Fundamental Tools

1
Fundamental Tools

• Chapter 1

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Fundamental Tools

• Expectations
• After this chapter, students will
• understand the basis of the SI system of units
• distinguish between units and dimensions
• be able to perform dimensional analyses
• distinguish between fundamental and derived units
• be able to convert a quantity to different units
• know standard powers-of-ten prefixes
• be able to solve right triangles

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Fundamental Tools

• Expectations (continued)
• After this chapter, students will
• distinguish between vector and scalar quantities
• be able to resolve vectors into orthogonal
  components
• be able to add and subtract vectors
• know how vectors can be multiplied by scalars
• know how vectors can be multiplied by vectors
• know how many significant figures are in a given
  number

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What is Physics?

• The mapping of mathematics onto the material
  world.
• A mathematical description of the interactions of
  space, time, matter, and energy.
• An experimental science theory is judged by how
  well it predicts the results of experiments.

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Numbers and Units

• I often say that when you can measure what you
  are speaking about, and express it in numbers,
  you know something about it but when you cannot
  express it in numbers, your knowledge is of a
  meager and unsatisfactory kind it may be the
  beginning of knowledge, but you have scarcely, in
  your thoughts, advanced to the stage of Science,
  whatever the matter may be.
• --- William Thomson (Lord Kelvin) 1824 - 1907

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Numbers and Units

• Quantity a characteristic of an object or
  material that can be expressed quantitatively (in
  numbers)
• examples height, weight, volume, density
• Dimension the name of the class or category of
  units that express a physical quantity
• examples length, mass, time, velocity

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Numbers and Units

• Unit a reference standard with an agreed-upon
  definition that allows quantities to be specified
  by comparison to it.
• examples meter, second, pound

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Types of Units

• Base unit the units of the quantities length,
  mass, and time.
• examples meter, kilogram, second
• Derived units units made by combining other
  units
• examples meters / second, kilogrammeters /
  (second)2

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Systems of Units

• SI (Système International dUnités)
• Base units length meter (m)
• mass kilogram (kg)
• time second (s)
• CGS (small metric)
• Base units length centimeter (cm)
• mass gram (g)
• time second (s)

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Systems of Units

• BE (British engineering)
• Base units length foot (ft)
• mass slug (sl)
• time second (s)
• In this class, well use SI units almost
  exclusively.

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Significant Figures

• The uncertainty in our knowledge of the numerical
  value of a physical quantity is indicated by the
  number of significant figures we use to express
  that value.
• To determine how many significant figures are in
  a number (example 0.0000149 m)
• Write the number in proper scientific notation
  1.4910-5 m.
• Count the digits in this part of the number 3
  digits
• Note for proper scientific notation a 10n,
• 1 a lt 10

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Unit Conversions

• Basic principles
• 1 is the multiplicative identity. Multiplying
  any quantity by one does not change the value of
  that quantity.
• If a fractions numerator and denominator are
  equal, the fraction is equal to one.
• Example convert 3.45 years to seconds

fractions equal to one
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Dimensional Analysis

• A consistency check for mathematical
  relationships in physics.
• A formula or equation that passes a dimensional
  analysis may or may not be correct.
• However
• A formula or equation that fails a dimensional
  analysis cannot be correct.

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Dimensional Analysis

• What is a dimensional analysis? How can I do
  one?
• Substitute the dimensions represented by each
  variable for that variable in the equation to be
  analyzed.
• Algebraically simplify the equation
  exponentiate, multiply, divide, add, subtract,
  cancel.
• In simplest terms, both sides of the equation
  should have the same dimensions.

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Dimensional Analysis

• (Simple) example your geometry is a little
  fuzzy. But youre pretty sure that the surface
  area of a sphere is given by
• Check it
• and the analysis fails. That formula cant be
  correct.

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Common Powers-of-Ten
power of ten prefix symbol example
109 giga - G GHz
106 mega - M MW
103 kilo - k km
10-2 centi - c cm
10-3 milli - m mm
10-6 micro - m mm
10-9 nano - n nm
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Right-Triangle Trigonometry

• Basic relationships

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Right-Triangle Trigonometry

• Basic relationships

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Vector and Scalar Quantities

• Scalar completely specified by a magnitude
  (size)
• Vector completely specified by both a magnitude
  and a direction
• Examples
• Distance (scalar) the airport is 15 km away from
  here.
• Displacement (vector) the airport is 15 km
  southwest from here.

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Vector and Scalar Quantities

• Scalar speed, temperature, time, mass, energy,
  volume, area, length
• Vector velocity, acceleration, momentum, force
• Note that the ability to take on or values
  does not make a quantity a vector. Example
  Celsius or Fahrenheit temperature.

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Vector Properties

• Symbol an arrow (line segment with a point)
• arrow length shows vector magnitude
• arrow points in vector direction
• Mathematical notation
• bold-font letter A
• arrow on top of letter
• hat on top of letter (usually a unit vector)

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Vector Mathematics

• Vectors can be
• Added
• Multiplied
• by a scalar
• by another vector (in two different ways)
• Subtracted
• Divided

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Vector Addition

• Heres a graphical look at vector addition we
  want to add A and B.

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Vector Addition

• First, we note that we can translate a vector to
  any other location without changing it (either
  magnitude or direction).

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Vector Addition

• So, we translate B so that its tail coincides
  with As point.

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Vector Addition

• Now, we draw a third vector from the beginning
  point of A (its tail) to the ending point of B
  (its point).
• That third vector is the sum A B.

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Vector Multiplication

• There are three kinds of multiplication that can
  be done with vectors.
• First multiplication by a scalar.
• Magnitude of the product vector magnitude of the
  factor vector times the scalar.
• Product vector direction same or opposite the
  factor vector direction

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Vector Multiplication

• There are three kinds of multiplication that can
  be done with vectors.
• Second scalar product of two vectors (dot
  product).
• The scalar product is zero
• if the vectors are perpendicular
• a maximum value when they
• are parallel

This kind of vector multiplication is commutative
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Vector Multiplication

• There are three kinds of multiplication that can
  be done with vectors.
• Third vector product of two vectors (cross
  product).
• Direction perpendicular to both
• A and B, and in accordance
• with the right-hand rule
• The vector product is zero
• if the vectors are parallel
• a maximum value when they
• are perpendicular

This kind of vector multiplication is NOT
commutative
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Vector Subtraction

• This time, we want
• A B.
• Graphically

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Vector Subtraction

• Our first step is to muliply
• B by the scalar -1,
• producing B

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Vector Subtraction

• And now we move B
• to the point of A, just as
• we did before

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Vector Subtraction

• And we draw in the
• sum A (-B)
• A B.

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Vector Addition by Components

• Any vector can be expressed as the sum of two
  vectors, both orthogonal to the coordinate axes.
• One is the X component, and one is the Y
  component.

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Vector Addition by Components

• Simple right-triangle trigonometry allows us to
  calculate the magnitudes of these components

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Vector Addition by Components

• Example we want to add vectors A and B.

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Vector Addition by Components

• First resolve A and B into components.
• (Replace A and B with component vectors AX, AY,
  BX, and BY, all orthogonal to the coordinate
  system.)

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Vector Addition by Components

• The components of the sum, C, are the sums of the
  components of A and B.
• Since the X components are either parallel or
  antiparallel, their magnitudes add algebraically.
• The same is true of the Y components.

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Vector Addition by Components
(magnitude)
(magnitude)
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Vector Addition by Components
(magnitude)
(magnitude)
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Vector Addition by Components

• Pythagoras theorem yields the magnitude of C
• The direction of C

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Vector Addition by Components

• A couple of things to remember
• You are free to define your coordinate system so
  that it makes your life easier.
• These are always correct
• as long as you measure q counterclockwise from
  the X direction.