Physics 220

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Physics 220

• Measurement and Data Analysis

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Fact

• A direct observation agreed upon by many people
• The earth revolves around the sun
• The earth is round
• Facts can change!

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Theory

• An explanation that has been confirmed by
  experimentation
• Atomic Theory
• Einsteins Theory of Relativity
• Can be disproved by a single crucial experiment

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Law

• A statement that describes a natural occurrence
• Law of Conservation of Mass
• Law of Conservation of Energy
• Boyles Law
• Based on observation first

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Essential Characteristics of Science

• It is guided by natural law
• It has to be explanatory in reference to natural
  law
• It is testable against the empirical world
• Its conclusions are tentative
• It is falsifiable

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Content of Physics

• Matter anything that has the properties of mass
  and inertia
• Mass The measure of the quantity of matter in
  an object The mass of an object is determined by
  comparing it to the mass of a known standard mass
• Inertia The property of matter that opposes any
  change in its state of motion. The inertia of an
  object can be used to measure mass by using an
  inertial balance

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Mass Density

• Mass per unit volume of a substance

g/cm3 kg/m3
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Energy

• Energy is the concept that unifies physics
• The capacity to do work
• Potential Energy (PE or U) Stored energy
• Kinetic Energy (KE or K) due to motion of
  matter

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Law of Conservation of Energy

• The total amount of energy in any system remains
  constant energy is never created or destroyed,
  it only changes forms
• Changes from PE to KE and back
• Examples a swinging pendulum, dropping a ball

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Mechanical Equivalent of Heat

• Count Rumford discovered heat and work are
  equivalent. When is used to do work on a system,
  the system can gain a proportional amount of
  heat. James Prescott Joule established the
  quantitative relationship between heat and
  mechanical energy.
• 1 calorie 4.1868 Joules (J)

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Matter and Energy

• Matter and energy are also related.
• Emc2
• cspeed of light constant, 3x108 m/s
• Einsteins equation lead physicists to believe
  matter and energy are different aspects of the
  same quantity. This equation also infers that as
  an object gains kinetic energy its mass should
  increase.

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Subdivisions of Physics

• Mechanics
• Motion
• Force
• Work
• Power
• Mechanics can be split in to divisions
• Kinematics conceptualization of motion
  (relative motion)
• Dynamics explanation of causes of motion

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Subdivisions of Physics

• Heat
• Waves
• Sound
• Light
• Electricity and Magnetism
• Nuclear Physics

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Measurement
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Measurement

• Physical Quantity measures something concrete
• Unit of Measure what is used to measure the
  physical quantity

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Definitions of Metric Units

• Meter the distance that light travels in a
  vacuum in of a second
• Kilogram the mass of a standard kilogram kept
  by the International Bureau of Measure
• Second 9,192,631,770 vibrations of a cesium-133
  atom
• These three units make up the mks or SI system.

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Derived Units

• Units that consist of combinations of fundamental
  units

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The Liter

• 1 Liter 1 cubic decimeter 1000 cubic
  centimeters
• 1 Liter of water has a mass of 1 kilogram
• 1 mL of water has a mass of 1 gram and a volume
  of 1 cm3

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Force and Weight

• Kilogram is the unit for mass
• Newton is the unit for force or weight
• Weight and mass are related by FWmg
• Where FW weight of an object
• m mass of an object
• g gravity (9.8 m/s2)

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Force and Weight

• Mass and weight are proportional, NOT equal
• So, 9.8 Newtons (N) will lift a mass of 1 kg on
  the surface of the earth.

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Converting Metric Measurements

• Factor Label / Dimensional Analysis
• Convert 132 kilometers to centimeters

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Converting Metric Measurements

• Convert 3.00 x 108 m/s to km/year

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Error in Measurement

• Types of Error
• Systematic one that always produces an error of
  the same sign positive is a reading too high and
  negative error is a reading too low
• Random occur as variations that are due to a
  large number of factors each of which adds to its
  own contribution of the total error. These
  errors are a matter of chance

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Types of Systematic Error

• Instrumental Error caused by faulty,
  inaccurate apparatus
• Personal Error caused by some peculiarity or
  bias of the observer
• External Error caused by external conditions
  (wind, temperature, humidity)

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Random Error

• Random errors are subject to the laws of chance.
  Taking a large number of observations may lessen
  their effect. When al errors are random, the
  value having the highest probability of being
  correct is the arithmetic mean or average.

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Propagation of Error

• Scientific measurements will always contain some
  degree of uncertainty. This uncertainty will
  depend on
• 1. The instrument(s) used to make measurements

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Propagation of Error

• 2. The object being measured
• 3. The proximity to the object being measured

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Variance

• The uncertainty of a measurement is indicated
  showing the possible variance with a plus and
  minus factor.
• Example You measure the length of an object
  five times and record the following measurements
• 53.33 cm, 53.36 cm, 53.32 cm, 53.34 cm, 53.38
  cm
• The average is 53.35 cm this should be written
  as
• 53.35 .03 cm

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Errors in Addition and Subtraction

• Example 13.02 ? .04 cm
• 23.04 ? .03 cm
• 14.36 ? .03 cm
• 26.89 ? .04 cm
•   77.31 ? .14 cm
• The variance of the result is equal to the sum of
  all the individual variances

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Errors in Multiplication and Division

• Example 13.2 ? .2 cm x 23.5 ? .3 cm
•   
• Maximum and Minimum
•  Maximum 13.4 cm x 23.7 cm 319 cm2
•  Minimum 13.0 cm x 23.2 cm 302 cm2
•  
•  Average 310. cm2
• Answer 310. ? 9 cm2
• The variance MUST be large enough to include
  both
• maximum and minimum

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Accuracy

• The closeness of a measurement to the accepted
  value for a specific physical quantity. Accuracy
  is indicated mathematically by a number referred
  to as error.
• Absolute Error (EA) (Average of observed
  values) (Accepted Value)
•  
•  Relative Error (ER) X 100

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Precision

• The agreement of several measures that have been
  made in the same way. Precision is indicated
  mathematically by a number referred to as
  deviation.
• Absolute Deviation (DA) (Each observed value)
  (Average of all values)
• Relative Deviation (DR) x 100

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Example for Measuring Error and Deviation

• Measured Values 893 cm/sec2 936 cm/sec2
• 1048 cm/sec2
• 915 cm/sec2
• 933 cm/sec2
•  Accepted Value 981 cm/sec2

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Example for Measuring Error and Deviation

• Step 1 Calculate the Average
• 893 cm/sec2
• 936 cm/sec2
• 1048 cm/sec2
• 915 cm/sec2
• 933 cm/sec2
• Average 945 cm/sec2

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Example for Measuring Error and Deviation

• Step 2 Calculate Absolute and Relative Error
•  
• Absolute Error (EA) (Average of observed
  values) (Accepted Value)
•   EA 945 cm/sec2 981 cm/sec2 36
  cm/sec2
•  
• Relative Error (ER)   x 100
• ER x 100 3.7

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Example for Measuring Error and Deviation

• Step 3 Calculate Absolute and Relative
  Deviations
•  Absolute Deviation (DA) (Each Observed Value)
  (Average of All Values)
•  DA 893 cm/sec2 945 cm/sec2 52 cm/sec2
•  DA 936 cm/sec2 945 cm/sec2 9 cm/sec2
•  DA 1048 cm/sec2 945 cm/sec2 103 cm/sec2
• DA 915 cm/sec2 945 cm/sec2 30 cm/sec2
•  DA 933 cm/sec2 945 cm/sec2 12 cm/sec2
•  
•  Average Absolute Deviation 206 cm/sec2 / 5
  41 cm/sec2

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Example for Measuring Error and Deviation

• Relative Deviation
•  
• Relative Deviation (DR) X 100
•  
• DR x 100 4.3

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Significant Figures

• Usually, you will estimate one digit beyond the
  smallest division on the measuring tool if the
  object you are measuring has a well defined edge.
• When reading a measurement that someone else has
  made, you must determine if the digits he/she has
  written down are significant to the measurement.

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Significant Figures

• Those digits in an observed quantity
  (measurement) that are known with certainty plus
  the one digit that is uncertain or estimated.
• The number of significant figures in a
  measurement depends on

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1. Smallest divisions on a measuring tool
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2. The size of the object being measured
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3. The difficulty in measuring a particular object
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Rules for Significant Figures

• Any digits which are NOT zero are significant
• A zero if significant if
• It lies between two other significant digits
• It lies to the right of the decimal point and it
  follows a significant digit
• NOTE Zeros are NOT SIGNIFICANT if they are used
  to hold place value, unless otherwise indicated

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Operations with Significant Digits

• Addition and Subtraction Add all numbers and
  round your answer off to the place value of the
  least precise measurement.
• Example 4.02 cm 10.1 cm 0.465 cm 14.585 cm
• 10.1 is the least precise value, so 14.6 cm

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Operations with Significant Digits

• Multiplication and Division Multiply all numbers
  and round off your answer to the same number of
  significant digits as the term with the least
  number of significant digits
• Example 6.98 cm x .23 cm 1.6054 cm2
• .23 has least number of significant figures, so
  1.6 cm2

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Scientific Notation

• A number expressed in the form M x 10n, in which
  M is a number between 1 and 10 and n is an
  integral power of 10. n is also known as the
  order of magnitude.
• Examples
• 29,900,000,000 cm/sec 2.99 x 1010 cm/sec
•   0.000034 g 3.4 x 10-5g
•  
• When doing operations with scientific notation,
  remember the rules for significant figures.
•  

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Examples of Scientific Notation

•   6.75 x 104 6.75 x 104
• 4.9 x 103 0.49 x 104
• 7.24 x 104
• 9.8 x 1010 8.6 x 1010
• 18.4 x 1010 1.84 x 1011
•  

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Examples of Scientific Notation

• 8.95 x 105 0.0895 x 107
• 4.3 x 107 4.3 x 107
•   4.3895 x 107 4.4 x
  107
• 4.2 x 103
• x 3.2 x 106
• 1.344 x 1010 1.3 x 1010

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Analyzing Data and Graphs

• Directly proportional As the independent
  variable increases or decreases, dependent
  variable increases or decreases proportionally.
  The slope of this line is straight and can be
  found using the equation y mx b.
• Inversely proportional As the independent
  variable increases, the dependent variable
  decreases disproportionately (and vice versa).
  This graph is a hyperbola and indicates that the
  product of the two graphs is a constant. xyk

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Directly Proportional Graph
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Inversely Proportional Graph
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Graphing Terms

• Interpolation estimating of values between two
  known data points
• Extrapolation estimating of values beyond the
  measurements obtained as known data points
• Best Line Fit - The curve line on a graph is
  drawn as a solid line. Because of the
  uncertainty involved in all measurement, it is
  possible to draw a smooth line through all data
  points. If a single dot deviates widely from the
  general trend, then it may be disregarded.

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Eight Steps of Problem Solving

• Step 1 Read the problem carefully and write
  down all the given data
• Step 2 Write down the symbol and unit of the
  physical quantity called for
• Step 3 Draw a sketch of the problem and write
  down the basic equation relating the known
  and unknown quantities
• Step 4 Obtain a working equation from the
  basic equation

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Eight Steps of Problem Solving

• Step 5 Substitute the given data in the working
  equation
• Step 6 Perform mathematical operations with
  units alone to make sure answer will be in
• required units (optional)
• Step 7 Perform the mathematical operations with
  numbers
• Step 8 Check to see if answer is reasonable

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Example

• Benjamin watches a thunderstorm from his
  apartment window. He sees the flash of a
  lightning bolt and begins counting the seconds
  until he hears a clap of thunder 10 seconds
  later. Assume the speed of sound in air is 340
  m/s. How far away is the lightning bolt in a) m?
  and b) km?

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Solution

• Given v 340 m/s
• ?t 10 s
• ?d ?
• ?d v?t (340 m/s)(10 s) 3400 m
•  3400 m 1 km
• 1000 m
•  
•  (A little over 2 miles)

3.4 km
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Measurement Worksheet

• What is accuracy? How is it expressed?
• Can a set of measurements be precise, but not
  accurate?
• What are sig figs?
• For each of the following measurements, express
  as scientific notation
• 300,000,000 0.053 x 105
• 25.030
• 0.0006070
• 1.004
• 91.09534 x 10-31

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Measurement Worksheet

• What is the order of magnitude for each of the
  measurements in the previous problem?
• The accepted value for the acceleration due to
  gravity is 9.81 m/s2. Find the following using
  the data below absolute and relative error and
  deviation

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• Trial Measurement (m/s2)
• 1 9.95
• 2 9.79
• 3 9.90
• 4 9.85
• 5 10.1
• 6 9.65
• 7 9.82

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• For each of the following sets of numbers,
  express in terms of the mean and variance
• 10.5, 9.2, 10.1
• .00234, .002, .00237
• 145, 102, 132, 135
• 1000, 1003, 1100, 1095, 1058
• 1.2, 1.3, 1.9