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.

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Scientific Notation

• 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
• b. 450,000 m
• c. 302,000,000 m
• d. 86,000,000,000 m

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Your turn to practice

• Express the following quantities in scientific
  notation
• a. 0.000,508 kg
• b. 0.000,000,45 kg
• c. 0.0003600 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. ?

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Converting Units with D. A.

• What is the equivalent in kg of 465 g?
• Recall that 1 kg 1000 g
• 465 g 1 kg 0.465 kg
• 1 1000g

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Your turn to practice

• Convert each of the following length measurements
  as directed.
• a. 1.1 cm to meters
• b. 76.2 pm to mm
• c. 2.1 km to meters
• d. 2.278 x 1011m to km

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Your turn to practice

• Convert each of the following mass measurements
  to kilograms.
• a. 147 g
• b. 11 Mg
• c. 7.23 µg
• 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) x10(311)kgm
• 20x1014kgm
• 2x1015 kgm

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Now Dividing

• 8x106 m3 / 2x10-2 m2
• 8/2 x 10(6-(-3)) m(3-2)
• 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)

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• 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 decimal place beyond that which
  you can measure with that tool.
• On a meter stick, you may record a pencils
  length to be 19.6cm.

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Significant Digits cont.

• If the pencils end is somewhere between 0.6 and
  0.7cm, then you should 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 trailing 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

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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
• EX 36.5m / 3.414s

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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

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Visualizing Data

• 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)

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Linear Relationships

• 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.

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Non-Linear Relationships

• 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

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Your turn to Practice

• 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 distance 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