Unit Systems

College and Engineering Physics

Mathematics in Physics

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Unit Systems

Conversions

Physical Quantities

Dimensions

Dimensional Analysis

Scientific Notation

Computer Notation

Calculator Notation

Significant Figures

Math Using Significant Figures

Order of Magnitude

Estimation

Fermi Problems

Powers of 10



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What is a physical quantity?

A physical quantity is any quantity that can be measured with a certain mathematical precision.

Example: Force

What is a dimension?

The product or quotient of fundamental physical quantities, raised to the appropriate powers, to form a derived physical quantity.

Example: mass x length / time2 (ML/T2)

What is a unit?

A precisely defined (standard) value of physical quantity against which any measurements of that quantity can be compared.

Example: Newton = kilogram x meters / second2



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TOC Sytème International

lengthmeters (m)

timeseconds (s)

masskilograms (kg)

temperatureKelvin (K)

currentAmperes (A)

amount of substanceMole (mol)

luminous intensitycandela (cd)



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TOC US Customary System

lengthinches (in, ")

timeseconds (s)

masspounds (lb)

temperatureFahrenheit (F)

currentAmperes (A)

amount of substanceMole (mol)

luminous intensitycandela (cd)



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Other unitslength

Feet (ft, '), mile (m, mi), furlong, hand

timeminute (m, min), hour (hr), second (s), fortnight, while

forceNewton (N)

temperatureCelsius (C)

energyJoule (J)

powerWatt (W)

pressurePascal (P)

magnetic fieldTesla (T), Gauss (G)



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Common conversion factors

length1 in = 2.54 cm

time60 s = 1 min, 60 min = 1 hr, 24 hours = 1 day, 365.25 days= 1 yr

force1 N = 0.2248 lb

energy1 J = 107 erg, 1 eV = 1.602x10-19 J

power1 hp = 746 W

pressure1 atm = 101.3 kPa

magnetic field1 T = 104 G



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How to convert units

day 1day 1

sec day 6060241

sec day 400,861

day 1

hr 24

hr 1

min 60

min 1

sec 60



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Most commonly used prefixes for powers of 10

tera T 1,000,000,000,000 1012

giga G 1,000,000,000 109

mega M 1,000,000 106

kilo k 1000 103

centi c 0.01 10-2

milli m 0.001 10-3

micro μ 0.000001 10-6

nano n 0.000000001 10-9



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TOCCommon physical quantities

MassDistanceTimeSpeed / VelocityAccelerationForce



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TOCDimensions of common physical quantities

Mass MDistance LTime TSpeed / Velocity L / TAcceleration L / T2

Force M L / T2



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What is the equation that relates force and mass?

Force M L / T2

Force = M (L / T2)= M (L/T)2 / L= M (L / T2) + M (L/T)2 / L

Possible Equations…

So how did Isaac Newton know which is correct?

maF l

vmF

2

l

vmmaF

2

l

vmmaF

2

164



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TOCWhat is scientific notation?

1,562,788. 1.562788x106

0.0012789 1.2789x10-3



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Using scientific notation in formulas

Addition and Subtraction• set both numbers to the same exponent• add or subtract the decimal numbers• the exponent of the sum is the same as that of

the numbers being added

Multiplication• multiply the decimal numbers• add the exponents

Division• divide the decimal numbers• subtract the exponents



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How do computer’s write scientific notation?

1,562,788. 1.562788e60.0012789 1.2789e-3



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Using scientific notation in calculators

When putting numbers in a calculator, it is best to covert them to non-prefixed units first (e.g. 1mm 1x10-3 m) and then convert back to the desired units when the problem is complete.

When using a calculator, it is also better to put in the full number (e.g. 1350x10-3 m instead of 1.35 m for 1350 mm). In this way you will avoid many of the “decimal place errors” so common in this class.



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Significant figures defined

Significant figures in a number indicate the certainty to which a number is known.(For example 1350 mm is known to within ~0.5 mm.)

Other than leading zeros, all digits in a number are significant.(For example 0.003450900000 has 10 significant figures.)

Numbers are rounded up or down to the nearest significant figure.



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Significant figures are used because any further digits added to your number have no physical meaning.

Any physically measured number (including physical constants) will be written with the correct number of significant figures unless otherwise noted.

Numerical constants, such as the 4 or the π in the equation

have an infinite number of significant figures because they are NOT measured.

3sphere 3

4rV



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Using significant figures

When multiplying or dividing numbers, the calculated number has the same number of significant figures as the number with the least significant figures used in the calculation.

98 103.310367.52.6

88

102.110367.5

2.6



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Using significant figures

When adding or subtracting numbers, the number of significant figures in the calculated number must be such that the decimal place of the result is not beyond the least decimal number in the numbers used in the calculation.

8

8

8

8

6.2 5.367 10

0.000000062 10

5.367 10

5.367 10



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Examples using significant figures

3107.40030.058.1

44.010688.510078.0 78

. . 998 1039.31026310281



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Why do we use order of magnitude?

Order of magnitude is used to make estimates.

For example: How many professors are there in the U.S.?(These kinds of questions are named for Enrico Fermi, who first proposed them.)

Order of magnitude is also used to check calculations.



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Order of magnitude defined

To determine the order of magnitude of a number, you must put the number in scientific notation using one digit before the decimal.(e.g. 1345 1.345x103

0.00845090 8.45090x10-3)

If the decimal number is less than five, the order of magnitude is then the exponent. If it is greater than or equal to five, the order of magnitude is the exponent plus one.

1345 has order of magnitude 30.00845090 has order of magnitude -2



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Estimation

Estimation is not the same as calculation. However, it will almost always be within one exponent of the calculated answer.

An estimate is a fast way to check a number or choose between two numbers given as an answer

Actual

Estimate

88

1012.710230

67.1235.11324

710

3

101101

11101



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An Example Fermi Problem

To within an order of magnitude how many bars of soap aresold in the United States each year?

1. There are about 1x108 people in the U.S. (the actual number is closer to 3x108).

2. Each person lives in a family of about 1 person (the average is really closer to 3).

3. Each family uses about 1 bar of soap each week (a better number would be 0.5).

4. There are about 100 weeks in a year (the number is actually 52).

Compare this to the actual answer

108 10110011101

108 1034.2525.03103



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