Physics 131
131- Fundamentals of
Physics for Biologists I
Professor: Arpita Upadhyaya
Quiz 7 (review)
 Random Motion
 Diffusion
 Kinetic Theory
y
 Fluids

1
Physics 131
Quiz 7
Quiz 7 Grades
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
10
Average = 5.5
Physics 131
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QUESTION 1
50
40
30
20
10
0
1
2
3
12
123
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Which ball will
k k the
knock
h bl
blockk over??
1.
2.
3
3.
4.
A superball
A clay
l ball
b ll off equall
mass
Both
Neither
Physics 131
Pivot
Ball
Rigid rod
Wooden block
4
QUESTION 2 40
35
30
25
20
15
10
5
0
A
B
C
D
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QUESTION 3
35
30
25
20
15
10
5
0
A
B
C
D
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QUESTION 4
35
30
25
20
15
10
5
0
A
B
C
D
Physics 131
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The diagram at the right depicts
the path of two colliding steel balls
rolling on a table. Which set of
arrows best represents the direction
of the change in momentum of
each ball?
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8
Random Motion
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2D Simulations:
Multiple representations
131 & Kerstin Nordstrom
Simulations by AlexPhysics
Morozov
10
2D Simulations:
Multiple representations
1.
2.
Watch all the particles.
Look at the density
y of the particles
p
– What do the colors represent?
3
3.
Look at a plot of the density along a slice
through the middle.
– What it will look like and what it will do.
do
4.
Look at the motion
of individual particles
particles.
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Start 200 random walkers in two dimension near 0
Sketch what you expect to see &answer clicker question!
Whiteboard,
TA & LA
1.
2.
3.
Particles will form a “wave
wave”” – a
gg ring
g of p
g
ragged
particles moving
outward.
They will be mostly stay near 0
no matter how long
g yyou wait.
Many particles will remain near
zero, but they will gradually
spread
p
out.
12
0
Physics 131
A simulation of many random walkers
Alex Morozov &
Kerstin Nordstrom
Density as a function of position
Gaussian Distribution
Measure width at half the peak
height. Measure the full
width.
Since we only have a fixed
number of walkers,, the line is
choppy
The variability is equal to
e.g. if on average we have 100
particles in a bin, variability
will
ill be
b 10
Physics 131
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Units analysis



Think about what p
particles are doing.
g
What do you think D depends on?
Units may help you a lot here.
[D] = L/T2.
Think about the “speed”
speed of diffusion:
what variables (position, velocity, ease
of motion) would change the time
scale? (Make diffusion faster, e.g.)
 
 r
2
 6Dt
6D
Whiteboard,
TA & LA
I wait a certain amount of time, t, and get a
distribution with width associated with x. If I wait
four times longer, the width of the distribution
y what factor?
increases by
A. 1
B 2
B.
C. ½
D. 4
E. 1/4
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Trajectory of individual particles
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Compared to a 1D random walk if the walker can in
addition also take a step in the second dimension
f ll i the
following
th same rules,
l the
th following
f ll i is
i true
t
about
b t
the Diffusion Constant D and the distance squared
traveled <
<r2>
1. <r2> is larger by factor 2
2. <r2> is larger by factor 2
3. <r2> is the same
4. <r2> is smaller by factor 2
5. <r2> is smaller by factor 2
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Random Motion in two dimensions
r
If I wait four times as long,
long the
trajectory is on average longer by a
factor ___?
If I wait four times as long, the distance
between start and end point r is on
average longer by a factor
f
_____?
?
Alex Morozov &
Kerstin Nordstrom
 r 
2
4
4D
D t
D is called the diffusion constant
and has dimensionality [D] = L2/T
r
Random walk in 2D





As a result of random motion, an initially localized
distribution will spread out, getting wider and wider. This
phenomenon is called diffusion
The square of the average distance traveled during random
motion will grow with time
time.
1 Dimension
 x 
2 Dimensions
 r 
3 Dimensions
 r 
10/22/12
2
2
2
 2 Dt

 x    y 

 x    y    z 
2
2
D is called the diffusion constant
and has dimensionality [D] = L2/T
2
2
 4 Dt
2
 6 Dt
For what biological systems or
situations might a 1D,
1D 2D,
2D or 3D
random model of diffusion
be relevant?
Physics 131
Whiteboard,
22LA
TA &
Sodium ions are at different densities on the inside
and outside of a cell. Assume each ion moves
randomly
d l as a result
lt off collisions
lli i
with
ith other
th atoms
t
and molecules.
A small patch of membrane (area A) is shown in
yellow. There are more ions on the left than on the
right.
right
What do you expect is true about the ions on the left
side
id off the
th membrane?
b
?
A.
B.
C.
D.
More ggo to the right
g
More go to the left
Equal amount goes left and right
There is not enough information to tell
23
Sodium ions are at different densities on the inside
and outside of a cell. Assume each ion moves
randomly
d l as a result
lt off collisions
lli i
with
ith other
th atoms
t
and molecules.
A small patch of membrane (area A) is shown in
yellow. There are more ions are on the left than on
the right.
right
What do you expect is true about the ions on the right
side
id off the
th membrane?
b
?
A.
B.
C.
D.
More go to the right
More go to the left
Equal
q amount goes
g
left and right
g
There is not enough information to tell
24
Sodium ions are at different densities on the inside
and outside of a cell. Assume each ion moves
randomly as a result of collisions with other atoms
and molecules.
A small
ll patchh off membrane
b
(area
(
A) is
i shown
h
in
i
yellow. There are more ions are on the left than
on the right.
g
If the membrane allows ions to pass through what
do you expect will be true?
A.
B
B.
C.
D.
There’ll be a net flow of ions to the right
There
There’ll
ll be a net flow of ions to the left
There’ll be no net flow. Equal amounts will go
left and right.
There is not enough information to tell
25
Foothold principles:
Randomness



Matter is made of molecules in constant motion
and interaction. This moves things around.
If the distribution of a chemical is nonnon-uniform,
uniform
the randomness of molecular motion will tend to
result in molecules moving from more dense
regions to less.
This is not directed! It is an emergent
phenomenon arising from the combination of
random motion and non
non--uniform concentration.
concentration
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Diffusion (simulation)
27
Diffusion:
Fick’’s law (1D analysis)
Fick


Uniform fluid (black)
containing
t i i (red)
( d) molecules
l l
with a varying concentration.
Fluid molecules jiggle the
(red) molecules around.
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Fick’’s law
Fick





What can the total flow depend on?
It depends
d
d on the
th difference
diff
between
b t
the concentration on either side of
th barrier:
the
b i dn
d /dx;
dn/dx;
/d what
h t else?
l ?
The average speed of the particles
<v0> which we’ll just call v0
How far they get before hitting other
particles  (“mean free path”)
Thee size
s e of
o the
t e opening
ope g A
Physics 131
Whiteboard,
TA & LA
Fick’’s law
Fick




J: “flux”
“flux” = number of particles
per unit area per unit time;
Total flow is flux times area times
amountt off time
ti
Break up our volume into equal
“bins” of width “mean free path”
n/x = n+ – n– , <v0>, 
A,
A, t
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How many cross A in a time Δt
Δt?

Total number hitting A from left bin

1
1
(n )Volume  (n ) A
2
2
…from right…
1
2

(n ) A
Net flow across A in the +dx
+dx direction:
1
1  n 
( n  n ) A    
x  A
2
2  x 

So the total flow in t becomes:
1 n x
JAt  
At
2 x t
Whiteboard,
31LA
TA &
Fick’’s law
Fick
1 dn
d dx
d
JAt  
At
 1D result
2 dx dt
1 dn
v0 
J 
2 dx
1
dn
J  D
D  v0
dx
2
Not the specific trajectory for individual
molecules
Tells us how a collection of molecules
distributes based on “mean
mean free path
path” and
average speed
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Fick’’s law
Fick

1D result
dn
J  D
dx

D  12  v0
For all directions (not just 1D) Fick’
Fick’s law
b
becomes


J   Dn
Physics 131
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The gradient



If we want to take the derivative of a function of one
variable, y = df
df/dx
/dx,, it’
it’s straightforward.
If we have a function of three variables –
f(x,y,z
f(
x,y,z)) – what do we do?
The gradient is the vector derivative.
To get it at a point (x,y,z
(x,y,z))
– Find the direction in which f is changing the fastest.
– Take the derivative by looking at the rate of change in that
direction.
– Put a vector in that direction with its magnitude equal to the
maximum rate of change.
change

f
– The result is the vector called
Physics 131
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