dimensional Analysis6) Taft football needs 550 cm for a 1st down.
How many yards is this?
550 cm 1 in
2.54 cm= 6.0 yd
cm yd
1 ft
12 in
1 yd
3 ft
Try: Dimensional Analysis5) Your European hairdresser wants to cut your hair
8.0 cm shorter. How many inches will he be cutting off?
8.0 cm 1 in
2.54 cm= 3.1 in
cm in
Simple Mathwith
Conversion FactorsDay 2
Measured dimensions of a rectangle:
Find area of rectangle.
A = L . W
= (9.70 cm)(4.25 cm)
length (L) = 9.70 cm
width (W) = 4.25 cm
L
W=
Example Problem
41.2 cm2
Convert 41.2 cm2 to m2.
100 cm1 m( )______
X m2 = 41.2 cm2
X m2 = 41.2 cm.cm
Recall that… 41.2 cm2 = 41.2 cm.cm
100 cm1 m( )______
X m2 = 41.2 cm2 = 0.412 m2
= 0.412 cm.m
WRONG!
( )______100 cm
1 m
= 0.00412 m2
( )______100 cm
1 m 2 = 0.00412 m2
Measured dimensions of a rectangular solid:
Find volume of solid.L
W
H
Length = 15.2 cm
Width = 3.7 cm
Height = 8.6 cm
V = L . W . H
= (15.2 cm)(3.7 cm)(8.6 cm)
= 480 cm3
Convert to m3.
X m3 = 480 cm3 = 0.000480 m3
100 cm
1 m 3
( )_____
X m3 = 480 cm3 =
X m3 = 480
100 cm
1 m( )_____100 cm
1 m( )_____100 cm
1 m( )_____ =
or
cm.cm.cm
1 m1000000 cm( )_________
3
34.80 x 10-4 m3
or
32cm
Measured dimensions of a rectangular solid:
Find volume of solid.L
W
H
Length = 15.2 cm
Width = 3.7 cm
Height = 8.6 cm
V = L . W . H
= (0.152 m)(0.037 m)(0.086 m)
= 0.000480 m3
0.152 m0.037 m
0.086 m
Convert to m3...
Try: Dimensional Analysis
• How many liters of water would fill a container that measures 75.0 in3?
75.0 in3 (2.34 cm)3
(1 in)3= 1.23 L
in3 L
1 L
1000 cm3
Try:Try:1. You want to buy 100 U.S. Dollars Candadian
dollars. If the exchange rate is 1 Can$ = 0.65 US$, how much will it cost?
# Can$ = 100 US$ x 1 Can$ 0.65 US$
= 153.85 Can$
Try:Try:3. There are 12 inches in a foot, 0.394 inches
in a centimeter, and 3 feet in a yard. How many cm are in one yard?
# cm = 1 yd x 3 ft 1 yd
= 91.37 cmx 12 in
1 ft x 1 cm
0.394 in
Challenge :Dimensional Analysis• You have 1.5 pounds of gold. Find its volume in
cm3 if the density of gold is 19.3 g/cm3.
lb cm3
1.5 lb 1 kg
2.2 lb= 35 cm3
1000 g
1 kg
1 cm3
19.3 g
