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Workshop: Mathematical Modeling
Edward (Joe) Redish
& Kimberly Moore
Department of Physics
University of Maryland
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Outline
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Meet
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Brief overview of NEXUS/Physics
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Introduction to Mathematical Modeling
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System Schema
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Activity:
How big is a worm?
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Activity:
Who has the more powerful pump?
Math Modeling Workshop
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Meet
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Introductions
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Name
Discipline
Institution & role
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Do you use math in your instruction?
How?
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Do you use math in your research?
How?
Math Modeling Workshop
+ In the summer of 2010, HHMI offered
four universities the opportunity to:
Develop prototype materials
for biologists and pre-meds in
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Chemistry (Purdue)
Math (UMBC)
Physics (UMCP)
Capstone case study course
(U of Miami)
that would
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take an interdisciplinary perspective
be competency based
Math Modeling Workshop
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+ Changing the course goals
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Explicitly serve biology students and faculty
by articulating with the biology curriculum
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Provide support for biology majors for difficult
physics concepts that they will encounter in
biology and chemistry classes, particularly those
that cannot be studied in depth in those classes.
Do this by using methods common
in intro physics
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Use simplified models to build understanding,
Build a sense of physical mechanism,
Develop coherences between things that might
seem contradictory, etc.
Math Modeling Workshop
Redish et al., Am. J. Phys 82:5 (2014) 368-377
+ New topics
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Focus on mathematical modeling and explicating
assumptions.
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Do micro and macro examples throughout
assuming students know about atoms.
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Include discussion of chemical energy and
reactions
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Treat random motion as well as coherent. (Labs!)
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Carefully build the basic statistical mechanics
support for thermodynamics (conceptually).
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Expand treatment of fluids and physics in fluids.
Math Modeling Workshop
Dreyfus et al., Am. J. Phys 82:5 (2014) 403-411
Geller et al., Am. J. Phys 82:5 (2014) 394-402
Moore et al., Am. J. Phys 82:5 (2014) 387-393
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Systems
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We will be considering situations in which many
things acting on each other.
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In order to make sense of what’s going on, we will
focus on a few at a time and create models of
what we think is happening.
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Sometimes we will focus on a set of things as our
“system” and consider the influence of everything
else as “external”.
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Some times we will consider something’s internal
structure; other times we will consider it as a
“black box”.
Math Modeling Workshop
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System schemas
Math Modeling Workshop
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What “things” should be
considered when thinking about
what influences the motion
– or non-motion – of the dogs?
How do they act on each other?
Math Modeling Workshop
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What if we only want to
consider the motion of dog 2?
Math Modeling Workshop
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What if we want to consider
the motion of both dogs?
Math Modeling Workshop
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A theoretical framework
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Tells us what we need to consider in
order to describe and think about
whatever system we are considering.
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Objects
Relationships
Structures
Math Modeling Workshop
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Math modeling in physics
Math Modeling Workshop
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Equations in physics
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Equations as a conceptual organizer
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Equations as specifying relationships
of measurements and functional
dependence
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Equations for calculating something
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Role of “toy models”
Math Modeling Workshop
+ Equations as a conceptual organizer
What does this equation tell you?
These relations
are independently
true for each direction.
Force is what
you have to pay
attention to when
considering motion
!
aA =
Forces change
an object’s
velocity
You have to pick
an object to pay
attention to
Math Modeling Workshop
! net
FA
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What matters is
the sum of the forces
on the object
being considered
mA
The total force
is “shared” to
all parts of
the object
Total force (shared over
the parts of the mass) causes
an object’s velocity to change
+ Activity:
How big is a worm?
The earthworm absorbs oxygen directly through its skin.
The worm does have a good circulatory system (with multiple
small hearts) that brings the oxygen to all the cells.
But the cells are distributed through the worm's volume
and the oxygen only gets to come in through the skin
-- so the surface to volume ratio plays an important role.
Let's see how this works.
Here are the worm's parameters. A typical specimen
of the common earthworm (Lumbricus terrestris)
has the following average dimensions:
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Mass — 3.7 g
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Length — 12 cm
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Width — 0.64 cm
The skin of the worm can absorb oxygen at a rate of
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A = 0.24 µmole/cm2/h.
The body of the worm needs to use oxygen at a rate of
approximately
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B = 0.98 µmole/g/h.
Math Modeling Workshop
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Activity:
Who has the more powerful pump?
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The giraffe and acacia
tree both have to
raise liquids
to a great height.
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The physics of fluid flow
is governed by
the Hagen-Poiseuille
equation.
Math Modeling Workshop
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Flow in a pipe: Viscous Drag
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A fluid flowing in a pipe doesn’t slip through
the pipe frictionlessly.
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The fluid sticks to the walls moves faster
at the middle of the pipe than at the edges.
As a result, it has to slide over itself (shear).
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There is friction between layers of fluid moving
at different speeds that creates a viscous drag
force, trying to reduce the sliding.
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The drag is proportional to the speed
and the length of pipe.
Fdrag = 8!µ Lv
Math Modeling Workshop
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Implication: Pressure drop
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If we have a fluid moving at a constant rate
and there is drag, N2 tells us there must be
another force to balance the drag
(no change in v)
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The internal pressure in the fluid must drop
in the direction of the flow to balance drag.
Drag force
Flow in
Math Modeling Workshop
Flow out
Pressure force
upstream
Pressure force
downstream
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The Hagen-Poiseuille Equation
If the velocity is constant, N2 tells us
pressure drop balances the drag:
!
Fdrag
!
Pupstream AL
Rate of flow
(volume per sec)
!P A = 8"µ Lv
!
Pdownstream AR
Q = Av
# Q&
!P A = 8"µ L % (
$ A'
# 8"µ L &
# 8µL &
!P = % 2 ( Q = %
Q
4 (
$ A '
$ "R '
Resistance
!P = ZQ
Math Modeling Workshop
Pressure
difference
Rate of flow
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Who has the more powerful pump?
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What are arguments for the
giraffe having the more
powerful pump?
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What are arguments for the
acacia having the more
powerful pump?
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How can we resolve this?
Math Modeling Workshop