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 mTime second sMass kilogram kgElectric current ampere ATemperature kelvin KAmount of substance mole molLuminous 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 Unit Symbol

Mass slug slug

Length foot ft

Time second s

Force pound lb

US Engineering System

Quantity Unit Symbol

Mass pound mass lbm

Length foot ft

Time second s

Force pound force lbf

Conversion of Units

• “Dimensional Analysis”

1 meter = 3.2808 feet x 1 minute = 0.05468 feetminute meter 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 <- less precise

- 0.0922 <- more precise

896.9078

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 400-600 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 station- keeping 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 360 lbs.

• Avionics 600 lbs.

• Batteries and solar cells 1200 lbs

• Station -keeping fuel 960 lbs.

• Spacecraft weight 3120 lbs.

• Apogee kick motor 3000 lbs.

• Total weight at launch 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