A Mathematical Toolkit

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A Mathematical Toolkit

• Introduction
• Chapter One Glencoe Physics 2006

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1. Errors in Measurement

• a. Personal Errors - due to bias or carelessness
• biased in favor of first observation
• parallax errors
• failure to follow significant figures
• b. Systematic Errors - due to instruments or
  techniques. Zeroing instrument, calibration
• c. Random Errors - unknown and unpredictable
  variations in experiments

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d. Accuracy and Precision

• (1) Accuracy - how close experimental results
  come to the true value
• correctness of the result
• depends on systematic errors
• (2) Precision - a measure of reliability, or how
  reproducible the result is
• measure of the magnitude of the uncertainty w/o
  reference to what results mean
• precision of experimental value depends on random
  errors

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• Good Accuracy - Poor Precision

x
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Good Accuracy - Good Precision
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Poor Accuracy - Good Precision
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2. Significant Digits

• a. Depends on measuring instrument - last digit
  written down is always an ESTIMATE
• b. Def - all the numbers that can be read
  directly from the instrument scale plus one
• c. all non-zero digits are significant
• d. not the same as decimal places

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e. Rules for Digits

• (1) leftmost non-zero digit is the MOST
  significant digit (.0028)
• (2) if there is no decimal point, the right most
  non-zero digit is the LEAST significant (58,000)
• (3) if there is a decimal point, the right most
  digit is the LEAST significant, even if it is
  zero (1.05060)
• (4) all digits between most and least are
  significant

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f. Operations

• (1) use scientific notation to show precision
• (2) Mult/Division - number of significant digits
  in the final answer is the same as the number of
  digits in the LEAST accurate of the
  operators/numbers being used
• (3) Add/Sub - add/subtract numbers, then round
  off answer to the first decimal position with a
  doubtful figure
• (4) 6.27 times 5.5

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g. Laboratory Measurements - set digits

• (1) Least Count - the smallest subdivision marked
  on an instrument scale.
• (a) No estimation
• (b) measured reading will have one more
  significant digit than the least count
• (2) Vernier Caliper - measures inside or outside
  of objects, such as pipe diameters
• (a) mm least count with vernier assisting in
  fractional part
• (b) check for zero and adjust as necessary
  

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• (3) micrometer (mike) - measure the diameter of a
  wire, thickness of small object
• (a) axle scale in mm
• (b) thimble scale in 0.01 mm
• (c) read to thousandth of a mm- 5.355mm
• (d) adjust for zero
• (4) read cylinders at bottom of meniscus

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3. Scientific Notation

• a. Most quantities found in physics are very
  large or very small. Get around using zeros by
  using scientific notation
• b. a.xxxx x 10 n 1lt a lt 9 , integer
• c. Powers of ten
• 10o 1 10 1 10 10 -1 1/10 10 2 100 10
  -2 1/100

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• d. When adding or subtracting - must have the
  same exponent
• EX - 6.5 x 10-4 - 7.2 x 10-6 650 x 10-6 -
  7.2 x 10-6 642.8 x 10-6 6.428 x 10-4

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4. Dimensional Analysis

• a. No number without a unit, UNLESS IT
  IS A RATIO
• b. Dimension - qualitative nature of a
  physical item
• (1) Ex - length - the separation between two
  points
• (2) can use brackets to denote dimensions

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• (3) use most convenient dimension/unit, trick is
  to make sure that you have the same units
• (4) dimensions are then handled algebraically -
  cancel out, mult., divide
• (5) can be used to verify the accuracy of a
  formula AND solution

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• c. EX PROBLEM

s distance m v m/s a m/s/s t sec
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d. Converting Units

• (1) Based on algebraic concept of the
  multiplicative identity. Multiply and non-zero
  number by ONE and get the number.
• (2) 5 x 1 5 a x 1 a a2b x 1
  a2b
• (3) use the over/under box method
• (4) determine how many seconds in a year

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5. Metric System

• a. Metric vs English (British Engineering)
• (1) Fundamental Units-length, mass, time
• (2) Supplemental Units - temperature, charge,
  illumination
• (3) Derived Units - such as the pound and newton
• Why metric (now called SI)?

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• (1) 1120 King of England - std length is yard
• (2) Foot - length of Louis XIVs foot
• (3) Meter - one ten-millionth of distance from
  equator to the north pole on line thru Paris
• (4) Inch - length of a knuckle

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• c. SI established in 1960 to create one system
  for worldwide use
• d. Prefixes - know prefixes and symbols
• 10 -12 Boo 10 -1 Mates
• 10 -6 Phones 10 1 Cards
• 10 -3 Tary 10 6 Phones
• 10 -2 Pedes 10 12 Bull

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6. Displaying Data

• a. Independent Variable - the item or value
  manipulated during the experiment
• b. Dependent Variable - the measured quantity
  taken as a result of the experiment
• c. Graphing
• (1) independent variable on the horizontal and
  dependent variable on the vertical

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• (2) Determine if the origin is a valid point, if
  so, include it in the graph
• (3) label axes, show numbers and units, and title
  the graph
• (4) use a sharp point to make the data points and
  then circle them
• (5) draw a smooth curve that best fits the data
  points. REMEMBER what you have studied and ask
  yourself what the shape is.

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7. Relationships

• a. linear y mx b
• (1) direct relationship
• (2) Hookes Law F k x
• (3) b 0 if it goes through the origin
• (4) use a straightedge to draw the graph - NO
  FREEHANDED

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b. Quadratic y a x 2 b x c

• (1) parabolic curve
• (2) y varies with the square of x
• What happens if you plot y vs x 2
• (3) use a french curve to draw the line
• (4)

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c. Inverse Relationship

• (1) hyperbola
• (2) y varies inversely with x
• (3) p V constant ( universal gas law with
  temperature held constant)

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8. Interpreting Graphs

• a. y k x 2 - parabola opens upward if k is
  positive opens downward for negative k
• b. y 2 k x - also a parabola, but opens left
  and right
• c. Interpolation - reading a graph between the
  data points
• d. Extrapolation - reading a graph beyond the
  limits of what you obtained experimentally. Use
  with caution.

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9. Equation Manipulation

• a. Solve for desired unknown before you
  substitute in known quantities
• b. Watch your units and dimensions
• c. How long does it take sunlight to reach the
  Earth? c 3.0 x 10 8 m/s and distance to sun is
  1.5 x 10 8 km
• d. Find the average density of the Earth if its
  mass is 6.0 x 10 24 kilograms and radius is 6.4 x
  10 6 meters. V 4/3 ??r3