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Dimensional Analysis

Dimensional analysis can be a useful tool for, at this stage, deriving units of measurement or checking formulas. Dimensional units, or units of measurement, can be treated as algebraic expressions (i.e., they are like variables). Take this example:
Given this formula: y=5gt2, where t=10 s and g=the acceleration due to gravity=9.80 m/s2
Find the units of y.
Solution by dimensional analysis:

Seconds are a measure of time and meters are a measurement of length aren't they? So t= Time, g= Length/Time2
This equation has the dimensions y=(L/T2)T2, which can be algebraically simplified to y=L because the T2 and the 1/T2 cancel each other out; the dimension for y in this example was Length, so the units were meters.

Scientific Notation

This notation is just another way to write numbers, sort of a scientist's "short hand." Take the number 6 million: 6,000,000. Instead, you can write it as 6 multiplied by a power of 10 (in this case, it is expressed as 6 x 106.

Here's another example: Express 25 in scientific notation. 2.5 x 101. Why can't you just write it like 25 x 100? Because scientific notation is always in the form a x 10b, where a is a number is a decimal less than 10. Examples of such: 4.5627, 3.00, 5.2, etc.

Significant Figures

Significant Figures, or the slang, "sig figs" are significant in determining the accuracy of measurements! If an engineer is a tenth of a degree off in determining a projectile's trajectory (path of flight), it could land quite far away from where the scientist had calculated, or in any calculation in which precision is important. Sometimes this could be a grievous error. Just think, if astronomers were "off" about the path of an asteroid approaching Earth...(well how MANY movies about that have you seen?? =])

Here are some rules for preserving significant digits in figures:

• Addition & Subtraction: Ever heard the phrase "a chain is as good as it's weakest link"? Well, when adding two numbers together, there is a similar rule. The precision (farthest decimal place) is as good as its least precise measurement (the least farthest decimal place) Take this example:
68.391+2.55=?
The first number has an accuracy to 3 decimal places, but the second number has only 2. So, the answer is precise to only 2 decimal places.
68.391+2.55=70.94 (70.941 rounded to the nearest hundredth)
• Multiplication & Division: The rule for multiplication and division is different from that for addition and subtraction. Try not to get the two mixed up because the results (obviously) may not always be the same. For multiplication and division, count the number of figures in the smallest number. The answer must have no more number of accurate figures than this number. If this sounds confusing, analyze these examples:
• 68.391·2.55=?
2.55 seems to have the least amount of digits (3), so the answer must have a maximum of 3-digit accuracy.
68.391·2.55=174 (174.39705 rounded to the nearest one's place)
• 0.00000000035·16.7=?
16.7 may seem to be the number with the least number of digits (3), HOWEVER,0.00000000035 has only 2-digit accuracy! How? What? Write it in scientific notation: 3.5 x 10-10. Only 3.5 are the significant digits. (lots of zeros in a long string don't count!
0.00000000035·16.7=(3.5 x 10-10)·16.7=5.8 x 10-9 (5.845 rounded to the nearest tenth)
• Let's compare the number 200. to 200 :
The first one has 3 sig. figs. whereas the second one only has one. Why?!? (They're the same number aren't they?) The value (200) is the same, but the decimal point at the end of the first one means that all three digits count as sig. figs. (all are certain to be accurate). For the second one, because there is no decimal point, the rule "lots of zeros in a long string don't count!" applys.

Unit Conversion

Now comes the conversion of units. Because there is more than one system of measurement, it would be useful to be able to "translate" certain units to those you are more familiar with.

Some Systems of Measurement:

 System Length Area (L2) Volume (L3) Velocity (L/T) Acceleration (L/T2) SI (International System) m m2 m3 m/s m/s2 British engineering (conventional system) ft ft2 ft3 ft/s ft/s2

One way to convert units is by using unit fractions (bizarre perhaps?--well, look back to dimensional analysis; weren't dimensions, and therefore units, considered like algebraic terms which could be multiplied and divided?). First consider the equivalence of this expression: 1 meter=100 centimeters. Say that a=1 m, and that b=100 cm. Thus, a=b.
Then what is a/b? a/b=1, because we stated that a=b earlier.
So, (1m/100cm)=1. Let's put this expression to use:
How do you convert 563 cm to an equivalent length in m?
Solution: Using the concept of (1m/100.cm)=1, multiply 563 by "1."
563 cm·1=563 cm ·(1m/100.cm)=5.63m

**Note: This is a common way to denote "the units of" x: [x]

If x was known to be a unit of length, [x] = m in the SI system.**

 Reference
 Conversion Factors Length Speed 1 m=39.37 in.=3.281 ft 1 in.=2.54 cm 1 km=0.621 mi 1 mi=5280 ft=1.609 km 1 lightyear=9.461 x 1015 m 1 angstrom (Å) =10-10 m 1 km/h=0.278 m/s=0.621 mi/h 1 m/s=2.237 mi/h=3.281 ft/s 1 mi/h=1.61 km/h=0.447 m/s=1.47 ft/s Mass Force 1 kg=6.85 x 10-2 slug 1 slug=14.59 kg 1 u=1.66 x 10-27 1 N=0.2248 lb=105 dynes 1 lb=4.448 N 1 dyne=10-5N=2.248 x 10-6 lb Time Work and Energy 1 min=60 s 1 h=3600 s 1 day=8.64 x 104 s 1 year=365 days 1 J=107 erg=0.738 ft·lb=0.239 cal 1 cal=4.186 J 1 ft·lb=1.356 J 1 Btu=1.054 x 103 J=252 cal 1 J=6.24 x 1018 eV 1 eV=1.6 x 10-19 1 kWh=3.60 x 106 Volume Pressure 1 liter=1000 cm3=3.531 x 10-2 ft3 1 ft3=2.832 x 10-2 m3 1 gallon=3.786 liter=231 in.3 1 atm=1.01 x 105 N/m2 (or Pa)=14.70 lb/in.2 1 Pa=1 N/m2=1.45 x 10-4 lb/in.2 1 lb/in.2=6.895 x 103 N/m2 Angle Power 180º=pi rad 1 rad=57.30º 1º=60 min=1.745 x 10-2 rad 1 hp=550 ft·lb/s=0.746 kW 1 W=1 J/s=0.738 ft·lb/s

Some Prefixes for Powers of Ten Used with Metric Units
 Power Prefix Abbreviation 10-18 atto- a 10-15 femto- f 10-12 pico- p 10-9 nano- n 10-6 micro- µ 10-3 milli- m 10-2 centi- c 10-1 deci- d 10 deka- da 103 kilo- k 106 mega- M 109 giga- G 1012 tera- T 1015 peta- P 1018 exa- E

Physical Constants
 Quantity Symbol Value SI unit Speed of light in a vacuum c 3.00 x 108 m/s Permittivity of a vacuum 8.85 x 10-12 F/m Coulomb's constant ke 8.99 x 109 N·m2/C2 Permeability of a vacuum 4pi x 10 N/A2 Elementary charge e 1.60 x 10-19 C Planck's constant h 6.63 x 10-34 J·s Electron rest mass me 9.11 x 10-31 kg Proton rest mass mp 1.67 x 10-27 kg Neutron rest mass mn 1.67 x 10-27 kg Avogadro's constant NA 6.02 x 1023 1/gmol Molar gas constant R 8.31 J/mol·K Boltzmann's constant kB 1.38 x 10-23 J/K Stefan-Boltzmann constant 5.67 x 10-8 W/m2·K4 Molar volume of ideal gas at STP V 22.4 L/mol Rydberg constant R 1.10 x 107 1/m Bohr radius a0 5.29 x 10-11 m Electron Compton wavelength h/mec 2.43 x 10-12 m Gravitational Constant G 6.67 x 10-11 m3/kg·s2 Standard gravity g 9.80 m/s2 Radius of the Earth (at the equator) RE 6.38 x 106 m Mass of the Earth ME 5.98 x 1024 kg Radius of the Moon Rm 1.74 x 106 m Mass of the Moon Mm 7.36 x 1022 kg

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Key Themes
• Dimensional Analysis
• Scientific Notation
• Significant Figures
• Unit Conversion
• Practice Questions

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