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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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lengthmeters (m)
timeseconds (s)
masskilograms (kg)
temperatureKelvin (K)
currentAmperes (A)
amount of substanceMole (mol)
luminous intensitycandela (cd)
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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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