Math: The language of Physics

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Math The language of Physics

• Measurement a comparison between an unknown
  quantity and a standard.

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Scientific Measurements

• Using the Metric System (SI) for continuity
• Being a decimal system, we use prefixes to change
  between powers of 10.
• Ie 1/1000 of a gram one milligram

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Scientific Notation

• We use Scientific notation for expressing numbers
  that are VERY LARGE or very small.
• We write the numerical part of a quantity as a
  number between 1 and 10 multiplied by a
  whole-number power of 10.
• The average distance from the sun to Mars is
  227,800,000,000m
• This is written as 2.278 x 1011m
• Moving the decimal to the LEFT a of places is
  POSITIVE.
• The average mass of an electron is about
• 0.000,000,000,000,000,000,000,000,000,000,9
  11 kg
• This is written as 9.11 x 10-31
  kg
• Moving the decimal to the RIGHT a of places is
  NEGATIVE.

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Your turn to practice

• Express the following quantities in scientific
  notation
• a. 5800 m c. 302,000,000 m
• b. 450,000 m d. 86,000,000,000 m
• Express the following quantities in scientific
  notation
• a. 0.000,508 kg c. 0.0003600 kg
• b. 0.000,000,45 kg d. 0.004 kg

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Converting Units with D. A.

• You can easily convert from a given unit to a
  needed unit using a series of Conversion Factors
  (fractions that equal one like
• 4 quarters / 1 or 5280 ft / 1 mile
• It is IMPERATIVE that you show your UNITS while
  you SHOW YOUR WORK. ?
• What is the equivalent in kg of 465 g?
• Recall that 1 kg 1000 g
• 456g/1 ( 1kg / 1000g) 456gkg / 1000g .456kg

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Your turn to practice

• Convert each of the following length measurements
  as directed.
• a. 1.1 cm to meters c. 2.1 km to meters
• b. 76.2 pm to mm d. 2.278 x 1011m to km
• Convert each of the following mass measurements
  to kilograms.
• a. 147 g c. 7.23 µg
• b. 11 Mg d. 478 mg

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Combinations with Scientific Notation

• Adding and subtracting
• (4 x 108m) (3 x 108m)
• (4 3) x 108m 7 x 108m
• (4.1x10-6kg) (3.0x10-7kg)
• 4.1x10-6kg 0.30x10-6kg
• (4.1-0.30)x10-6kg
• 3.8x10-6kg

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Multiplying and Dividing with Scientific Notation

• First Multiplying
• (4x103 kg) (5x1011m)
• (4 x 5) x 10311 kgm
• 20x1014kgm
• 2x1015 kgm

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Now Dividing

• 8x106 m3 / 2x10-2 m2
• 8/2 x 106-(-3) m3-2
• 4 x 109 m
• See? Easy! ?

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Measurement Uncertainties

• Scientific results need to be reproducible.
• All measurements have a degree of uncertainty.
• Precision- the degree of exactness of a
  measurement. (to within 1/2 of the smallest
  measurement increment)
• Accuracy-how close results compare to a standard.
  Be sure to calibrate (zero) your instrument
  before using it.

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Significant Digits

• When making a measurement, record your quantity
  by estimating 1 position beyond that which you
  can measure with that tool. In a meter stick, a
  pencils length might be recorded as 19.6cm. If
  the pencils end is somewhere between 0.6 and
  0.7cm, then you estimate how far between and
  record the measurement as 19.62cm. All nonzero
  digits are significant. This has 4 significant
  digits.

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What about Zeros?

• All final zeros after the decimal are
  significant.
• Zeros between 2 significant digits are always
  significant.
• Zeros used only as place holders are NOT
  significant.
• All of the following have 3 significant digits
• 245m 18.0g 308km 0.00623g

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Your turn to Practice

• State the number of significant digits in each of
  the following measurements
• 2804 m e. 0.003,068 m
• 2.84 km f. 4.6 x 105 m
• 0.007060 m g. 4.06 x 10-5 m
• 75.00 m h. 1.20 x 10-4 m

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Math functions with sig. figs

• In recording results of experiments, the answer
  can never be more precise than any individual
  measurement involved in calculating that answer.
• For adding and subtracting, first perform the
  function and then round to the appropriate
  decimal having the least precise value.
• EX 24.686 m 2.343 m 3.21 m 30.239 m
  rounded off correctly to 30.24 m

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Multiplying Dividing

• Perform the calculation, note the factor with
  the least significant digits, then round this
  answer to the same number of significant digits.
• EX 3.22 cm x 2.1 cm 6.762 cm2 6.8 cm2
• EX 36.5m / 3.414s 10.691m/s 10.7m/s

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Your turn to Practice

• Solve the following problems
• Add 1.6 km 1.62 m 1200 cm
• 10.8 g 8.264 g
• 3.2145 km x 4.23 km
• 18.21 g / 4.4 cm3

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Answers

• Answers
• 1. 1.6 km 2. 2.5 g
• 3. 13.6 km2 4. 4.1g/cm3

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Sig Figs summarized

• There are no significant digits for counting.
• Only measurements have uncertainty.
• Significant digits are important to determine
  meaning in your calculations.

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Visualizing Data

• Graphs should tell the whole story in a picture
  format.
• Line graphs can be linear or non-linear.
• A variable that is changed or manipulated is an
  independent variable. (One you can control
  directly) plot on x-axis of graph
• Dependent variables change as a result of the
  independent variable. plot on y-axis
• Always draw a line of best fit to show the
  relationship of data measured (not dot-to-dot)

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Line Graph Guidelines

• Label both axis with name (and unit)
• Plot s evenly distributed on each axis
• Decide if the origin (0,0) is a valid point
• Spread out the graph as much as possible
• Draw the best fit straight line or smooth curve
  that passes through as many data points as
  possible.
• Title your graph

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Linear Relationships

• In a linear relationship, two variables are
  directly proportional.
• The relationship is y mx b
• The slope, m, is ?y / ?x (AKA rise/run)
• The 2 data points to determine m, MUST be ON the
  line of best fit. (and should be as far apart as
  possible)
• The y-intercept, b, is the point where the line
  crosses the y-axis when x zero.
• When b 0, then the equation is y mx
• When y gets smaller if x gets bigger, then slope
  is negative.

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Non-Linear Relationships

• In a parabola, gentle curve upward, variables are
  related by a quadratic relationship
• y ax2 bx c
• One variable depends on the square of the
  other.
• In a hyperbola, gentle curve downward, variables
  are related by an inverse relationship
• y a/x or xy a
• One variable depends on the inverse of the
  other.

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Your turn to Practice

• The total distance a lab cart travels during
  specified lengths of time is given in the
  following data table
• 1. Plot the dist vs time and best fit line for
  the points
• 2. Describe the curve
• 3. What is the slope of
• 
  the line?
• 4. Write an equation
• relating dist and time
• for this data.

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Key Equations

• y mx b
• m rise / run ?y / ?x
• y ax2 bx c
• xy a