Units & Estimation
Freshman Clinic I
Units
• Physical Quantities
• Dimensions
• Units
Physical Quantities
• Measurement of physical quantities, e.g.,
length, time, temperature, force
• To specify a physical quantity, compare
measured numerical value to a reference
quantity called a unit
• A measurement is a comparison of how
many units constitute a physical quantity
Physical Quantities
• If we measure length (L) and use meters as units,
and L is 20 of these meter units, we say that
L=20.0 meters (m)
• For this relationship to be valid, an exact copy of
the unit must be available, i.e., a standard
– Standards: set of fundamental unit quantities kept under
normalized conditions to preserve their values as
accurately as possible
Dimensions
• Used to derive physical quantities
NOTE: Dimensions are independent of units;
for a given dimension there may be many
different units
• Length is represented by the dimension L
• Others physical quantities are time T, force
F, mass m
Kinds of Dimensions
• Fundamental dimension – can be
conveniently and usefully manipulated
when expressing physical quantities for a
particular field of science or engineering
• More simply, a basic dimension
• Velocity, e.g., can be considered a
fundamental dimension but we customarily
treat it as a derived dimension (V=L/T)
Units
• Each fundamental dimension requires a
base unit
• BUT (!), there are many unit systems that
can be used with a given dimension system
Units
• The International System of Units (SI)
serves as an international standard to
provide worldwide consistency
• Two fundamental unit systems exist today –
the meter-kilogram-second (MKS) used
worldwide and the Engineering System –
foot, pound force, second used in the US
SI Units
• Seven base units are defined so that they can be
reproduced
Length
meter
m
Time
second
s
Mass
kilogram
kg
Electric current
ampere
A
Temperature
kelvin
K
Amount of substance
mole
mol
Luminous intensity
candela
cd
SI Units
• Table 6.4 lists derived units with special names
• Table 6.5 lists derived units that are combinations
of units with special names and base units
• Unit Prefixes are listed in Table 6.6. They can be
used to eliminate non-significant zeros and leading
zeros
• It is customary to express a numerical value as a
number between 0.1 and 1000 with a prefix
More About Prefixes
• Use prefixes or scientific notation to
indicate significance
10.000 km
5SF 9999.5-10000.5 m
10.00 km
4SF 9995-10005 m
10.0 km
3 SF 9950-10050 m
10 km
2 SF 5000-15000 m
Rules for SI Units
• Periods not used
• Lower case unless derived from proper
name
• Do not add “s” to pluralize symbols
• Leave a space between numerical value and
symbol (except degrees, minutes, and
seconds of angles and degrees Celsius)
More Rules
• Plurals of the unit name (not the symbol) are
formed as necessary except for lux, hertz, and
siemens
• No hyphens or spaces between prefix and unit
name
• Omit final vowel in megohm, kilohm, and hectare
• Use symbols with numerical values; use names
with numerical value written in words
Multiplication/Division
• For unit products leave one space between
units or use a hyphen. For symbol products
use a center dot.
• Use the word “per” in a quotient; use the
slash (/) with symbols or unit-1
• For powers use “squared” or “cubed” after
the unit name. For area or volume, place
the modifier before the unit name.
US Customary System
Quantity
Mass
Length
Time
Force
Unit
slug
foot
second
pound
Symbol
slug
ft
s
lb
US Engineering System
Quantity
Mass
Length
Time
Force
Unit
Symbol
pound mass lbm
foot
ft
second
s
pound force lbf
Conversion of Units
• “Dimensional Analysis”
1 meter = 3.2808 feet
minute
meter
x 1 minute = 0.05468 feet
60 seconds
second
Estimation
• Significant Digits (Significant Figures)
• Accuracy and Precision
• Approximations
Significant Digits
(www.batesville.k12.in.us/Physics)
• All non-zero digits are significant digits.
– 4 has one significant digit
– 1.3 has two significant digits
– 4,325.334 has seven significant digits
Significant Digits
(www.batesville.k12.in.us/Physics)
• Zeros that occur between significant
digits are significant digits.
– 109 has three significant digits
– 3.005 has four significant digits
– 40.001 has five significant digits
Significant Digits
(www.batesville.k12.in.us/Physics)
• Zeros to the right of the decimal point and to
the right of a non-zero digit are significant.
– 0.10 has two significant digits
• leading zero is not significant, but the trailing zero is significant)
– 0.0010 has two significant digits (the last two)
– 3.20 has three significant digits
– 320 has two significant digits
• zero is to the left of the decimal point - not significant.)
– 14.3000 has six significant digits
– 400.00 has five significant digits
• two zeros to the right of the decimal point are significant because they are to the right of
the "4". The two zeros to the left of the decimal point are significant because they lie
between significant digits.
Significant Digits
(www.batesville.k12.in.us/Physics)
• The second and third rules above can also
be thought of like this:
– If a zero is to the left of the decimal point, it has
to be between two non-zero digits to be
significant.
– If a zero is to the right of the decimal point, it
has to be to the right of a non-zero digit to be
significant,
Significant Digits
(www.batesville.k12.in.us/Physics)
• These three rules have the effect that all
digits of the mantissa (number part) are
always significant in a number written in
scientific notation.
– 2.00 x 107 has three significant digits
– 1.500 x 10-2 has four significant digits
Multiplication and Division
• Answer should have same number of
significant digits as in number with fewest
significant digits.
• e.g., (2.43)(17.675)=42.95025 should be
expressed as 43.0 (3 significant digits, same
as 2.43, not 7-the actual product)
More Examples
• Using an exact conversion factor
(2.479 hr)(60 min/hr)=148.74 minutes (5SF?)
Express the answer as 148.7 minutes (4SF, same as
in the number 2.479)
• Conversion factor not exact
(4.00x102 kg)(2.2046lbm/kg)=881.84 lbm (5SF?)
Express the answer as 882 lbm (3 SF as in 4.00x102
kg)
One More…
• Quotient
589.62/1.246=473.21027 (Should this be 8
SF?)
Express the answer as 473.2 which is correct
to 4SF, the number of SF in 1.246)
Addition and Subtraction
• Show significant digits only as far to the
right as is seen in the least precise number
in the calculation (the last number may be
an estimate).
1725.463
189.2 (least precise)
16.73
1931.393 Report as 1931.4
More on Addition and
Subtraction
897.0
- 0.0922
896.9078
<- less precise
<- more precise
Report as 896.9
Combined Operations
• When adding products or quotients, perform the
multiplication/division first, establish the correct
number of significant figures, and then
add/subtract and round properly.
• If results of additions/subtractions are to be
multiplied/divided, determine significant figures
as operations are performed. If using a calculator,
report a reasonable number of significant figures.
Rules for Rounding
• Increase the last digit by 1 if the first digit
dropped is 5 or greater
827.48 becomes 827.5 for 4 SF
827.48 becomes 827 for 3 SF
23.650 becomes 23.7 for 3 SF
0.0143 becomes 0.014 for 2 SF
Accuracy and Precision
• Accuracy is the measure of the nearness of a given
value to the correct or true value.
• Precision is the repeatability of a measurement,
i.e., how close successive measurements are to
each other.
• Accuracy can be expressed as a range of values
around the true value, usually shown as a value
with a +/- range. 32.3+0.2 means that the true
value lies between 32.1 and 32.5
Accuracy and Precision
• The range of a permissible error can also be
expressed as a percentage of the value.
Consider a thermometer where the accuracy is
given as + 1% of full scale. If the full scale
reading is 220oF then readings should be
within + 2.2o of the true value, i.e.,
220x0.01=2.2
Approximations
• Precision is a desirable attribute of
engineering work
• You do not always have time to be precise
• You need to be able to estimate
(approximate) an answer to a given problem
within tight time and cost constraints.
Approximations
• A civil engineer is asked to estimate the
amount of land required for a landfill. This
landfill will need to operate for the coming
ten years for a city of 12000 people.
• How would you approach this estimation
problem?
Approximations
• The engineer knows that the national
average solid waste production is 2.75 kg
per person per day. He then estimates that
each person will generate
(2.75 kg/day)(365 days/year) = 1000 kg/year
• The engineer’s experience with landfills
says that refuse can be compacted to 400600 kg/m3.
Approximations
• This leads to the conclusion that the per
person landfill volume will be 2 m3 per
year.
• One acre filled 1 m deep will hold one
year’s refuse of 2000 people. (We get this
from 1 acre =4047 m2).
• The area requirement would then be 1 acre
filled to a depth of 6 meters.
Approximations
• But the engineer knows that bedrock exists
at the proposed site at a depth of 6 feet. So
the estimated depth needs to be reduced to 4
feet and the area needs to be increased to
1.5 acres for 1 year, or 15 acres for a 10
year landfill life.
Approximations
• To allow for expected population growth
the engineer revises the final estimate to 20
acres for a landfill life of 10 years.
Now It’s Your Turn…
• Estimate the cost to launch a
communications satellite. The satellite
should have a life of 12 years.
• The satellite has 24 transponders plus 6
spares that weigh 12 pounds each.
Communications Satellite
Each transponder requires:
• 20 lbs. of avionics
• 40 lbs. of batteries and solar cells
The satellite uses 80 pounds of stationkeeping fuel per year
The satellite carries an apogee kick motor that
weighs 3000 lbs.
Launch Vehicle
• Cost to launch on a Delta rocket is
$8000/lb. per lb. up to 6000 lb. and
$10000/lb. for each pound over 6000 lbs.
• Cost to launch on an Atlas-Centaur rocket is
$9000 per lb.
• Which is the more economical launch
vehicle for this spacecraft?
Solution
• 24 transponders plus 6 spares at 12 lbs. each
weighs 360 lbs.
• 20 lbs. of avionics per transponder (30)
weighs 600 lbs.
• 40 lbs. of batteries and solar cells per
transponder (30) weighs 1200 lbs.
• 80 lbs. of station-keeping fuel per year (12)
weighs 960 lbs.
Spacecraft Total Weight
•
•
•
•
•
•
•
Transponders
Avionics
Batteries and solar cells
Station -keeping fuel
Spacecraft weight
Apogee kick motor
Total weight at launch
360 lbs.
600 lbs.
1200 lbs
960 lbs.
3120 lbs.
3000 lbs.
6120 lbs.
Launch Costs
• For Delta: (6000 lbs.)($8000)
+(120 lbs.)($10000) = $49.2M
• For Atlas-Centaur: (6120 lbs.)($9000) =
$55.08M
• Launching on Delta is cheaper by $5.88M