Dimensional Reasoning

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Dimensional Reasoning
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Dimensions and Measurements

• Dimension is characteristic of the object,
  condition, or event and is described
  quantitatively in terms of defined units.
• A physical quantity is equal to the product of
  two elements
• A quality or dimension
• A quantity expressed in terms of units
• Dimensions
• Physical things are measurable in terms of three
  primitive qualities (Maxwell 1871)
• Mass (M)
• Length (L)
• Time (T)
• Note (Temperature, electrical charge, chemical
  quantity, and luminosity were added as
  primitives some years later.)

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Dimensions and Measurements (cont.)

• Examples
• Length (L)
• Velocity (L/T)
• Force (ML/T2)
• Units
• Measurements systems--cgs, MKS, SI--define units
• SI units are now the international standard
  (although many engineers continue to use Imperial
  or U.S.)

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SI Primitives
DIMENSION UNIT SYMBOL for UNIT
Length meter m
Mass kilogram kg
Time second s
Elec. Current ampere A
luminous intensity candela cd
amount of substance mole mol
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SI Derived units
DESCRIPTION DERIVED UNIT SYMBOL DIMENSION
Force newton N mkg/s2
Energy joule J m2kg/s2
Pressure pascal Pa kg/(ms2)
Power watt W m2kg/s3
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Dimensional analysis

• Fundamental rules
• All terms in an equation must reduce to identical
  primitive dimensions
• Dimensions can be algebraically manipulated, e.g.
• Example
• Uses
• Check consistency of equations
• Deduce expression for physical phenomenon

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Dimensional analysis
distance s s0 vt2 0.5at3 constant p
?gh ?v2/2 volume of a torus 2p2(Rr)2
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Deduce expressions

• Example What is the period of oscillation for a
  pendulum?
• Possible variables length l L, mass m M,
  gravity g
• i.e. period P f(l , m, g)
• Period T , so combinations of variables
  must be equivalent to T .

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Dimensional analysis (cont.)

• Example What is the period of oscillation for a
  pendulum?
• Possible variables length l L, mass m M,
  gravity g
• i.e. period P f(l , m, g)
• Period T , so combinations of variables
  must be equivalent to T .
• Only possible combination is
• Note mass is not involved

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Quantitative considerations

• Each measurement carries a unit of measurement
• Example it is meaningless to say that a board is
  3 long. 3 what? Perhaps 3 meters long.
• Units can be algebraically manipulated (like
  dimensions)
• Conversions between measurement systems can be
  accommodated, e.g., 1 m 100 cm,
• or or
• Example

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Quantitative considerations (cont.)

• Arithmetic manipulations between terms can take
  place only with identical units.
• Example
• but,

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Buckingham Pi Theorem (1915)

• Pi theorem tells how many dimensionless groups
  define a problem.
• Theorem If n variables are involved in a
  problem and these are expressed using k primitive
  dimensions, then (n-k) dimensionless groups are
  required to characterize the problem.
• Example in the pendulum, the variables were
  time T,
• gravity L/T2, length L, mass M . So, n
  4 k 3. So, only one dimensionless group
  describes the system.

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Buckingham Pi Theorem (cont.)

• How to find the dimensional groups
• Pendulum example
• where a,b,c,d are coefficients to be determined.
• In terms of dimensions

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Buckingham Pi Theorem (cont.)

• Therefore
• a - 2c 0
• b c 0
• d 0
• Arbitrarily choose a 1. Then b -1/2, c
  1/2, d 0.
• This yields

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Buckingham Pi Theorem (cont.) Oscillations of a
star
A star undergoes some mode of oscillation. How
does the frequency of oscillation ? depend upon
the properties of the star? Certainly the density
? and the radius R are important we'll also need
the gravitational constant G which appears in
Newton's law of universal gravitation. We could
add the mass m to the list, but if we assume that
the density is constant, then m ?(4pR3/3) and
the mass is redundant. Therefore, ? is the
governed parameter, with dimensions ? T-1,
and (? R G) are the governing parameters, with
dimensions ? ML-3, R L, and G
M-1L3T-2 (check the last one). You can easily
check that (? RG) have independent dimensions
therefore, n 3 k 3, so the function F is
simply a constant in this case. Next, determine
the exponents ? T-1 ?aRbGc
Ma-cL-3ab3cT-2c Equating exponents on both
sides, we have a - c 0 -3a b 3c 0 -2c
-1 Solving, we find a c 1/2, b 0, so that ?
C(Gs)1/2, with C a constant. We see that the
frequency of oscillation is proportional to the
square root of the density, and independent of
the radius.
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Dimensionless Quantities

• Dimensional quantities can be made
  dimensionless by normalizing them with
  respect to another dimensional quantity of the
  same dimensionality.
• Example speed V (m/s) can be made
  "dimensionless by dividing by the velocity of
  sound c (m/s) to obtain M V/c, a dimensionless
  speed known as the Mach number. Mgt1 is faster
  than the speed of sound Mlt1 is slower than the
  speed of sound.
• Other examples percent, relative humidity,
  efficiency
• Equations and variables can be made
  dimensionless, e.g., Cd 2D/(?v2A)
• Useful properties
• Dimensionless equations and variable are
  independent of units.
• Relative importance of terms can be easily
  estimated.
• Scale (battleship or model ship) is automatically
  built into the dimensionless expression.

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Dimensionless quantities (cont.)

• Reduces many problems to a single problem through
  normalization.
• Example Convert a dimensional stochastic
  variable x to a
• non-dimensional variable
• to represent its position with respect to a
  Gaussian curve--N(0,1),
• e.g., grades on an exam

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Proof of the Pythagorean Theorem
The area of any triangle depends on its size and
shape, which can be unambiguously identified by
the length of one of its edges (for example, the
largest) and by any two of its angles (the third
being determined by the fact that the sum of all
three is p). Thus, recalling that an area has the
dimensions of a length squared, we can
write area largest edge2 f (angle1,
angle2), where f is an nondimensional function of
the angles. Now, referring to the figure at
right, if we divide a right triangle in two
smaller ones by tracing the segment perpendicular
to its hypotenuse and passing by the opposite
vertex, and express the obvious fact that the
total area is the sum of the two smaller ones, by
applying the previous equation we have c2 f
(a, p/2) a2 f (a, p/2) b2 f (a,
p/2). And, eliminating f c2 a2 b2, Q.E.D.
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Scaling, modeling, similarity

• Types of similarity between two
  objects/processes.
• Geometric similarity linear dimensions are
  proportional angles are the same.
• Kinematic similarity includes proportional time
  scales, i.e., velocity, which are similar.
• Dynamic similarity includes force scale
  similarity, i.e., equality of Reynolds number
  (inertial/viscous), Froud number
  (inertial/buoyancy), Rossby number
  (inertial/Coriolis), Euler number
  (inertial/surface tension).

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Scaling, modeling, similarity

• Distorted models
• Sometimes its necessary to violate geometric
  similarity A 1/1000 scale model of the
  Chesapeake Bay is ten times as deep as it should
  be, because the real Bay is so shallow that, with
  proportional depths, the average model depth
  would be 6mm, too shallow to exhibit stratified
  flow.

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Scaling, modeling, similarity

• Scaling
• Whats the biggest elephant? If one tries to
  keep similar geometric proportions, weight ? L3,
  where L is a characteristic length, say height.
• However, an elephants ability to support his
  weight is proportional to the cross-sectional
  area of his bones, say R2.
• Therefore, if his height doubles, his bones would
  have to increase in radius as 2?2 R, not 2R.
• Note A cross-section of 8 R2 (2?2 R)2.
  So, with increasing size, an elephant will
  eventually have legs whose cross-sectional area
  will extend beyond its body

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Biological scaling