Chapter 1: Mathematical Preliminaries

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Chapter 1 Mathematical Preliminaries

• Dimensional Analysis
• Unit Conversion
• Significant Figures
• Order-of-Magnitude Estimates
• Coordinate Systems
• Trigonometry

2
Fundamental Quantities and Their Dimension

• Length L
• Mass M
• Time T
• Other physical quantities can be constructed from
  these three
• Example - Force F M L/T2

3
Dimensional Analysis

• Technique to check the correctness of an equation
• Can guess approximately correct equations
• Dimensions (length, mass, time, combinations) can
  be treated as algebraic quantities
• Add, subtract, multiply, divide
• Both sides of equation must have the same
  dimensions

4
Dimensional Analysis

• Example Relate energy to velocity.
• Now, .
• So we can have, E mv2 (kinetic energy,
  incorrect by a factor of 1/2).
• If v c, we have E mc2 for rest mass energy.
• Example Problem 1-6

5
Units

• To communicate the result of a measurement for a
  quantity, a unit must be defined
• Defining units allows everyone to relate to the
  same fundamental quantity

6
Systems of Units

• Standardized systems
• Agreed upon by some authority, usually a
  governmental body
• SI Systéme International
• Agreed to in 1960 by an international committee
• Main system used in this text and rest of world

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Length

• Units
• SI meter, m
• Defined in terms of the distance traveled by
  light in (1/299 792 458) s in a vacuum
• Also establishes the value for the speed of light
  in a vacuum as 299 792 458 m/s

8
Mass

• Units
• SI kilogram, kg
• Based on a specific cylindrical mass kept at the
  International Bureau of Weights and Measures

9
Time

• Units
• SI - seconds, s
• Defined in terms of the frequency of radiation
  from a cesium-133 atom
• 9 192 631 700 ? period of oscillation

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Other Systems of Measurements

• cgs Gaussian system
• Named for the first letters of the units it uses
  for fundamental quantities
• US
• Common units
• Feet, miles, pounds etc.

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Prefixes

• Prefixes correspond to powers of 10
• Each has a specific name and abbreviation

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Conversions

• When units are not consistent, you need to
  convert to appropriate ones
• Inside of the front cover of text has a list of
  conversion factors
• Units can be treated like algebraic quantities
  that can cancel each other
• Example
• Example 1-15

13
Uncertainty in Measurements

• There is uncertainty in every measurement this
  uncertainty carries over through the calculations
• Need a technique to account for this uncertainty
• Use significant figures to approximate the
  uncertainty in results of calculations

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Significant Figures

• A significant figure is a reliably known digit
• All non-zero digits are significant
• Zeros are significant when
• Between other non-zero digits
• After the decimal point and another significant
  figure
• This can be clarified by using scientific notation

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Operations with Significant Figures

• Accuracy number of significant figures
• Least accurate number has the lowest number of
  significant figures
• Multiplying or dividing two or more quantities
• The number of significant figures in the result
  is the same as the number of significant figures
  in the least accurate of the factors being
  combined

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Operations with Significant Figures, cont.

• Adding or subtracting
• Round the result to the smallest number of
  decimal places of any term in the sum
• If the last digit to be dropped is less than 5,
  drop the digit
• If the last digit dropped is greater than or
  equal to 5, raise the last retained digit by 1

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Operations with Significant Figures, cont.

• Examples
• 1.736 ? 8.32 14.4 (not 14.44)
• 0.859 ? 1.222 0.703 (not 0.7029)
• 12.0 0.357 12.4
• 7.898 - 5.6248 2.273 (not 2.2732)

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Estimates

• Can yield useful approximate answers
• An exact answer may be difficult or impossible
• Mathematical reasons
• Limited information available
• Can serve as a partial check for exact
  calculations

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Order of Magnitude

• Approximation based on a number of assumptions
• May need to modify assumptions if more precise
  results are needed
• Order of magnitude is the power of 10 that applies

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Coordinate Systems

• Used to describe the position of a point in space
• Coordinate system consists of
• A fixed reference point called the origin, O
• Specified axes with scales and labels
• Instructions on how to label a point relative to
  the origin and the axes

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Types of Coordinate Systems

• Cartesian (x, y) or (x, y, z)
• Plane polar (r, ?)
• Cylindrical (r, ?, z)
• Spherical (r, ?, ?)
• Use trigonometry to convert from one to another

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Cartesian coordinate system

• Also called rectangular coordinate system
• x- and y- axes
• Points are labeled (x,y)

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Plane polar coordinate system

• Origin and reference line are noted
• Point is distance r from the origin in the
  direction of angle ?, ccw from reference line
• Points are labeled (r,?)

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Trigonometry Review
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More Trigonometry

• Pythagorean Theorem
• r2 x2 y2
• To find an angle, you need the inverse trig
  function
• For example, q sin-1 0.707 45
• Active Figure 1.6
• Example 1-42

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Degrees vs. Radians

• Be sure your calculator is set for the
  appropriate angular units for the problem
• For example
• tan -1 0.5774 30.0
• tan -1 0.5774 0.5236 rad
• Conversion factor ? rad 180?

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Rectangular ? Polar

• Rectangular to polar
• Given x and y, use Pythagorean theorem to find r
• Use x and y and the inverse tangent to find angle
• Polar to rectangular
• x r cos ?
• y r sin ?